Scalar, Vector, and Matrix Mathematics: Theory, Facts, and Formulas - Revised and Expanded Edition 9781400888252

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Scalar, Vector, and Matrix Mathematics

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Scalar, Vector, and Matrix Mathematics

Theory, Facts, and Formulas Revised and Expanded Edition

Dennis S. Bernstein

PRI NC ET ON U NI VER SI TY P RE SS PRINCETON AND OXFORD

c Copyright ⃝2018 by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire, 0X20 1TW All Rights Reserved

Library of Congress Cataloging-in-Publication Data Names: Bernstein, Dennis S., 1954– | Bernstein, Dennis S., 1954– Matrix mathematics. Title: Scalar, vector, and matrix mathematics: theory, facts, and formulas / Dennis S. Bernstein. Other titles: Matrix mathematics Description: Revised and expanded edition. | Princeton: Princeton University Press, [2018] | “Revised and expanded edition of Matrix mathematics, retitled Scalar, vector, and matrix mathematics”–Preface. | Includes bibliographical references and index. Identifiers: LCCN 2017009620 | ISBN 9780691151205 (hardcover: alk. paper) | ISBN 9780691176536 (pbk.) Subjects: LCSH: Matrices. | Linear systems. | Vector analysis. Classification: LCC QA188 .B475 2018 | DDC 512.9/434–dc23 LC record available at https://lccn.loc.gov/2017009620 British Library Cataloging-in-Publication Data is available This book has been composed in Times New Roman and Helvetica. The publisher would like to acknowledge the author of this volume for providing the camera-ready copy from which this book was printed. Printed on acid-free paper. ∞ press.princeton.edu Printed in the United States of America 10

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To the memory of my parents, Irma Shorrie (Hirshon) Bernstein and Milton Bernstein, whose love and guidance are everlasting

To Susan, with love and gratitude

. . . vessels, unable to contain the great light flowing into them, shatter and break. . . . the remains of the broken vessels fall . . . into the lowest world, where they remain scattered and hidden — D. W. Menzi and Z. Padeh, The Tree of Life: Chayyim Vital’s Introduction to the Kabbalah of Isaac Luria, Jason Aaronson, Northvale, 1999

Thor . . . placed the horn to his lips . . . He drank with all his might and kept drinking as long as ever he was able; when he paused to look, he could see that the level had sunk a little, . . . for the other end lay out in the ocean itself. — P. A. Munch, Norse Mythology, AMS Press, New York, 1970

Contents

Preface to the Revised and Expanded Edition

xvii

Preface to the Second Edition

xix

Preface to the First Edition

xxi

Special Symbols

xxv

Conventions, Notation, and Terminology 1. Sets, Logic, Numbers, Relations, Orderings, Graphs, and Functions 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18 1.19 1.20 1.21 1.22

Sets Logic Relations and Orderings Directed and Symmetric Graphs Numbers Functions and Their Inverses Facts on Logic Facts on Sets Facts on Graphs Facts on Functions Facts on Integers Facts on Finite Sums Facts on Factorials Facts on Finite Products Facts on Numbers Facts on Binomial Coefficients Facts on Fibonacci, Lucas, and Pell Numbers Facts on Arrangement, Derangement, and Catalan Numbers Facts on Cycle, Subset, Eulerian, Bell, and Ordered Bell Numbers Facts on Partition Numbers, the Totient Function, and Divisor Sums Facts on Convex Functions Notes

2. Equalities and Inequalities 2.1 2.2 2.3 2.4

Facts on Equalities and Inequalities in One Variable Facts on Equalities and Inequalities in Two Variables Facts on Equalities and Inequalities in Three Variables Facts on Equalities and Inequalities in Four Variables

xxxvii 1

1 2 5 9 12 16 21 22 25 26 28 36 49 52 52 54 95 103 105 113 116 118 119

119 129 146 177

x

CONTENTS

2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13 2.14 2.15 2.16 2.17 2.18 2.19 2.20 2.21 2.22

Facts on Equalities and Inequalities in Five Variables Facts on Equalities and Inequalities in Six Variables Facts on Equalities and Inequalities in Seven Variables Facts on Equalities and Inequalities in Eight Variables Facts on Equalities and Inequalities in Nine Variables Facts on Equalities and Inequalities in Sixteen Variables Facts on Equalities and Inequalities in n Variables Facts on Equalities and Inequalities in 2n Variables Facts on Equalities and Inequalities in 3n Variables Facts on Equalities and Inequalities in 4n Variables Facts on Equalities and Inequalities for the Logarithm Function Facts on Equalities for Trigonometric Functions Facts on Inequalities for Trigonometric Functions Facts on Equalities and Inequalities for Inverse Trigonometric Functions Facts on Equalities and Inequalities for Hyperbolic Functions Facts on Equalities and Inequalities for Inverse Hyperbolic Functions Facts on Equalities and Inequalities in Complex Variables Notes

3. Basic Matrix Properties 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14 3.15 3.16 3.17 3.18 3.19 3.20 3.21 3.22 3.23 3.24 3.25 3.26

Vectors Matrices Transpose and Inner Product Geometrically Defined Sets Range and Null Space Rank and Defect Invertibility The Determinant Partitioned Matrices Majorization Facts on One Set Facts on Two or More Sets Facts on Range, Null Space, Rank, and Defect Facts on the Range, Rank, Null Space, and Defect of Partitioned Matrices Facts on the Inner Product, Outer Product, Trace, and Matrix Powers Facts on the Determinant Facts on the Determinant of Partitioned Matrices Facts on Left and Right Inverses Facts on the Adjugate Facts on the Inverse Facts on Bordered Matrices Facts on the Inverse of Partitioned Matrices Facts on Commutators Facts on Complex Matrices Facts on Majorization Notes

183 184 186 187 187 187 188 215 226 226 226 231 246 254 261 264 266 276 277

277 280 285 290 290 292 294 299 302 305 306 310 315 320 326 329 334 342 345 348 351 352 354 356 359 362

CONTENTS

4. Matrix Classes and Transformations 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14 4.15 4.16 4.17 4.18 4.19 4.20 4.21 4.22 4.23 4.24 4.25 4.26 4.27 4.28 4.29 4.30 4.31 4.32 4.33

Types of Matrices Matrices Related to Graphs Lie Algebras Abstract Groups Addition Groups Multiplication Groups Matrix Transformations Projectors, Idempotent Matrices, and Subspaces Facts on Elementary, Group-Invertible, Range-Hermitian, Range-Disjoint, and Range-Spanning Matrices Facts on Normal, Hermitian, and Skew-Hermitian Matrices Facts on Linear Interpolation Facts on the Cross Product Facts on Inner, Unitary, and Shifted-Unitary Matrices Facts on Rotation Matrices Facts on One Idempotent Matrix Facts on Two or More Idempotent Matrices Facts on One Projector Facts on Two or More Projectors Facts on Reflectors Facts on Involutory Matrices Facts on Tripotent Matrices Facts on Nilpotent Matrices Facts on Hankel and Toeplitz Matrices Facts on Tridiagonal Matrices Facts on Triangular, Hessenberg, and Irreducible Matrices Facts on Matrices Related to Graphs Facts on Dissipative, Contractive, Cauchy, and Centrosymmetric Matrices Facts on Hamiltonian and Symplectic Matrices Facts on Commutators Facts on Partial Orderings Facts on Groups Facts on Quaternions Notes

5. Geometry 5.1 5.2 5.3 5.4 5.5

xi 363

363 367 368 369 371 371 373 374 376 377 383 384 387 391 396 398 407 409 416 417 417 418 420 422 424 426 427 427 428 430 432 437 440 441

Facts on Angles, Lines, and Planes Facts on Triangles Facts on Polygons and Polyhedra Facts on Polytopes Facts on Circles, Ellipses, Spheres, and Ellipsoids

441 443 489 493 495

6. Polynomial Matrices and Rational Transfer Functions

499

6.1 6.2 6.3 6.4 6.5

Polynomials Polynomial Matrices The Smith Form and Similarity Invariants Eigenvalues Eigenvectors

499 501 503 506 511

xii

CONTENTS

6.6 6.7 6.8 6.9 6.10 6.11 6.12

The Minimal Polynomial Rational Transfer Functions and the Smith-McMillan Form Facts on Polynomials and Rational Functions Facts on the Characteristic and Minimal Polynomials Facts on the Spectrum Facts on Graphs and Nonnegative Matrices Notes

7. Matrix Decompositions 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10 7.11 7.12 7.13 7.14 7.15 7.16 7.17 7.18 7.19 7.20 7.21

Smith Decomposition Reduced Row Echelon Decomposition Multicompanion and Elementary Multicompanion Decompositions Jordan Decomposition Schur Decomposition Singular Value Decomposition, Polar Decomposition, and Full-Rank Factorization Eigenstructure Properties Pencils and the Kronecker Canonical Form Facts on the Inertia Facts on Matrix Transformations for One Matrix Facts on Matrix Transformations for Two or More Matrices Facts on Eigenvalues and Singular Values for One Matrix Facts on Eigenvalues and Singular Values for Two or More Matrices Facts on Matrix Pencils Facts on Eigenstructure for One Matrix Facts on Eigenstructure for Two or More Matrices Facts on Matrix Factorizations Facts on Companion, Vandermonde, Circulant, Permutation, and Hadamard Matrices Facts on Simultaneous Transformations Facts on Additive Decompositions Notes

8. Generalized Inverses 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 8.10 8.11 8.12 8.13

Moore-Penrose Generalized Inverse Drazin Generalized Inverse Facts on the Moore-Penrose Generalized Inverse for One Matrix Facts on the Moore-Penrose Generalized Inverse for Two or More Matrices Facts on the Moore-Penrose Generalized Inverse for Range-Hermitian, Range-Disjoint, and Range-Spanning Matrices Facts on the Moore-Penrose Generalized Inverse for Normal Matrices, Hermitian Matrices, and Partial Isometries Facts on the Moore-Penrose Generalized Inverse for Idempotent Matrices Facts on the Moore-Penrose Generalized Inverse for Projectors Facts on the Moore-Penrose Generalized Inverse for Partitioned Matrices Facts on the Drazin and Group Generalized Inverses for One Matrix Facts on the Drazin and Group Generalized Inverses for Two or More Matrices Facts on the Drazin and Group Generalized Inverses for Partitioned Matrices Notes

512 513 517 524 530 537 544 545

545 545 546 549 553 555 558 563 565 569 575 579 589 597 597 603 605 610 617 618 619 621

621 625 628 632 641 649 650 652 659 669 674 678 679

CONTENTS

9. Kronecker and Schur Algebra 9.1 9.2 9.3 9.4 9.5 9.6 9.7

Kronecker Product Kronecker Sum and Linear Matrix Equations Schur Product Facts on the Kronecker Product Facts on the Kronecker Sum Facts on the Schur Product Notes

10. Positive-Semidefinite Matrices 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9 10.10 10.11 10.12 10.13 10.14 10.15 10.16 10.17 10.18 10.19 10.20 10.21 10.22 10.23 10.24 10.25 10.26

Positive-Semidefinite and Positive-Definite Orderings Submatrices and Schur Complements Simultaneous Diagonalization Eigenvalue Inequalities Exponential, Square Root, and Logarithm of Hermitian Matrices Matrix Inequalities Facts on Range and Rank Facts on Unitary Matrices and the Polar Decomposition Facts on Structured Positive-Semidefinite Matrices Facts on Equalities and Inequalities for One Matrix Facts on Equalities and Inequalities for Two or More Matrices Facts on Equalities and Inequalities for Partitioned Matrices Facts on the Trace for One Matrix Facts on the Trace for Two or More Matrices Facts on the Determinant for One Matrix Facts on the Determinant for Two or More Matrices Facts on Convex Sets and Convex Functions Facts on Quadratic Forms for One Matrix Facts on Quadratic Forms for Two or More Matrices Facts on Simultaneous Diagonalization Facts on Eigenvalues and Singular Values for One Matrix Facts on Eigenvalues and Singular Values for Two or More Matrices Facts on Alternative Partial Orderings Facts on Generalized Inverses Facts on the Kronecker and Schur Products Notes

xiii 681

681 683 685 685 691 697 701 703

703 704 707 709 713 714 722 723 724 730 735 749 761 763 774 776 785 792 795 799 800 804 813 815 820 831

11. Norms

833

11.1 11.2 11.3 11.4 11.5 11.6 11.7 11.8 11.9 11.10 11.11

833 835 838 841 845 847 849 853 860 868 884

Vector Norms Matrix Norms Compatible Norms Induced Norms Induced Lower Bound Singular Value Inequalities Facts on Vector Norms Facts on Vector p-Norms Facts on Matrix Norms for One Matrix Facts on Matrix Norms for Two or More Matrices Facts on Matrix Norms for Commutators

xiv 11.12 11.13 11.14 11.15 11.16 11.17 11.18

CONTENTS

Facts on Matrix Norms for Partitioned Matrices Facts on Matrix Norms and Eigenvalues for One Matrix Facts on Matrix Norms and Eigenvalues for Two or More Matrices Facts on Matrix Norms and Singular Values for One Matrix Facts on Matrix Norms and Singular Values for Two or More Matrices Facts on Linear Equations and Least Squares Notes

12. Functions, Limits, Sequences, Series, Infinite Products, and Derivatives 12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8 12.9 12.10 12.11 12.12 12.13 12.14 12.15 12.16 12.17 12.18 12.19

Open Sets and Closed Sets Limits of Sequences Series, Power Series, and Bi-power Series Continuity Derivatives Complex-Valued Functions Infinite Products Functions of a Matrix Matrix Square Root and Matrix Sign Functions Vector and Matrix Derivatives Facts on One Set Facts on Two or More Sets Facts on Functions Facts on Functions of a Complex Variable Facts on Functions of a Matrix Facts on Derivatives Facts on Limits of Functions Facts on Limits of Sequences and Series Notes

13. Infinite Series, Infinite Products, and Special Functions Facts on Series for Subset, Eulerian, Partition, Bell, Ordered Bell, Bernoulli, Euler, and Up/Down Numbers 13.2 Facts on Bernoulli, Euler, Chebyshev, Legendre, Laguerre, Hermite, Bell, Ordered Bell, Harmonic, Fibonacci, and Lucas Polynomials 13.3 Facts on the Zeta, Gamma, Digamma, Generalized Harmonic, Dilogarithm, and Dirichlet L Functions 13.4 Facts on Power Series, Laurent Series, and Partial Fraction Expansions 13.5 Facts on Series of Rational Functions 13.6 Facts on Series of Trigonometric and Hyperbolic Functions 13.7 Facts on Series of Binomial Coefficients 13.8 Facts on Double-Summation Series 13.9 Facts on Miscellaneous Series 13.10 Facts on Infinite Products 13.11 Notes

885 890 892 895 899 909 912 913

913 915 919 921 924 926 929 930 932 932 934 937 941 945 948 949 954 957 974 975

13.1

975 981 994 1004 1021 1057 1063 1071 1074 1080 1092

CONTENTS

14. Integrals 14.1 14.2 14.3 14.4 14.5 14.6 14.7 14.8 14.9 14.10 14.11 14.12 14.13 14.14

Facts on Indefinite Integrals Facts on Definite Integrals of Rational Functions Facts on Definite Integrals of Radicals Facts on Definite Integrals of Trigonometric Functions Facts on Definite Integrals of Inverse Trigonometric Functions Facts on Definite Integrals of Logarithmic Functions Facts on Definite Integrals of Logarithmic, Trigonometric, and Hyperbolic Functions Facts on Definite Integrals of Exponential Functions Facts on Integral Representations of G and γ Facts on Definite Integrals of the Gamma Function Facts on Integral Inequalities Facts on the Gaussian Density Facts on Multiple Integrals Notes

15. The Matrix Exponential and Stability Theory 15.1 15.2 15.3 15.4 15.5 15.6 15.7 15.8 15.9 15.10 15.11 15.12 15.13 15.14 15.15 15.16 15.17 15.18 15.19 15.20 15.21 15.22 15.23 15.24 15.25

Definition of the Matrix Exponential Structure of the Matrix Exponential Explicit Expressions Matrix Logarithms Principal Logarithm Lie Groups Linear Time-Varying Differential Equations Lyapunov Stability Theory Linear Stability Theory The Lyapunov Equation Discrete-Time Stability Theory Facts on Matrix Exponential Formulas Facts on the Matrix Sine and Cosine Facts on the Matrix Exponential for One Matrix Facts on the Matrix Exponential for Two or More Matrices Facts on the Matrix Exponential and Eigenvalues, Singular Values, and Norms for One Matrix Facts on the Matrix Exponential and Eigenvalues, Singular Values, and Norms for Two or More Matrices Facts on Stable Polynomials Facts on Stable Matrices Facts on Almost Nonnegative Matrices Facts on Discrete-Time-Stable Polynomials Facts on Discrete-Time-Stable Matrices Facts on Lie Groups Facts on Subspace Decomposition Notes

xv 1093

1093 1096 1111 1114 1130 1132 1150 1157 1169 1171 1171 1172 1173 1178 1179

1179 1181 1185 1187 1190 1191 1193 1195 1198 1201 1203 1204 1209 1209 1211 1217 1220 1223 1226 1232 1234 1239 1243 1243 1247

xvi

CONTENTS

16. Linear Systems and Control Theory 16.1 16.2 16.3 16.4 16.5 16.6 16.7 16.8 16.9 16.10 16.11 16.12 16.13 16.14 16.15 16.16 16.17 16.18 16.19 16.20 16.21 16.22 16.23 16.24 16.25 16.26

State Space Models Laplace Transform Analysis and Transfer Functions The Unobservable Subspace and Observability Observable Asymptotic Stability Detectability The Controllable Subspace and Controllability Controllable Asymptotic Stability Stabilizability Realization Theory Zeros H2 System Norm Harmonic Steady-State Response System Interconnections Standard Control Problem Linear-Quadratic Control Solutions of the Riccati Equation The Stabilizing Solution of the Riccati Equation The Maximal Solution of the Riccati Equation Positive-Semidefinite and Positive-Definite Solutions of the Riccati Equation Facts on Linear Differential Equations Facts on Stability, Observability, and Controllability Facts on the Lyapunov Equation and Inertia Facts on the Discrete-Time Lyapunov Equation Facts on Realizations and the H2 System Norm Facts on the Riccati Equation Notes

1249

1249 1252 1253 1257 1259 1259 1266 1268 1270 1278 1285 1288 1289 1291 1293 1295 1298 1302 1304 1305 1307 1309 1313 1313 1316 1319

Bibliography

1321

Author Index

1433

Subject Index

1449

Preface to the Revised and Expanded Edition

This third edition of Matrix Mathematics, retitled Scalar, Vector, and Matrix Mathematics, is the culmination of seven years of effort to expand the scope of the second edition of this work. In contrast to the first two editions, which were typeset in Computer Modern, this edition is typeset in New Times Roman, which, as a more compact font, accommodates more material per line. The resulting horizontal compression along with tighter line spacing facilitates one of the goals of this edition, which is to substantially expand the scope of the work to include more scalar and vector mathematics than was envisioned in the original “matrix” area. To this end, this edition includes extensive material on scalar inequalities, graphs, groups, geometry, combinatorics, number theory, finite sums and products, special functions, series, and integrals. As an indication of this augmented scope, the second edition of Matrix Mathematics cited 1540 references, whereas the present volume cites 3024. After three editions and almost three decades of organizational and lexicographical labor, it is perhaps fitting to reflect on mathematics as an artistic and utilitarian endeavor. Mathematics is about the creation of concepts, ideas, and idealizations that are natural, attractive, interesting, and powerful. These attributes may or may not be motivated by physical applications, but what is natural, attractive, interesting, and powerful often turns out to be useful, and vice versa. Mathematical ideas can be appreciated for either their inner beauty or their usefulness. In science and engineering, idealizations serve as approximations; applying and interpreting these approximations is an art guided by analysis, experience, and insight. The ability to think about realworld phenomena in terms of idealizations is essential to the ability of scientists and engineers to interpret and analyze data. The “gap” between idealizations and reality is the space inhabited by mathematically oriented scientists and engineers, who must reconcile the tangibility of data and reality with the ethereal nature of ideas and concepts. Beyond approximation, mathematics is about characterization, classification, and connection. Characterization is the elucidation of properties possessed by an object such as a number or a function; classification is the construction of taxonomies and hierarchies of concepts and structures; and connections between concepts and constructions reveal deeper and hidden properties. Among the most surprising results are connections that link seemingly unrelated objects in elegant and unexpected ways. eπ ȷ = −1 is the classic example, but is only one of many that can be found in this book. Each serendipitous connection reinforces the belief that mathematical objects have an independent existence that transcends the ink on the page and the thoughts in our heads. Mathematics is also about composition and decomposition—putting things together and taking things apart. How can a matrix of one type be factored into matrices of another type? How can matrices of one type be combined to form a matrix of another type? Through its focus on conceptual atoms and their agglomeration into complex molecules, mathematics is the chemistry of ideas. Mathematics strives for abstraction. However, the original motivation for this book was to minimize abstraction so that a user of mathematics could find a much-needed result in a sufficiently

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PREFACE TO THE SECOND EDITION

concrete form to facilitate its correct application. To achieve concreteness, this book avoids abstract structures such as fields and vector spaces. But abstractions are powerful. A result proved for topological/metric/normed/inner-product/symplectic spaces is valid for all such spaces, whether they are defined in terms of scalars, vectors, matrices, or functions in finite or (with suitable restrictions) infinite dimensions. Abstraction provides efficiency of effort, unity of thought, and depth of understanding. With a few exceptions, however, this book intentionally avoids abstraction in order to facilitate the accessibility of the material. But the price paid for this accessibility is to some extent a tunnel view of the larger picture of common ideas and structures. The perfect “handbook of mathematics” would embrace just enough abstraction to unify a huge body of results, and then systematically specialize those results to a multitude of accessible cases that mathematically oriented scientists and engineers might find useful. The perfect handbook remains to be written. Why devote three decades to writing a book such as this one? The main goal is to provide a convenient resource for users of mathematics. This collection complements existing compendia, with coverage and organization that are unique and, hopefully, useful. An unexpected benefit of collecting and organizing this diverse material is the connections that are uncovered. In addition to connections, this collection reveals gaps in knowledge left for researchers to fill and explore. The inclusion of conjectures and problems is a reminder that much remains to be done. As in all worthwhile endeavors, we have finally reached the beginning.

Acknowledgments A published review of the second edition of this book aptly suggested that a work of this scope would ideally be the labor of a team of authors. In fact, I have relied heavily on advice and input from many individuals, with 33 acknowledged in the first edition and 44 in the second. For this edition I am indebted to numerous individuals who answered my queries about their books and papers, provided feedback on portions of the manuscript, and contacted me with valuable suggestions. These include Shoshana Abramovich, Khaled Aljanaideh, Ovidiu Bagdasar, Oskar Baksalary, Ravindra Bapat, Sanjay Bhat, JC Bourin, Ryan Caverly, Naveen Crasta, Marco Cuturi, Anton de Ruiter, Ayhan Dil, Justin Edmondson, James Forbes, Daniel Franco, Ovidiu Furdui, Michael Gil, Chris Gilbreth, Ankit Goel, Wassim Haddad, Nicholas Higham, Jesse Hoagg, Matthew Holzel, Qing Hui, Jeffrey Humphery, Gidado-Yisa Immanuel, Fuad Kittaneh, Omran Kouba, Peter Larcombe, Minghua Lin, Zongli Lin, Florian Luca, Victor Moll, Robert Piziak, Olivier Ramare, Ranjan Roy, Sneha Sanjeevini, Meiyue Shao, Joseph Silverman, Alina Sintamarian, Wasin So, Valeriu Soltan, Yongge Tian, G¨otz Trenkler, Antai Xie, Doron Zeilberger, Fuzhen Zhang, Xuan Zhou, and Limin Zou. I am especially indebted to Oskar Baksalary, Omran Kouba, Minghua Lin, G¨otz Trenkler, and Yongge Tian for their substantial advice, encouragement, and assistance. Finally, I take full responsibility for the inevitable errors. I encourage you to inform me of any that you may find, and I will post them on my website. Dennis S. Bernstein Ann Arbor, Michigan [email protected] November 2017

Preface to the Second Edition

This second edition of Matrix Mathematics represents a major expansion of the original work. While the total number of pages is increased 57% from 752 to 1181, the increase is actually greater since this edition is typeset in a smaller font to facilitate a manageable physical size. The second edition expands on the first edition in several ways. For example, the new version includes material on graphs (developed within the framework of relations and partially ordered sets), as well as alternative partial orderings of matrices, such as rank subtractivity, star, and generalized L¨owner. This edition also includes additional material on the Kronecker canonical form and matrix pencils; matrix representations of finite groups; zeros of multi-input, multi-output transfer functions; equalities and inequalities for real and complex numbers; bounds on the roots of polynomials; convex functions; and vector and matrix norms. The additional material as well as works published subsequent to the first edition increased the number of cited works from 820 to 1540, an increase of 87%. To increase the utility of the bibliography, this edition uses the “back reference” feature of LATEX, which indicates where each reference is cited in the text. As in the first edition, the second edition includes an author index. The expansion of the first edition resulted in an increase in the size of the index from 108 pages to 161 pages. The first edition included 57 problems, while the current edition has 74. These problems represent extensions or generalizations of known results, sometimes motivated by gaps in the literature. In this edition, I have attempted to correct all errors that appeared in the first edition. As with the first edition, readers are encouraged to contact me about errors or omissions in the current edition, which I will periodically update on my home page.

Acknowledgments I am grateful to many individuals who kindly provided advice and material for this edition. Some readers alerted me to errors, while others suggested additional material. In other cases I sought out researchers to help me understand the precise nature of interesting results. At the risk of omitting those who were helpful, I am pleased to acknowledge the following: Mark Balas, Jason Bernstein, Sanjay Bhat, Gerald Bourgeois, Adam Brzezinski, Francesco Bullo, Vijay Chellaboina, Naveena Crasta, Anthony D’Amato, Sever Dragomir, Bojana Drincic, Harry Dym, Matthew Fledderjohn, Haoyun Fu, Masatoshi Fujii, Takayumi Furuta, Steven Gillijns, Rishi Graham, Wassim Haddad, Nicholas Higham, Diederich Hinrichsen, Matthew Holzel, Qing Hui, Masatoshi Ito, Iman Izadi, Pierre Kabamba, Marthe Kassouf, Christopher King, Siddharth Kirtikar, Michael Margliot, Roy Mathias, Peter Mercer, Alex Olshevsky, Paul Otanez, Bela Palancz, Harish PalanthandalamMadapusi, Fotios Paliogiannis, Isaiah Pantelis, Wei Ren, Ricardo Sanfelice, Mario Santillo, Amit Sanyal, Christoph Schmoeger, Demetrios Serakos, Wasin So, Robert Sullivan, Dogan Sumer, Yongge Tian, G¨otz Trenkler, Panagiotis Tsiotras, Takeaki Yamazaki, Jin Yan, Masahiro Yanagida, Vera Zeidan, Chenwei Zhang, Fuzhen Zhang, and Qing-Chang Zhong. As with the first edition, I am especially indebted to my family, who endured four more

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PREFACE TO THE SECOND EDITION

years of my consistent absence to make this revision a reality. It is clear that any attempt to fully embrace the enormous body of mathematics known as matrix theory is a neverending task. After devoting more than two decades to this project of reassembling the scattered shards, I remain, like Thor, barely able to perceive a dent in the vast knowledge that resides in the hundreds of thousands of pages devoted to this fascinating and incredibly useful subject. Yet, it is my hope that this book will prove to be valuable to everyone who uses matrices, and will inspire interest in a mathematical construction whose secrets and mysteries have no bounds. Dennis S. Bernstein Ann Arbor, Michigan [email protected] March 2009

Preface to the First Edition The idea for this book began with the realization that at the heart of the solution to many problems in science, mathematics, and engineering often lies a “matrix fact,” that is, an identity, inequality, or property of matrices that is crucial to the solution of the problem. Although there are numerous excellent books on linear algebra and matrix theory, no one book contains all or even most of the vast number of matrix facts that appear throughout the scientific, mathematical, and engineering literature. This book is an attempt to organize many of these facts into a reference source for users of matrix theory in diverse applications areas. Viewed as an extension of scalar mathematics, matrix mathematics provides the means to manipulate and analyze multidimensional quantities. Matrix mathematics thus provides powerful tools for a broad range of problems in science and engineering. For example, the matrix-based analysis of systems of ordinary differential equations accounts for interaction among all of the state variables. The discretization of partial differential equations by means of finite differences and finite elements yields linear algebraic or differential equations whose matrix structure reflects the nature of physical solutions [2553]. Multivariate probability theory and statistical analysis use matrix methods to represent probability distributions, to compute moments, and to perform linear regression for data analysis [1072, 1276, 1343, 1431, 1923, 2415]. The study of linear differential equations [1401, 1402, 1474] depends heavily on matrix analysis, while linear systems and control theory are matrix-intensive areas of engineering [5, 151, 298, 302, 688, 696, 782, 832, 834, 964, 1070, 1287, 1503, 1714, 1736, 1894, 2257, 2340, 2357, 2451, 2469, 2496, 2726, 2784, 2912, 2999]. In addition, matrices are widely used in rigid body dynamics [57, 1473, 1486, 1578, 1606, 1709, 1968, 2104, 2195, 2196, 2421, 2459, 2530, 2758], structural mechanics [1734, 2010, 2269], computational fluid dynamics [680, 1038, 2872], circuit theory [72], queuing and stochastic systems [1328, 1859, 2122], econometrics [884, 1924, 2300], geodesy [2556], game theory [488, 1802, 2542], computer graphics [122, 1062], computer vision [1912], optimization [558, 835, 1939], signal processing [1431, 2382, 2776], classical and quantum information theory [789, 1431, 2131, 2237], communications systems [1556, 1557], statistics [1221, 1343, 1924, 2300, 2403], statistical mechanics [34, 337, 338, 2789], demography [655, 1605], combinatorics, networks, and graph theory [275, 345, 418, 486, 503, 586, 588, 591, 677, 678, 745, 606, 805, 897, 932, 1040, 1067, 1183, 1256, 1323, 1431, 1700, 1861, 1884, 2338, 2809], optics [1163, 1359, 1593], dimensional analysis [1327, 2582], and number theory [1674]. In all applications involving matrices, computational techniques are essential for obtaining numerical solutions. The development of efficient and reliable algorithms for matrix computations is therefore an important area of research that has been extensively developed [201, 679, 869, 1196, 1389, 1391, 1467, 1526, 2533, 2534, 2536, 2538, 2694, 2785, 2874, 2878, 2882, 2958]. To facilitate the solution of matrix problems, entire computer packages have been developed using the language of matrices. However, this book is concerned with the analytical properties of matrices rather than their computational aspects. This book encompasses a broad range of fundamental questions in matrix theory, which, in many cases can be viewed as extensions of related questions in scalar mathematics. A few such

xxii

PREFACE TO THE FIRST EDITION

questions follow. What are the basic properties of matrices? How can matrices be characterized, classified, and quantified? How can a matrix be decomposed into simpler matrices? A matrix decomposition may involve addition, multiplication, and partition. Decomposing a matrix into its fundamental components provides insight into its algebraic and geometric properties. For example, the polar decomposition states that every square matrix can be written as the product of a rotation and a dilation analogous to the polar representation of a complex number. Given a pair of matrices having certain properties, what can be inferred about the sum, product, and concatenation of these matrices? In particular, if a matrix has a given property, to what extent does that property change or remain unchanged if the matrix is perturbed by another matrix of a certain type by means of addition, multiplication, or concatenation? For example, if a matrix is nonsingular, how large can an additive perturbation to that matrix be without the sum becoming singular? How can properties of a matrix be determined by means of simple operations? For example, how can the location of the eigenvalues of a matrix be estimated directly in terms of the entries of the matrix? To what extent do matrices satisfy the formal properties of the real numbers? For example, while 0 ≤ a ≤ b implies that ar ≤ br for real numbers a, b and a positive integer r, when does 0 ≤ A ≤ B imply Ar ≤ Br for positivesemidefinite matrices A and B and with the positive-semidefinite ordering? Questions of these types have occupied matrix theorists for at least a century, with motivation from diverse applications. The existing scope and depth of knowledge are enormous. Taken together, this body of knowledge provides a powerful framework for developing and analyzing models for scientific and engineering applications. This book is intended to be useful to at least four groups of readers. Since linear algebra is a standard course in the mathematical sciences and engineering, graduate students in these fields can use this book to expand the scope of their linear algebra text. For instructors, many of the facts can be used as exercises to augment standard material in matrix courses. For researchers in the mathematical sciences, including statistics, physics, and engineering, this book can be used as a general reference on matrix theory. Finally, for users of matrices in the applied sciences, this book will provide access to a large body of results in matrix theory. By collecting these results in a single source, it is my hope that this book will prove to be convenient and useful for a broad range of applications. The material in this book is thus intended to complement the large number of classical and modern texts and reference works on linear algebra and matrix theory [21, 838, 1071, 1139, 1140, 1184, 1228, 1429, 1580, 1762, 1909, 1943, 1952, 2053, 2139, 2169, 2263, 2338, 2445, 2553]. After a review of mathematical preliminaries in Chapter 1, fundamental properties of matrices are described in Chapter 2. Chapter 3 summarizes the major classes of matrices and various matrix transformations. In Chapter 4 we turn to polynomial and rational matrices whose basic properties are essential for understanding the structure of constant matrices. Chapter 5 is concerned with various decompositions of matrices including the Jordan, Schur, and singular value decompositions. Chapter 6 provides a brief treatment of generalized inverses, while Chapter 7 describes the Kronecker and Schur product operations. Chapter 8 is concerned with the properties of positivesemidefinite matrices. A detailed treatment of vector and matrix norms is given in Chapter 9, while formulas for matrix derivatives are given in Chapter 10. Next, Chapter 11 focuses on the matrix

xxiii

PREFACE TO THE FIRST EDITION

exponential and stability theory, which are central to the study of linear differential equations. In Chapter 12 we apply matrix theory to the analysis of linear systems, their state space realizations, and their transfer function representation. This chapter also includes a discussion of the matrix Riccati equation of control theory. Each chapter provides a core of results with, in many cases, complete proofs. Sections at the end of each chapter provide a collection of Facts organized to correspond to the order of topics in the chapter. These Facts include corollaries and special cases of results presented in the chapter, as well as related results that go beyond the results of the chapter. In some cases the Facts include open problems, illuminating remarks, and hints regarding proofs. The Facts are intended to provide the reader with a useful reference collection of matrix results as well as a gateway to the matrix theory literature.

Acknowledgments The writing of this book spanned more than a decade and a half, during which time numerous individuals contributed both directly and indirectly. I am grateful for the helpful comments of many people who contributed technical material and insightful suggestions, all of which greatly improved the presentation and content of the book. In addition, numerous individuals generously agreed to read sections or chapters of the book for clarity and accuracy. I wish to thank Jasim Ahmed, Suhail Akhtar, David Bayard, Sanjay Bhat, Tony Bloch, Peter Bullen, Steve Campbell, Agostino Capponi, Ramu Chandra, Jaganath Chandrasekhar, Nalin Chaturvedi, Vijay Chellaboina, Jie Chen, David Clements, Dan Davison, Dimitris Dimogianopoulos, Jiu Ding, D. Z. Djokovic, R. Scott Erwin, R. W. Farebrother, Danny Georgiev, Joseph Grcar, Wassim Haddad, Yoram Halevi, Jesse Hoagg, Roger Horn, David Hyland, Iman Izadi, Pierre Kabamba, Vikram Kapila, Fuad Kittaneh, Seth Lacy, Thomas Laffey, Cedric Langbort, Alan Laub, Alexander Leonessa, Kai-Yew Lum, Pertti Makila, Roy Mathias, N. Harris McClamroch, Boris Mordukhovich, Sergei Nersesov, JinHyoung Oh, Concetta Pilotto, Harish Palanthandalum-Madapusi, Michael Piovoso, Leiba Rodman, Phil Roe, Carsten Scherer, Wasin So, Andy Sparks, Edward Tate, Yongge Tian, Panagiotis Tsiotras, Feng Tyan, Ravi Venugopal, Jan Willems, Hong Wong, Vera Zeidan, Xingzhi Zhan, and Fuzhen Zhang for their assistance. Nevertheless, I take full responsibility for any remaining errors, and I encourage readers to alert me to any mistakes, corrections of which will be posted on the web. Solutions to the open problems are also welcome. Portions of the manuscript were typed by Jill Straehla and Linda Smith at Harris Corporation, and by Debbie Laird, Kathy Stolaruk, and Suzanne Smith at the University of Michigan. John Rogosich of Techsetters, Inc., provided invaluable assistance with LATEX issues, and Jennifer Slater carefully copyedited the entire manuscript. I also thank JinHyoung Oh and Joshua Kang for writing C code to refine the index. I especially thank Vickie Kearn of Princeton University Press for her wise guidance and constant encouragement. Vickie managed to address all of my concerns and anxieties, and helped me improve the manuscript in many ways. Finally, I extend my greatest appreciation for the (uncountably) infinite patience of my family, who endured the days, weeks, months, and years that this project consumed. The writing of this book began with toddlers and ended with a teenager and a twenty-year old. We can all be thankful it is finally finished. Dennis S. Bernstein Ann Arbor, Michigan [email protected] January 2005

Special Symbols General Notation

π

3.1415926535897932384626433832795028841971693993751058 . . .

e

2.7182818284590452353602874713526624977572470936999595 . . .

γ (Euler’s constant)

0.5772156649015328606065120900824024310421593359399235 . . .

G (Catalan’s constant) 0.9159655941772190150546035149323841107741493742816721 . . . △

=

equals by definition

limε↑0

limit from the left

limε↓0

limit from the right

limε→0

limit

0!

1

n!

n(n − 1) · · · (2)(1)

k

z(z − 1) · · · (z − k + 1)

k

z

z(z + 1) · · · (z + k − 1)

z

0!!

1

(−1)!!

1

n!!

n(n − 2)(n − 4) · · · (2) = 2n/2 (n/2)! if n is even; (n + 1)! if n is odd n(n − 2)(n − 4) · · · (3)(1) = (n+1)/2 2 [ 12 (n + 1)]! α(α − 1) · · · (α − m + 1) (p. 14) m! n! (p. 15) m!(n − m)! ∑ n! , where li=1 ki = n (p. 15) k1 ! · · · kl !

( ) α m ( ) n m (

n k1 , . . . , kl    n    k      n     k 

)

cycle number (p. 105, Fact 1.19.1)

subset number (p. 107, Fact 1.19.3)

⌊a⌋

largest integer less than or equal to (floor of) a

⌈a⌉

smallest integer greater than or equal to (ceiling of) a

δi, j

1 if i = j, 0 if i , j (Kronecker delta)

log

logarithm with base e

xxvi

SPECIAL SYMBOLS

sign α

1 if α > 0, −1 if α < 0, 0 if α = 0

dn

derangement number (p. 104, Fact 1.18.2)

Cn

Catalan number (p. 104, Fact 1.18.4)

Bn

Bell number (p. 112, Fact 1.19.6)

pn

partition number (p. 113, Fact 1.20.1)

Bn

Bernoulli number (p. 977, Fact 13.1.6)

En

Euler number (p. 980, Fact 13.1.8) ∞ ∑ 1 , where n ≥ 2 (p. 994, Fact 13.3.1) in i=1

ζ(n) Chapter 1

{}

set (p. 1)

∈

is an element of (p. 1)


> y

x(i) > y(i) for all i (x − y is positive) (p. 277)

conv S

convex hull of S (p. 279)

cone S

conical hull of S (p. 279)

coco S

convex conical hull of S (p. 279)

span S

span of S (p. 279)

affin S

affine hull of S (p. 279)

dim S

dimension of S (p. 279)

R

n×m

n × m real matrices (p. 280)

C

n×m

n × m complex matrices (p. 280)

Fn×m

Rn×m or Cn×m (p. 280)

rowi (A)

ith row of A (p. 280)

coli (A)

ith column of A (p. 280)

A(i, j)

(i, j) entry of A (p. 280)

i

A←b

matrix obtained from A ∈ Fn×m by replacing coli (A) with b ∈ Fn or rowi (A) with b ∈ F1×m (p. 280)

dmax (A)

largest diagonal entry of A ∈ Fn×m having real diagonal entries (p. 280)

di(A)

ith largest diagonal entry of A ∈ Fn×m having real diagonal entries (p. 280)

dmin (A)

smallest diagonal entry of A ∈ Fn×m having real diagonal entries (p. 280)

xxix

SPECIAL SYMBOLS

d(A)

   d1 (A)    ..  of diagonal entries of A ∈ Fn×m having real diagonal vector   .   dmin {n,m} entries (p. 281)

A(S1 ,S2 )

submatrix of A formed by retaining the rows of A listed in S1 and the columns of A listed in S2 (p. 281)

A(S)

A(S,S) (p. 281)

A(S1 ,·)

submatrix of A formed by retaining the rows of A listed in S1 (p. 281)

A(·,S2 )

submatrix of A formed by retaining the columns of A listed in S2 (p. 281)

A[S1 ,S2 ]

submatrix of A formed by deleting the rows of A listed in S1 and the columns of A listed in S2 (p. 281)

A[S]

A[S,S] (p. 281)

A[S1 ,·]

submatrix of A obtained by deleting the rows of A listed in S1 (p. 281)

A[·,S2 ]

submatrix of A obtained by deleting the columns of A listed in S2 (p. 281)

A[i, j]

A[{i},{ j}] (p. 281)

A ≥≥ B

A(i, j) ≥ B(i, j) for all i, j (A − B is nonnegative) (p. 281)

A >> B

A(i, j) > B(i, j) for all i, j (A − B is positive) (p. 281)

A (S)

image of the set S by the inverse of the map f (x) = Ax (p. 282)

[A, B]

commutator AB − BA (p. 283)

inv

adA(X)

adjoint operator [A, X] (p. 283)

x×y

cross product of vectors x, y ∈ R3 (p. 283)

K(x)

cross-product matrix for x ∈ R3 (p. 283)

0n×m , 0

n × m zero matrix (p. 283)

In , I

n × n identity matrix (p. 284)

1

Iˆn , Iˆ

0  n × n reverse permutation matrix 

Pn

n × n cyclic permutation matrix (p. 284)

..

.

1

Nn , N

n × n standard nilpotent matrix (p. 284)

ei,n , ei

coli (In ) (p. 285)

Ei, j,n×m , Ei, j

ei,n eTj,m (p. 285)

1n×m

n × m ones matrix (p. 285)

∥x∥2

Euclidean norm of x ∈ Fn (p. 286)

AT

transpose of A (p. 286)

tr A

trace of A (p. 287)

C

complex conjugate of C ∈ Cn×m (p. 287)

ST

{AT : A ∈ S} or {AT : A ∈ S}ms (p. 287)

A∗

A conjugate transpose of A (p. 288)

Re A

real part of A ∈ Fn×m (p. 288)

Im A

imaginary part of A ∈ Fn×m (p. 288)

T

0

  (p. 284) 

xxx

S

SPECIAL SYMBOLS

{A : A ∈ S} or {A : A ∈ S}ms (p. 288)

∗

AT

{A∗ : A ∈ S} or {A∗ : A ∈ S}ms (p. 288) ˆ TIˆ reverse transpose of A (p. 288) IA

A∗ˆ

ˆ ∗Iˆ reverse complex conjugate transpose of A (p. 288) IA

S

ˆ

Tˆ

S

{AT : A ∈ S} or {AT : A ∈ S}ms (p. 289)

ˆ

S∗ˆ

{A∗ˆ : A ∈ S} or {A∗ˆ : A ∈ S}ms (p. 289)

|x|

absolute value of x ∈ Fn (p. 289)

|A|

absolute value of A ∈ Fn×n (p. 289)

sign x

sign of x ∈ Rn (p. 289)

sign A

sign of A ∈ Rn×n (p. 289)

S⊥

orthogonal complement of S (p. 289)

polar S

polar of S (p. 290)

dcone S

dual cone of S (p. 290)

R(A)

range of A (p. 290)

N(A)

null space of A (p. 291)

rank A

rank of A (p. 292)

def A

defect of A (p. 292)

AL

left inverse of A (p. 294)

R

A

ˆ

right inverse of A (p. 294)

−1

inverse of A (p. 297)

−T

(AT )−1 (p. 298)

−∗

A

(A∗ )−1 (p. 298)

det A

determinant of A (p. 299)

A

A

A

A

adjugate of A (p. 301)

↓

vector with components of x in decreasing order (p. 305)

x↑

vector with components of x in increasing order (p. 305)

x

w

x≺y s

x≺y wlog

x ≺ y slog

x ≺ y Chapter 4

diag(a1 , . . . , an )

y weakly majorizes x (p. 305) y strongly majorizes x (p. 305) y weakly log majorizes x (p. 305) y strongly log majorizes x (p. 305)

 a1   

0 ..

.

0

an

   (p. 365) 

diag(x)

diag(x(1) , . . . , x(n) ), where x ∈ Fn (p. 365)

revdiag(a1 , . . . , an )

0   

an

a1  .

.. 0

  (p. 365) 

xxxi

SPECIAL SYMBOLS

revdiag(x)

revdiag(x(1) , . . . , x(n) ), where x ∈ Fn (p. 365)

diag(A1 , . . . , Ak )

A  1  block-diagonal matrix   0 [0 I] n −In 0 (p. 367)

J2n , J

0 ..

. Ak

   n ×m , where Ai ∈ F i i (p. 365)

glF (n), plC (n), slF (n), u(n), su(n), so(n), sympF (2n), osympF (2n), affF (n), seF (n), transF (n)

Lie algebras (p. 369)

S1 ≃ S2

the groups S1 and S2 are isomorphic (p. 372)

GLF (n), PLF (n), groups (p. 372) SLF (n), U(n), O(n), U(n, m), O(n, m), SU(n), SO(n), P(n), A(n), D(n), C(n), SympF (2n), OSympF (2n), AffF (n), SEF (n), TransF (n) A⊥

complementary idempotent matrix or projector I − A of A (p. 374)

ind A

index of A (p. 375)

rs

A≤B

rank subtractivity partial ordering (p. 430, Fact 4.30.3)

A≤B

star partial ordering (p. 431, Fact 4.30.8)

H

quaternions (p. 437, Fact 4.32.1)

Sp(n)

symplectic group in H (p. 438, Fact 4.32.4)

∗

Chapter 6

F[s]

polynomials with coefficients in F (p. 499)

deg p

degree of p ∈ F[s] (p. 499)

mroots(p)

multiset of roots of p ∈ F[s] (p. 499)

roots(p)

set of roots of p ∈ F[s] (p. 499)

mult p (λ)

multiplicity of λ as a root of p ∈ F[s] (p. 499)

δ(p)

spread of p (p. 500)

ρi (p)

ith largest root modulus of p (p. 500)

ρmin (p)

minimum root modulus of p (p. 500)

ρmax (p)

root radius of p (p. 500)

αi (p)

ith largest root real part of p (p. 500)

αmin (p)

minimum root real part of p (p. 500)

αmax (p)

root real abscissa of p (p. 500)

βi (p)

ith largest root imaginary part of p (p. 500)

βmin (p)

minimum root imaginary part of p (p. 500)

xxxii

SPECIAL SYMBOLS

βmax (p)

root imaginary abscissa of p (p. 500)

F[s]

n × m matrices with entries in F[s] (n × m polynomial matrices with coefficients in F) (p. 501)

n×m

rank P

rank of P ∈ F[s]n×m (p. 502)

Szeros(P)

set of Smith zeros of P ∈ F[s]n×m (p. 504)

mSzeros(P)

multiset of Smith zeros of P ∈ F[s]n×m (p. 504)

χA

characteristic polynomial of A (p. 506)

λi (A)

ith largest eigenvalue of A ∈ Fn×n having real eigenvalues (p. 506)

λmax (A)

largest eigenvalue λ1 (A) of A ∈ Fn×n having real eigenvalues (p. 506)

λmin (A)

smallest eigenvalue λn (A) of A ∈ Fn×n having real eigenvalues (p. 506)    λ1 (A)   .  vector  ..  of eigenvalues of A ∈ Fn×n having real eigenvalues (p. 506)   λn (A)

λ(A) amultA (λ)

algebraic multiplicity of λ ∈ spec(A) (p. 506)

spec(A)

spectrum of A (p. 506)

mspec(A)

multispectrum of A (p. 506)

δ(A)

spread of A (p. 510)

ρi (A)

ith largest spectral modulus of A (p. 510)

ρmax (A)

spectral radius of A (p. 510)

ρmin (A)

minimum spectral modulus of A (p. 510)    ρ1 (A)   .  vector  ..  of spectral moduli of A ∈ Fn×n (p. 510)   ρn (A)

ρ(A) gmultA (λ)

geometric multiplicity of λ ∈ spec(A) (p. 511)

αi (A)

ith largest spectral real part of A (p. 510)

αmax (A)

spectral abscissa of A (p. 510)

αmin (A)

minimum spectral real part of A (p. 510)    α1 (A)    .  vector  ..  of spectral real parts of A ∈ Fn×n (p. 510)   αn (A)

α(A) βi (A)

ith largest spectral imaginary part of A (p. 510)

βmax (A)

spectral imaginary abscissa of A (p. 511)

βmin (A)

minimum spectral imaginary part of A (p. 510)    β1 (A)   .  vector  ..  of spectral imaginary parts of A ∈ Fn×n (p. 511)   βn (A)

β(A) ν−(A), ν0 (A), ν+(A)

number of eigenvalues of A counting algebraic multiplicity having negative, zero, and positive real part, respectively (p. 511)

xxxiii

SPECIAL SYMBOLS

In A

   ν−(A)    inertia  ν0 (A)  of A (p. 511)   ν+(A)

sig A

signature of A; that is, ν+(A) − ν−(A) (p. 511)

µA

minimal polynomial of A (p. 512)

F(s)

rational functions with coefficients in F (SISO rational transfer functions) (p. 513)

F(s)prop

proper rational functions with coefficients in F (SISO proper rational transfer functions) (p. 513)

reldeg g

relative degree of g ∈ F(s)prop (p. 513)

F(s)n×m

n × m matrices with entries in F(s) (MIMO rational transfer functions) (p. 514)

n×m F(s)prop

n × m matrices with entries in F(s)prop (MIMO proper rational transfer functions) (p. 514)

reldeg G

relative degree of G ∈ F(s)n×m prop (p. 514)

rank G

rank of G ∈ F(s)n×m (p. 514)

poles(G)

set of poles of G ∈ F(s)n×m (p. 514)

bzeros(G)

set of blocking zeros of G ∈ F(s)n×m (p. 514)

McdegG

McMillan degree of G ∈ F(s)n×m (p. 515)

tzeros(G)

set of transmission zeros of G ∈ F(s)n×m (p. 515)

mpoles(G)

multiset of poles of G ∈ F(s)n×m (p. 515)

mtzeros(G)

multiset of transmission zeros of G ∈ F(s)n×m (p. 515)

mbzeros(G)

multiset of blocking zeros of G ∈ F(s)n×m (p. 515)

B(p, q)

Bezout matrix of p, q ∈ F[s] (p. 519, Fact 6.8.8)

H(g)

Hankel matrix of g ∈ F(s) (p. 520, Fact 6.8.10)

Chapter 7

C(p)

companion matrix for monic polynomial p (p. 546)

Jl (q)

l × l or 2l × 2l real Jordan matrix (p. 549)

Hl (q)

l × l or 2l × 2l hypercompanion matrix (p. 550)

σi (A)

ith largest singular value of A ∈ Fn×m (p. 555)    σ1 (A)    ..  of singular values of A ∈ Fn×m (p. 556) vector  .    σmin {n,m} (A)

σ(A) msval(A)

multiset {σ1 (A), . . . , σmin {n,m} (A)}ms of singular values of A ∈ Fn×m (p. 556)

σmax (A)

largest singular value σ1 (A) of A ∈ Fn×m (p. 556)

σmin (A)

minimum singular value σn (A) of a square matrix A ∈ Fn×n (p. 556)

indA (λ)

index of λ with respect to A (p. 558)

PA,B

pencil of (A, B), where A, B ∈ Fn×n (p. 563)

spec(A, B)

generalized spectrum of (A, B), where A, B ∈ Fn×n (p. 563)

xxxiv

SPECIAL SYMBOLS

mspec(A, B)

generalized multispectrum of (A, B), where A, B ∈ Fn×n (p. 563)

χA,B

characteristic polynomial of (A, B), where A, B ∈ Fn×n (p. 564)

V(λ1, . . . , λn )

Vandermonde matrix (p. 613, Fact 7.18.3)

circ(a0 , . . . , an−1 )

circulant matrix of a0 , . . . , an−1 ∈ F (p. 614, Fact 7.18.13)

Chapter 8

A+

(Moore-Penrose) generalized inverse of A (p. 621)

D|A

Schur complement of D with respect to A (p. 625)

AD

Drazin generalized inverse of A (p. 625)

#

A

group generalized inverse of A (p. 627) #

A≤B c

A≤B

sharp partial ordering (p. 648, Fact 8.5.16) core partial ordering (p. 648, Fact 8.5.17)

Chapter 9

vec A −1

vec

vector formed by stacking columns of A (p. 681) A

inverse vec operator (p. 681)

⊗

Kronecker product (p. 681)

Pn,m

Kronecker permutation matrix (p. 683)

⊕

Kronecker sum (p. 683)

A⊙B

Schur product of A and B (p. 685)

A⊙α

Schur power of A, (A⊙α )(i, j) = (A(i, j) )α (p. 685)

Chapter 10

Hn

n × n Hermitian matrices (p. 703)

Nn

n × n positive-semidefinite matrices (p. 703)

P

n

n × n positive-definite matrices (p. 703)

A≥B

A − B ∈ Nn (p. 703)

A>B

A − B ∈ Pn (p. 703)

⟨A⟩

(A∗A)1/2 (p. 714)

A#B

geometric mean of A and B (p. 743, Fact 10.11.68)

A#α B

generalized geometric mean of A and B (p. 745, Fact 10.11.72)

A:B

parallel sum of A and B (p. 818, Fact 10.24.20)

sh(A, B)

shorted operator (p. 819, Fact 10.24.21)

Chapter 11

(

∥x∥ p

H¨older norm

∥A∥ p

H¨older norm

n ∑

)1/p |x(i) | p

(i=1 n,m ∑

i, j=1

(p. 833) )1/p |A(i, j) | p (p. 836)

xxxv

SPECIAL SYMBOLS

∥A∥F ∥A∥ p|q ∥A∥σp

√ tr A∗A (p. 836)

 

 ∥col1 (A)∥ p 



 .. 

(p. 836) mixed H¨older norm

  .

 

∥colm (A)∥ p 

q

Frobenius norm

Schatten norm ∥σ(A)∥ p (p. 837)

∥A∥q,p

H¨older-induced norm (p. 842)

∥A∥ p,q,F

H¨older-induced norm over F (p. 842)

∥A∥col

column norm ∥A∥1,1 = ∥A∥1|∞ = max j∈{1,...,m} ∥col j (A)∥1 (p. 844)

∥A∥row

row norm ∥A∥∞,∞ = ∥AT ∥1|∞ = maxi∈{1,...,n} ∥rowi (A)∥1 (p. 844)

ℓ(A)

induced lower bound of A (p. 845)

ℓq,p (A)

H¨older-induced lower bound of A (p. 846)

∥ · ∥D

dual norm (p. 859, Fact 11.8.24)

Chapter 12

Bε (x)

open ball of radius ε centered at x (p. 913)

Sε (x)

sphere of radius ε centered at x (p. 913)

int S

interior of S (p. 913)

intS′ S

interior of S relative to S′ (p. 913)

relint S

interior of S relative to affin S (p. 913)

cl S

closure of S (p. 914)

lim S

set of limit points of S (p. 914)

clS′ S

closure of S relative to S′ (p. 914)

bd S

boundary of S (p. 915)

bdS′ S

boundary of S relative to S′ (p. 915)

relbd S

boundary of S relative to affin S (p. 915)

fcone D

feasible cone of D (p. 924)

D+ f (x0 ; ξ)

one-sided directional derivative of f at x0 in the direction ξ (p. 924)

∂f (x0 ) ∂x(i) ′

partial derivative of f with respect to x(i) at x0 (p. 924)

f (x)

derivative of f at x (p. 925)

df (x0 ) dx(i) (k)

f ′(x0 ) (p. 925)

f (x)

d+ f (x0 ) dx(i) d− f (x0 ) dx(i)

Sign(A)

kth derivative of f at x (p. 926) right one-sided derivative (p. 926) left one-sided derivative (p. 926) matrix sign of A ∈ Cn×n (p. 932)

xxxvi

SPECIAL SYMBOLS

Chapter 14

eA or exp(A)

matrix exponential (p. 1179)

L

Laplace transform (p. 1181)

Ss(A)

asymptotically stable subspace of A (p. 1199)

Su(A)

unstable subspace of A (p. 1199)

Chapter 15

U(A, C)

unobservable subspace of (A, C) (p. 1254)  C   CA   CA. 2  (p. 1254)  .  .n−1

O(A, C)

CA

C(A, B)

controllable subspace of (A, B) (p. 1260)

K(A, B) [ A G∼

B

C

[B AB A2B · · · An−1B] (p. 1260)

]

state space realization of G ∈ F(s)l×m prop (p. 1270)

0

Hi, j,k (G) H(G) min

G ∼ H

[

Markov block-Hankel matrix Oi (A, C)K j (A, B) (p. 1274)

A

B

C

0

]

Markov block-Hankel matrix O(A, C)K(A, B) (p. 1275) minimal state space realization of G ∈ F(s)l×m prop (p. 1276) [A Σ ] Hamiltonian R1 −AT (p. 1297)

Conventions, Notation, and Terminology For convenience and clarity, this section summarizes conventions, notation, and terminology used in this book. Precise definitions are given in the main text. Italic font is used to indicate that a word is being defined. All definitions of words, phrases, and symbols are “if and only if” statements, although for brevity △ △ “only if” is omitted. The symbol = means equal by definition, where A = B means that the left-hand expression A is defined to be the right-hand expression B. A mathematical object defined by means of a constructive procedure is well-defined if the construction produces a unique object. A hypothesis established by the words “let” or “assume” is valid for all statements in the remainder of the paragraph, which may be a theorem, proposition, lemma, corollary, or fact. A hypothesis established by the word “if” is valid only within that sentence. This convention applies to both implications in an “if and only if” statement. Every theorem, proposition, lemma, corollary, and fact consists of exactly one paragraph. A proof may consist of multiple paragraphs. The statements in a proof are valid only within the proof. No statement in a proof is accessed from outside the proof. The end of a proof is denoted by “.” An example may consist of multiple paragraphs. The end of an example is denoted by “^.” Three types of existence statements are used. Existence is the statement “there exists x ∈ X”; equivalently, “there exists at least one x ∈ X.” Existence with uniqueness is the statement “there exists a unique x ∈ X”; equivalently, “there exists exactly one x ∈ X.” Existence with pre-uniqueness is the statement “there exists at most one x ∈ X.” The phrases “for all,” “for every,” and “for each” are synonymous. The words “always,” “any,” “provided,” “some,” “unless,” “when,” and “whenever” are not used for mathematical statements in this book. Analogous statements are written in parallel using the following style: If n is (even, odd), then n + 1 is (odd, even). If n is (even, odd), then so is n + 2. n is (even, odd) if and only if n + 2 is. N denotes {0, 1, 2, 3, . . .}, and P denotes {1, 2, 3, . . .}, where “N” and “P” denote “nonnegative” and “positive,” respectively. Traditionally, “N” denotes the natural numbers {1, 2, 3, . . .}. Unless stated otherwise, the variables i, j, k, l, m, n denote integers. Hence, k ≥ 0 denotes a nonnegative integer, k ≥ 1 denotes a positive integer, and the limit limk→∞ Ak is taken over positive integers. √ ˆ The imaginary unit −1 is denoted by dotless ȷ. The unit quaternions are denoted by 1, ıˆ, ȷˆ, k. Unless stated otherwise, the letter s represents a complex scalar. The letter z may or may not represent a complex scalar.

xxxviii

CONVENTIONS, NOTATION, AND TERMINOLOGY

A line over a variable has dual meaning, distinguishable from context. In particular, z is the complex conjugate of z, whereas xn denotes the n-term rising factorial. The inequalities c ≤ a ≤ d and c ≤ b ≤ d are written simultaneously as { } a c≤ ≤ d. b The inequalities a ≤ b ≤ c ≤ d ≤ e may be written as a≤b≤c ≤ d ≤ e. z

z

xy + z denotes (xy) + z, x/y + z denotes (x/y) + z, xy denotes x(y ) , sin x + y denotes (sin x) + y, ∑ sin xy + z denotes (sin xy) + z, log x + y denotes (log x) + y, log xy + z denotes (log xy) + z, xi yi ∑ ∑ ∑ denotes (xi yi ), and xi + y denotes ( xi ) + y. x/yz is ambiguous and is not used. For clarity, we sometimes write log(x sin y) for log x sin y, although the parentheses are superfluous. sin(x+y) sin(x−y) denotes [sin(x+y)] sin(x−y); det(A+ B) det(A− B) denotes [det(A+ B)] det(A− B). ∑ ∏ △ △ i∈∅ = 0, and i∈∅ = 1. The prefix “non” means “not” in the words nonconstant, nonempty, nonintegral, nonnegative, nonreal, nonrepeated, nonsingular, nonsquare, nonunique, and nonzero. In some traditional usage, “non” may mean “not necessarily.” “Increasing” and “decreasing” indicate strict change for a change in the argument. The word “strict” is superfluous, and thus is omitted. Nonincreasing means nowhere increasing, while nondecreasing means nowhere decreasing. A set can have a finite or infinite number of elements. A finite set has a finite number of elements. A multiset can have repeated elements with at most finite multiplicity. Hence, the multisets {x}ms and {x, x}ms are different. The listed elements α, β, γ of the set {α, β, γ} need not be distinct. For example, {α, β, α} = {α, β}. However, in a definition such as “Let spec(A) = {λ1 , . . . , λr },” the listed elements λ1 , . . . , λr are assumed to be distinct. A set is a collection of distinct objects without a specified ordering. A multiset is a collection of possibly repeated objects without a specified ordering. A tuple can be viewed as an ordered multiset, which is a collection of possibly repeated objects with a specified ordering. Consequently, the order in which the elements of the set {x1 , . . . , xn } and the elements of the multiset {x1 , . . . , xn }ms are listed has no significance. The components of the n-tuple (x1 , . . . , xn ) are ordered. (xi )∞ i=1 denotes the sequence (x1 , x2 , . . .). A sequence can be viewed as a tuple with a countably infinite number of components, where the order of the components is specified and the components need not be distinct. For clarity, square brackets alternate with parentheses. For example, f [g(x)] denotes f (g(x)). S1 ⊂ S2 means that S1 is a proper subset of S2 , whereas S1 ⊆ S2 means that S1 is either a proper subset of S2 or is equal to S2 . Hence, S1 ⊂ S2 is equivalent to both S1 ⊆ S2 and S1 , S2 , while S1 ⊆ S2 is equivalent to either S1 ⊂ S2 or S1 = S2 . The word “graph” corresponds to what is commonly called a “simple directed graph,” while “symmetric graph” corresponds to a “simple undirected graph.”

xxxix

CONVENTIONS, NOTATION, AND TERMINOLOGY

The set-valued inverse of the function f : X 7→ Y is denoted by f inv . If f is invertible, then its inverse is denoted by f Inv . The traditional notation f −1 is not used in order to avoid ambiguity with 1/f. △ If f : X 7→ Y is one-to-one, then fˆ: X 7→ f (X), where, for all x ∈ X, fˆ(x) = f (x), is onto and thus invertible. For convenience and with a slight abuse of notation, we write f Inv : f (X) 7→ X. If f is not onto, then it does not have a unique left inverse. However, each left inverse restricted to f (X) is an inverse of fˆ. Consequently, all left inverses of f are identical on f (X). This observation explains why, if A is left invertible and b ∈ R(A), then, despite the fact that A has infinitely many left inverses, every left inverse of A yields the unique solution of Ax = b.

The matrix A ∈ Fn×m may be square, tall, or wide depending on whether m = n, m < n, or m > n, respectively. If A is tall, then it is left invertible if and only if it has m linearly independent rows or columns; if A is wide, then it is right invertible if and only if it has n linearly independent rows or columns. It is helpful to keep this distinction in mind. The inverse trigonometric functions are denoted by asin, acos, atan, acsc, asec, acot, while the inverse hyperbolic functions are denoted by asinh, acosh, atanh, acsch, asech, acoth . The inverse tangent function with two arguments is denoted by atan2. For real arguments, the domain of asin and acos is [−1, 1], the domain of acsc and asec is (−∞, −1]∪ [1, ∞), and the domain of atan and acot is R. Furthermore, the range of asin is [− π2 , π2 ], the range of acos is [0, π], the range of atan is (− π2 , π2 ), the range of acsc is [− π2 , 0) ∪ (0, π2 ], the range of asec is [0, π2 ) ∪ ( π2 , π], and the range of acot is (− π2 , 0) ∪ (0, π2 ]. The principal argument of the nonzero complex number z is denoted by arg z ∈ (−π, π]. Likewise, the principal logarithm of the nonzero complex number z is denoted by log z. Traditional notation (not followed in this book) is to use “arg” and “log” to denote set-valued functions, and “Arg” and “Log” to denote principal functions. Set-valued inverse functions are denoted by f inv . For example, sininv (0) = {iπ : i ∈ Z}. √ △ √ For n ≥√1, (−1)1/n = e(π/n) ȷ . However, for all a < 0 and odd n ≥ 1, n a = − n |a|. Hence, √3 all a < 0 and −1 = −1 , 21 (1 + 3 ȷ) = e(π/3) ȷ = (−1)1/3 . The angle between two vectors is an element of [0, π]. Therefore, by using acos, the inner product of two vectors can be used to compute the angle between two vectors. △

△

△

0! = 1, 0/0 = (sin 0)/0 = (1 − cos 0)/0 = (sinh 0)/0 = 1, and 1/∞ = 0. For all z ∈ C and k ∈ Z,

In particular, if n, k ∈ N, then

Hence,

 z(z − 1) · · · (z − k + 1)   ( )  , k > 0,   k! z △   = 1, k = 0,   k    0, k < 0. n! ( )     , n ≥ k ≥ 0, n  (n − k)!k! =   k 0, k > n ≥ 0. ( )    n 1, n ≥ 0, =  0, n < 0. n

xl

In particular,

CONVENTIONS, NOTATION, AND TERMINOLOGY

(0) 0

= 1. △

△

For all square matrices A, A0 = I . In particular, 00n×n = In . With this convention, we can write ∞ ∑ i=0

αi =

1 1−α

for all α ∈ (−1, 1). Of course, lim x↓0 0 = 0, lim x↑0 x0 = lim x↓0 x0 = 1, and lim x↑0 x x = lim x↓0 x x = 1. x

The symbols ∞ and −∞ are defined as limits of sequences of real numbers, and neither ∞ nor −∞ is a real number. The set of extended real numbers is R ∪ {−∞, ∞}. On the set of extended real numbers, we define ∞ + ∞ = ∞, −∞ − ∞ = −∞, ∞∞ = ∞, (−∞)∞ = ∞(−∞) = −∞, and (−∞)(−∞) = ∞. Furthermore, for all real numbers α, α − ∞ = −∞, α + ∞ = ∞, and α/∞ = 0, whereas, for all nonzero real numbers α, α∞ = sign(α)∞. Hence, for all α ∈ (0, ∞], α∞ = ∞, whereas, for all α ∈ [−∞, 0), α∞ = −∞. The expressions 0∞, ∞ − ∞, and ∞∞ are not defined. Finally, for all α ∈ (0, ∞), we define α/0 = ∞, whereas, for all α ∈ (−∞, 0), we define α/0 = −∞. See [154, pp. 14, 15] and [2249, p. 44]. Let a and b be real numbers such that a < b. A finite interval is of the form (a, b), [a, b), (a, b], or [a, b], whereas an infinite interval is of the form (−∞, a), (−∞, a], (a, ∞), [a, ∞), or (−∞, ∞). An interval is either a finite interval or an infinite interval. Neither the empty set nor a single point is an interval. An extended infinite interval includes either ∞ or −∞. For example, [−∞, a) = {−∞} ∪ (−∞, a), [−∞, a] = {−∞} ∪ (−∞, a], (a, ∞] = (a, ∞) ∪ {∞}, [a, ∞] = [a, ∞) ∪ {∞}, and [−∞, ∞] = {−∞} ∪ (−∞, ∞) ∪ {∞}. The symbol F denotes either R or C consistently in each theorem, proposition, lemma, corollary, and fact. For example, in Theorem 7.6.3, the three appearances of “F” can be read as either all “C” or all “R.” The imaginary numbers are denoted by IA. Hence, 0 is both a real number and an imaginary number. The notation Re A and Im A represents the real and imaginary parts of A, respectively. Some books use Re A and Im A to denote the Hermitian and skew-Hermitian matrices 12 (A + A∗ ) and 21 (A − A∗ ). For the scalar ordering “≤,” if x ≤ y, then x , y if and only if x < y. For vector and matrix orderings, the conditions x ≤ y and x , y do not imply x < y. Operations denoted by superscripts are applied before operations represented by preceding operators. For example, tr (A + B)2 means tr[(A + B)2 ], and cl S∼ means cl(S∼ ). This convention simplifies many expressions. An element of Fn is a column vector, which is also a matrix with one column. The components of x ∈ Fn can be viewed as coordinates. In more general settings, “vector” typically refers to a coordinate-free object. Sets have elements; vectors, tuples, and sequences have components; and matrices have entries. This terminology has no mathematical consequence. All matrices have nonnegative integral dimensions. A matrix that has either zero rows or zero columns is an empty matrix. The notation x(i) denotes the ith component of the vector x. The entries of a submatrix Aˆ of a matrix A are the entries of A located in specified rows and columns of A. The submatrix Aˆ of A is a block of A if Aˆ is a submatrix of A whose entries are entries of adjacent rows and adjacent columns of A. Every matrix is both a submatrix and block of itself.

xli

CONVENTIONS, NOTATION, AND TERMINOLOGY

A(i, j) denotes the scalar (i, j) entry of A. Ai, j or Ai j denotes a block or submatrix of A. A(S1 ,S2 ) denotes the submatrix of A formed by retaining the rows of A listed in S1 and the columns of A listed in S2 . A(S) denotes A(S,S) . A[S1 ,S2 ] denotes the submatrix of A formed by deleting the rows of A listed in S1 and the columns of A listed in S2 . A[S] denotes A[S,S] . A[i, j] denotes the submatrix of A obtained by deleting rowi (A) and col j (A). A[S,·] denotes the submatrix of A obtained by deleting the rows of A listed in S, and A[·,S] denotes the submatrix of A obtained by deleting the columns of A listed in S. A[i,·] denotes the submatrix of A obtained by deleting rowi (A), and A[·, j] denotes the submatrix of A obtained by deleting col j (A). The determinant of a square submatrix is a subdeterminant. Some books use “minor.” The determinant of a matrix is also a subdeterminant. The dimension of the null space of a matrix is its defect. Some books use “nullity.” A block of a square matrix is diagonally located if the block is square and the diagonal entries of the block are also diagonal entries of the matrix; otherwise, the block is off-diagonally located. This terminology avoids confusion with a “diagonal block,” which is a block that is also a square, diagonal submatrix. [ ] For the partitioned matrix CA DB ∈ F(n+m)×(k+l), it can be inferred that A ∈ Fn×k and similarly for B, C, and D. The Schur product of matrices A and B is denoted by A ⊙ B. Matrix multiplication is given priority over Schur multiplication; that is, A ⊙ BC means A ⊙ (BC). The adjugate of A ∈ Fn×n is denoted by AA . The traditional notation is adj A, while the notation AA is used in [2537]. If A ∈ F is a scalar, then AA = 1. In particular, 0A1×1 = 1. However, for all n ≥ 2, 0An×n = 0n×n . If F = R, then A becomes A, A∗ becomes AT, “Hermitian” becomes “symmetric,” “unitary” becomes “orthogonal,” “unitarily” becomes “orthogonally,” and “congruence” becomes “T-congruence.” The square complex matrix A is symmetric if AT = A and orthogonal if ATA = I. The diagonal entries of A ∈ Fn×n , all of whose diagonal entries are real, are ordered as dmax (A) = d1 (A) ≥ d2(A) ≥ · · · ≥ dn (A) = dmin (A). Every n × n matrix has n eigenvalues. Hence, eigenvalues are counted in accordance with their algebraic multiplicity. The phrase “distinct eigenvalues” ignores algebraic multiplicity. The eigenvalues of A ∈ Fn×n , all of whose eigenvalues are real, are ordered as λmax (A) = λ1 (A) ≥ λ2(A) ≥ · · · ≥ λn (A) = λmin (A). The inertia of A ∈ Fn×n is written as

   ν− (A)    In A =  ν0 (A)  .   ν+ (A) △

Some books use the notation (ν(A), δ(A), π(A)). For A ∈ Fn×n, amultA(λ) is the number of copies of λ in the multispectrum of A, gmultA (λ) is the number of Jordan blocks of A associated with λ, and indA(λ) is the size of the largest Jordan block of A associated with λ. The index of A, denoted by ind A = indA(0), is the size of the largest Jordan block of A associated with the eigenvalue 0.

xlii

CONVENTIONS, NOTATION, AND TERMINOLOGY

A ∈ Cn×n is semisimple if the size of every Jordan block of A is 1. Defective means not semisimple. A ∈ Cn×n is cyclic if A has exactly one Jordan block associated with each distinct eigenvalue. Derogatory means not cyclic. A ∈ Fn×n is diagonalizable over F if it can be transformed into a diagonal matrix whose entries are in F by means of a similarity transformation whose entries are in F. Therefore, A ∈ Cn×n is diagonalizable over C if and only if it is semisimple. Furthermore, A ∈ Rn×n is diagonalizable [ ] 0 1 is over R if and only if it is semisimple and all of its eigenvalues are real. The real matrix −1 0 diagonalizable over C, but not diagonalizable over R. A ∈ Fn×m has exactly min {n, m} singular values, exactly rank A of which are positive. △

The min {n, m} singular values of the matrix A ∈ Fn×m are ordered as σmax (A) = σ1 (A) ≥ σ2 (A) ≥ △ · · · ≥ σmin {n,m} (A) ≥ 0. If n = m, then σmin(A) = σn (A). The notation σmin(A) is defined only for square matrices. By definition, positive-semidefinite and positive-definite matrices are Hermitian. An idempotent matrix A ∈ Fn×n satisfies A2 = A, while a projector is a Hermitian, idempotent matrix. Some books use “projector” for idempotent and “orthogonal projector” for projector. A reflector is a Hermitian, involutory matrix. A projector is a normal matrix each of whose eigenvalues is 1 or 0, while a reflector is a normal matrix each of whose eigenvalues is 1 or −1. An elementary matrix is a nonsingular matrix formed by adding an outer-product matrix to the identity matrix. An elementary reflector is a reflector exactly one of whose eigenvalues is −1. An elementary projector is a projector exactly one of whose eigenvalues is 0. Elementary reflectors are elementary matrices. However, elementary projectors are not elementary matrices since elementary projectors are singular. A range-Hermitian matrix is a square matrix whose range is equal to the range of its complex conjugate transpose. These matrices are sometimes called EP matrices. The polynomials 1 and s3 + 5s2 − 4 are monic. The zero polynomial is not monic. The rank of the polynomial matrix P is the maximum rank of P(s) over all s ∈ C. This quantity is also called the normal rank. We denote this quantity by rank P as distinct from rank P(s), which denotes the rank of the matrix P(s). The rank of the rational transfer function G is the maximum rank of G(s) over all s ∈ C excluding poles of the entries of G. This quantity is also called the normal rank. We denote this quantity by rank G as distinct from rank G(s), which denotes the rank of the matrix G(s). ⊕ denotes the Kronecker sum. Some books use ⊕ to denote a direct sum of matrices or subspaces. |A| represents the matrix obtained by replacing every entry of A by its absolute value. ⟨A⟩ represents the matrix (A∗A)1/2. Some books use |A| to denote this matrix. Statements about vector norms on Fnm apply to matrix norms on Fn×m , and vice versa. The H¨older norm of A ∈ Fn×m is denoted by ∥A∥ p . The matrix norm induced by ∥ · ∥q on the domain and ∥ · ∥ p on the codomain is denoted by ∥ · ∥ p,q . The Schatten p norm of A ∈ Fn×m is denoted by ∥A∥σp , and the Frobenius norm of A is denoted by ∥A∥F . Hence, ∥A∥σ∞ = ∥A∥2,2 = σmax(A), ∥A∥σ2 = ∥A∥F , and ∥A∥σ1 = tr ⟨A⟩. Unitarily invariant norms are not necessarily normalized.

CONVENTIONS, NOTATION, AND TERMINOLOGY

xliii

Terminology Relating to Inequalities Let “≤” be a partial ordering, let X be a set, and consider the inequality f (x) ≤ g(x) for all x ∈ X.

(1)

Inequality (1) is sharp if there exists x0 ∈ X such that f (x0 ) = g(x0 ). The inequality f (x) ≤ f (y) for all x ≤ y

(2)

f (x) ≤ p(x) ≤ g(x) for all x ∈ X,

(3)

is a monotonicity result. The inequality

where p is not identically equal to either f or g on X, is an interpolation or refinement of (1). The inequality g(x) ≤ α f (x) for all x ∈ X,

(4)

where α > 1, is a reversal of (1). The inequality g(x) ≤ f (x) + α for all x ∈ X,

(5)

where α > 0, is a reversal of (1). △

Defining h(x) = g(x) − f (x), it follows that (1) is equivalent to h(x) ≥ 0 for all x ∈ X.

(5)

Now, suppose that h has a global minimizer x0 ∈ X. Then, (5) implies that 0 ≤ h(x0 ) = min h(x) ≤ h(y) for all y ∈ X. x∈X

(6)

Consequently, inequalities are often expressed equivalently in terms of minimizers, and vice versa. Many inequalities are based on a function that is either monotonic or convex.

Scalar, Vector, and Matrix Mathematics

Chapter One Sets, Logic, Numbers, Relations, Orderings, Graphs, and Functions In this chapter we review basic terminology and results concerning sets, logic, numbers, relations, orderings, graphs, and functions. This material is used throughout the book.

1.1 Sets A set {x, y, . . .} is a collection of elements. A set can include either a finite or infinite number of elements. The set X is finite if it has a finite number of elements; otherwise, X is infinite. The set X is countably infinite if X is infinite and its elements are in one-to-one correspondence with the positive integers. The set X is countable if it is either finite or countably infinite. Let X be a set. Then, x∈X

(1.1.1)

means that x is an element of X. If w is not an element of X, then we write w < X.

(1.1.2)

No set can be an element of itself. Therefore, there does not exist a set that includes every set. The set with no elements, denoted by ∅, is the empty set. If X , ∅, then X is nonempty. Let X and Y be sets. The intersection of X and Y is the set of common elements of X and Y, which is given by △

X ∩ Y = {x: x ∈ X and x ∈ Y} = {x ∈ X: x ∈ Y} = {x ∈ Y: x ∈ X} = Y ∩ X,

(1.1.3)

The union of X and Y is the set of elements in either X or Y, which is the set △

X ∪ Y = {x: x ∈ X or x ∈ Y} = Y ∪ X.

(1.1.4)

The complement of X relative to Y is △

Y\X = {x ∈ Y: x < X}.

(1.1.5)

If Y is specified, then the complement of X is △

X∼ = Y\X.

(1.1.6)

The symmetric difference of X and Y is the set of elements that are in either X or Y but not both, which is given by △

X ⊖ Y = (X ∪ Y)\(X ∩ Y).

(1.1.7)

If x ∈ X implies that x ∈ Y, then X is a subset of Y (equivalently, Y contains X), which is written as

Equivalently,

X ⊆ Y.

(1.1.8)

Y ⊇ X.

(1.1.9)

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Note that X ⊆ Y if and only if X\Y = ∅. Furthermore, X = Y if and only if X ⊆ Y and Y ⊆ X. If X ⊆ Y and X , Y, then X is a proper subset of Y and we write X ⊂ Y. The sets X and Y are disjoint if X ∩ Y = ∅. A partition of X is a set of pairwise-disjoint and nonempty subsets of X whose union is equal to X. The symbols N, P, Z, Q, and R denote the sets of nonnegative integers, positive integers, integers, rational numbers, and real numbers, respectively. A set cannot have repeated elements. Therefore, {x, x} = {x}. A multiset is a finite collection of elements that allows for repetition. The multiset consisting of two copies of x is written as {x, x}ms . For example, the roots of the polynomial p(x) = (x − 1)2 are the elements of the multiset {1, 1}ms , while the prime factors of 72 are the elements of the multiset {2, 2, 2, 3, 3}ms . The operations “∩,” “∪,” “\,” “⊖,” and “×” and the relations “⊂” and “⊆” extend to multisets. For example, {x, x}ms ∪ {x}ms = {x, x, x}ms .

(1.1.10)

By ignoring repetitions, a multiset can be converted to a set, while a set can be viewed as a multiset with distinct elements. The Cartesian product X1 × · · · × Xn of sets X1, . . . , Xn is the set consisting of tuples of the form (x1 , . . . , xn ), where, for all i ∈ {1, . . . , n}, xi ∈ Xi . A tuple with n components is an n-tuple. The components of a tuple are ordered but need not be distinct. Therefore, a tuple can be viewed as an ordered multiset. We thus write  △ (x1 , . . . , xn ) ∈ ni=1 Xi = X1 × · · · × Xn . (1.1.11)  Xn denotes ni=1 X. Definition 1.1.1. A sequence (xi )∞ i=1 = (x1 , x2 , . . .) is a tuple with a countably infinite number of components. Now, let i1 < i2 < · · · . Then, (xi j )∞j=1 is a subsequence of (xi )∞ i=1 . △

Let X be a set, and let X = (xi )∞ i=1 be a sequence whose components are elements of X; that is, {x1 , x2 , . . .} ⊆ X. For convenience, we write either X ⊆ X or X ⊂ X, where X is viewed as a set and the multiplicity of the components of the sequence is ignored. For sequences X, Y ⊂ Fn , define △ △ ∞ X + Y = (xi + yi )∞ i=1 and X ⊙ Y = (xi ⊙ yi )i=1 , where “⊙” denotes component-wise multiplication. In △ the case n = 1, we define XY = (xi yi )∞ i=1 .

1.2 Logic Every statement is either true or false, and no statement is both true and false. A proof is a collection of statements that verify that a statement is true. A conjecture is a statement that is believed to be true but whose proof is not known. Let A and B be statements. The not of A is the statement (not A), the and of A and B is the statement (A and B), and the or of A and B is the statement (A or B). The statement (A or B) does not contradict the statement (A and B); hence, the word “or” is inclusive. The exclusive or of A and B is the statement (A xor B), which is [(A and not B) or (B and not A)]. Equivalently, (A xor B) is the statement [(A or B) and not(A and B)], that is, A or B, but not both. Note that (A and B) = (B and A), (A or B) = (B or A), and (A xor B) = (B xor A). Let A, B, and C be statements. Then, the statements (A and B or C) and (A or B and C) are ambiguous. For clarity, we thus write, for example, [A and (B or C)] and [A or (B and C)]. In words, we write “A and either B or C” and “A or both B and C,” respectively, where “either” and “both” signify parentheses. Furthermore, (A and B) or C = (A and C) or (B and C),

(1.2.1)

(A or B) and C = (A or C) and (B or C).

(1.2.2)

SETS, LOGIC, NUMBERS, RELATIONS, ORDERINGS, GRAPHS, AND FUNCTIONS

3

Let A be a statement. To analyze statements involving logic operators, define truth(A) = 1 if A is true, and truth(A) = 0 if A is false. Then, truth(not A) = truth(A) + 1,

(1.2.3)

where 0 + 0 = 0, 1 + 0 = 0 + 1 = 1, and 1 + 1 = 0. Therefore, A is true if and only if (not A) is false, while A is false if and only if (not A) is true. Note that truth[not(not A)] = truth(not A) + 1 = [truth(A) + 1] + 1 = truth(A). Furthermore, note that truth(A) + truth(A) = 0 and truth(A) truth(A) = truth(A). Let A and B be statements. Then, truth(A and B) = truth(A) truth(B), truth(A or B) = truth(A) truth(B) + truth(A) + truth(B), truth(A xor B) = truth(A) + truth(B).

(1.2.4) (1.2.5) (1.2.6)

truth(A and B) = min {truth(A), truth(B)}, truth(A or B) = max {truth(A), truth(B)}.

(1.2.7) (1.2.8)

Hence,

Consequently, truth(A and B) = truth(B and A), truth(A or B) = truth(B or A), and truth(A xor B) = truth(B xor A). Furthermore, truth(A and A) = truth(A or A) = truth(A), and truth(A xor A) = 0. Let A and B be statements. The implication (A =⇒ B) is the statement [(not A) or B]. Therefore, truth(A =⇒ B) = truth(A) truth(B) + truth(A) + 1.

(1.2.9)

The implication (A =⇒ B) is read as either “if A, then B,” “if A holds, then B holds,” or “A implies B.” The statement A is the hypothesis, while the statement B is the conclusion. If (A =⇒ B), then A is a sufficient condition for B, and B is a necessary condition for A. It follows from (1.2.9) that, if A and B are true, then (A =⇒ B) is true; if A is true and B is false, then (A =⇒ B) is false; and, if A is false, then (A =⇒ B) is true whether or not B is true. For example, both implications [(2 + 2 = 5) =⇒ (3 + 3 = 6)] and [(2 + 2 = 5) =⇒ (3 + 3 = 8)] are true. Finally, note that [(A =⇒ B) and A] = A and B. A predicate is a statement that depends on a variable. Let X be a set, let x ∈ X, and let A(x) be a predicate. There are two ways to use a predicate to create a statement. An existential statement has the form there exists x ∈ X such that A(x) holds,

(1.2.10)

whereas a universal statement has the form for all x ∈ X, A(x) holds.

(1.2.11)

Note that truth[there exists x ∈ X such that A(x) holds] = max truth[A(x)],

(1.2.12)

truth[for all x ∈ X, A(x) holds] = min truth[A(x)].

(1.2.13)

x∈X

x∈X

An argument is an implication whose hypothesis and conclusion are predicates that depend on the same variable. In particular, letting x denote a variable, and letting A(x) and B(x) be predicates,

4

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the implication [A(x) =⇒ B(x)] is an argument. For example, for each real number x, the implication [(x = 1) =⇒ (x + 1 = 2)] is an argument. Note that the variable x links the hypothesis and the conclusion, thereby making this implication useful for the purpose of inference. In particular, for all real numbers x, truth[(x = 1) =⇒ (x + 1 = 2)] = 1. The statements (for all x, [A(x) =⇒ B(x)] holds) and (there exists x such that [A(x) =⇒ B(x)] holds) are inferences. Let A and B be statements. The bidirectional implication (A ⇐⇒ B) is the statement [(A =⇒ B) and (A ⇐= B)], where (A ⇐= B) means (B =⇒ A). If (A ⇐⇒ B), then A and B are equivalent. Furthermore, truth(A ⇐⇒ B) = truth(A) + truth(B) + 1.

(1.2.14)

Therefore, A and B are equivalent if and only if either both A and B are true or both A and B are false. Let A and B be statements, and assume that (A ⇐⇒ B). Then, A holds if and only if B holds. The implication A =⇒ B (the “only if” part) is necessity, while B =⇒ A (the “if” part) is sufficiency. Let A and B be statements. The converse of (A =⇒ B) is (B =⇒ A). Note that (A =⇒ B) ⇐⇒ [(not A) or B] ⇐⇒ [(not A) or not(not B)] ⇐⇒ [not(not B) or not A] ⇐⇒ (not B =⇒ not A). Therefore, the statement (A =⇒ B) is equivalent to its contrapositive [(not B) =⇒ (not A)]. Let A, B, A′ , and B′ be statements, and assume that (A′ =⇒ A =⇒ B =⇒ B′ ). Then, (A′ =⇒ B′ ) is a corollary of (A =⇒ B). Let A, B, and A′ be statements, and assume that A =⇒ B. Then, (A =⇒ B) is a strengthening of [(A and A′ ) =⇒ B]. If, in addition, (A =⇒ A′ ), then the statement [(A and A′ ) =⇒ B] has a redundant assumption. An interpretation is a feasible assignment of true or false to all statements that comprise a statement. For example, there are four interpretations of the statement (A and B), depending on whether A is assigned to be true or false and B is assigned to be true or false. Likewise, [(x = 1) and (x = 2)] has three interpretations, which depend on the value of x. Let A1 , A2 , . . . be statements, and let B be a statement that depends on A1 , A2 , . . . Then, B is a tautology if B is true whether or not A1 , A2 , . . . are true. For example, let B denote the statement (A or not A). Then, truth(A or not A) = 1,

(1.2.15)

and thus the statement (A or not A) is true whether or not A is true. Hence, (A or not A) is a tautology. Likewise, (A =⇒ A) is a tautology. Furthermore, since truth[(A and B) =⇒ A] = truth(A)2 truth(B) + truth(A) truth(B) + 1 = 1,

(1.2.16)

it follows that [(A and B) =⇒ A] is a tautology. Likewise, truth([A and not A] =⇒ B) = 1, and thus ([A and not A] =⇒ B) is a tautology. Let A1 , A2 , . . . be statements, and let B be a statement that depends on A1 , A2 , . . . Then, B is a contradiction if B is false whether or not A1 , A2 , . . . are true. For example, let B denote the statement (A and not A). Then, truth(A and not A) = 0,

(1.2.17)

SETS, LOGIC, NUMBERS, RELATIONS, ORDERINGS, GRAPHS, AND FUNCTIONS

5

and thus the statement (A and not A) is false whether or not A is true. Hence, (A and not A) is a contradiction. Let A and B be statements. If the implication (A =⇒ B) is neither a tautology nor a contradiction, then truth(A =⇒ B) depends on the truth of the statements that comprise A and B. For example, truth(A =⇒ not A) = truth(A) + 1, and thus the statement (A =⇒ not A) is true if and only if A is false, and false if and only if A is true. Hence, (A =⇒ not A) is neither a tautology nor a contradiction. A statement that is neither a tautology nor a contradiction is a contingency. For example, the implication [A =⇒ (A and B)] is a contingency. Likewise, for each real number x, truth[(x = 1) =⇒ (x = 2)] = truth(x , 1), and thus the statement [(x = 1) =⇒ (x = 2)] is a contingency. An argument that is a contingency is a theorem, proposition, corollary, or lemma. A theorem is a significant result; a proposition is a theorem of less significance. The primary role of a lemma is to support the proof of a theorem or a proposition. A corollary is a consequence of a theorem or a proposition. A fact is either a theorem, proposition, lemma, or corollary. In order to visualize logic operations on predicates, it is helpful to replace statements with sets and logic operations by set operations; the truth of a statement can then be visualized in terms of Venn diagrams. To do this, let X be a set, for all x ∈ X, let A(x) and B(x) be predicates, and define △ △ A = {x ∈ X : truth[A(x)] = 1} and B = {x ∈ X : truth[B(x)] = 1}. Then, the logic operations “and,” “or,” “xor,” and “not” are equivalent to “∩,” “∪,” “⊖,” and “∼ ,” respectively. For example, {x ∈ X : truth[(not A(x)) and B(x)] = 1} = A∼ ∩ B. Furthermore, since [A(x) =⇒ B(x)] is equivalent to [(not A(x)) or B(x)], it follows that {x ∈ X : truth[A(x) =⇒ B(x)] = 1} = A∼ ∪ B. Similarly, since [A(x) ⇐⇒ B(x)] is equivalent to [(A(x) or not B(x)) and ([not A(x)] or B(x))], it follows that {x ∈ X : A(x) ⇐⇒ B(x)} = (A ∪ B∼ ) ∩ (A∼ ∪ B) = (A ∩ B) ∪ (A ∪ B)∼ . Now, define X, A(x), B(x), A, and B as in the previous paragraph, and assume that, for all x ∈ X, A(x) =⇒ B(x). Therefore, A∼ ∪ B = {x ∈ X : truth[(not A(x)) or B(x)] = 1} = X, and thus A\B = (A∼ ∪ B)∼ = {x ∈ X : truth[(not A(x)) or B(x)] = 0} = ∅. Consequently, A ⊆ B. This means that the logic operator “=⇒” is represented by “⊆.” For example, for all x ∈ X, let C(x) be a predicate, and △ define C = {x ∈ X : truth[C(x)] = 1}. Then, for all x ∈ X, truth[(A(x) and B(x)) =⇒ C(x)] = 1 if and only if A ∩ B ⊆ C. Likewise, for all x ∈ X, truth([A(x) and (B(x) or C(x))] ⇐⇒ [(A(x) and B(x)) or (A(x) and C(x))]) = 1

(1.2.18)

if and only if A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C).

(1.2.19)

Note that (1.2.19) represents a tautology.

1.3 Relations and Orderings Let X, X1, and X2 be sets. A relation R on (X1 , X2 ) is a subset of X1 × X2 . A relation R on X is a subset of X × X. Likewise, a multirelation R on (X1 , X2 ) is a multisubset of X1 × X2 , while a multirelation R on X is a multisubset of X × X. Let X be a set, and let R1 and R2 be relations on X. Then, the sets R1 ∩ R2 , R1\R2 , and R1 ∪ R2 are relations on X. Furthermore, if R is a relation on X and X0 ⊆ X, then we define the restricted △ relation R|X0 = R ∩ (X0 × X0 ), which is a relation on X0 . Definition 1.3.1. Let R be a relation on the set X. Then, the following terminology is defined: i) R is reflexive if, for all x ∈ X, it follows that (x, x) ∈ R. ii) R is symmetric if, for all (x1 , x2 ) ∈ R, it follows that (x2 , x1 ) ∈ R. iii) R is transitive if, for all (x1 , x2 ) ∈ R and (x2 , x3 ) ∈ R, it follows that (x1 , x3 ) ∈ R. iv) R is an equivalence relation if R is reflexive, symmetric, and transitive.

6

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Proposition 1.3.2. Let R1 and R2 be relations on the set X. If R1 and R2 are (reflexive, symmetric) relations, then so are R1 ∩ R2 and R1 ∪ R2 . If R1 and R2 are (transitive, equivalence) relations, then so is R1 ∩ R2 . Definition 1.3.3. Let R be a relation on the set X. Then, the following terminology is defined: △ i) The complement R∼ of R is the relation R∼ = (X × X)\R. ii) The support supp(R) of R is the smallest subset X0 of X such that R is a relation on X0 . △ iii) The reversal rev(R) of R is the relation rev(R) = {(y, x) : (x, y) ∈ R}. △ iv) The shortcut shortcut(R) of R is the relation shortcut(R) = {(x, y) ∈ X × X: x and y are distinct and there exist k ≥ 1 and x1 , . . . , xk ∈ X such that (x, x1 ), (x1 , x2 ), . . . , (xk , y) ∈ R}. v) The reflexive hull ref(R) of R is the smallest reflexive relation on X that contains R. vi) The symmetric hull sym(R) of R is the smallest symmetric relation on X that contains R. vii) The transitive hull trans(R) of R is the smallest transitive relation on X that contains R. viii) The equivalence hull equiv(R) of R is the smallest equivalence relation on X that contains R. Proposition 1.3.4. Let R be a relation on the set X. Then, the following statements hold: i) ref(R) = R ∪ {(x, x) : x ∈ X}. ii) sym(R) = R ∪ rev(R). iii) trans(R) = R ∪ shortcut(R). iv) If R is symmetric, then trans(R) = sym(trans(R)). v) equiv(R) = trans(sym(ref(R))). Furthermore, the following statements hold: vi) R is reflexive if and only if R = ref(R). vii) The following statements are equivalent: a) R is symmetric. b) R = sym(R). c) R = rev(R). viii) R is transitive if and only if R = trans(R). ix) R is an equivalence relation if and only if R = equiv(R). For an equivalence relation R on the set X, (x1 , x2 ) ∈ R is denoted by x1 ≡ x2 . If R is an △ equivalence relation and x ∈ X, then the subset E x = {y ∈ X: y ≡ x} of X is the equivalence class of x induced by R. Theorem 1.3.5. Let R be an equivalence relation on a set X. Then, the set {E x : x ∈ X} of equivalence classes induced by R is a partition of X. ∪ Proof. Since X = x∈X E x , it suffices to show that, if x, y ∈ X, then either E x = Ey or E x ∩Ey = ∅. Hence, let x, y ∈ X, and suppose that E x and Ey are not disjoint so that there exists z ∈ E x ∩ Ey . Thus, (x, z) ∈ R and (z, y) ∈ R. Now, let w ∈ E x . Then, (w, x) ∈ R, (x, z) ∈ R, and (z, y) ∈ R imply that (w, y) ∈ R. Hence, w ∈ Ey , which implies that E x ⊆ Ey . By a similar argument, Ey ⊆ E x . Consequently, E x = Ey .  The following result, which is the converse of Theorem 1.3.5, shows that a partition of a set X defines an equivalence relation on X. Theorem 1.3.6. Let X be a set, let P be a partition of X, and define the relation R on X by (x, y) ∈ R if and only if x and y belong to the same element of P. Then, R is an equivalence relation on X. Theorem 1.3.5 shows that every equivalence relation induces a partition, while Theorem 1.3.6

SETS, LOGIC, NUMBERS, RELATIONS, ORDERINGS, GRAPHS, AND FUNCTIONS

7

shows that every partition induces an equivalence relation. Definition 1.3.7. Let X be a set, let P be a partition of X, and let X0 ⊆ X. Then, X0 is a representative subset of X relative to P if, for all X∈P, exactly one element of X0 is an element of X. Definition 1.3.8. Let R be a relation on the set X. Then, the following terminology is defined: i) R is antisymmetric if (x1 , x2 ) ∈ R and (x2 , x1 ) ∈ R imply that x1 = x2 . ii) R is a partial ordering if R is reflexive, antisymmetric, and transitive. iii) (X, R) is a partially ordered set if R is a partial ordering. Let (X, R) be a partially ordered set. Then, (x1 , x2 ) ∈ R is denoted by x1 ≼ x2 . If x1 ≼ x2 and x2 ≼ x1 , then, since R is antisymmetric, it follows that x1 = x2 . Furthermore, if x1 ≼ x2 and x2 ≼ x3 , then, since R is transitive, it follows that x1 ≼ x3 . Definition 1.3.9. Let (X, R) be a partially ordered set. Then, the following terminology is defined: i) Let S ⊆ X. Then, y ∈ X is a lower bound for S if, for all x ∈ S, it follows that y ≼ x. ii) Let S ⊆ X. Then, y ∈ X is an upper bound for S if, for all x ∈ S, it follows that x ≼ y. The following result shows that every partially ordered set has at most one lower bound that is “greatest” and at most one upper bound that is “least.” Lemma 1.3.10. Let (X, R) be a partially ordered set, and let S ⊆ X. Then, there exists at most one lower bound y ∈ X for S such that every lower bound x ∈ X for S satisfies x ≼ y. Furthermore, there exists at most one upper bound y ∈ X for S such that every upper bound x ∈ X for S satisfies y ≼ x. Proof. For i = 1, 2, let yi ∈ X be such that yi is a lower bound for S and, for all x ∈ X, x ≼ yi . Therefore, y1 ≤ y2 and y2 ≤ y1 . Since “≼” is antisymmetric, it follows that y1 = y2 .  Definition 1.3.11. Let (X, R) be a partially ordered set. Then, the following terminology is defined: i) Let S ⊆ X. Then, y ∈ X is the greatest lower bound for S if y is a lower bound for S and every lower bound x ∈ X for S satisfies x ≼ y. In this case, we write y = glb(S). ii) Let S ⊆ X. Then, y ∈ X is the least upper bound for S if y is an upper bound for S and every upper bound x ∈ X for S satisfies y ≼ x. In this case, we write y = lub(S). iii) (X, ≼) is a lattice if, for all distinct x, y ∈ X, the set {x, y} has a least upper bound and a greatest lower bound. iv) (X, ≼) is a complete lattice on X if every subset S of X has a least upper bound and a greatest lower bound. Example 1.3.12. Consider the partially ordered set (P, ≼), where m ≼ n indicates that n is an integer multiple of m. For example, 3 ≼ 21, but it is not true that 2 ≼ 3. Next, note that the greatest lower bound of a subset S of P is the greatest common divisor of the elements of S. For example, glb {9, 21} = 3. Likewise, the least upper bound of a subset S of P is the least common multiple of the elements of S. For example, lub {2, 3, 4} = 12. Therefore, (P, ≼) is a lattice. Next, note that 1 is a lower bound for every subset of P. Since every subset of P has a smallest element in the usual ordering, it follows that every subset of P has a greatest lower bound. In particular, glb(P) = 1. However, no subset of P that has an infinite number of elements has an upper bound. Therefore, (P, ≼) is not a complete lattice. Now, consider (N, ≼). Note that 1 is a lower bound for every subset of N. Since every subset of N has a smallest element in the usual ordering, it follows that every subset of N has a greatest lower bound. In particular, glb(N) = 1. Furthermore, for all m ∈ N, 0 = 0 · m, and thus 0 is an upper bound for every subset of N. In particular, since 0 is the unique upper bound of N, it follows that 0 is the least upper bound of N. Hence, (N, ≼) is a complete lattice. ^

8

CHAPTER 1

Proposition 1.3.13. Let (X, ≼) be a lattice, and let S1 , S2 ⊆ X. Then,

glb(S1 ∪ S2 ) = glb[S1 ∪ {glb(S2 )}],

lub(S1 ∪ S2 ) = lub[S1 ∪ {lub(S2 )}].

(1.3.1)

Definition 1.3.14. Let (X, R) be a partially ordered set. Then, R is a total ordering on X if, for all x, y ∈ X, either (x, y) ∈ R or (y, x) ∈ R. Let S ⊆ R. Then, it is traditional to write inf S and sup S for glb(S) and lub(S), respectively, where “inf” and “sup” denote infimum and supremum, respectively. If S = ∅, then we define △ △ inf ∅ = ∞ and sup ∅ = −∞. Finally, if S has no lower bound, then we write inf S = −∞, whereas, if S has no upper bound, then we write sup S = ∞. The following result uses the fact that “⊆” is a partial ordering on every collection of sets. Proposition 1.3.15. Let S be a collection of sets. Then, ∩ ∪ glb(S) = S , lub(S) = S. (1.3.2) S ∈S

S ∈S

Hence, for all S ∈ S, glb(S) ⊆ S ⊆ lub(S).

(1.3.3)

△

Let S = (S i )∞ i=1 be a sequence of sets. Then, by viewing S as the collection of sets {S 1 , S 2 , . . .}, it follows that ∞ ∞ ∩ ∪ glb(S) = S i , lub(S) = Si . (1.3.4) i=1

i=1

Hence, for all i ≥ 1, glb(S) ⊆ S i ⊆ lub(S).

(1.3.5)

Note that glb(S) and lub(S) are independent of the ordering of the sequence S. △ Proposition 1.3.16. Let S be a collection of sets, let A be a set, let S0 = {S ∈ S : A ⊆ S }, and assume that S0 , ∅. Then, A ⊆ glb(S0 ). If, in addition, glb(S0 ) ∈ S0 , then glb(S0 ) is the smallest element of S that contains A in the sense that, if S ∈ S and A ⊆ S , then glb(S0 ) ⊆ S . △ Proposition 1.3.17. Let S be a collection of sets, let A be a set, and let S0 = {S ∈ S : S ⊆ A}. Then, lub(S0 ) ⊆ A. If, in addition, lub(S0 ) ∈ S0 , then lub(S0 ) is the largest element of S that is contained in A in the sense that, if S ∈ S and S ⊆ A, then S ⊆ lub(S0 ). △ Definition 1.3.18. Let S = (S i )∞ i=1 be a sequence of sets. Then, the essential greatest lower bound of S is defined by △

essglb(S) =

∞ ∩ ∞ ∪

Si ,

(1.3.6)

Si .

(1.3.7)

j=1 i= j

and the essential least upper bound of S is defined by △

esslub(S) =

∞ ∪ ∞ ∩ j=1 i= j

△

∞ Let S = (S i )∞ i=1 be a sequence of sets. Then, the set essglb(S) consists of all elements of ∪i=1 S i that belong to all but finitely many of the sets in S. Furthermore, the set esslub(S) consists of all elements of ∪∞ i=1 S i that belong to infinitely many of the sets in S. Therefore, essglb(S) and esslub(S) are independent of the ordering of the sequence S, and

glb(S) ⊆ essglb(S) ⊆ esslub(S) ⊆ lub(S).

(1.3.8)

9

SETS, LOGIC, NUMBERS, RELATIONS, ORDERINGS, GRAPHS, AND FUNCTIONS

Note that lub(S)\esslub(S) is the set of elements of ∪∞ i=1 S i that belong to at most finitely many of the sets in S. Example 1.3.19. Consider the sequence of sets given by ({1, 4}, {1, 2}, {1, 2, 3}, {1, 2}, {1, 2, 3}, {1, 2}, {1, 2, 3}, . . .). Then, (1.3.8) becomes {1} ⊆ {1, 2} ⊆ {1, 2, 3} ⊆ {1, 2, 3, 4}. ^ △ Definition 1.3.20. Let S = (S i )∞ be a sequence of sets, and assume that essglb(S) = esslub(S). i=1 Then, the essential limit of S is defined by △

esslim(S) = essglb(S) = esslub(S).

(1.3.9)

△

Let S = (S i )∞ i=1 be a sequence of sets. Then, S is nonincreasing if, for all i ∈ P, S i+1 ⊆ S i . Furthermore, S is nondecreasing if, for all i ∈ P, S i ⊆ S i+1 . △ Proposition 1.3.21. Let S = (S i )∞ i=1 be a sequence of sets. If S is nonincreasing, then esslim(S) = glb(S) = essglb(S) = esslub(S).

(1.3.10)

Furthermore, if S is nondecreasing, then esslim(S) = essglb(S) = esslub(S) = lub(S).

(1.3.11)

Example 1.3.22. Consider the nonincreasing sequence of sets

(N, N\{1}, N\{1, 2}, N\{1, 2, 3}, . . .). Then, (1.3.8) becomes {0} = {0} = {0} ⊆ N. Now, consider the nondecreasing sequence of subsets of R given by ({1}, {1, 2}, {1, 2, 3}, {1, 2, 3, 4}, . . .). Then, (1.3.8) becomes {1} ⊆ P = P = P, where P is the set of positive integers. ^ △ △ ∞ ∞ ∞ k ∞ ˆ = S ]) = (∪ S ) Let S = (S i )∞ be a sequence of sets. Then, the sequence S (∩ [∪ i=1 i=k i k=1 = j=1 i= j i k=1 ∞ (Sˆk )i=1 is nonincreasing. Hence, ˆ = glb(S) ˆ = essglb(S) ˆ = esslub(S). ˆ esslub(S) = esslim(S)

(1.3.12)

△ ∞ ∞ ∞ ˜ ∞ Furthermore, the sequence S˜ = (∪kj=1 [∩∞ i= j S i ])k=1 = (∩i=k S i )k=1 = (S k )i=1 is nondecreasing. Hence,

˜ = essglb(S) ˜ = esslub(S) ˜ = lub(S). ˜ essglb(S) = esslim(S)

(1.3.13)

1.4 Directed and Symmetric Graphs Let X be a finite, nonempty set, and let R be a multirelation on X. Then, the pair G = (X, R) is a directed multigraph. The elements of X are the nodes of G, while the elements of R are the directed edges of G. If R is a relation on X, then G = (X, R) is a directed graph. We focus on directed graphs, which have distinct (that is, nonrepeated) directed edges. The directed graph G = (X, R) can be visualized as a set of points in the plane representing the nodes in X connected by the directed edges in R. Specifically, the directed edge (x, y) ∈ R from x to y can be visualized as a directed line segment or curve connecting node x to node y. The direction of a directed edge can be denoted by an arrowhead. A directed edge of the form (x, x) is a self-directed edge. If the relation R is symmetric, then G is a symmetric graph. In this case, it is convenient to represent the pair of directed edges (x, y) and (y, x) in R by a single edge {x, y}, which is a subset of X. For the self-directed edge (x, x), the corresponding edge is the single-element self-edge {x}. To illustrate these notions, consider a directed graph that represents a city with streets (directed edges)

10

CHAPTER 1

connecting intersections (nodes). Each directed edge represents a one-way street, while the presence of the one-way street (x, y) and its reverse (y, x) represents a two-way street. A symmetric relation is a street plan consisting entirely of two-way streets (that is, edges) and thus no one-way streets (directed edges), whereas an antisymmetric relation is a street plan consisting entirely of one-way streets (directed edges) and thus no two-way streets (edges). Definition 1.4.1. Let G = (X, R) be a directed graph. Then, the following terminology is defined: i) If x, y ∈ X are distinct and (x, y) ∈ R, then y is the head of (x, y) and x is the tail of (x, y). ii) If x, y ∈ X are distinct and (x, y) ∈ R, then x is a parent of y, and y is a child of x. iii) If x, y ∈ X are distinct and either (x, y) ∈ R or (y, x) ∈ R, then x and y are adjacent. iv) If x ∈ X has no parent, then x is a root. v) If x ∈ X has no child, then x is a leaf. Definition 1.4.2. Let G = (X, R) be a directed graph. Then, the following terminology is defined: △ i) The reversal of G is the graph rev(G) = (X, rev(R)). △ ii) The complement of G is the graph G∼ = (X, R∼ ). △ iii) The reflexive hull of G is the graph ref(G) = (X, ref(R)). △ iv) The symmetric hull of G is the graph sym(G) = (X, sym(R)). △ v) The transitive hull of G is the graph trans(G) = (X, trans(R)). △ vi) The equivalence hull of G is the graph equiv(G) = (X, equiv(R)). vii) G is reflexive if R is reflexive. viii) G is transitive if R is transitive. ix) G is an equivalence graph if R is an equivalence relation. x) G is antisymmetric if R is antisymmetric. xi) G is partially ordered if R is a partial ordering on X. xii) G is totally ordered if R is a total ordering on X. xiii) G is a tournament if G is antisymmetric and sym(R) = X × X\{(x, x) : x ∈ X}. Definition 1.4.3. Let G = (X, R) be a directed graph. Then, the following terminology is defined: i) The directed graph G′ = (X′ , R′ ) is a directed subgraph of G if X′ ⊆ X and R′ ⊆ R. ii) The directed subgraph G′ = (X′ , R′ ) of G is a spanning directed subgraph of G if supp(R) = supp(R′ ). △ iii) If X0 ⊆ X, then G|X0 = (X0 , R|X0 ). △

△

iv) If G′ = (X′ , R′ ) is a directed graph, then G ∪ G′ = (X ∪ X′ , R ∪ R′ ) and G ∩ G′ = (X ∩ X′ , R ∩ R′ ). v) For x, y ∈ X, a directed walk in G from x to y is an n-tuple of directed edges of G of the form ((x, y)) ∈ R for n = 1 and ((x, x1 ), (x1 , x2 ), . . . , (xn−1 , y)) ∈ Rn for all n ≥ 2. The length of the directed walk is n. The nodes x, x1 , . . . , xn−1 , y are the nodes of the walk. Furthermore, if n ≥ 2, then the nodes x1 , . . . , xn−1 are the intermediate nodes of the walk. vi) For x, y ∈ X, a directed trail in G from x to y is a directed walk in G from x to y whose directed edges are distinct. vii) For x, y ∈ X, a directed path in G from x to y is a directed trail in G from x to y whose intermediate nodes are distinct and do not include x and y.

SETS, LOGIC, NUMBERS, RELATIONS, ORDERINGS, GRAPHS, AND FUNCTIONS

11

viii) For x ∈ X, a directed cycle in G at x is a directed path in G from x to x whose length is at least 2. ix) G is directionally acyclic if G has no directed cycles. x) If G has at least one directed cycle, then the directed period of G is the greatest common divisor of the lengths of the directed cycles of G. xi) G is directionally aperiodic if it has at least one directed cycle and the greatest common divisor of the lengths of the directed cycles in G is 1. xii) A directed Hamiltonian path is a directed path whose nodes include all of the nodes of X. xiii) A directed Hamiltonian cycle is a directed cycle whose nodes include every node in X. xiv) G is a directed tree if G has exactly one root x and, for all y ∈ X such that y , x, y has exactly one parent. xv) G is a directed forest if G is a union of disjoint directed trees. xvi) G is a directed chain if G is a tree and has exactly one leaf. xvii) G is directionally connected if, for all distinct x, y ∈ X, there exist directed walks in G from x to y and from y to x. xviii) G is bipartite if there exist nonempty, disjoint sets X1 and X2 such that X = X1 ∪ X2 and R ∩ (X1 × X1 ) = R ∩ (X2 × X2 ) = ∅. △ xix) The indegree of x ∈ X is indeg(x) = card {y ∈ X: y is a parent of x}. △ xx) The outdegree of x ∈ X is outdeg(x) = card {y ∈ X: y is a child of x}. xxi) Let X = X1∪X2 , where X1 and X2 are nonempty and disjoint, and assume that X = supp(G). Then, (X1, X2 ) is a directed cut of G if, for all x1 ∈ X1 and x2 ∈ X2 , there does not exist a directed walk from x1 to x2 . A self-directed edge is a directed path; however, a self-directed edge is not a directed cycle. A directed Hamiltonian cycle is both a directed Hamiltonian path and a directed cycle, both of which are directed paths. Definition 1.4.4. Let G = (X, R) be a symmetric graph. Then, the following terminology is defined: i) For x, y ∈ X, a walk in G connecting x and y is an n-tuple of edges of G of the form ({x, y}) ∈ E for n = 1 and ( {x, x1 }, {x1 , x2 }, . . . , {xn−1 , y}) ∈ En for n ≥ 2. The length of the walk is n. The nodes x, x1 , . . . , xn−1 , y are the nodes of the walk. Furthermore, if n ≥ 2, then the nodes x1 , . . . , xn−1 are the intermediate nodes of the walk. ii) For x, y ∈ X, a trail in G connecting x and y is a walk in G connecting x to y whose edges are distinct. iii) For x, y ∈ X, a path in G connecting x and y is a trail in G connecting x and y whose intermediate nodes are distinct and do not include x and y. iv) For x ∈ X, a cycle in G at x is a path in G connecting x and x whose length is at least 3. v) G is acyclic if G has no cycles. vi) If G has at least one cycle, then the period of G is the greatest common divisor of the lengths of the cycles of G. vii) G is aperiodic if the period of G is 1. viii) A Hamiltonian path is a path whose nodes include every node in X. ix) G is Hamiltonian if G has a Hamiltonian cycle P, which is a cycle such that every node in X is a node of P. x) G is a tree if there exists a directed tree G′ = (X, R′ ) such that G = sym(G′ ).

12

CHAPTER 1

G is a forest if G is a union of disjoint trees. G is a chain if there exists a directed chain G′ = (X, R′ ) such that G = sym(G′ ). G is connected if, for all distinct x, y ∈ X, there exists a walk in G connecting x and y. G is bipartite if there exist nonempty, disjoint sets X1 and X2 such that X = X1 ∪ X2 and {{x, y} ∈ R : x ∈ X1 and y ∈ X2 } = ∅. △ xv) The degree of x ∈ X is deg(x) = indeg(x) = outdeg(x). A self-edge is a path; however, a self-edge is not a cycle. A Hamiltonian cycle is both a Hamiltonian path and a cycle, both of which are paths. Let G = (X, R) be a directed graph, and let w : X × X 7→ [0, ∞), where w(x, y) > 0 if (x, y) ∈ R and w(x, y) = 0 if (x, y) < R. For each directed edge (x, y) ∈ R, w(x, y) is the weight associated with the directed edge (x, y), and the triple G = (X, R, w) is a weighted directed graph. The graph G′ = (X′ , R′ , w′ ) is a weighted directed subgraph of G if X′ ⊆ X, R′ is a relation on X′ , R′ ⊆ R, and w′ is the restriction of w to R′ . Finally, if G is symmetric, then w is symmetric if, for all (x, y) ∈ R, w(x, y) = w(y, x). In this case, w is defined on each edge {x, y} of G. xi) xii) xiii) xiv)

1.5 Numbers Let x and y be real numbers. Then, x divides y if there exists an integer n such that y = nx, In this case, we write x|y. For example, 6|12, 3| − 9, π| − 2π, 3|0, and 0|0. The notation x - y means that x does not divide y. Let n1 , . . . , nk be integers, not all of which are zero. Then, the greatest common divisor of the set {n1 , . . . , nk } is the positive integer defined by △

gcd {n1 , . . . , nk } = max{i ∈ P : i divides n1 , . . . , nk }. For example, gcd {5, 10} = 5, and gcd {0, 2} = 2. The set {n1 , . . . , nk } is coprime if gcd {n1 , . . . , nk } = 1. For example, gcd {−3, −7} = 1, and thus {−3, −7} is coprime. Let n1 , . . . , nk be nonzero integers. Then, the least common multiple of the set {n1 , . . . , nk } is the positive integer defined by △

lcm {n1 , . . . , nk } = min{i ∈ P : n1 , . . . , nk divide i}. For example, lcm {−3, −7} = 21, and lcm {−2, 3} = 6. Let m be a nonzero integer, and let n be an integer. Then, m|n if and only if gcd {m, n} = |m|. Let n be an integer, and let k be a positive integer. Furthermore, let l be an integer, and let r ∈ [0, k − 1] be an integer satisfying n = kl + r. Then, we write r = remk (n).

(1.5.1)

where r is the remainder after dividing n by k. For example, rem3 (−11) = 1 and rem3 (11) = 2. Furthermore, k|n if and only if remk (n) = 0. Proposition 1.5.1. Let m and n be integers, and let k be a positive integer. Then, remk (n − m) = remk [remk (n) − remk (m)].

(1.5.2)

Furthermore, k|n − m if and only if remk (n) = remk (m). Definition 1.5.2. Let n and m be integers, and let k be a positive integer. Then, n and m are congruent modulo k if k divides n − m. In this case, we write k

n ≡ m. k

(1.5.3)

Proposition 1.5.1 implies that n ≡ m if and only if the remainders of n and m after dividing by k

13

SETS, LOGIC, NUMBERS, RELATIONS, ORDERINGS, GRAPHS, AND FUNCTIONS 3

3

3

3

differ by a multiple of k. For example, −1 ≡ 2 ≡ 8 ≡ 26 ≡ 29. Let n be an integer. Then, n is even if 2 divides n, whereas n is odd if 2 does not divide n. Now, assume that n ≥ 2. Then, n is prime if, for all integers m such that 2 ≤ m < n, m does not divide n. Note that 2 is prime, but 1 is not prime. Letting pn denote the nth prime, it follows that (pi )25 i=1 = (2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97). The nth harmonic number is denoted by △

Hn =

n ∑ 1 i=1

Then,

i

.

(1.5.4)

( ) 3 11 25 137 49 363 761 7129 7381 83711 86021 (Hi )12 = 0, 1, , , , , , , , , , , . i=0 2 6 12 60 20 140 280 2520 2520 27720 27720

For all α ∈ R, the nth generalized harmonic number of order α is denoted by △

Hn,α = △

n ∑ 1 . α i i=1

(1.5.5)

△

Define H0 = H0,α = 0. Then, ( ) 5 49 205 5269 5369 266681 1077749 9778141 1968329 10 (Hi,2 )i=0 = 0, 1, , , , , , , , , . 4 36 144 3600 3600 176400 705600 6350400 1270080 The symbol C denotes the set of complex numbers. The elements of R and C are scalars. Define √ △ (1.5.6) ȷ = −1. Let z ∈ C. Then, z = x + y ȷ, where x, y ∈ R. Define the complex conjugate z of z by △

z = x − yȷ

(1.5.7)

and the real part Re z of z and the imaginary part Im z of z by △

Re z = 12 (z + z) = x,

△

Im z =

1 2 ȷ (z

− z) = 21 (z − z) ȷ = y.

Furthermore, the absolute value |z| of z is defined by √ △ |z| = x2 + y2 . Finally, the argument arg z ∈ (−π, π] of z is defined by   0,        atan yx ,       π △ − 2 , arg z =   π    2,      −π + atan yx ,     π + atan y , x

y = x = 0, x > 0, y < 0, x = 0, y > 0, x = 0, y < 0, x < 0, y ≥ 0, x < 0,

(1.5.8)

(1.5.9)

(1.5.10)

where atan: R 7→ (− π2 , π2 ). Let z be a complex number. Then, z = |z|e(arg z) ȷ .

(1.5.11)

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CHAPTER 1

z is a nonnegative number if and only if arg z = 0, and z is a negative number if and only if arg z = −π. If z is not a nonnegative number, then arg z ∈ (−π, 0) ∪ (0, π] is the angle from the positive real axis to the line segment connecting z to the origin in the complex plane, where clockwise angles are negative and confined to the set (−π, 0), and counterclockwise angles are positive and confined to the set (0, π]. Furthermore, if z is nonzero, then   1  − arg z, arg z ∈ (−π, π), (1.5.12) arg =  π, z  arg z = π. Let z1 and z2 be nonzero complex numbers. Then, there exists k ∈ {−1, 0, 1} such that arg z1 z2 = arg z1 + arg z2 + 2kπ.

(1.5.13)

Hence, 2π| arg z1 z2 − arg z1 − arg z2 . For example, arg (−1)(−1) = arg 1 = 0 = π + π − 2π = arg −1 + arg −1 − 2π, arg (1)(−1) = arg −1 = π = 0 + π = arg 1 + arg −1, arg (− ȷ)(− ȷ) = arg −1 = π = −π/2 − π/2 + 2π = arg − ȷ + arg − ȷ + 2π. The closed left half plane (CLHP), open left half plane (OLHP), closed right half plane (CRHP), and open right half plane (ORHP) are the subsets of C defined by △

OLHP = {x ∈ C: Re x < 0}, △

CLHP = {x ∈ C: Re x ≤ 0},

△

ORHP = {x ∈ C: Re x > 0}, △

CRHP = {x ∈ C: Re x ≥ 0}.

(1.5.14) (1.5.15)

The imaginary numbers are represented by IA . Note that 0 is a real number, an imaginary number, and a complex number. Next, we define the open inside unit disk (OIUD) and the closed inside unit disk (CIUD) by △

OIUD = {x ∈ C: |x| < 1},

△

CIUD = {x ∈ C: |x| ≤ 1}.

(1.5.16)

The complements of the open inside unit disk and the closed inside unit disk are given, respectively, by the closed outside unit disk (COUD) and the open outside unit disk, which are defined by △

COUD = {x ∈ C: |x| ≥ 1},

△

OOUD = {x ∈ C: |x| > 1}.

(1.5.17)

The unit circle in C is denoted by UC . Since R is a proper subset of C, we state many results for C. In other cases, we treat R and C separately. To do this efficiently, we use the symbol F to consistently denote either R or C. Let n ∈ N. Then,    n ≥ 1, △ n(n − 1) · · · (2)(1), n! =  (1.5.18)  1, n = 0. Then, (i!)12 i=0 = (1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916800, 479001600). Let z ∈ C and k ∈ Z. Then,

 z(z − 1) · · · (z − k + 1)   ( )  ,    k! z △  = 1,   k    0,

k > 0, k = 0, k < 0.

(1.5.19)

SETS, LOGIC, NUMBERS, RELATIONS, ORDERINGS, GRAPHS, AND FUNCTIONS

In particular, if n, k ∈ N, then

n! ( )     , n ≥ k ≥ 0, n  (n − k)!k! =   k 0, k > n ≥ 0.

Hence,

( )    n 1, n ≥ 0, =  0, n < 0. n

(1.5.20)

(1.5.21)

) ( ) ( ) ( ) ( ) −1 −1 1 −1 0 = 0, = −1, = 0, = 1, = 1, −1 1 −1 0 0 ( ) ( 1) ( ) (1) ( ) −2 −1 0 1 −5 1 2 = −1, , = 0, , = 0. = = 3 16 3 16 3 3 3 () Note that, for all n ≥ k ≥ 1, nk is the number of k-element subsets of {1, . . . , n}. Let z, w ∈ C, and assume that z < −P, w < −P, and z − w < −P. Then, ( ) z △ Γ(z + 1) = . w Γ(w + 1)Γ(z − w + 1) ∑ For k1 , . . . , kl ∈ N, where li=1 ki = n, we define the multinomial coefficient ( ) n n! △ = . k1 , . . . , kl k1 ! · · · kl ! For example,

15

(

Note that, if 1 ≤ m ≤ n, then

(1.5.22)

(1.5.23)

( ) ( ) n n = . m m, n − m

For z ∈ C and k ∈ N, we define the falling factorial    k △ z(z − 1) · · · (z − k + 1), z =  1,

k ≥ 0, k = 0.

In particular, if n ∈ N, then nn = n!. Hence, if z ∈ C and k ∈ Z, then  k ( )  z   z △   , k ≥ 0, = k!   k  0, k < 0. Furthermore, for all z ∈ C and k ∈ N, we define the rising factorial    k △ z(z + 1) · · · (z + k − 1), k ≥ 1, z =  1, k = 0.

(1.5.24)

(1.5.25)

(1.5.26)

In particular, if n ∈ N, then 1n = n!. Finally, if z ∈ C and k ∈ N, then zk = (z − k + 1)k ,

zk = (z + k − 1)k ,

zk = (−1)k (−z)k .

(1.5.27)

16

CHAPTER 1

The double factorial is defined by    n(n − 2)(n − 4) · · · (2) = 2n/2 (n/2)!, n even,    △  n!! =  (n + 1)!     n(n − 2)(n − 4) · · · (3)(1) = 2(n+1)/2 [ 1 (n + 1)]! , n odd. 2

(1.5.28)

By convention, (−1)!! = 0!! = 1. Finally, if n ≥ 1, then (2n)!!(2n−1)!! = (2n)! and (2n+1)!!(2n)!! = (2n + 1)!.

1.6 Functions and Their Inverses Let X and Y be nonempty sets. Then, a function f that maps X into Y is a rule f : X 7→ Y that assigns a unique element f (x) (the image of x) of Y to each element x of X. Equivalently, a function f : X 7→ Y can be viewed as a subset F of X × Y such that, for each x ∈ X, there exists a unique y ∈ Y such that (x, y) ∈ F. In this case, △

F = Graph( f ) = {(x, f (x)): x ∈ X}.

(1.6.1)

The set X is the domain of f, while the set Y is the codomain of f. For X1 ⊆ X, it is convenient to define △

f (X1 ) = { f (x): x ∈ X1 }.

(1.6.2)

△

The range of f is the set R( f ) = f (X). The function f is one-to-one if, for all x1 , x2 ∈ X such that f (x1 ) = f (x2 ), it follows that x1 = x2 . The function f is onto if R( f ) = Y. The function IX : X 7→ X △ defined by IX (x) = x for all x ∈ X is the identity mapping on X. Finally, if S ⊆ X, fS : S 7→ Y, and, ˆ fS (x) = f (x), then fS is the restriction of f to S. for all x ∈ X, Note that the subset F of X × Y can be viewed as a relation on (X, Y). Consequently, a function can be viewed as a special case of a relation. ˆ be a partition of X. Furthermore, let f : X ˆ 7→ X, where, for all S ∈ X, ˆ Let X be a set, and let X it follows that f (S) ∈ S. Then, f is a canonical mapping, and f (S) is a canonical form. That is, for ˆ of X, the function f assigns an element of S to the set S. For each element S ⊆ X in the partition X △ △ ˆ = example, let S = {1, 2, 3, 4}, X {{1, 3}, {2, 4}}, f ({1, 3}) = 1, and f ({2, 4}) = 2. Let X and Y be sets. If f : X 7→ Y is one-to-one and onto, then X and Y have the same cardinality, which is written as card(X) = card(Y). Consequently, if X is finite, then card(X) is the number of elements of X. If f : X 7→ Y is one-to-one, then card(X) ≤ card(Y). If every function f : X 7→ Y that is one-to-one is not onto, then card(X) < card(Y). If card(X) = card(P), then X is countable. Note that card(N) = card(P) = card(Z) = card(Q) < card([0, 1]) = card(R) = card(R2 ). Let X be a finite multiset. Then, card(X) is the number of elements in X. Cardinality is not defined for infinite multisets. Let X be a set, and let f : X 7→ X. Then, f is a function on X. The element x ∈ X is a fixed point of f if f (x) = x. Let X, Y, and Z be sets, let f : X 7→ Y, and let g: f (X) 7→ Z. Then, the composition of g and △ f is the function g ◦ f : X 7→ Z defined by (g ◦ f )(x) = g[ f (x)]. The following result shows that function composition is associative. Proposition 1.6.1. Let X, Y, Z, and W be sets, and let f : X 7→ Y, g : Y 7→ Z, h : Z 7→ W. Then, h ◦ (g ◦ f ) = (h ◦ g) ◦ f.

(1.6.3)

Hence, we write h ◦ g ◦ f for h ◦ (g ◦ f ) and (h ◦ g) ◦ f. Proposition 1.6.2. Let X, Y, and Z be sets, and let f : X 7→ Y and g : Y 7→ Z. Then, the

SETS, LOGIC, NUMBERS, RELATIONS, ORDERINGS, GRAPHS, AND FUNCTIONS

17

following statements hold: i) If g ◦ f is onto, then g is onto. ii) If g ◦ f is one-to-one, then f is one-to-one. Proof. To prove i), note that Z = g( f (X)) ⊆ g(Y) ⊆ Z. Hence, g(Y) = Z. To prove ii), suppose that f is not one-to-one. Then, there exist distinct x1 , x2 ∈ X such that f (x1 ) = f (x2 ). Therefore, g( f (x1 )) = g( f (x2 )), and thus g ◦ f is not one-to-one.  Let f : X 7→ Y. Then, f is left invertible if there exists a function f L : Y 7→ X (a left inverse of f ) such that f L ◦ f = IX , whereas f is right invertible if there exists a function f R : Y 7→ X (a right inverse of f ) such that f ◦ f R = IY. In addition, the function f : X 7→ Y is invertible if there exists a function f Inv : Y 7→ X (the inverse of f ) such that f Inv ◦ f = IX and f ◦ f Inv = IY; that is, f Inv is both a left inverse of f and a right inverse of f. ˜ denote the set of subsets of X. Then, for all y ∈ Y, the set-valued inverse Let f : X 7→ Y, and let X △ inv ˜ f : Y 7→ X is defined by f inv (y) = {x ∈ X : f (x) = y}. If f is one-to-one, then, for all y ∈ R( f ), the set f inv (y) has a single element, and thus f inv : R( f ) 7→ X is a function. If f is invertible, then, for all y ∈ Y, f inv (y) = { f Inv (y)}. The inverse image f inv (S) of S ⊆ Y is the set ∪ △ f inv (S) = f inv (y) = {x ∈ X: f (x) ∈ S}. (1.6.4) y∈S

Note that f inv (S) is defined whether or not f is invertible. In fact, f inv (Y) = f inv [ f (X)] = X and f [ f inv (Y)] = f (X). Proposition 1.6.3. Let X and Y be sets, let f : X 7→ Y, and let g: Y 7→ X. Then, the following statements are equivalent: i) f is a left inverse of g. ii) g is a right inverse of f. Proposition 1.6.4. Let X and Y be sets, let f : X 7→ Y, and assume that f is invertible. Then, f has a unique inverse. Now, let g: Y 7→ X. Then, the following statements are equivalent: i) g is the inverse of f. ii) f is the inverse of g. Theorem 1.6.5. Let X and Y be sets, and let f : X 7→ Y. Then, the following statements hold: i) f is left invertible if and only if f is one-to-one. ii) f is right invertible if and only if f is onto. Furthermore, the following statements are equivalent: iii) f is invertible. iv) f has a unique inverse. v) f is one-to-one and onto. vi) f is left invertible and right invertible. vii) f has a unique right inverse. viii) f has a one-to-one left inverse. ix) f has an onto right inverse. If, in addition, card(X) ≥ 2, then the following statement is equivalent to iii)–ix): x) f has a unique left inverse. Proof. To prove i), suppose that f is left invertible with left inverse g: Y 7→ X. Furthermore, suppose that x1 , x2 ∈ X satisfy f (x1 ) = f (x2 ). Then, x1 = g[ f (x1 )] = g[ f (x2 )] = x2 , which shows that f is one-to-one. Conversely, suppose that f is one-to-one so that, for all y ∈ R( f ), there exists △ a unique x ∈ X such that f (x) = y. Hence, define the function g: Y 7→ X by g(y) = x for all

18

CHAPTER 1

y = f (x) ∈ R( f ) and by g(y) arbitrary for all y ∈ Y\R( f ). Consequently, g[ f (x)] = x for all x ∈ X, which shows that g is a left inverse of f. To prove ii), suppose that f is right invertible with right inverse g: Y 7→ X. Then, for all y ∈ Y, it follows that f [g(y)] = y, which shows that f is onto. Conversely, suppose that f is onto so that, for all y ∈ Y, there exists at least one x ∈ X such that f (x) = y. Selecting one such x arbitrarily, △ define g: Y 7→ X by g(y) = x. Consequently, f [g(y)] = y for all y ∈ Y, which shows that g is a right inverse of f.  Let f : X 7→ Y, and assume that f is one-to-one. Then, the function fˆ : X 7→ R( f ) defined by △ fˆ(x) = f (x) is one-to-one and onto and thus invertible. For convenience, we write f Inv : R( f ) 7→ X. The sine and cosine functions sin : R 7→ [−1, 1] and cos : R 7→ [−1, 1] can be defined in an elementary way in terms of ratios of sides of triangles. The additional trigonometric functions tan : R\π( 12 + Z) 7→ R, csc : R\πZ 7→ R, sec : R\π( 12 + Z) 7→ R, and cot : R\πZ 7→ R are defined by △

tan x =

sin x , cos x

1 , sin x

△

csc x =

△

sec x =

1 , cos x

△

cot x =

cos x . sin x

(1.6.5)

The exponential function exp : R 7→ (0, ∞) is defined by △

exp(x) = e x ,

(1.6.6)

△

where e = lim x→∞ (1 + 1/x) x ≈ 2.71828 . . .. The exponential function can be extended to complex arguments as follows. For all x ∈ R, the power series for “exp” is given by exp(x) =

∞ ∑ xi . i! i=0

(1.6.7)

Hence, for all y ∈ R, we define △

exp(y ȷ) = ey ȷ =

∞ ∑ (y ȷ)i

i!

i=0

=

∞ ∞ ∑ ∑ y2i y2i+1 (−1)i + (−1)2i+1 ȷ = cos y + (sin y) ȷ. (2i)! i=0 (2i + 1)! i=0

(1.6.8)

Thus, for all y ∈ R, sin y =

1 yȷ (e − e−y ȷ ), 2ȷ

cos y = 12 (ey ȷ + e−y ȷ ).

(1.6.9)

Now, let z = x + y ȷ, where x, y ∈ R. Then, exp : C 7→ C\{0} is defined by △

exp(z) = exp(x + y ȷ) = e x+y ȷ = e x ey ȷ = e x [cos x + (sin x) ȷ].

(1.6.10)

In particular, eπ ȷ = −1. The six trigonometric functions can now be extended to complex arguments. In particular, by replacing y ∈ R in (1.6.9) by z ∈ C, we define sin : C 7→ C and cos : C 7→ C by △

sin z =

1 zȷ (e − e−z ȷ ), 2ȷ

△

cos z = 21 (ez ȷ + e−z ȷ ).

(1.6.11)

e−z ȷ = cos z − (sin z) ȷ.

(1.6.12)

Hence, ez ȷ = cos z + (sin z) ȷ, Likewise, tan : defined by

C\π( 21

+ Z) 7→ R, csc : C\πZ 7→ R, sec : △

tan z =

sin z , cos z

△

csc z =

1 , sin z

C\π( 12 △

sec z =

+ Z) 7→ R, and cot : C\πZ 7→ R are

1 , cos z

△

cot z =

cos z . sin z

(1.6.13)

19

SETS, LOGIC, NUMBERS, RELATIONS, ORDERINGS, GRAPHS, AND FUNCTIONS

Let f : X 7→ Y. If f is not one-to-one, then f is not invertible. This is the case, for example, for a periodic function such as sin : R 7→ [−1, 1], respectively. In particular, sininv (1) = {(4k + 1)π/2 : k ∈ Z}. However, it is convenient to define a principal inverse asin of sin by choosing an element of the set sininv (y) for each y ∈ [−1, 1]. Although this choice can be made arbitrarily, it is traditional to define asin: [−1, 1] 7→ [− π2 , π2 ].

(1.6.14)

Similarly, atan: R 7→ (− π2 , π2 ),

acos: [−1, 1] 7→ [0, π],

(1.6.15)

acsc: (−∞, −1] ∪ [1, ∞) 7→ [− π2 , 0) ∪ (0, π2 ], asec: (−∞, −1] ∪ [1, ∞) 7→ [0, π2 ) ∪ ( π2 , π],

(1.6.16) (1.6.17)

acot: R 7→ (− π2 , 0) ∪ (0, π2 ].

(1.6.18)

An analogous situation arises for the exponential function f (z) = ez , which is not one-to-one and thus requires a principal inverse in the form of a logarithm defined on C\{0}. Let w be a nonzero complex number, and, for all i ∈ Z, define △

zi = log |w| + (arg w + 2iπ) ȷ.

(1.6.19)

ezi = |w|e(arg w) ȷ e2iπȷ = |w|e(arg w) ȷ = w.

(1.6.20)

Then, for all i ∈ Z, Consequently, f inv (w) = {zi : i ∈ Z}. For example, f inv (1) = {2iπȷ: i ∈ Z}, and f inv (−1) = {(2i + 1)πȷ: i ∈ Z}. The principal logarithm log w of w is defined by choosing z0 , which yields △

log w = z0 = log |w| + (arg w) ȷ.

(1.6.21)

log: C\{0} 7→ {z : Re z , 0 and − π < Im z ≤ π}.

(1.6.22)

Therefore,

Hence, Re log w = log |w|,

Im log w = arg w.

(1.6.23)

Let w1 and w2 be nonzero complex numbers. Then, with f : C 7→ C\{0} given by (1.6.10), f inv (w1 w2 ) = f inv (w1 ) + f inv (w2 ).

(1.6.24)

log w1 w2 = log w1 + log w2

(1.6.25)

arg w1 w2 = arg w1 + arg w2 .

(1.6.26)

However,

if and only if

For example, (√ arg

2 2

+

√ 2 2

(√ )2 π π π ȷ = arg ȷ = = + = arg 22 + 2 4 4 (√

and thus log However,

2 2

+

√

2 2

)2 (√ ȷ = log 22 +

√ 2 2

√ 2 2

) (√ ȷ + arg 22 +

) (√ ȷ + log 22 +

√

2 2

) ȷ .

arg (−1)2 = arg 1 = 0 , 2π = π + π = arg(−1) + arg(−1),

√ 2 2

) ȷ ,

20

and thus

CHAPTER 1

log (−1)2 = log 1 = 0 , 2πȷ = πȷ + πȷ = log(−1) + log(−1).

Therefore, there exist nonzero complex numbers w1 and w2 such that the principal logarithm does not satisfy (1.6.25). Let w be a nonzero complex number. Then, w = elog w .

(1.6.27)

( ) Im z log ez = z − round 2πȷ, 2π

(1.6.28)

Now, let z be a complex number. Then,

where, for all x ∈ R, round(x) denotes the closest integer to x except in the case where 2x is an integer, in which case round(x) = ⌊x⌋. Therefore, log ez = z if and only if Im z ∈ (−π, π]. An analogous situation arises for nth roots. Consider f : R 7→ [0, ∞) defined by f (x) = x2 . √ √ √ Then, for all y ∈ [0, ∞), it follows that f inv (y) = {− y, y}, where y represents the nonnegative square root of y ≥ 0. For complex-valued extensions, let n ≥ 1, and define f : C 7→ C by f (z) = zn . Let w be a nonzero complex number. If z satisfies zn = w, then log zn = log w = log |w| + (arg w) ȷ, where “log” is the principal log. Furthermore, z satisfies zn = w if and only if there exists an integer i such that n log z = log |w| + (arg w + 2iπ) ȷ. Therefore, for all i ∈ Z, define △

zi = e n [log |w|+(arg w+2iπ) ȷ] ,

(1.6.29)

zni = w.

(1.6.30)

1

which satisfies Note that, for all i ∈ Z, zn+i = zi . Therefore, for all i ∈ {0, . . . , n − 1}, define the n distinct numbers √ arg w 2iπ △ n zi = |w|e n ȷ e n ȷ , (1.6.31) √n where |w| is the nonnegative nth root of |w|. Consequently, f inv (w) = {z0 , . . . , zn−1 }. The principal nth root w1/n of w is defined by choosing z0 , which yields √n arg w △ w1/n = z0 = |w|e n ȷ . (1.6.32) √ n In particular, if w is a positive number, then w1/n = w, which is the positive nth root of w. However, for an odd integer n and a negative number a, a notational conflict arises between the principal nth root of a and the negative nth √ root of a. For example, (−1)1/3 = e(π/3) ȷ , whereas, for all √ odd integers √ n △ n, it is traditional to interpret −1 as −1. In other words, for all a < 0 and odd n ≥ 1, n a = − n |a|, and thus √n √ a1/n = |a|e(π/n) ȷ = n ae[(1/n−1)π] ȷ . (1.6.33) Let z and α be complex numbers, and assume that z is not zero. As an extension of the functions f (z) = zn and f (z) = z1/n , define △

zα = eα log z ,

(1.6.34)

where log z is the principal logarithm of z. For example, 1 = e−2 ȷ log ȷ = e−2 ȷ(π/2) ȷ = eπ . ȷ2 ȷ Next, let z1 and z2 be complex numbers, and let α be a real number. Then, (z1 z2 )α = zα1 zα2 . Now, let α be a complex number. Then, αz1 αz2 = αz1 +z2 . However, (z1 z2 )α and zα1 zα2 are not necessarily

SETS, LOGIC, NUMBERS, RELATIONS, ORDERINGS, GRAPHS, AND FUNCTIONS

21

equal. For example, (−1) ȷ (−1) ȷ = e−π e−π = e−2π , 1 = 1 ȷ = [(−1)(−1)] ȷ . However, (z1 z2 )α = zα1 zα2 e2nπα ȷ , where

   1, −2π < arg z1 + arg z2 ≤ −π,     n= 0, −π < arg z1 + arg z2 ≤ π,     −1, π < arg z1 + arg z2 ≤ 2π.

(1.6.35)

(1.6.36)

Finally,

where

(αz1 )z2 = αz1 z2 e2nπz2 ȷ ,

(1.6.37)

⌋ 1 (Im z1 ) log |α| + (Re z1 ) arg α − . n= 2 2π

(1.6.38)

⌊

For example, setting α = −1, z1 = −1, and z2 = (−1)−1/2 enπ ȷ = (1/ ȷ)(−1) = ȷ. Furthermore,

where n =

⌊

1 2

−

Im z1 2π

⌋

1 2

yields n = 1, and thus ȷ = (−1)1/2 = [(−1)−1 ]1/2 =

(ez1 )z2 = ez1 z2 e2nπz2 ȷ ,

(1.6.39)

. See [2216, pp. 108–114] and [2249, pp. 91, 114–119].

Finally, let z, α, and β be complex numbers. Then, (zα )β , (zβ )α , and zαβ may be different as can be seen from the example z = 21 ȷ, α = 2 − ȷ, and β = −3 − ȷ, where (zα )β ≈ 0.03 + 0.04 ȷ, (zβ )α ≈ 9104 + 10961 ȷ, and zαβ ≈ 17 + 20 ȷ. A similar situation can occur in the case where z, α, and β are real. For example, if z = −1, α = 1/2, and β = 2, then (zα )β = zαβ = −1 , 1 = (zβ )α . As π a final example, let z = e, α = 2πi ȷ, where i ≥ 1, and β = π. Then, (zβ )α = (eπ )2πi ȷ = e2πi ȷ log e = 2π2 i ȷ αβ 2 2 α β 2πi ȷ π π π log 1 π0 e = z = cos 2π i + ȷ sin 2π i and (z ) = (e ) = 1 = e = e = 1. Since, for all i ≥ 1, cos 2π2 i + ȷ sin 2π2 i , 1, it follows that (zβ )α = zαβ , (zα )β . See [2107, pp. 166, 167]. Definition 1.6.6. Let I ⊂ R be a finite or infinite interval, and let f : I 7→ R. Then, f is convex if, for all α ∈ [0, 1] and x, y ∈ I, f [αx + (1 − α)y] ≤ α f (x) + (1 − α) f (y).

(1.6.40)

Furthermore, f is strictly convex if, for all α ∈ (0, 1) and distinct x, y ∈ I, f [αx + (1 − α)y] < α f (x) + (1 − α) f (y).

(1.6.41)

Finally, f is (concave, strictly convex) if − f is (convex, strictly convex). A more general definition of a convex function is given by Definition 10.6.14. Let X be a set, and let σ : X × · · · × X 7→ X × · · · × X, where each Cartesian product has n factors. Then, σ is a permutation if, for all (x1 , . . . , xn ) ∈ X × · · · × X, the tuples (x1 , . . . , xn ) and σ[(x1 , . . . , xn )] have the same components with the same multiplicity but possibly in a different order. For convenience, we write (σ(x1 ), . . . , σ(xn )) for σ[(x1 , . . . , xn )]. In particular, we write (σ(1), . . . , σ(n)) for σ[(1, . . . , n)]. The permutation σ is a transposition if (σ(x1 ), . . . , σ(xn )) and (x1 , . . . , xn ) differ by exactly two distinct interchanged components. Finally, let sign(σ) denote −1 raised to the smallest number of transpositions needed to transform (σ(1), . . . , σ(n)) to (1, . . . , n). Note that, if σ1 and σ2 are permutations of (1, . . . , n), then sign(σ1 ◦ σ2 ) = sign(σ1 ) sign(σ2 ).

1.7 Facts on Logic Fact 1.7.1. Let A and B be statements. Then, the following statements hold: i) [A and (A =⇒ B)] =⇒ B.

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not(A and B) ⇐⇒ [(not A) or not B]. not(A or B) ⇐⇒ [(not A) and not B]. (A or B) ⇐⇒ [(not A) =⇒ B] ⇐⇒ [(A and B) xor (A xor B)]. (A =⇒ B) ⇐⇒ [(not A) or B] ⇐⇒ not(A and not B)] ⇐⇒ [(A and B) xor not A]. not(A and B) ⇐⇒ (A =⇒ not B) ⇐⇒ (B =⇒ not A). [A and not B] ⇐⇒ [not(A =⇒ B)]. Remark: Each statement is a tautology. Remark: ii) and iii) are De Morgan’s laws. See [493, p. 24]. See Fact 1.8.1. Fact 1.7.2. Let A and B be statements. Then, the following statements are equivalent: i) A ⇐⇒ B. ii) (A or not B) and not(A and not B). iii) (A or not B) and [(not A) or B]. iv) (A and B) or [(not A) and not B]. v) not(A xor B). Remark: The equivalence of each pair of statements is a tautology. Fact 1.7.3. Let A, B, and C be statements. Then, ii) iii) iv) v) vi) vii)

[(A =⇒ B) and (B =⇒ C)] =⇒ (A =⇒ C). Fact 1.7.4. Let A, B, and C be statements. Then, the following statements are equivalent:

i) A =⇒ (B or C). ii) [A and (not B)] =⇒ C. Remark: The statement that i) and ii) are equivalent is a tautology. Fact 1.7.5. Let A, B, and C be statements. Then, the following statements are equivalent: i) (A and B) =⇒ C. ii) [B and (not C)] =⇒ (not A). iii) [A and (not C)] =⇒ (not B). Source: To prove i) =⇒ ii), note that [(A and B) or (not B)] =⇒ [C or (not B)], that is, [A or (not B)] =⇒ [C or (not B)], and thus A =⇒ [C or (not B)]. Hence, [B and (not C)] =⇒ (not A). Conversely, to prove ii) =⇒ i), note that [(B and (not C)) or (not B)] =⇒ [(not A) or (not B)], that is, [(not C) or (not B)] =⇒ [(not A) or (not B)], and thus (not C) =⇒ [(not A) or (not B)]. Hence, (A and B) =⇒ C. Fact 1.7.6. Let X and Y be sets, and let Z be a statement that depends on elements of X and Y. Then, the following statements are equivalent: i) Not[for all x ∈ X, Z holds]. ii) There exists x ∈ X such that Z does not hold. Furthermore, the following statements are equivalent: iii) Not[there exists y ∈ Y such that Z holds]. iv) For all y ∈ Y, Z does not hold. Finally, the following statements are equivalent: v) Not[for all x ∈ X, there exists y ∈ Y such that Z holds]. vi) There exists x ∈ X such that, for all y ∈ Y, Z does not hold.

1.8 Facts on Sets Fact 1.8.1. Let A and B be subsets of a set X. Then, the following statements hold:

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23

A ∩ A = A ∪ A = A. A\B = A ∩ B∼ . (A ∪ B)∼ = A∼ ∩ B∼ . (A ∩ B)∼ = A∼ ∪ B∼ . (A\B) ∪ (A ∩ B) = A. A\(A ∩ B) = A ∩ B∼ . A ∩ (A∼ ∪ B) = A ∩ B. (A ∪ B) ∩ (A ∪ B∼ ) = A. [A\(A ∩ B)] ∪ B = A ∪ B. (A ∪ B) ∩ (A∼ ∪ B) ∩ (A ∪ B∼ ) = A ∩ B. (A∼ ∪B)∩(A∪B∼ ) = (A∩B)∪(A∼ ∩B∼ ) = [(A∪B)\(A∩B)]∼ = [(A∩B∼ )∪(A∼ ∩B)]∼ . Remark: iii) and iv) are De Morgan’s laws. See Fact 1.7.1. Fact 1.8.2. Let A, B, and C be subsets of a set X. Then, the following statements hold: i) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C). ii) A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C). iii) (A\B)\C = A\(B ∪ C). iv) (A ∩ B)\C = (A\C) ∩ (B\C). v) (A ∩ B)\(C ∩ B) = (A\C) ∩ B. vi) (A ∪ B)\C = (A\C) ∪ (B\C) = [A\(B ∪ C)] ∪ (B\C). vii) (A ∪ B)\(C ∩ B) = (A\B) ∪ (B\C). viii) A\(B ∪ C) = (A\B) ∩ A\B). ix) A\(B ∩ C) = (A\B) ∪ A\B). Fact 1.8.3. Let A, B, and C be subsets of a set X. Then, the following statements hold: i) A ⊖ ∅ = ∅ ⊖ A = A, A ⊖ A = ∅. ii) A ⊖ B = B ⊖ A. iii) A ⊖ B = (A ∩ B∼ ) ∪ (B ∩ A∼ ) = (A\B) ∪ (B\A) = (A ∪ B)\(A ∩ B). iv) A ⊖ B = {x ∈ X : (x ∈ A) xor (x ∈ B)}. v) A ⊖ B = ∅ if and only if A = B. vi) A ⊖ (B ⊖ C) = (A ⊖ B) ⊖ C. vii) (A ⊖ B) ⊖ (B ⊖ C) = A ⊖ C. viii) A ∩ (B ⊖ C) = (A ∩ B) ⊖ (A ∩ C). If, in addition, A and B are finite, then i) ii) iii) iv) v) vi) vii) viii) ix) x) xi)

card(A ⊖ B) = card(A) + card(B) − 2 card(A ∩ B). Fact 1.8.4. Let A, B, and C be finite sets. Then,

card(A × B) = card(A) card(B), card(A ∪ B) = card(A) + card(B) − card(A ∩ B), card(A ∪ B ∪ C) = card(A) + card(B) + card(C) − card(A ∩ B) − card(A ∩ C) − card(B ∩ C) + card(A ∩ B ∩ C). Remark: The second and third equalities are versions of the inclusion-exclusion principle. See [411, p. 82], [1372, p. 67], and [2520, pp. 64–67]. Remark: The inclusion-exclusion principle

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holds for multisets A and B with “A ∪ B” defined as the smallest multiset that contains both A and B. For example, card({1, 1, 2, 2}) = card({1, 1, 2} ∪ {1, 2, 2}) = card({1, 1, 2}) + card({1, 2, 2}) − card({1, 2}); that is, 4 = 3 + 3 − 2. See [2879]. △ Fact 1.8.5. Define A = {x1 , . . . , x1 , . . . , xn , . . . , xn }ms , where, for all i ∈ {1, . . . , n}, ki is the ∏ number of repetitions of xi . Then, the number of multisubsets of A is ni=1 (ki + 1). Source: [2460]. Fact 1.8.6. Let A, B ⊆ R. Then, the following statements hold: i) sup(−A) = − inf A. ii) inf(−A) = − sup A. iii) sup(A + B) = sup A + sup B. iv) sup(A − B) = sup A − inf B. v) inf(A + B) = inf A + inf B. vi) inf(A − B) = inf A − sup B. vii) sup(A ∪ B) = max {sup A, sup B}. viii) inf(A ∪ B) = min {inf A, inf B}. ix) If 0 < A, then } { } { 1 1 1 : x ∈ A = max , . sup x inf[A ∩ (−∞, 0)] inf[A ∩ (0, ∞)] x) sup {xy : x ∈ A, y ∈ B} = max {(inf A) inf B, (inf A) sup B, (sup A) inf B, (sup A) sup B}. Source: [1566, p. 3]. △ ∑ Fact 1.8.7. Let S 1 , . . . , S m be finite sets, and let n = m i=1 card(S i ). Then, ⌈n⌉ ≤ max card(S i ). i∈{1,...,m} m In particular, if m < n, then there exists i ∈ {1, . . . , m} such that card(S i ) ≥ 2. Remark: This is the pigeonhole principle. Fact 1.8.8. Let S 1 , . . . , S m be sets, assume that, for all i ∈ {1, . . . , m}, card(S i ) = n, and assume that, for all distinct i, j ∈ {1, . . . , m}, card(S i ∩ S j ) ≤ k. Then, ( ) n2 m ≤ card ∪m i=1 S i . n + (m − 1)k Source: [1561, p. 23]. △ Fact 1.8.9. Let X be a set, let n = card(X), let S 1 , . . . , (S m ⊆) X, and assume that, for all distinct n i, j ∈ {1, . . . , m}, S i \S j and S j \S i are nonempty. Then, m ≤ ⌊n/2⌋ . Source: [1992, p. 57]. Remark:

This is a Sperner lemma. △ Fact 1.8.10. Let X be a set, let n = card(X), let S 1 , . . . , S m ⊆ X, let k ≤ n/2, assume that, for all i (∈ {1, ) . . . , m}, card(S i ) = k, and, for all distinct i, j ∈ {1, . . . , m}, S i ∩ S j is nonempty. Then, n−1 m ≤ k−1 . Source: [1992, p. 57]. Remark: This is the Erd¨os-Ko-Rado theorem. △

Fact 1.8.11. Let X be a set, let n = card(X), let S 1 , . . . , S m ⊆ X, assume that, for all i ∈

{1, . . . , m}, card(S i ) is odd, and, for all distinct i, j ∈ {1, . . . , m}, card(S i ∩ S j ) is even. Then, m ≤ n. Source: [1992, p. 57]. Remark: This is the oddtown theorem. △ Fact 1.8.12. Let X be a set, let n = card(X), let S 1 , . . . , S m ⊆ X, let p ≥ 2 be prime, and assume that, for all i ∈ {1,( . ). . , m}, card(S i ) = 2p−1, and, for all distinct i, j ∈ {1, . . . , m}, card(S i ∩S j ) , p−1. ∑ p−1 n Then, m ≤ i=1 i . Source: [1992, p. 58]. Remark: Excluding intersections of cardinality p − 1 restricts the number of possible subsets of X.

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Fact 1.8.13. Let X be a set, let S 1 , . . . , S m , T 1 , . . . , T m ⊆ X, let k ≥ 1 and l ≥ 1, and assume that, for all i ∈ {1, . . . , m}, card(S i ) = k,( card(T ) i ) = l, and S i ∩ T i = ∅, and, for all i, j ∈ {1, . . . , m} such that i < j, S i ∩ T j , ∅. Then, m ≤ k+l l . Source: [1992, pp. 171–173]. Fact 1.8.14. Let S be a set, and let S denote the set of all subsets of S . Then, “⊂” and “⊆” are transitive relations on S, and “⊆” is a partial ordering on S. Fact 1.8.15. Define the relation R on R × R by △

R = {((x1 , y1 ), (x2 , y2 )) ∈ (R × R) × (R × R) : x1 ≤ x2 and y1 ≤ y2 }. Then, R is a partial ordering. Fact 1.8.16. Define the relation L on R × R by △

L = {((x1 , y1 ), (x2 , y2 )) ∈ (R × R) × (R × R) : x1 ≤ x2 and, if x1 = x2 , then y1 ≤ y2 }. Then, L is a total ordering on R × R. Remark: Denoting this total ordering by “≼,” note that (1, 4) ≼ (2, 3) and (1, 4) ≼ (1, 5). Remark: This ordering is the lexicographic ordering or dictionary ordering, where “book” ≼ “box”. Note that the ordering of words in a dictionary is reflexive, antisymmetric, and transitive, and that every pair of words can be ordered. Related: Fact 3.11.23. Fact 1.8.17. Let n≥1 and x1 , . . . , xn2 +1 ∈R. Then, at least one of the following statements holds: i) There exist 1 ≤ i1 ≤ · · · ≤ in+1 ≤ n2 + 1 such that xi1 ≤ · · · ≤ xin+1 . ii) There exist 1 ≤ i1 ≤ · · · ≤ in+1 ≤ n2 + 1 such that xi1 ≥ · · · ≥ xin+1 . Source: [2294, p. 53] and [2526]. Remark: This is the Erd¨os-Szekeres theorem.

1.9 Facts on Graphs Fact 1.9.1. Let G = (X, R) be a directed graph. Then, the following statements hold: i) R is the graph of a function on X if and only if every node in X has exactly one child. Furthermore, the following statements are equivalent: ii) R is the graph of a one-to-one function on X. iii) R is the graph of an onto function on X. iv) R is the graph of a one-to-one and onto function on X. v) Every node in X has exactly one child and not more than one parent. vi) Every node in X has exactly one child and at least one parent. vii) Every node in X has exactly one child and exactly one parent. Related: Fact 1.10.1. Fact 1.9.2. Let G = (X, R) be a directed graph, and assume that R is the graph of a function f : X 7→ X. Then, either f is the identity function or G has a directed cycle. Fact 1.9.3. Let G = (X, R) be a directed graph, and assume that G has a directed Hamiltonian cycle. Then, G has no roots and no leaves. Fact 1.9.4. Let G = (X, R) be a directed graph. Then, G has either a root or a directed cycle. Fact 1.9.5. Let G = (X, R) be a directed graph. If G is a directed tree, then it is not transitive. Fact 1.9.6. Let G = (X, R) be a directed graph, and assume that G is directionally acyclic. Furthermore, for all x, y ∈ X, let “x ≼ y” denote the existence of directional path from x to y. Then, “≼” is a partial ordering on X. Remark: This result provides the foundation for the Hasse diagram, which illustrates the structure of a partially ordered set. See [2405, 2734]. Fact 1.9.7. Let G = (X, R) be a directed graph. If G is a directed forest, then G is directionally acyclic.

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Fact 1.9.8. Let G = (X, R) be a symmetric graph, and let n = card(X). Then, the following statements are equivalent: i) G is a forest. ii) G is acyclic. iii) No pair of nodes in X is connected by more than one path. Furthermore, the following statements are equivalent: iv) G is a tree. v) G is a connected forest. vi) G is connected and has no cycles. vii) G is connected and has n − 1 edges. viii) G has no cycles and has n − 1 edges. ix) Every pair of nodes in X is connected by exactly one path. Fact 1.9.9. Let G = (X, R) be a tournament. Then, G has a directed Hamiltonian path. If, in addition, G is directionally connected, then G has a directed Hamiltonian cycle. Remark: The second statement is Camion’s theorem. See [276, p. 16]. Remark: The directed edges in a tournament distinguish winners and losers in a contest where every player (that is, node) encounters every other player exactly once. Fact 1.9.10. Let G = (X, R) be a symmetric graph without self-edges, where X ⊂ R2, assume △ that v = card(X) ≥ 3, assume that G is connected, and assume that the edges in R can be represented by line segments that lie in the same plane and that pairwise either are disjoint or intersect at a node. Furthermore, let e denote the number of edges of G, and let f denote the number of disjoint regions in R2 whose boundaries are the edges of G. Then,

3 f ≤ e ≤ 3v − 6, f ≤ 2v − 4. 2 If, in addition, G has no triangles, then e ≤ 2v − 4. Source: [754, pp. 162–166] and [2735, pp. 97–116]. Remark: The equality gives the Euler characteristic for a planar graph. A related result for the surfaces of a convex polyhedron is given by Fact 5.4.8. See [2307]. f + v − e = 2,

1.10 Facts on Functions Fact 1.10.1. Let X and Y be finite sets, and let f : X 7→ Y. Then, the following statements hold:

i) If card(X) < card(Y), then f is not onto. ii) If card(Y) < card(X), then f is not one-to-one. iii) If f is one-to-one and onto, then card(X) = card(Y). Now, assume that card(X) = card(Y). Then, the following statements are equivalent: iv) f is one-to-one. v) f is onto. vi) card[ f (X)] = card(X). Related: Fact 1.9.1. Fact 1.10.2. Let f : X 7→ Y be invertible. Then, f Inv is invertible, and ( f Inv )Inv = f. Fact 1.10.3. Let f : X 7→ Y. Then, for all A, B ⊆ X, the following statements hold: i) A ⊆ f inv [ f (A)] ⊆ X. ii) f inv [ f (X)] = X = f inv (Y). iii) If A ⊆ B, then f (A) ⊆ f (B). iv) f (A ∩ B) ⊆ f (A) ∩ f (B).

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v) f (A ∪ B) = f (A) ∪ f (B). vi) f (A)\ f (B) ⊆ f (A\B). Furthermore, the following statements are equivalent: vii) f is one-to-one. viii) For all A ⊆ X, f inv [ f (A)] = A. ix) For all A, B ⊆ X, f (A ∩ B) = f (A) ∩ f (B). x) For all disjoint A, B ⊆ X, f (A) and f (B) are disjoint. xi) For all A, B ⊆ X, f (A)\ f (B) = f (A\B). Source: [154, pp. 44, 45] and [643, p. 64]. Remark: To show that equality does not necessarily hold in iv), let f (x) = x2 , A = [−2, 1], and B = [−1, 2]. Then, f (A∩B) = [0, 1] ⊂ [0, 4] = f (A)∩ f (B). Related: Fact 3.12.7. Fact 1.10.4. Let f : X 7→ Y. Then, for all A, B ⊆ Y, the following statements hold: i) f [ f inv (A)] = A ∩ f (X) ⊆ A. ii) f [ f inv (Y)] = f (X). iii) If A ⊆ B, then f inv (A) ⊆ f inv (B). iv) f inv (A ∩ B) = f inv (A) ∩ f inv (B). v) f inv (A ∪ B) = f inv (A) ∪ f inv (B). vi) f inv (A)\ f inv (B) = f inv (A\B). In addition, the following statements are equivalent: vii) f is onto. viii) For all A ⊆ Y, f [ f inv (A)] = A. Source: [154, pp. 44, 45] and [643, p. 64]. Related: Fact 3.12.8. Fact 1.10.5. Let f : X 7→ Y. Then, the following statements hold: i) If f is invertible, then, for all y ∈ Y, f inv (y) = { f Inv (y)}. △ ii) Assume that f is left invertible, and define fˆ : X 7→ R( f ), where, for all x ∈ X, fˆ(x) = f (x). Then, fˆ is invertible, and, for all y ∈ R( f ), f inv (y) = { fˆInv (y)}. iii) If f is left invertible and f L is a left inverse of f , then, for all y ∈ R( f ), f inv (y) = { f L (y)}. iv) If f is right invertible and f R is a right inverse of f , then, for all y ∈ Y, f R (y) ∈ f inv (y). Related: Fact 3.18.8. Fact 1.10.6. Let g: X 7→ Y and f : Y 7→ Z. Then, the following statements hold: i) If A ⊆ Z, then ( f ◦ g)inv (A) = ginv [ f inv (A)]. ii) f ◦ g is one-to-one if and only if g is one-to-one and the restriction fˆ: g(X) 7→ Z of f is one-to-one. If these conditions hold and gL and fˆL are left inverses of g and fˆ, respectively, then gL ◦ fˆL is a left inverse of f ◦ g. iii) f ◦ g is onto if and only if the restriction fˆ: g(X) 7→ Z of f is onto. Let gˆ : X 7→ g(X), where, for all x ∈ X, gˆ (x) = g(x). If these conditions hold and gˆ R and fˆR are right inverses of gˆ and fˆ, respectively, then gˆ R ◦ fˆR is a right inverse of f ◦ g. iv) f ◦ g is invertible if and only if g is one-to-one and the restriction fˆ: g(X) 7→ Z of f is oneto-one and onto. If these conditions hold, gL is a left inverse of g, and fˆInv is the inverse of fˆ, then ( f ◦ g)Inv = gL ◦ fˆInv . Remark: A matrix version of this result is given by Fact 3.18.9 and Fact 3.18.10. Fact 1.10.7. Let f : X 7→ Y, let g: Y 7→ X, and assume that f and g are one-to-one. Then, there exists h: X 7→ Y such that h is one-to-one and onto. Source: [968, pp. 311, 312] and [2092,

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pp. 16, 17]. Remark: This is the Schroeder-Bernstein theorem. Fact 1.10.8. Let X and Y be sets, let f : X 7→ Y, and, for i ∈ {1, 2}, let gi : R( f ) 7→ Fn and αi ∈ F. Then, (α1 g1 + α2 g2 ) ◦ f = α1 (g1 ◦ f ) + α2 (g2 ◦ f ). Remark: The composition operator △ C(g, f ) = g ◦ f is linear in its first argument.

1.11 Facts on Integers Fact 1.11.1. Let n, m ≥ 0 and k, l ≥ 2. Then,

∏k

+ i) , ml . Source: [997]. Remark: A product of consecutive integers cannot be a power of an integer. Fact 1.11.2. Let n be an integer. Then, n(n + 1)(n + 2)(n + 3) + 1 = (n2 + 3n + 1)2 . Hence, n(n + 1)(n + 2)(n + 3) + 1 is a square. Example: 5(6)(7)(8) + 1 = 412 . Related: Fact 2.1.2. Fact 1.11.3. Let x be a real number, and assume that x + 12 is not an integer. Then, the integer closest to x is ⌊x + 12 ⌋. Fact 1.11.4. Let w, x, y, and z be real numbers, and let n and m be integers. Then, the following statements hold: i) If w|x and y|z, then wy|xz. ii) If x|y and x|z, then x2 |yz. iii) If x|y, then x|ny. iv) If x|y and y|z, then x|z. v) If x|y and x|z, then x|my + nz. Fact 1.11.5. Let n and m be integers, at least one of which is nonzero. Then, the following statements hold: i) Assume that m is positive. Then, there exist unique integers q and r ∈ [0, m − 1] such that n = qm + r. In particular, q = ⌊n/m⌋ and r = remm (n) = n − qm = n − m⌊n/m⌋ ∈ [0, m − 1]. ii) If m is positive, then ⌈n/m⌉ = ⌊(n + m − 1)/m⌋. iii) If n|m, then gcd {n, m} = |n|. iv) If k is prime and k|mn, then either k|m or k|n. v) gcd {n/ gcd {n, m}, m/ gcd {n, m}} = 1. vi) If both n and m are prime and m , n, then n and m are coprime. vii) If n > 0 and m > 0, then 1 ≤ gcd {n, m} ≤ min {n, m, |n − m|}. viii) (lcm {n, m}) gcd {n, m} = |nm|. ix) n and m are coprime if and only if lcm {n, m} = |nm|. x) There exist integers k, l such that gcd {n, m} = kn + lm. Now, assume that n and m are coprime, and let k be an integer. Then, the following statements hold: xi) gcd {n − m, n + m, nm} = 1. xii) gcd {nk − mk , nk + mk } ≤ 2. xiii) gcd {(n − m)k , (n + m)k } ≤ 2k . xiv) gcd {n2 − nm + m2 , n + m} ≤ 3. xv) gcd {nk, m} = gcd {k, m}. Finally, let n1 , . . . , nk and m1 , . . . , ml be integers. Then, the following statement holds: xvi) gcd {n1 m1 , n1 m2 , . . . , nk ml } = (gcd {n1 , . . . , nk }) gcd {m1 , . . . , ml }. Source: [2380, p. 12]. x)–xiv) are given in [1757, pp. 86, 89, 105]; xv) is given in [1241, p. 123]. Example: gcd {221, 754} = 13 = −17(221) + 5(754). See [1757, pp. 86, 87]. Remark: The first set in xvi) contains kl products. Remark: x) is the GCD identity. See [79, p. 17]. Fact 1.11.6. Let l, m, n ≥ 1. Then, the following statements hold: i=1 (n

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i) ii) iii) iv) v) vi) vii) viii)

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gcd {l, m, n} = gcd {gcd {l, m}, gcd {m, n}, gcd {n, l}}. lmn = (gcd {lm, mn, nl}) lcm {l, m, n}. gcd {l, lcm {m, n}} = lcm {gcd {l, m}, gcd {l, n}}. lcm {l, gcd {m, n}} = gcd {lcm {l, m}, lcm {l, n}}. gcd {lcm {l, m}, lcm {m, n}, lcm {n, l}} = lcm {gcd {l, m}, gcd {m, n}, gcd {n, l}}. lmn gcd {l, m, n} = (lcm {l, m, n})(gcd {l, m})(gcd {m, n}) gcd {n, l}. gcd {l, m} = gcd {l + m, lcm {l, m}}. (lcm {l, m, n})2 (gcd {l, m, n})2 = . gcd {l, m} gcd {m, n} gcd {n, l} lcm {l, m} lcm {m, n} lcm {n, l}

Source: [1757, p. 105]. i) is given in [289, pp. 25, 144]; viii) is given in [1158, p. 310]. Fact 1.11.7. Let n ≥ 1. Then, gcd {n2 + 1, (n + 1)2 + 1} ∈ {1, 5}. Furthermore, gcd {n2 + 1, (n + 5

1)2 + 1} = 5 if and only if n ≡ 2. Source: [289, pp. 31, 165]. Fact 1.11.8. Let k1 , . . . , kn be positive integers, and assume that k1 < · · · < kn . Then, n−1 ∑ i=1

1 1 ≤ 1 − n−1 . lcm {ki , ki+1 } 2

Source: [2380, p. 12]. Fact 1.11.9. Let m and n be integers. Then, the following statements are equivalent:

i) Either both m and n are even or both m and n are odd. 2

ii) n ≡ m. Furthermore, the following statements are equivalent: iii) m|n. |m|

iv) n ≡ 0. |m|

v) n ≡ m. Fact 1.11.10. Let k ≥ 1, and let m, n, p, q be integers. Then, the following statements hold: k

i) If n = m, then n ≡ m. k

ii) n ≡ n. Furthermore, the following statements are equivalent: iii) k|(n − m). k

iv) n ≡ m. k

v) m ≡ n. k

vi) −n ≡ −m. k

vii) n − m ≡ 0. Furthermore, the following statement holds: k

k

k

viii) If n ≡ m and m ≡ p, then n ≡ p. k

k

Next, if p ≡ q and n ≡ m, then the following statements hold: k

ix) n + p ≡ m + q. k

x) n − p ≡ m − q.

30

CHAPTER 1 k

xi) np ≡ mq. Finally, the following statements hold: k

k

k

k

xii) If n ≡ m, and p is a positive integer, then pn ≡ pm. xiii) If n ≡ m, and p is a positive integer, then n p ≡ m p . k

xiv) If pn ≡ pm, then n

k/ gcd {k,p}

≡

m.

k

k

xv) If pn ≡ pm and gcd {k, p} = 1, then n ≡ m. ∏ xvi) k!| k−1 i=0 (n + i). For example, 11(12)(13) = 6(286) and (22)(23) · · · (28) = 5040(1184040). k

k

k

xvii) If n ≡ n0 and m ≡ m0 , then nm ≡ remk (n0 m0 ). k

Source: xiv) is given in [2763, pp. 30, 31]. Remark: “≡” is an equivalence relation on Z, which

partitions Z into residue classes. Fact 1.11.11. Let n ≥ 1, and let m be the sum of the decimal digits of n. Then, the following statements hold: i) 3|n if and only if 3|m. 9

ii) n ≡ m. Source: [2763, pp. 31, 32]. Fact 1.11.12. Let n be a positive integer. Then, the following statements hold: 3

3

i) n2 ≡ 0 if and only if n ≡ 0. 3

3

3

ii) n2 ≡ 1 if and only if either n ≡ 1 or n ≡ 2. 3

3

3

3

3

3

3

3

3

3

Source: [2114]. Example: 3 ≡ 6 ≡ 9 ≡ 12 ≡ 15 ≡ 0, 9 ≡ 36 ≡ 81 ≡ 144 ≡ 225 ≡ 0, 3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

1 ≡ 4 ≡ 7 ≡ 10 ≡ 13 ≡ 1, 2 ≡ 5 ≡ 8 ≡ 11 ≡ 14 ≡ 2, and 1 ≡ 4 ≡ 16 ≡ 25 ≡ 49 ≡ 64 ≡ 100 ≡ 3 3 3 121 ≡ 169 ≡ 196 ≡ 1. Fact 1.11.13. Let k, l, m, n ≥ 1. Then, the following statements hold: i) If m ≤ n is prime, then m does not divide n!+1. Hence, there exists a prime k ∈ [n+1, n!+1] such that k|n! + 1. ii) None of the integers n! + 2, n! + 3, . . . , n! + n are prime. √ iii) Assume that n ≥ √3 2 is not prime, and let k be the smallest prime such that k|n. Then, k ≤ n. If, in addition, n < k, then n/k is prime. iv) If n is prime, then (2n−1 − 1)/n is an integer. 8

v) If n ≥ 3 is odd, then n2 ≡ 1. 6

6

vi) If n is prime and n ≥ 5, then either n ≡ 1 or n ≡ 5. 8

vii) If n ≡ 7, then n is not the sum of three squares of integers. viii) ix) x) xi) xii) xiii) xiv)

9

If n ≡ 4, then n is not the sum of three cubes of integers. The last digit of n2 is neither 2, 3, 7, nor 8. Neither 3 nor 5 divides (n + 1)3 − n3 . If n ≥ 2, then n4 + 4n is not prime. 3|n(n2 − 3n + 8), 6|n3 + 5n, 8|(n − 1)(n3 − 5n2 + 18n − 8). 9|4n + 15n − 1, 30|n5 − n, 120|n5 − 5n3 + 4n. 121 does not divide n2 + 3n + 5.

SETS, LOGIC, NUMBERS, RELATIONS, ORDERINGS, GRAPHS, AND FUNCTIONS

xv) xvi) xvii) xviii) xix) xx) xxi) xxii) xxiii) xxiv) xxv) xxvi) xxvii) xxviii) xxix) xxx) xxxi) xxxii) xxxiii)

31

n

3n+1 |23 + 1. 2n does not divide n!. If m ≤ n, then m!|nm . gcd {2m − 1, 2n − 1} = 2gcd {m,n} − 1. Hence, n|m if and only if 2n − 1|2m − 1. If n and 6 are coprime, then 24|n2 − 1. If n is even, then n2 − 1|2n! − 1. If 6|k + l + m, then 6|k3 + l3 + m3 . If n ≥ 4 and m ≥ 4 are prime, then 24|n2 − m2 . If n is not prime, then 2n − 1 is not prime. Furthermore, if n ∈ {2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279}, then 2n − 1 is prime. If n ≥ 1 and 2n + 1 is prime, then there exists k ≥ 0 such that n = 2k . If k ∈ {0, 1, 2, 3, 4} k k then 22 + 1 is prime. If 5 ≤ k ≤ 32, then 22 + 1 is not prime. If n ≥ 3 is odd, then 3 - 2n − 1. If n ≥ 4 is even, then 3 - 2n + 1. If 4n + 2n + 1 is prime, then there exists a positive integer m such that n = 3m . √ If n ≥ 5 is prime, then there exists a positive integer m such that n = 24m + 1. n5 5

+

n4 2

+

n3 3

−

n 30

is a positive integer. √ If n ≥ 12, then n2 − 19n + 89 is not an integer. If n = k2 + l2 + m2 , then there exist positive integers p, q, r such that n2 = p2 + q2 + r2 . There exist infinitely many multisets of integers {x, y, z, p, q, r}ms such that x2 + y2 + z2 = p3 + q3 + r3 . If m and n are coprime, then {m + in : i ≥ 1} contains an infinite number of primes. n

If n is prime, then (k + l)n ≡ kn + ln . If kl = mn, then k + l + m + n is not prime. 2(k4 − l4 ) , m2 . If k, l, m, n are nonnegative, {k, l} , {m, n}, and k2 + l2 = m2 + n2 , then k2 + l2 is not prime. ( ) (n) xxxviii) If m and n are prime and m < n, then mn| n+m m − m − 1. xxxiv) xxxv) xxxvi) xxxvii)

n

If n ≥ 3 is prime, then ([(n − 1)/2]!)2 ≡ (−1)(n+1)/2 . 10|1 + 8n − 3n − 6n . 5|1 + 2n + 3n + 4n if and only if n/4 is not an integer. If k ≥ 3 and l ≥ 5 are consecutive primes, then k + l is the product of at least three primes. √ xliii) n4 + 2n3 + 2n2 + 2n + 1 is not an integer. Source: [107, pp. 595–598], [993, pp. 118, 131–137, 208]. iii) is given in [2763, pp. 13, 19]; viii) and ix) are given in [2763, pp. 31, 33]; xi) is given in [993, p. 120]; xx) is given in [1158, p. 266]; xxix) is given in [1757, p. 64]; xxxiii) is given in [2529, chapter 8]; xxxiv) is given in [2068, p. 68]; xxxv)–xxxvii) are given in [107, pp. 595–599]; xlviii) is given in [108, pp. 51, 294, 295]; xxxix) is given in [14]; xl)–xliii) are given in [289, pp. 7, 11, 32, 36, 72, 73, 82, 167, 178]. Remark: vi) implies that, if n is prime and n ≥ 5, then there exists a positive integer k such that either n = 6k − 1

xxxix) xl) xli) xlii)

6

6

or n = 6k + 1. For example, 23 ≡ 5 and 31 ≡ 1. For k = 20, neither n = 6k − 1 = 119 = 7(17) nor n = 6k + 1 = 121 = 112 is prime. Remark: i) and xxxiii) imply that there are an infinite number of primes. Remark: xxxiv) is Dirichlet’s theorem. Remark: The prime numbers 2n − 1 listed in xxiii) are Mersenne primes. It is unknown whether or not there exist infinitely many Mersenne

32

CHAPTER 1

primes. Remark: The prime numbers 2n + 1 listed in xxiv), namely, 3, 5, 17, 257, 65537, are Fermat primes. These are the only known Fermat primes. Example: In xxxvii), 12 + 72 = 52 + 52 = 50, 2 2 2 12 +(8)2 =( 4)2 + 72 = 65, and 02 +( 10 Example: ) (= ) 6 + 8 = 100 are not prime. (24 ) ( ) In xlviii), 5 3 8 5 2(3)| 2 − 2 − 1; that is, 6|6; 3(5)| 3 − 3 − 1, that is, 15|45; and 11(13)| 11 − 13 11 − 1, that is, 143|2496065. Fact 1.11.14. Let n ≥ 1. Then, there exist n consecutive positive integers whose sum of squares is prime if and only if n ∈ {2, 3, 6}. Source: [109, pp. 74, 75]. Example: 12 + 22 = 5, 22 + 32 + 42 = 29, and 22 + 32 + 42 + 52 + 62 + 72 = 139. n Fact 1.11.15. Let n ≥ 2 be prime, and let k ≥ 1. Then, n|kn − k. Equivalently, kn ≡ k. Source: [411, p. 115], [947], [993, p. 119]. Remark: This is Fermat’s little theorem. Remark: An equivalent statement is the following: Let n be prime, let k be a positive integer, and assume that n n and k are coprime. Then, kn−1 ≡ 1. See [2763, p. 42]. Example: 47 − 4 = 7(2340) and 133 − 13 = 3(728). Remark: 341|2341 − 2, but 341 = 11(31) is not prime. See [993, p. 120]. n

Fact 1.11.16. Let n ≥ 2 be prime, and let k and l be positive integers. Then, (k + l)n ≡ kn + ln . Source: [149]. Fact 1.11.17. Let n ≥ 2. Then, n is prime if and only if

∑n−1

n

in−1 ≡ n − 1. Remark: Necessity follows from Fermat’s little theorem given by Fact 1.11.15. Sufficiency is a conjecture. Example: i=1

7

16 + 26 + 36 + 46 + 56 + 66 = 67171 = 7(9595) + 6 ≡ 6. Fact 1.11.18. Let n be prime, and let k ≥ 1. Then,  n  ∑  k p −1, n − 1|k, i ≡  0, n − 1 - k. i=1 3

Source: [1918]. Example: Let n = 3 and k = 2. Then, 12 + 22 + 32 = 14 ≡ −1. Fact 1.11.19. Let n ≥ 5 be prime. Then,

    1, ≡   i −1,

n−1 ( ) ∑ 2i i=0

n

3

n ≡ 1, 3

n ≡ 2.

Source: [149]. Fact 1.11.20. Let n ≥ 2. Then, n is prime if and only if n|(n − 1)! + 1. Remark: This is Wilson’s n theorem. Remark: n|(n − 1)! + 1 is equivalent to (n − 1)! ≡ −1. Example: 4! + 1 = 5(5) and

12! + 1 = 13(36846277).

∏

2n −1

i Fact 1.11.21. Let n ≥ 3. Then, n is prime if and only if n−1 i=1 (2 − 1) ≡ n. Remark: This is Vantieghem’s theorem. Example: 4! + 1 = 5(5). Source: [1132]. Fact 1.11.22. Let p ≥ 2 be prime and let 1 ≤ n ≤ p. Then, p|(n − 1)!(p − n)! + (−1)n+1 . Source: [2068, p. 67]. Remark: This is an extension of Wilson’s theorem given by Fact 1.11.20. Example:

4!6! + 1 = 11(1571) and 13!9! − 1 = 23(98246143821913). Fact 1.11.23. Let m, n ≥ 1. Then, (m2 − n2 )2 + (2mn)2 = (m2 + n2 )2. Remark: This result characterizes all Pythagorean triples within an integer multiple. Example: If m = 2 and n = 1, then 32 + 42 = 52 ; if m = 3 and n = 2, then 52 + 122 = 132 ; if m = 4 and n = 1, then 82 + 152 = 172 ; if m = 4 and n = 3, then 72 + 242 = 252 . ∑ Fact 1.11.24. Let n ≥ 1. Then, there exist k ≥ 1 and δ1 , . . . , δk ∈ {−1, 1} such that n = ki=1 δi i2 . Source: [289, pp. 33, 171] and [1158, p. 9]. Example: 7 = 1−4−9+16+25−36, 12 = −1+4+9, and 18 = 1 − 4 − 9 + 16 + 25 − 36 − 49 − 64 + 81 − 100 + 121. Fact 1.11.25. Let n be a positive integer. Then, the number of 4-tuples of integers ( j, k, l, m) such that j2 + k2 + l2 + m2 = n is equal to 8 times the sum of the distinct divisors of n that are

SETS, LOGIC, NUMBERS, RELATIONS, ORDERINGS, GRAPHS, AND FUNCTIONS

33

not divisible by 4. Source: Fact 13.5.5 and [117, 2970]. Remark: This is Jacobi’s four-square theorem. Example: The distinct divisors of 4 that are not divisible by 4 are 1 and 2. Accordingly, the number of ways of writing 4 as a sum of squares of the components of a 4-tuple of integers is 24. Two of these are 02 + 02 + 02 + 22 and 12 + (−1)2 + 12 + 12 . Fact 1.11.26. Let n ≥ 0. Then, the following statements hold: ∑ i) There exist nonnegative integers m1 , . . . , m4 such that n = 4i=1 m2i . ∑ ii) There exist nonnegative integers m1 , . . . , m9 such that n = 9i=1 m3i . ∑ 4 iii) There exist nonnegative integers m1 , . . . , m19 such that n = 19 i=1 mi . ∑37 5 iv) There exist nonnegative integers m1 , . . . , m37 such that n = i=1 mi . ∑ 6 v) There exist nonnegative integers m1 , . . . , m73 such that n = 73 i=1 mi . Source: [1260, pp. 372, 373]. Remark: These are solutions of Waring’s problem. The first result is Lagrange’s four-square theorem. For example, 3 = 02 + 12 + 12 + 12 and 310 = 12 + 22 + 42 + 172 . Fact 1.11.27. Let n ≥ 0. Then, the following statements hold: i) There exist nonnegative integers m1 , . . . , m4 such that n = m21 + m22 + m23 + m24 . ii) There exist nonnegative integers m1 , . . . , m4 such that n = m21 + m22 + 2m23 + 2m24 . iii) There exist nonnegative integers m1 , . . . , m4 such that n = m21 + 2m22 + 4m23 + 14m24 . iv) rem(n, 4) , 3 if and only if there exist nonnegative integers m1 , . . . , m4 such that n = m21 + m22 + 4m23 + 4m24 . v) If n ≥ 2, then there exist nonnegative integers m1 , . . . , m4 such that n = 2m21 + 3m22 + 4m23 + 5m24 . vi) Let k1 , k2 , k3 , k4 be positive integers, and assume that, for all k ∈ {1, 2, 3, 5, 6, 7, 10, 14, 15}, there exist nonnegative integers m1 , . . . , m4 such that k = k1 m21 + k2 m22 + k3 m23 + k4 m24 . Then, there exist nonnegative integers m1 , . . . , m4 such that n = k1 m21 + k2 m22 + k3 m23 + k4 m24 . Remark: i)–iii) are universal positive integer-matrix quaternary quadratic forms. There are 54 such forms. See [2316, pp. 123–125] and [2886]. Related: Fact 10.18.27. Fact 1.11.28. Let i, j, k, l be odd positive integers. Then, there exist even nonnegative integers q, r, s, t such that q2 + r2 + s2 + t2 = i2 + j2 + k2 + l2 . If, in addition, i, j, k, l are distinct, then so are q, r, s, t. Example: 12 + 32 + 52 + 72 = 02 + 22 + 42 + 82 . Source: [2116]. Related: Fact 2.4.8. Fact 1.11.29. Let n ≥ 1, let d1 , . . . , dl be the distinct positive divisors of n, and, for all i ∈ {1, . . . , l}, let ai denote the number of distinct positive divisors of di . Then,  l 2 l ∑ ∑  3 ai =  ai  . i=1

i=1

Source: [2380, p. 64]. Remark: This is Liouville’s theorem. Related: Fact 1.12.1. Example:

Let n = 8 so that d1 = 1, d2 = 2, d3 = 4, d4 = 8, a1 = 1, a2 = 2, a3 = 3, and a4 = 4. Then, 13 + 23 + 33 + 43 = (1 + 2 + 3 + 4)2 . Let n = 15 so that d1 = 1, d2 = 3, d3 = 5, d4 = 15, a1 = 1, a2 = 2, a3 = 2, and a4 = 4. Then, 13 + 23 + 23 + 43 = (1 + 2 + 2 + 4)2 . Fact 1.11.30. The following statements hold: i) 12 +72 = 52 +52 = 50, 12 +82 = 42 +72 = 65, 22 +92 = 62 +72 = 85, 22 +112 = 52 +102 = 125. ii) 52 + 142 = 102 + 112 = 221, 42 + 192 = 112 + 162 = 377, 72 + 242 = 152 + 202 = 252 = 625. iii) 12 + 182 = 62 + 172 = 102 + 152 = 325, 202 + 1072 = 432 + 1002 = 682 + 852 = 11849. iv) 152 + 702 = 302 + 652 = 342 + 632 = 472 + 542 = 5125, 102 + 112 + 122 = 132 + 142 = 365. v) 252 + 602 = 332 + 562 = 162 + 632 = 392 + 522 = 652 = 4225, 1 + 3 + 32 + 33 + 34 = 112 . vi) 72 + 742 = 142 + 732 = 222 + 712 = 252 + 702 = 412 + 622 = 502 + 552 = 5525.

34

CHAPTER 1

52 + 172 + 182 = 92 + 142 + 192 = 638, 212 + 222 + 232 + 242 = 252 + 262 + 272 = 2030. 362 + 372 + 382 + 393 + 402 = 412 + 422 + 432 + 442 = 7230. 552 + 562 + 572 + 582 + 592 + 602 = 612 + 622 + 632 + 642 + 652 = 19855. 2972 = (88 + 209)2 = 88209, 77772 = (6048 + 1729)2 = 60481729. 33 + 43 + 53 = 63 = 216, 583 + 593 + 693 = 903 = 729000, 13 + 123 = 93 + 103 = 1729. 103 + 273 = 193 + 243 = 20683, 43 + 483 = 363 + 403 = 110656, 1 + 18 + 182 = 73 . 1673 + 4363 = 2283 + 4233 = 2553 + 4143 = 87539319, 113 + 123 + 133 + 143 = 203 = 8000. 313 + 333 + 353 + 373 + 393 + 413 = 663 = 287496, 24 + 24 + 34 + 44 + 44 = 54 . 594 + 1584 = 1334 + 1344 = 635318657, 304 + 1204 + 2724 + 3154 = 3534 = 15527402881. 2404 + 3404 + 4304 + 5994 = 6514 = 179607287601. 275 + 845 + 1105 + 1335 = 1445 = 61917364224, 1 + 7 + 72 + 73 = 202 . 16 − 26 + 36 = 3(6 + 63 ) = 22 + 32 + 52 + 72 + 112 + 132 + 172 = 666. 958004 + 2175194 + 4145604 = 4224814 = 31858749840007945920321. 36 + 196 + 226 = 106 + 156 + 236 = 160426514, 132 + 73 = 29 , 27 + 173 = 712 . 107 + 147 + 1237 + 1497 = 157 + 907 + 1297 + 1467 = 2056364173794800. 818 + 5398 + 9668 = 1588 + 3108 + 4818 + 7258 + 9548 = 765381793634649192581218. 429 + 999 + 1799 + 4759 + 5429 + 5749 + 6259 + 6689 + 8229 + 8519 = 9179 = 458483827502199203411828597. xxiv) 6210 + 11510 + 17210 + 24510 + 29510 + 53310 + 68910 + 92710 + 101110 + 123410 + 160310 + 168410 = 177210 = 303518810756415395921574821458201. xxv) For all i ∈ {1, 2, 3}, 1i + 21i + 36i + 56i = 2i + 18i + 39i + 55i . xxvi) For all i ∈ {1, 3, 9}, 1i + 13i + 13i + 14i + 18i + 23i = 5i + 9i + 10i + 15i + 21i + 22i . xxvii) For all i ∈ {−1, 1}, 4i + 10i + 12i = 5i + 6i + 15i , 6i + 14i + 14i = 7i + 9i + 18i , and 3i + 40i = 4i + 15i + 24i = 5i + 8i + 30i . xxviii) For all i ∈ {−2, −1, 1, 2}, (−230)i + (−92)i + 23i + 46i = (−220)i + (−110)i + 22i + 55i . xxix) For all i ∈ {1, 2, 6}, 83i +211i +(−300)i = (−124)i +(−185)i +303i , and 43i +371i +(−372)i = 140i + 307i + (−405)i . xxx) For all i ∈ {1, 3, 5}, (−51)i + (−33)i + (−24)i + 7i + 13i + 38i + 50i = (−134)i + (−75)i + (−66)i + 8i + 47i + 87i + 133i = 0. xxxi) For all i ∈ {1, 2, 3, 9}, (−621)i +51i +253i +412i +600i = (−624)i +187i +100i +429i +603i . xxxii) For all i ∈ {1, 3, 5, 7}, (−98)i + (−82)i + (−58)i + (−34)i + 13i + 16i + 69i + 75i + 99i = (−169)i + (−161)i + (−119)i + (−63)i + 8i + 50i + 132i + 148i + 174i = 0. xxxiii) For all i ∈ {1, 2, 3, 4, 5}, (−461)i + (−233)i + (−199)i + 465i + 237i + 203i = (−435)i + (−343)i + 1i + 3i + 347i + 439i . xxxiv) 13! = 1122962 − 798962 = 6227020800. Source: [564, 651, 731, 981, 1321, 2000, 2232] and [2259, pp. 48, 49]. Remark: xvii) and xix) are counterexamples to Euler’s conjecture, which states that, for all n ≥ 4, the nth power of a positive integer cannot be decomposed into the sum of n − 1 or fewer nth powers of integers. Euler’s conjecture is true in the case n = 3; that is, the cube of a positive integer cannot be the sum of the cubes of two positive integers. This case is given by Fact 1.11.39. Fact 1.11.31. Let i, j, k, l be positive integers. Then, there exist positive integers m, n, r, s such that {m, n} , {r, s} and (i2 + j2 )(k2 + l2 ) = m2 + n2 = r2 + s2 . vii) viii) ix) x) xi) xii) xiii) xiv) xv) xvi) xvii) xviii) xix) xx) xxi) xxii) xxiii)

35

SETS, LOGIC, NUMBERS, RELATIONS, ORDERINGS, GRAPHS, AND FUNCTIONS

In particular, m = |ik − jl|, n = jk + il, r = ik + jl, and s = |il − jk|. Source: Fact 2.4.7 and [2107, pp. 25, 26]. Example: (22 + 32 )(42 + 52 ) = 533 = 72 + 222 = 232 + 22 . Fact 1.11.32. Let k, m, n ≥ 1, assume that k > m + n, let x1 , . . . , xm , y1 , . . . , yn be integers, ∑ ∑n k k ↓ ↓ and assume that m i=1 xi = i=1 yi . Then, m = n and x = y . Remark: This is Euler’s extended conjecture. See [982, 1740]. Fact 1.11.33. Let n ≥ 0. Then, there exist k, l ≥ 0 such that n = k2 + l2 if and only if n does not have a prime factor of the form 4k + 3 raised to an odd exponent. Source: [20, Chapter 4] and [2450, p. 378]. Remark: 29 = 22 + 52 , but neither 27, 71, nor 243 is the sum of two squares. Fact 1.11.34. Let n ≥ 0. Then, there exist k, l, m ≥ 0 such that n = k2 + l2 + m2 if and only if 8

8

8

there do not exist i, j ≥ 0 such that n = 4i (8 j + 7). Hence, if k, l ≥ 1, k ≡ 3, and l ≡ 5, then kl ≡ 7, and thus kl is not the sum of three squares. Source: [1258, p. 38] and [2316, p. 59]. Remark: 14 = 12 + 22 + 32 , but 15 = 40 (8 · 1 + 7) is not the sum of three squares. Fact 1.11.35. Let n ≥ 0. Then, there exist positive integers k, l, m such that k < l < m and n = k2 + l2 − m2 . Source: [2380, pp. 56, 57]. Example: 0 = 32 + 42 − 52 , 1 = 42 + 72 − 82 , and 2 = 52 + 112 − 122 . Fact 1.11.36. Let l, m, n ≥ 1. Then, there exist integers j, k such that j2 + k2 = (l2 + m2 )n . Source: [1757, p. 115]. Example: (22 + 32 )3 = 2197 = 92 + 462 . Fact 1.11.37. Let n ≥ 1. Then, the following statements are equivalent: i) There exist k, l ≥ 1 such that n = k3 + l3 . √ √ √ 2 4n ii) There exists a divisor m of n such that 3 n ≤ m ≤ 22/3 3 n, 3|m2 − n/m, and 3m − m3 is an integer. Furthermore, the following statements are equivalent: iii) There exist k, l ≥ 1 such that n = k3 − l3 . √ √ 2 4n iv) There exists a divisor m of n such that 1 ≤ m ≤ 3 n, 3|m2 − mn , and 3m − m3 is an integer. Source: [580]. Example: 91 = 33 + 43 and m = 7. Fact 1.11.38. Let n ≥ 2. Then, Hn is not an integer. Source: [1757, p. 105]. Fact 1.11.39. Let k, l, m ≥ 1 and n ≥ 3. Then, kn + ln , mn. Remark: This is Fermat’s last theorem. Credit: A. Wiles. 4

Fact 1.11.40. Let n ≥ 2 be prime, and assume that n ≡ 1. Then, there exist k, l ≥ 1 such that n = k2 + l2 . Source: [116, p. 41] and [2963]. Credit: P. de Fermat. Example: 29 = 4 + 25 and

89 = 25 + 64.

Fact 1.11.41. Let k, l, m, n ≥ 2, and assume that kl − mn = 1. Then, k = 3, l = 2, m = 2, and n = 3. Remark: This is Catalan’s conjecture. Credit: P. Mih˘ailescu. Fact 1.11.42. Let n ≥ 1. Then, there exists a prime m ∈ (n, 2n]. If, in addition, n ≥ 2898242, then there exists a prime m ∈ (n, n + n/(111 log2 n)]. Source: [20, Chapter 2] and [202, 2736]. Remark: The first statement is Bertrand’s postulate. Fact 1.11.43. Let n ≥ 20, and, for all i ≥ 1, let pi denote the ith prime. Then,

n(log n + log log n − 32 ) < pn < n(log n + log log n − 21 ). Source: [1350, p. 183] and [2342]. ∏ Fact 1.11.44. Let n ≥ 1, and, for all i ≥ 1, let pi denote the ith prime. Then, pi ≤ 4n , where the product is taken over all i such that pi ≤ n. Source: [2068, p. 90]. ∏ Fact 1.11.45. For all i ≥ 1, let pi denote the ith prime. Then, for all k ≥ 4, p2k+1 < ki=1 pi . Remark: This is Bonse’s inequality. Remark: 121 < 210 and 169 < 2310. Fact 1.11.46. Let n ≥ 4 be even. Then, there exist primes k and l such that n = k + l. Remark:

36

CHAPTER 1

This is the Goldbach conjecture. Example: 44 = 13 + 31 and 100 = 17 + 83. Remark: An incomplete proof is given in [594]. Fact 1.11.47. Let n ≥ 1, and let dn denote the sum of all positive integers (not counting multiplicity) that divide n. Then, dn ≤ Hn + eHn log Hn . Remark: This result is equivalent to the Riemann hypothesis. See [524, p. 48] and [1724]. Equivalent statements are given by Fact 13.3.1. Remark: △ Let rn = dn /(Hn + eHn log Hn ). Then, r12 ≈ .98864, r120 ≈ .98344, r360 ≈ .97111, and r2520 ≈ .97831. Fact 1.11.48. Let n ≥ 1, let {a1 , . . . , an } ∪ {b1 , . . . , bn } = {1, . . . , 2n}, and assume that a1 < · · · < ∑ an and bn < · · · < b1 . Then, ni=1 |ai − bi | = n2 . Source: [2380, p. 66]. Fact 1.11.49. If n ≥ 1, then there exist finitely many multisets {k1 , . . . , kn }ms of positive integers ∑ ∏ △ △ such that ni=1 k1i = 1. Now, define S 1 = 2 and, for all n ≥ 2, define S n = 1 + n−1 i=1 S i . In particular, ∑ 6 (S i )i=1 = (2, 3, 7, 43, 1807, 3263443). If n ≥ 2 and the positive integers k1 , . . . , kn satisfy ni=1 k1i = 1, then max {k1 , . . . , kn } ≤ S n−1 − 1. Source: [2336, p. 288] and [2494]. Fact 1.11.50. Let n ≥ 1. Then, 4 1 1 = − , 4n + 1 n n(4n + 1) If n is odd, then

4 1 1 = + . 4n − 1 n n(4n − 1)

4 2 2 4 . = + − 2 n n − 1 n + 1 n(n − 1)

3

If n ≡ 2, then

4 1 3 3 = + + . n n n + 1 n(n + 1)

Source: [131]. Remark: These equalities concern Egyptian fractions and are associated with the Erd¨os-Straus conjecture. See Fact 1.11.51. Fact 1.11.51. Let n ≥ 2. Then, there exist k, l, m ≥ 1 such that 4/n = 1/k+1/l+1/m. Example: 4/5 = 1/2 + 1/4 + 1/20 = 1/2 + 1/5 + 1/10. Remark: This is the Erd¨os-Straus conjecture. Related: Fact 1.11.50. √ √ √ Fact 1.11.52. Let n ≥ 1. Then, ⌊ n + n + 1⌋ = ⌊ 4n + 2⌋. Source: [289, pp. 19, 119].

1.12 Facts on Finite Sums Fact 1.12.1. Let n, k ≥ 1. Then, n ∑ i=1

   k+1 ( ) ( ) k ∑   ∑ 1 k + 1 k + 1 1 i  .   ik = − B B (n + 1) Bi (n + 1)k+1−i =  k+1 k+1−i   i k + 1 i=0 k + 1  i=0 i

In particular,

( ) n+1 i= = 12 n(n + 1) = 21 n2 + 21 n, 2 i=1 ( ) ( ) ( ) n ∑ 1 2n + 2 n+1 n+1 i2 = = +2 = 16 n(n + 1)(2n + 1) = 31 n3 + 21 n2 + 16 n, 4 3 2 3 i=1  n 2 ( )2 n ∑ ∑  n+1 3   i =  i = = 41 n2 (n + 1)2 = 41 n4 + 12 n3 + 41 n2 , 2 i=1 i=1 n ∑

n ∑ i=1

i4 =

1 30 n(n

+ 1)(2n + 1)(3n2 + 3n − 1) = 51 n5 + 12 n4 + 31 n3 −

1 30 n,

37

SETS, LOGIC, NUMBERS, RELATIONS, ORDERINGS, GRAPHS, AND FUNCTIONS n ∑

i5 =

1 2 12 n (n

+ 1)2 (2n2 + 2n − 1) = 16 n6 + 21 n5 +

5 4 12 n

−

1 2 12 n ,

i=1 n ∑

i6 =

1 42 n(n

+ 1)(2n + 1)(3n4 + 6n3 − 3n + 1) = 17 n7 + 21 n6 + 12 n5 − 61 n3 +

1 42 n,

i=1 n ∑

i7 =

1 2 24 n (n

+ 1)2 (3n4 + 6n3 − 4n + 2),

i=1 n ∑

i8 =

1 90 n(n

+ 1)(2n + 1)(5n6 + 15n5 + 5n4 − 15n3 − n2 + 9n − 3),

i=1 n ∑

n ∑

i(i + 1) = 31 n(n + 1)(n + 2),

i=1 n ∑

i(i + 1)2 =

1 12 n(n

i(i + 1)(i + 2) = 14 n(n + 1)(n + 2)(n + 3),

i=1 n ∑

+ 1)(n + 2)(3n + 5),

i=1

1 60 n(n

+ 1)(n + 2)(12n2 + 39n + 29),

i=1 n−1 ∑ (2i + 1) = n2 ,

n−1 ∑ (2i + 1)2 = 31 n(4n2 − 1),

n−1 ∑ (2i + 1)3 = n2 (2n2 − 1),

i=0

i=0

i=0

n−1 ∑ (2i + 1)4 =

4 1 15 n(48n

− 40n2 + 7),

n−1 ∑ (2i + 1)5 = 31 n2 (16n4 − 20n2 + 7), i=0

i=0 n−1 ∑

i(i + 1)3 =

(2i + 1)6 =

2 1 21 n(4n

− 1)(48n4 − 72n2 + 31),

i=0 △

∑n

n−1 ∑ (2i + 1)7 = 13 n2 (48n6 − 112n4 + 98n2 − 31). i=0

Now, let k ≥ 1 and n ≥ 1, and define pk (n) = i=1 ik . Then, the following statements hold: i) pk (n) is a polynomial whose degree is k + 1 and whose leading coefficient is 1/(k + 1). ii) The coefficient of n in pk (n) is (−1)k Bk . iii) pk (1) = 1. iv) For all z ∈ C, p′k (z) = kpk−1 (z) + (−1)k Bk . v) p2 divides p2k , and p3 divides p2k+1 . vi) p1 (n) divides p2k+1 (n). ∑ ( ) k+1 vii) ki=1 k+1 − n − 1. i pi (n) = (n + 1) Source: The first equality is the Bernoulli formula, where Bi is the ith Bernoulli number. See Fact 13.1.6. See [771, pp. 153–155], [1217, pp. 2, 3], [1219, pp. 283, 284], and [2689]. i)–iv) are given in [2915]; v) is given in [771, p. 155]; vi) is given in [1919]; and vii) is given in [2504, p. 135]. Remark: v) is a statement about polynomials, whereas vi) is a statement about integers. Remark: A matrix approach to sums of powers of integers is given in [959]. The expressions involving binomial coefficients are given in [410] and [411, pp. 109–112]. See also [1917]. Related: Fact 1.11.29, Fact 1.12.2, Fact 2.11.27, and [1371, p. 11]. △ ∑ Fact 1.12.2. Let n ≥ 1, let k ≥ 0, and define σk = ni=1 ik . Then, σ1 = 21 (n + 12 )2 − 18 ,

σ2 = 13 (n + 21 )3 −

1 1 2σ41 = σ5 + σ7 , 12 (n + 2 ), 2 6 1 σ3 = σ21 , σ4 = ( 56 σ1 − 15 )σ2 , σ5 = 34 σ31 − 13 σ21 , σ6 = ( 12 7 σ1 − 7 σ1 + 7 )σ2 , 1 5 σ7 = 2σ41 − 43 σ31 + 13 σ21 , σ31 = 41 σ3 + 43 σ5 , σ51 = 16 σ5 + 58 σ7 + 16 σ9 , 3 2 3 2 2 4 2 2 3 8σ1 + σ1 − 9σ2 = 0, 81σ2 − 18σ2 σ3 + σ3 − 64σ3 = 0, 16σ3 − σ3 − 6σ3 σ5 − 9σ25

= 0.

38

CHAPTER 1

Furthermore,

) k ( ∑ k+1 i=0

i

σi = (n + 1)k+1 − 1.

Next, define the polynomial △

Fk (s) =

( ) k 1 ∑ k+1 Bi (s + 1)k+1−i . k + 1 i=0 i

Then, F3 (s) = s2 , F4 (s) = 65 s − 51 , F5 (s) = 43 s3 − 13 s2 , F6 (s) =

12 2 7 s

− 67 s + 71 , F7 (s) = 2s4 − 43 s3 + 13 s2 .

If k ≥ 3 is odd, then σk = Fk (σ1 ) and deg Fk = 12 (k + 1). If k ≥ 2 is even, then σk = σ2 Fk (σ1 ) and deg Fk = 21 (k − 2). Source: [334]. Remark: Fk is a Faulhaber polynomial. Generating functions are given in [334]. Remark: Bi is the ith Bernoulli number. See Fact 13.1.6. Related: Fact 1.12.1. △ Fact 1.12.3. For all n ≥ 0, define the nth triangular number by T n = 21 n(n + 1). Then, the following statements hold: i) (T i )20 i=0 = (0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210). ( ) √∑n ∑ 1 3 ii) If n ≥ 1, then T n = ni=1 i = n+1 i=1 i . 2 = 2 n(n + 1) = iii) iv) v) vi) vii) viii) ix) x) xi) xii) xiii) xiv) xv) xvi) xvii) xviii) xix)

2

2T n T n+1 = T n2 +2n , T n T n+2 = 2T (n2 +3n)/2 , and T n ≡ ⌊(n + 1)/2⌋. If n ≥ 1, then 8T n + 1 = (2n + 1)2 and T n + T n+1 = (n + 1)2 . If n ≥ 2, then 8T n + 1 = T n+1 + 6T n + T n−1 . ∑ ∑ ∑ i+1 2 If n ≥ 1, then T n2 = ni=1 i3 , T 2n2 −1 = ni=1 (2i − 1)3 , and 2n−1 i=1 (−1) T i = n . ∑ 2 2 2 If n ≥ 1, then 9T n + 1 = T 3n+1 , T (n+1)2 = T n2 + T n+1 = T n2 + T n+1 − T n−1 , and ni=0 9i = T (3n+1 −1)/2 . ∑ If n ≥ 1, then ni=1 T i = n3 T n+1 = 31 (n + 2)T n = 16 n(n + 1)(n + 2) and 2T n2 = T n2 + n3 . If n ≥ 2, then T n2 = T n + T n−1 T n+1 . √ √ √ If n ≥ 2, then none of 3 T n , 4 T n , 5 T n are integers. If n ≥ 1, then the last digit of T n is not an element of {2, 4, 7, 9}. If n ≥ 1 and T n is prime, then n = 2 and T n = 3. If n, m ≥ 1, then T m+n = T m + T n + mn, T mn = T m T n + T m−1 T n−1 , T mn−1 = T m−1 T n + T m T n−1 . For all n ≥ 1, let τn be the nth positive integer such that T τn −1 is square. Then, (τi )12 i=1 = (1, 2, 9, 50, 289, 1682, 9801, 57122, 332929, 1940450, 11309769, 65918162). There are infinitely many square triangular numbers. For all n ≥ 1, τn+2 = 6τn+1 − τn − 2. √ √ For all n ≥ 1, τn = 12 [(3 − 2 2)n−1 + 1]2 /(3 − 2 2)n−1 . √ √ √ For all n ≥ 1, τn − τn − 1 = ( 2 − 1)n−1 . For all n ≥ 2, ⌊(n−1)/2⌋ ( )2 ⌊(n−2)/2⌋ ( )2  ∑ n − 1   ∑ i n − 1  i     . T τn −1 =  2 2 2i   i=0 2i + 1  i=0

In particular, (T τi −1 )9i=1 = (0, 1, 36, 1225, 41616, 1413721, 48024900, 1631432881). √ √ xx) For all n ≥ 1, T τn+2 = (6 T τn+1 − T τn )2 . √ xxi) ( T τi −1 )12 i=1 = (0, 1, 6, 35, 204, 1189, 6930, 40391, 235416, 1372105, 7997214, 46611179).

SETS, LOGIC, NUMBERS, RELATIONS, ORDERINGS, GRAPHS, AND FUNCTIONS

39

xxii) For all n ≥ 1, T τn+2 −1 = 34T τn+1 −1 − T τn −1 + 2. √ √ 1 [(1 + 2)2n−2 − (1 − 2)2n−2 ]2 . xxiii) For all n ≥ 1, T τn −1 = 32 xxiv) n is a triangular number and a Fibonacci number if and only if n ∈ {1, 3, 21, 55}. xxv) Every nonnegative integer is the sum of three triangular numbers. xxvi) Every triangular number except T 1 and T 3 is the sum of three positive triangular numbers. 2 2 2 xxvii) T 132 + T 143 = T 164 . ∏n xxviii) If n ≥ 3, then i=1 T i < T n! . ∑ () xxix) If n ≥ 0, then ni=0 ni T i = 2n−2 (T n+1 − 1). xxx) If n ≥ 1, then T n2 +n−1 + T n2 +3n+1 = (n + 1)4 . Source: [132, 2193, 2733]. xiii) and xxviii) are given in [177]; xviii) is given in [1497]; xix) is given in [108, pp. 55, 312, 313]; xx) is given in [1598]; xxv) is given in [1258, p. 25] and [2569]; xxix) is given in [1197]. xxx) is given in [1958]. Remark: T n is given by P2 (n) in Fact 1.12.6. See Fact 1.20.1. Related: Fact 13.4.6. △ Fact 1.12.4. For all n ∈ N, define the nth pentagonal number by Pn = 21 n(3n − 1). Then, the following statements hold: i) (Pi )18 i=0 = (0, 1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176, 210, 247, 287, 330, 376, 425, 477). ∑ ii) If n ≥ 1, then 1n ni=1 Pi = T n and 3Pn = T 3n−1 . √ iii) If n is a pentagonal number, then 61 ( 24n + 1 + 1) = n. iv) Let n ≥ 1. Then, the following statements are equivalent: a) n is a pentagonal number. √ 6 b) 24n + 1 is a square, and 24n + 1 ≡ 5. √ c) 16 ( 24n + 1 + 1) is an integer. v) Every nonnegative integer is the sum of five pentagonal numbers. vi) If n ≥ 1 and n < {9, 21, 31, 43, 55, 89}, then n is the sum of four pentagonal numbers. △ Finally, for all n ≥ 1, define the nth dual pentagonal number by P′n = 12 n(3n+1). Then, the following statements hold: vii) For all n ≥ 1, Pn < P′n < Pn+1 . viii) (P′i )18 i=1 = (2, 7, 15, 26, 40, 57, 77, 100, 126, 155, 187, 222, 260, 301, 345, 392, 442, 495). Source: [2569, 2862]. Remark: For all n ≥ 1, P′n = P−n . Remark: See [1083, pp. 45–47]. Related: Fact 13.4.7. Fact 1.12.5. For all n ≥ 0, define the nth generalized pentagonal number by     1 (n + 1)(3n + 1), n odd, △ 8 gn =     1 n(3n + 2), n even. 8

Then, the following statements hold: i) (gi )21 i=0 = (0, 1, 2, 5, 7, 12, 15, 22, 26, 35, 40, 51, 57, 70, 77, 92, 100, 117, 126, 145, 155, 176). ii) For all n ≥ 1, g2n−1 = 21 n(3n − 1) and g2n = 21 n(3n + 1). 1 ∞ iii) (gi )∞ i=−∞ = ( 2 i(3i − 1))i=−∞ . iv) For all n ≥ 0,

   n odd, P(n+1)/2 , gn =   ′ P−n/2 = P , n even. n/2 △

40

CHAPTER 1

v) (g0 , g1 , g2 , g3 , g4 , . . .) = (P0 , P1 , P′1 , P2 , P′2 , . . .), where Pn is the nth pentagonal number and P′n is the nth dual pentagonal number. vi) Every nonnegative integer is the sum of three generalized pentagonal numbers. Source: [136, 1282, 2569]. Related: Fact 1.12.4 and Fact 13.10.27. △ Fact 1.12.6. Let n, k ≥ 1, and define Pk (n) = card {(i1 , . . . , ik ) : 1 ≤ i1 ≤ · · · ≤ ik ≤ n}. Then, ( ) n+k−1 Pk (n) = . k In particular, P2 (n) =

n ∑

( i=

i=1

) n+1 n(n + 1) = , 2 2 (

P3 (n) =

P2 (n) =

( ) ∑ n ∑ i n+2 n(n + 1)(n + 2) , = j= 6 3 i=1 j=1

) ∑ j n ∑ i ∑ n(n + 1)(n + 2)(n + 3) n+3 . l= = 24 4 i=1 j=1 l=1

Remark: P2 (n), P3 (n), and P4 (n) are the triangular, tetrahedral, and pentatopic ( )numbers. Remark:

Pk (n) is the number of k-element multisubsets of {1, . . . , n}; that is, Pk (n) = nk . See Fact 1.16.16. r Related: Fact 1.12.3. () △ Fact 1.12.7. Let n ≥ 0 and k ≥ 3, and define the (n, k) polygonal number pk (n) = (k − 2) n2 + n. Then, the following statements hold: i) pk (n) = 12 n[(k − 2)n + 4 − k]. ii) p3 (n) = 21 n(n + 1) is the nth triangular number. iii) p4 (n) = n2 . iv) p5 (n) = 12 n(3n − 1) is the nth pentagonal number. dn x[(k − 3)x + 1] . v) pk (n) = dxn n!(1 − x)3 x=0 ∑ vi) Let m ≥ 0. Then, there exist nonnegative integers n1 , . . . , nk such that m = ki=1 pk (ni ). Source: [115, 1282, 2569, 2863]. Credit: The last statement is due to A. L. Cauchy. Fact 1.12.8. Let n ≥ 1. Then, n n ∑ ∑ 1 1 (i − j)2 = n2 (n2 − 1). |i − j| = n(n2 − 1), 3 6 i, j=1 i, j=1 △

Now, let k ≥ 1, and define σk = n ∑

∑n i=1

ik . Then,

|ik − jk | = 4σk+1 − 2(n + 1)σk ,

i, j=1

n ∑

|(i − j)(ik − jk )| = 2nσk+1 − n(n + 1)σk .

i, j=1

Source: [394]. Related: Fact 2.11.10. Fact 1.12.9. Let n ≥ 1. Then,

exp

n ∑ log

i, j=1

∏n 4i i i = i=1 , j (n!)2n+2

Source: [394]. Related: Fact 2.11.10.

exp

n ∑ (i − j) log

i, j=1

( ∏n 2i )n i i=1 i = . j (n!)n+1

41

SETS, LOGIC, NUMBERS, RELATIONS, ORDERINGS, GRAPHS, AND FUNCTIONS

Fact 1.12.10. Let 1 ≤ k ≤ n, let r ∈ R, and define △

S k,r =

k ∑∏

irj ,

j=1

where the sum is taken over all k-tuples (i1 , . . . , ik ) such that 1 ≤ i1 < · · · < ik ≤ n. Then, 1 1 n(n2 − 1)(3n + 2), S 3,1 = (n − 2)(n − 1)n2 (n + 1)2 , 24 48 1 1 S 1,2 = n(n + 1)(2n + 1), S 2,2 = n(n + 1)(2n + 1)(10n3 − 3n2 − 13n + 6). 6 432 Furthermore, for all r ∈ R, 1 2 S 3,r = (2S 1,3r − 3S 1,r S 1,2r + S 1,r ). 6 Source: [366]. √ 3 3 2 2 Fact √3 1.12.11. Let k ≥ 1 and n ≥ 1. If n ≤ k ≤ (n+1) −1, then ⌊ k⌋ = n. If n ≤ k ≤ (n+1) −1, then ⌊ k⌋ = n. Now, assume that n ≥ 2. Then, S 1,1 = 12 n(n + 1),

S 2,1 =

3 n∑ −1

2 n∑ −1

√3 1 ⌊ i⌋ = (n − 1)n2 (3n + 1). 4 i=1

√ 1 ⌊ i⌋ = n(n − 1)(4n + 1), 6 i=1

Source: [289, pp. 39, 187]. Fact 1.12.12. Let n ≥ 1. Then, n ⌊ ⌋ ∑ i

2

i=1

=

n2 (−1)n − 1 + , 4 8

⌊n/2⌋ ∑⌊ i=1

⌊ ⌋ i ⌋ ⌊n⌋ n + 2 = . 2 4 4

Source: [335]. Fact 1.12.13. Let n ≥ 3 and m ≥ 1, assume that n is prime, and assume that n - m. Then, n−1 ⌊ ∑ im ⌋ n−1 ⌊ 3 ⌋ ∑ im i=1

n

i=1

n

= 12 (n − 1)(m − 1),

⌋ n−1 ⌊ ∑ (−1)i i2 m n

i=1

= 41 (n − 1)(n2 m − nm − 2),

⌋ n−1 ⌊ ∑ (−1)i i4 m i=1

n

= 21 (n − 1)(m − 1),

= 21 (n − 1)[m(n2 − n − 1) − 1].

In particular, n−1 ⌊ ⌋ ∑ i i=1

n

= 0,

⌋ n−1 ⌊ ∑ (−1)i i2 i=1

n

If n is odd, then

If n is even and

= 0,

n−1 ⌊ 3 ⌋ ∑ i i=1

n

= 41 (n − 2)(n2 − 1),

⌊ 2⌋ n−1 ∑ i = 21 (n − 1). (−1)i n i=1 n 4 2 ≡

1, then

⌊ 2⌋ n−1 ∑ i n (−1)i =1− . n 2 i=1

⌋ n−1 ⌊ ∑ (−1)i i4 i=1

n

= 12 (n − 2)(n2 − 1).

42

CHAPTER 1

Finally, if n is an odd prime, then n−1 ⌊ n ⌋ ∑ i

n2

i=1

1 = 2 n

 n−1  ∑   in  − 1 (n − 1).  2  i=1

Source: [107, pp. 428–432] and [1673]. Fact 1.12.14. Let 1 ≤ m ≤ n. Then, m ∑∏

∏ n (n2 − i2 ), (2m − 1)! i=1 m−1

ij =

j=1

∑ where the sum is taken over all m-tuples (i1 , . . . , im ) of positive integers such that mj=1 i j = n. In particular, ∑ 1 i j = n(n2 − 1), 6 where the sum is taken over all ordered pairs (i, j) of positive integers such that i + j = n. Source: [771, pp. 33, 85]. Fact 1.12.15. Let 1 ≤ m < n. Then, n m ∑ ∏ (n + m + 1)! . i (i2 − j2 ) = 2(m + 1)(n − m − 1)! i=m+1 j=1 Source: [108, pp. 31, 188]. Fact 1.12.16. Let 1 ≤ k ≤ n. Then,

∑

card(∩ki=1 Si ) = 2k(n−1) n,

where the sum is taken over all k-tuples (S1 , . . . , Sk ) of subsets of {1, . . . , n}. In particular, ∑ card(S1 ∩ S2 ) = 4n−1 n, where the sum is taken over all ordered pairs (S1 , S2 ) of subsets of {1, . . . , n}. Source: [771, pp. 33, 34]. Fact 1.12.17. Let n ≥ 2. Then, ( ) n card({(i, j) : i, j ∈ {1, . . . , n} and i < j}) = . 2 Fact 1.12.18. Let n ≥ 1. Then, n ∑ (2i − 1) = n2 ,

2n ∑

i=1

i=1

i=

n ∑ (4i − 1) = (2n + 1)n, i=1

2n−1 ∑ i=1

i=

n ∑ (4i − 3) = (2n − 1)n. i=1

Fact 1.12.19. Let m, n ≥ 1. Then, n ∑ (mi − 1) = 21 mn(n + 1) − n,

n ∑ (mi − 1)2 = 16 m2 n(n + 1)(2n + 1) − mn(n + 1) + n,

i=1

i=1 n ∑ (mi − 1)3 = 41 m3 n2 (n + 1)2 − 12 m2 n(n + 1)(2n + 1) + 23 mn(n + 1) − n. i=1

In particular, n ∑ (2i − 1) = n2 ,

n ∑ (3i − 1) = 23 n2 + 21 n,

n ∑

i=1

i=1

i=1

(2i − 1)2 = 43 n3 − 31 n,

SETS, LOGIC, NUMBERS, RELATIONS, ORDERINGS, GRAPHS, AND FUNCTIONS n ∑ (3i − 1)2 = 3n3 − 23 n2 − 12 n,

n ∑ (2i − 1)3 = 2n4 − n2 .

i=1

i=1

43

Source: [1217, pp. 2, 3] and [1524, p. 37]. Fact 1.12.20. Let m ≥ n ≥ 1. Then, m,n ∑

1 n(n + 1)(3m − n + 1), 6

min {i, j} =

i, j=1

m,n ∑

max {i, j} =

i, j=1

1 1 2 n(n − 1) + mn(m + 1). 6 2

Source: [771, p. 168]. Fact 1.12.21. Let n ≥ 1. Then, n ∑

2i i = 2n+1 (n − 1) + 2,

i=1

n ∑

2i i2 = 2n+1 (n2 − 2n + 3) − 6,

i=1 n ∑

2i i3 = 2n+1 (n3 − 3n2 + 9n − 13) + 26.

i=1

If n ≥ 2, then

n−1 ∑

2i−1 (n − i) = 2n − n − 1.

i=1

Source: [2228, pp. 95, 97]. Fact 1.12.22. Let n ≥ 1, let x be a complex number, and assume that x , 1. Then, n ∑ i=0

xi =

n ∑

1 − xn+1 , 1−x

i=1

x − xn+1 , 1−x

n−1 ∑ xn+1 − (n + 1)x + n , (n − i)xi = (x − 1)2 i=0

ixi =

∑ i [nxn+1 − (n + 1)xn + 1]x (nxn − n−1 i=0 x )x = , x−1 (x − 1)2

i2 x i =

([n(x − 1) − 1]2 + x)xn+1 − x2 − x (x − 1)3

n ∑ i=1 n ∑

xi =

i=1

[n2 (x − 1)2 − 2n(x − 1) + x + 1]xn+1 − x2 − x (x − 1)3 ∑ i [n2 xn+1 − (n2 + 2n − 1)xn + 2 n−1 i=1 x + 1]x = (x − 1)2 ∑n−1 2 n [n x − i=0 (2i + 1)xi ]x . = x−1 =

In particular, n ∑ 2n+1 − n − 2 i = , i 2 2n i=1

n ∑ n2 + 4n + 6 i2 =6− . i 2 2n i=1

Source: [289, pp. 22, 132], [1757, pp. 54, 55], [1937], and [2228, pp. 95, 97]. Related: Fact

13.5.34.

44

CHAPTER 1

Fact 1.12.23. Let n ≥ 1. Then,

 1 n  ∑  − 2 n(n + 1), i+1 2 (−1) i =  1  n(n + 1),

 1 n  ∑  n even, − 2 n, i+1 (−1) i =  1  i=1 2 (n + 1), n odd,

i=1

2

n even, n odd.

Now, let m ≥ 1. Then, n ∑ (−1)i+1 (mi − 1) = 14 (−1)n [2 − m(2n + 1)] + 41 (m − 2), i=1 n ∑ (−1)i+1 (mi − 1)2 = 12 (−1)n+1 [m2 n(n + 1) − m(2n + 1) + 1] + 21 (1 − m). i=1

Source: [867] and [1217, pp. 2, 3]. Fact 1.12.24. Let n ≥ 2. Then,

( )2 i n−1 ∑ 1 n−1−i n2 −i2 ( 2n ) . = 4 (−1) i2 i=1 i=n+1 2n−1 ∑

n−i

+ = − Source: [42]. Example: Fact 1.12.25. Let n, m, k ≥ 1. Then, 1 16

n ∑ i=1

1 25

8 75

1 240

=

1 n = , [m + k(i − 1)](m + ki) m(kn + m)

41 400 . n ∑

1 n+1 = . (ki + m)(ki + m + k) m(kn + m + k)

i=0

In particular, n ∑ i=1 n ∑ i=1

n ∑

1 n = , i(i + 1) n + 1

1 n = , (i + 2)(i + 3) 3n + 9

i=1 n ∑ i=1

n ∑

n 1 = , 4i2 − 1 2n + 1

i=1

1 n = , (3i + 1)(3i − 2) 3n + 1

n 1 = , (i + 1)(i + 2) 2n + 4 n ∑ i=1

1 n = . (5i + 2)(5i − 3) 10n + 4

Source: [1217, p. 3]. Related: Fact 13.5.27. Fact 1.12.26. Let n, k ≥ 1. Then, n ∑ i=0 n ∑ i=0

n+1 1 = , (i + k)(i + k + 1) k(n + k + 1)

1 (n + 1)[(2k + 1)n + 2(k + 1)2 ] = . (i + k)(i + k + 2) 2k(k + 1)(n + k + 1)(n + k + 2)

Related: Fact 13.5.68. Fact 1.12.27. Let n ≥ 1. Then, n ∑ i=1

∑ 1 1 =2−2 (−1)i+1 , i(2i + 1) i i=1

n ∑ 2i2 − 1 i=1 n ∑ i=1

4i4

n ∑

2n+1

+1

=

1 2n + 1 − , 2 2(2n2 + 2n + 1)

i=1 n ∑ i=1

i 1 1 1 = − ∏n , 2 2 (2 i=0 j + 1) j=0 (2 j + 1)

∏i

4i4

i4

i 1 1 = − , 2 + 1 4 4(2n + 2n + 1)

i 3 2n2 + 2n + 3 = − , 2 + 4 8 4(n + 1)(n2 + 2n + 2)

n ∑ i=1

1 n2 + 3n = , i(i + 1)(i + 2) 4(n + 1)(n + 2)

45

SETS, LOGIC, NUMBERS, RELATIONS, ORDERINGS, GRAPHS, AND FUNCTIONS

∑ 1 1 1 =1+4 (−1)i + , i(i + 1)(2i + 1) i n+1 i=3

n ∑ i=1

n ∑ 3i2 + 3i + 1 i=1 n ∑ i=1

i3 (i + 1)3

n ∑

−1 + 1, (n + 1)3

=

i=1 2

i3 (i

i=1

If n ≥ 2, then

+

1)3 n ∑ i=2

If n ≥ 3, then n ∑ 1 i2 − 4

=

i2

=

n

2 , n3

i=1

i2 (i

+

1)2

=

2n+1 − 2, (n + 1)2

n2 + 2n 6i + 3 = , 4i4 + 8i3 + 8i2 + 4i + 3 2n2 + 4n + 3

i + 3i + 3 2n + 5n =− 2 , 4 3 2 i + 2i − 3i − 4i + 2 n + 2n − 1 2

n ∑ 2i (i3 − 3i2 − 3i − 1)

i=1, i,2

n ∑ 2i (i2 − 2i − 1)

2n+1

n ∑ i=1

4i i2 2 4n+1 (n − 1) = + , (i + 1)(i + 2) 3 3(n + 2)

n ∑ i3 + 6i2 + 11i + 5

(i + 3)!

i=0

=

5 n2 + 6n + 10 − . 2 (n + 3)!

3n2 − n − 2 3 1 2n + 1 = = − . 2 4 2n(n + 1) −1 4n + 4n

3 1 1 1 1 3 2n3 + 3n2 − n − 1 − − − − = − . 16 4(n − 1) 4n 4(n + 1) 4(n + 2) 16 2(n − 1)n(n + 1)(n + 2)

Source: [1217, pp. 2, 3]. The first equality is given in [506, p. 119]. The second equality is given

in [112, p. 41]. The third equality is given in [506, p. 235]. The fourth equality is given in [506, p. 122]. The fifth equality is given in [1757, p. 171]. The sixth equality is given in [506, p. 118]. The penultimate equality is given in [1217, p. 3]. Related: Fact 13.5.29 and Fact 13.5.71. Fact 1.12.28. Let n ≥ 1. Then, n ∑ i=1

∑1 1 2 1 =2 −3+ + , 2 2 2 2 n + 1 i (i + 1) i (n + 1) i=1 n

n n ∑ ∑ (−1)i+1 (−1)n+1 (−1)n+1 2 (−1)i+1 + . = 3 − 4 + i n+1 i2 (i + 1)2 (n + 1)2 i=1 i=1

Source: [108, pp. 34, 203–205]. Related: Fact 13.5.103. Fact 1.12.29. Let n ≥ 1. Then,

∏n−1

n ∑ i=1

j=1 (4i

i

∏n 2

4

+ j4 )

j=1, j,i (i

4

− j4 )

=

( ) 1 2n . 2n2 n

Source: [42]. Remark: For n = 1, both products are set to 1. Fact 1.12.30. Let n ≥ 1. Then,

√

1+ n ∑

√

i=1

Source: [384].

1 1 1 1 , + =1+ − 2 2 n n+1 n (n + 1)

∑ i2 + i + 1 n(n + 2) 1 1 + = = , i(i + 1) n+1 i2 (i + 1)2 i=1 n

1+

n ∑ i2 + i − 1 i=1

i(i + 1)

=

n2 . n+1

46

CHAPTER 1

Fact 1.12.31. Let n ≥ 1. Then, n ∑ i=1

√ 1 = n + 1 − 1. √ √ i+ i+1

Source: [1158, p. 121]. Fact 1.12.32. Let n ≥ 1. Then, n ∑ i=1

√

1 1 + (1 + 1/i)2 +

√

1 + (1 − 1/i)2

=

1 √ ( (n + 1)2 + n2 − 1). 4

Source: [107, pp. 3, 70, 71]. Fact 1.12.33. Let n ≥ 1. Then, n ∑ i=1

√4 1 = n + 1 − 1. √ √4 √ √4 ( i + i + 1)( i + i + 1)

Source: [107, pp. 4, 73]. Fact 1.12.34. Let n ≥ 1. Then,

∑ 1 ∑ 1 ∑ 1 1 ∏n =n+1− + + ··· + , i 1≤i< j≤n+1 i j (n + 1)! j=1 i j 1≤i≤n+1

where the last sum is taken over all n-tuples (i1 , . . . , in ) such that 1 ≤ i1 < · · · < in ≤ n + 1. Furthermore, n ∑ ∑ 1 = n, ∏i j=1 k j i=1 where the last sum is taken over all i-tuples (k1 , . . . , ki ) such that 1 ≤ k1 < · · · < ki ≤ n. Now, let n ≥ 2. Then, n−1 ∑ ∑ 1 (−1)i+1 ∏i

j=1 k j

i=1 n−1 ∑

(−1)i+1

i=1

=

∑

n−1 , n

n−1 ∑ ∑ 1 (−1)i+1 ∏i

2 j=1 k j

i=1

=

n−1 , 2n

(n − 1)(n + 2) 2i = , 3 3n(n + 1) (k + 1) j=1 j

∏i

where the second sum in each equality is taken over all i-tuples (k1 , . . . , ki ) such that 2 ≤ k1 < · · · < ki ≤ n. Source: [398, 894]. Fact 1.12.35. Let n ≥ 1. Then, ( ) ( ) n √ ∑ √ √ 2 3/2 2n 1 1 2n 1 1 2 1√ n < + − √ n+1< i< + − √ n + 1 < n3/2 + n. 3 3 8 8 n+1 3 6 3 2 6 n+1 i=1 Source: [2020]. Remark: It is conjectured in [2020] that

  ⌊( ⌋ ) n √  1 ∑  2 1 √    i = n+1 . + n i=1  3 6n

Fact 1.12.36. Let n ≥ 1. Then, n−1 ∑ i=1

n3 ∑ 2 < i, 3 i=1 n

i2
k ≥ 0 and z ∈ C. ) n ( )( ∑ z 1−z i=0

lvii) Let n ≥ 1. Then,

i n−i

=

( )( ) −z (n − 1)(1 − z) − k z − 1 . n(n − 1) k n−k−1

)2 n ( )2 ( ∑ n n+i i=0

i

i

=

)( )3 n ∑ i ( )( ∑ n n+i i . i i j i=0 j=0

lviii) Let n ≥ 1 and z ∈ C. Then, )( )( ) ) ]2 n ( n [( ∑ ∑ n + i 2i 2i i−1 n+i i+1 i z (z + 1) = n(n + 1) Ci z . 2i i i+1 2i i=1 i=0 lix) Let n ≥ 1. Then, ( )2 ( ) ∑ ( ) n n ∑ n 2n 1 2n − i Hi = Hn − , i n i n−i i=1 i=1 lx) Let n ≥ 0 and z1 , . . . , zm ∈ C. Then, ) m ( ∑∏ ij + zj j=1

ij

n ∑ i=1

( =

Hi

( )( ) ( ) ∑ ( ) n n 2n 3n 1 3n − i = Hn − . i i n i n−i i=1

n+m−1+ n

∑m

i=1 zi

) ,

where the sum is taken over all multisets {i1 , . . . , im }ms of nonnegative integers such that ∑m j=1 i j = n.

84

CHAPTER 1

lxi) Let n ≥ 1 and k ≥ 1. Then,

( ∑n )∏ ( ) n ( )i j ∑ ∑n n n+k−1 j=1 i j (−1) j=1 i j = , k i1 , . . . , in j=1 j

where the sum is over all n-tuples (i1 , . . . , in ) of nonnegative integers such that lxii) Let n ≥ 1. Then, )2 ( ) ( )2 n ∑ n ( ∑ i + j 4n − 2i − 2 j 2n = (2n + 1) . i 2n − 2i n i=0 j=0 lxiii) Let n ≥ 1. Then,

∑n j=1

ji j = n.

)( )( ) ∑ n ( ) n ∑ n ( ∑ 2i i+ j n−i n− j . = i i j n − i − j i=0 i=0 j=0

lxiv) Let m, n ≥ 1. Then, ( )( ) )( )( )( ) min {m,n} m ∑ n ( ∑ 1 m n i+ j m−i+ j n− j+i m+n−i− j (m + n + 1)! ∑ . = m!n! 2i + 1 i i i j i m−i i=0 i=0 j=0 lxv) Let n ≥ 0. Then,

lxvi) Let n ≥ 1. Then,

( )( ) ∑ ( )( ) ( ) n n ∑ 1 4n + 1 1 2i 2n + 1 1 2i 2n − i = = . 4i i n 4i i 2i 4n 2n i=0 i=0 ) m ( ∑∏ 2i j j=1

ij

=

4n Γ(n + m/2) , Γ(m/2)

where the sum is taken over all m-tuples (i1 , . . . , im ) of nonnegative integers such that ∑m j=1 i j = n. lxvii) Let n ≥ 0 and z1 , . . . , zm ∈ C. Then, (∑m ) m ( ) ∑∏ zi zj = i=1 , n ij j=1 where the sum is taken over all m-tuples (i1 , . . . , im ) of nonnegative integers such that ∑m j=1 i j = n. lxviii) Let n, m ≥ 1. Then, )( ) ( )2 n ( ∑ 2n 2n 2 2 2 2n |i − j | = 2n . n+i n+ j n i, j=−n lxix) Let n ≥ 1. Then,

) ( 1 )( 1 )( ( )2 2n ∑ − 2 −2i 4n + 1 2n = 4i 2 . 16n n i i 2n − i i=1

lxx) Let n ≥ 0, 0 ≤ k ≤ 2n + 1, and x ∈ C, and assume that −x < {0, 1, . . . , n}. Then, ] ( )2 n ( )2 [ ∑ n! (−i)k−1 n (−i)k [k − 2i(H − H )] = + xk , i n−i 2 n+1 x + i i (x + i) x i=0 ] ( )2 n ( )2 [ ∑ n 1 2 n! + (Hi − Hn−i ) = , i (x + i)2 x + i xn+1 i=0

SETS, LOGIC, NUMBERS, RELATIONS, ORDERINGS, GRAPHS, AND FUNCTIONS

85

)[ n ( )2 ( ∑ n n+i

] 1 n!(1 − x)n 1 + , (3H − 2H − H ) = i n−i n+i i i (x + i)2 x + i (xn+1 )2 i=0 )2 [ )2 ] ( n ( )2 ( ∑ n n+i 1 2 (1 − x)n + , (2Hi − Hn−i − Hn+i ) = i i (x + i)2 x + i xn+1 i=0 ) n ( )2 ( n ( )2 ∑ ∑ n n+i n (3Hi − 2Hn−i − Hn+i ) = 0, (Hi − Hn−i ) = 0, i i i i=0 i=0 )2 ( )2 ( ) n ( )2 ( n ∑ ∑ n n+i n 2n (2Hi − Hn−i − Hn+i ) = 0, (2i − n) (Hi − Hn−i ) = . i i i n i=0 i=0 lxxi) Let n ≥ 0 and x ∈ C, and assume that −x < {0, 1, . . . , n}. Then, ]2 )2 [ ] [ n ( )2 ( −i (1 − x)n 1 ∑ n n+i 1 + 2iHn+i + 2iHn−i − 4Hi , + =x + i x i=1 i x+i (x + i)2 xn+1 )2 [ ]2 ( )] [ n ( )2 ( ∑ n n+i i2 2i2 1 (1 − x)n 1+ − . + H + H − 2H = n+i n−i i i i (x + i)2 x + i i (1 + x)n i=1 Furthermore,

)2 n ( )2 ( ∑ n n+i

(1 + 2iHn+i + 2iHn−i − 4Hi ) = 0, i i ( )2 ( )2 ( ) n ∑ n+i 1 2 n i + Hn+i + Hn−i − 2Hi = n(n + 1). i i i i=1 i=1

lxxii) Let n ≥ 1. Then,

n ( )2 ∑ n

1 [2Hi + (n − 2i)(2Hi2 + Hi,2 )] = − , n i=1 n ( )3 n ( )3 ∑ ∑ n n [1 + 3(n − 2i)Hi ] = (−1)n , [2Hi + (n − 2i)(3Hi2 + Hi,2 )] = (−1)n 2Hn , i i i=0 i=1 ( ) ∑ ) n ( )4 n ( )( )( ∑ n n 2n 2n n 2n [1 + 4(n − 2i)Hi ] = (−1) , [1 + (n − 2i)(2Hi + Hn+i ] = (−1)n . i n i i n+i i=1 i=0 i

lxxiii) Let n ≥ 0 and 0 ≤ k ≤ 2n + 1. Then,  ( )2 n  ∑  0 ≤ k ≤ 2n, n 0, ik−1 [k − 2i(Hi − Hn−i )] =   (n!)2 , k = 2n + 1. i i=0 lxxiv) Let n ≥ 1. Then,

( )( ) 1 2i 2(n − i − 1) 16n ( ). = (2i + 1)(2n − 2i − 1) i n−i−1 8n2 2n i=0 n △ ∑m lxxv) Let n, m, k1 , . . . , km be positive integers, and define k = i=1 ki . Then, ) ( ) m ( ∑∏ ij + kj − 1 n+k−1 = , ij n j=1 n ∑

86

CHAPTER 1

( ) where the sum is taken over all n+m−1 m-tuples (i1 , . . . , im ) of nonnegative integers such n ∑m that j=1 i j = n. Source: i), iii), iv), xi), and lxvii) are given in [1219, pp. 167, 169, 171, 172]; ii) is given in [504, p. 75]; v), xii), xv), xvii), xix), xxiii), and xxiv) are given in [411, pp. 64–68, 78]; vi) is given in [1197, 2134]; vii) is given in [1225, p. 2] and [2228, p. 31]; viii) is given in [411, p. 78] and [2228, p. 130]; ix) is given in [2880, p. 138]; x) is given in [2799]; xiii) is given in [785]; xiv) is given in [1371, p. 9]; xvi) is given in [1372, p. 62]; xviii) is given in [2434]; xx) is given in [2402]; xxi) is given in [2228, p. 31]; xxii) is given in [2228, p. 138]; xxiv) follows from [1757, p. 163]; xxv) and xxvi) are given in [2228, pp. 95, 96]; xxvii) is given in [2243]; xxviii) is given in [666]; xxix) is given in [599, 671, 1154, 2799] and [1219, p. 187]; xxx) is given in [411, p. 79] and [666, 2255]; xxxi) is given in [1675, p. 84] and [2509]; xxxii) is given in [1675, p. 140]; xxxiii) is given in [69], [411, p. 78], and [1675, p. 97]; xxxiv) is given in [2228, p. 113]; xxxv) is given in [738], [1219, p. 201], [1274], and [2228, p. 142]; xxxvi) and xxxvii) are given in [1372, p. 66]; xxxviii) is given in [1757, pp. 161, 162]; xxxix)–xl) are given in [1158, pp. 300, 303]; xli) is given in [511, p. 156]; xlii) is given in [618]; xliv) is given in [771, p. 173]; xlv) is given in [616]; xlvi) is given in [1405, 2353] and [1675, p. 96]; xlvii) is given in [1210]; xlix) is given in [1166] and [2228, p. 22]; l) is given in [1158, p. 304]; li) is given in [771, p. 90], [2068, p. 171], and [2558]; lii) is given in [614] liii) is given in [116, pp. 399, 400] and [2068, p. 171]; liv) is given in [771, p. 90], [2068, p. 171], and [2228, p. 33]; lv) and lvi) are given in [771, p. 169]; lvii) is given in [2558]; lviii) is given in [2570]; lix) is given in [2100]; lx) is given in [2]; lxi) is given in [2016]; lxii)–lxiv) are given in [711]; lxv) is given in [67]; lxvi)–lxvii) are given in [2799]; lxviii) is given in [2744]; lxix) is given in [1164]; lxx) and lxxi) are given in [736, 737]; lxxii) is given in [2830]; lxxiii) is given in [737]; lxxiv) is given in [700]; lxxv) is given in [1587]. Remark: v) is Vandermonde’s convolution. xxxv) is Rothe’s identity; see [738, 1274]. S (n) in l) is the nth Franel number. See Fact 13.2.8. Fact 1.16.14. The following statements hold: i) Let n, m ≥ 0. Then, ( )( ) ( ) ( ) min {n,m} ∑ n+m m+n n+m m+n (−1)i = = . n+i m+i n m i=0 ii) Let n, m ≥ 0. Then, ( )( ) ∑ ( )( ) (2n)(2m) (2n)(2m) m n ∑ (2n)!(2m)! 2n 2m 2n 2m n m n m . (−1)i = (−1)i = (n+m) = (n+m) = n!m!(n + m)! n − i m − i n − i m − i i=−m i=−n n

iii) Let 0 ≤ n ≤ k ≤ m. Then,

( )( ) ( ) n m+i m n (−1) = (−1) . i k k−n i=0

n ∑

In particular,

iv) Let 0 ≤ k ≤ n. Then,

i

( )( ) k ∑ k m+i (−1)i = (−1)k . i k i=0 ( )( )  n  ∑  k < n, 0, i n i (−1) =  n (−1) , k = n. i k i=k

m

SETS, LOGIC, NUMBERS, RELATIONS, ORDERINGS, GRAPHS, AND FUNCTIONS

v) Let n, m, k ≥ 0. Then,

   0,  ( )( )     n m + i  (−1)n , (−1)i =  k−n  i k   n m i=0   (−1) (k − n)! ,

k < n, k = n,

n ∑

vi) Let n, m, k, l ≥ 0. Then,

k > n.

( ) )( ) l k+i l+m k − m . = (−1) (−1) n−l m+i n i=max {−m,n−k} (

l−m ∑

i

vii) Let n, m, k, l ≥ 0. Then, min {l−m,k+n} ∑ i=n

viii) Let n ≥ k ≥ 1. Then,

( (−1)i

( ) )( ) k−m−1 l−i k . = (−1)l+m l−m−n m i−n

( )( ) k ∑ n n−i (−1)i+1 = 0. i k−i i=0

ix) Let n ≥ 0. Then,

 0, ( )( )   n  ∑  1 2i n  ( ) i (−1) i = 1 n   2 i i   2n n/2 , i=0

x) Let n ≥ 0 and x ∈ C. Then,

xi) Let n ≥ 1.

n even, n odd.

( )( ) ( ) n ∑ i n i+x n x (−1) = (−1) . i i n i=0 ( )( ) n ∑ 2i n + i i 1 (−1) = 0. i+1 i 2i i=0

xii) Let n ≥ k ≥ 1. Then,

( )( ) ( ) n−k ∑ 2i n + i n−1 i 1 = . (−1) i + 1 i k + 2i k−1 i=0

xiii) Let n, k ≥ 0. Then,

(2n)(2k) ( )( ) 2n ∑ 2k k i 2n n n (−1) = (−1) (n+k) . i k − n + i i=0 n

In particular,

( )2 ( ) 2n ∑ 2n 2n (−1)i = (−1)n . i n i=0

xiv) Let n ≥ 0. Then, ⌊n/2⌋ ∑

(−1)i

i=0

( )( ) ⌊n/2⌋ ( )( ) ∑ n 2n − 2i n − i 2n − 2i = (−1)i = 2n , i n i n − i i=0

87

88

CHAPTER 1

(n) ( )( )  n  ∑  2n n/2 , n even, i 2i 2n − 2i (−1) =  0, i n−i n odd. i=0

(

)( ) n + 1 2n − 2i (−1) = n + 1, i n i=0

⌊n/2⌋ ∑

i

xv) Let n, k ≥ 1. Then,

( ) ( )( ) n ∑ k n (n − i)k = nkn−1 . (−1)i 2 i n+1 i=0

xvi) Let n ≥ 0. Then,

2n+1 ∑

(−1)i

i=0

xvii) Let n, k ≥ 0. Then,

( )2 2n + 1 = 0. i

( )( ) n ∑ n+k+1 k+i = 1. (−1)i k+i+l k i=0

xviii) Let n, m ≥ 0. Then,

) ( ) ( )( n n 2n − i . = (−1) m i m−i i=0

n ∑

xix) Let n ≥ 1 and 1 ≤ m ≤ 2n. Then, min {n,2n−m} ∑

(−4)i

i=0

i

( )( ) ( ) n 2n − 2i 2n = (−1)m . i m−i m

xx) Let n ≥ 1 and 1 ≤ m ≤ n. Then, ( )( ) ( ) n ∑ n 2i 2n (−1)i 4n−i = . i i−m n−m i=m In particular,

( )( ) ( ) n ∑ 2n i n−i n 2i (−1) 4 = . i i n i=0

xxi) Let n, k ≥ 0. Then,

( )( ) ( ) n ∑ n+1 i+k n−2i n − i i (−1) 2 = . i k 2k + 1 i=0

xxii) Let n ≥ 0 and k ≥ n + 1. Then, ( )( ) [( ) ] n ∑ k k−1−i 1 k i 1 n (−1) = + (−1) . i+1 i n−i k+1 n+1 i=0 xxiii) Let n ≥ k ≥ 0. Then,

In particular,

xxiv) Let n ≥ 1. Then,

( )( ) n− j n ∑ ∑ i i+ j j (−1) = 1. j k j=k i=0 ( ) ∑ ( ) n− j n− j n ∑ n ∑ ∑ i i+ j i i+ j (−1) = (−1) j = 1. j j j=0 i=0 j=1 i=0 () n ∑ 4i ni (−1)i (2i) = i=0

i

1 . 1 − 2n

SETS, LOGIC, NUMBERS, RELATIONS, ORDERINGS, GRAPHS, AND FUNCTIONS

xxv) Let n ≥ 1. Then,

89

(n)2 n ∑ 1 i (−1)i (2n) = (2n) . i=0

i

n

xxvi) Let n ≥ 0 and x ∈ C, and, if n ≥ 1, assume that −x < {1, . . . , n}. Then, (n) n ∑ x i i (−1) ( x+i) = . x + n i=0 i

xxvii) Let n ≥ 1. Then,

(n) (n) n n ∑ ∑ 1 i+1 i i i (−1) (n+i) = . (−1) (n+i) = 2 i=1 i=0 i

xxviii) Let n ≥ 1. Then,

i

( )3 ∑ ( )3 2n n ∑ (3n)! i 2n i 2n (−1) (−1) = (−1) = . i n+i (n!)3 i=−n i=0 n

xxix) Let n ≥ 0. Then,

( )( )( ) ( )2 2n ∑ 2n 2i 4n − 2i 2n (−1)i = . i i 2n − i n i=0

xxx) Let k, m, n ≥ 0. Then, k ∑ i=−k

(−1)i

(

)( )( ) 2k 2m 2n (k + m + n)!(2k)!(2m)!(2n)! = . k+i m+i n+i (k + m)!(m + n)!(n + k)!k!m!n!

xxxi) Let k, l, m, n, p ≥ 0. Then, ( )( )( )( ) ( )( ) min {k,n} ∑ p−k i+ j i + j k n p + n − i − j l n+k (−1) = (−1) . j+l i j m−i n+l m−n−l i, j=0 xxxii) Let n ≥ 1 and x, y ∈ C. Then, ( )( ) ( ) n ∑ y−x+n−1 i x+i y+n (−1) = , i n−i n i=0

(n)( x+i) n ∑ (y − x)n i i . (−1)i (y+i) = (y + 1)n i=0 i

xxxiii) Let n, k ≥ 1 and x ∈ C, where x < {−n, . . . , −1, 0}. Then, (n) n k ∑ ∑∏ 1 x i i (−1) ( x+i) = , x + i (x + n)k+1 j i=0 j=1 i

where the second sum is taken over all k-tuples of integers (i1 , . . . , ik ) such that 0 ≤ i1 ≤ · · · ≤ ik ≤ i. In particular, ( )∑∏ ( )∑ j n k n i ∑ ∑ ∑ n 1 1 n 1 1 (−1)i+1 = k, (−1)i+1 = 2, i i i jl n n j i=1 j=1 i=1 j=1 l=1 (n) j n i ∑ ∑ ∑ x 1 i = , (−1)i ( x+i) (x + j)(x + l) (x + n)3 i=0 j=1 l=1 i

90

CHAPTER 1

(n) j n i ∑ ∑ ∑ i+1 i (−1) ( x+i)

1 n , = (x + j)(x + l) (x + n)3 i=1 j=1 l=1 i  ( ) ∑ n i ∑ ∑ ∑   1 1 n 1   = 1 . + + (−1)i+1   n3 3 jk( j + k) jkl i j i=1 j=1 1≤ j 0. Remark: This result is equivalent to Bernoulli’s inequality. See Fact 2.1.21. Remark: For α ∈ [0, 1], a matrix version is given by Fact 10.10.47. Problem: Compare the second inequality to Fact 2.2.54 with y = 1. Fact 2.1.24. Let x and y be positive numbers. If x, y ∈ (0, 1], then ( )y y 1 ≤1+ . 1+ x x

Equality holds if and only if either y = 0 or x = y = 1. If x ∈ (0, 1), then )x ( 1 < 2. 1+ x

124

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If x > 1 and y ∈ [1, x], then 1+

( )y y 1 y y2 ≤ 1+ < 1+ + 2. x x x x

The left-hand inequality is an equality if and only if y = 1. Furthermore, if x > 1, then )x ( 1 < 3. 2< 1+ x Source: Fact 2.1.21 and [1371, p. 137].

√ √ √ a+1 a = a + 1 a . Then, a ≈ 6.91156. Furthermore, if √ √ √ √ √ √ < x + 1 x , whereas, if x ∈ (a, ∞), then x + 1 x < x x+1 . Source:

Fact 2.1.25. Let a > 0 satisfy √

√

√ x ∈ (0, a), then x x+1 [289, pp. 3, 54]. Fact 2.1.26. Let x ∈ [0, ∞) and 1 ≤ k ≤ l ≤ n. Then, √ √ √n l k 1 l k 1 + x ≤ 1 + x ≤ 1 + x ≤ 1 + x. n n n Source: [356]. Fact 2.1.27. Let x and α be real numbers, assume that either α ≤ 0 or α ≥ 1, and assume that x ∈ [0, 1]. Then, (1 + x)α ≤ 1 + (2α − 1)x. Equality holds if and only if either α = 0, α = 1, x = 0, or x = 1. Source: [77]. Fact 2.1.28. Let x ∈ R and n ≥ 1. Then, n ∑ x[nxn+1 − (n + 1)xn + 1] = (x − 1)2 ixi . i=1

Now, assume that x is nonnegative. Then, (n + 1)xn ≤ nxn+1 + 1,

(n + 1)xn/(n+1) ≤ nx + 1.

Furthermore, each inequality is an equality if and only if x = 1. Source: [658]. Remark: For x > 0, setting n = 3 and replacing x by 1/x yields 4x ≤ x4 + 3. Related: Fact 2.1.29 and Fact 2.2.39. Fact 2.1.29. Let x ∈ R and n ≥ 2. Then,   n−2  ∑   n−1 i n−1 n  x  . n(1 − x ) − (n − 1)(1 − x ) = (x − 1) (n − 1)x − i=0

Now, assume that x is nonnegative. Then, 1 − xn 1 − xn−1 ≤ . n n−1 Furthermore, equality holds if and only if x = 1. Source: [2332]. Related: This result follows from Fact 2.1.28 by replacing n with n − 1. Fact 2.1.30. Let x ∈ [−1, 1] and n ≥ 1. Then, (1 + x)n + (1 − x)n ≤ 2n . Source: [1158, p. 146]. Fact 2.1.31. Let a, b, c, x be real numbers, and assume that a2 + c2 ≤ 4b. Then, x4 + ax3 + bx2 + cx + 1 ≥ 0. Source: [112, p. 35]. Fact 2.1.32. Let x ≥ 0 and n ≥ 2. Then, nx ≤ xn + n − 1. Source: [2527, p. 34].

125

EQUALITIES AND INEQUALITIES

Fact 2.1.33. Let x be a nonnegative number. Then,

8x < x4 + 9, 3x2 ≤ x3 + 4, 27x3 < (x + 1)5 , 3x5 ≤ x11 + x4 + 1, 2x3 + x2 ≤ 2x4 + 1, x9 + x < x12 + x4 + 1, 4x2 ≤ x4 + x3 + x + 1, 8x2 < x4 + x3 + 4x + 4, 3(2x27 + x6 ) ≤ (2x15 + 1)2 . Source: [806, p. 5], [1033], and [1371, pp. 117, 123, 152, 153, 155]. Fact 2.1.34. Let n ≥ 1, 1 ≤ k ≤ n, and x ≥ 0. Then,

xk + xn−k ≤ xn + 1, (

x+1 (n + 1) 2

2

n ∑

xi ≤ n(xn+1 + 1),

(2n + 1)xn ≤

i=1

)n ≤

n ∑ i=0

2n ∑

xi =

i=0

x +1 x ≤ (n + 1) , 2 n

i

n−1 ∑ i=0

(

xn + 1 x ≤n 2 i

1 − x2n+1 , 1−x

)1−1/n .

Source: [393] and [1566, p. 10]. Related: Fact 2.2.38. Fact 2.1.35. Let n ≥ 1 and x > 0. Then,

xn + · · · + x2 + x + 1 (n + 1)(x + 1) n + 1 x2n + · · · + x4 + x2 + 1 ≤ n−1 ≤ 2n−1 , . 2n n x + · · · + x2 + x + 1 x + · · · + x3 + x If n ≥ 2, then xn + · · · + x2 + x + 1 (n + 1)x ≤ n−2 . n−1 x + · · · + x2 + x + 1 Source: [378]. Remark: For n = 3, the last inequality implies that, for all x ≥ 0, x2 + x ≤ x3 + 1. Fact 2.1.36. Let n ≥ 1 and x ∈ (0, 1). Then,

 i−1 2 n  ∑ n−1 ∑ ∑   1  xi  ≤ 4(log 2) x2i . i i=1

j=0

i=0

Source: [2527, p. 177]. Related: Fact 2.11.132. Fact 2.1.37. Let x be a positive number. Then,

1 + 12 x − 18 x2
0 and z < y < x. Then,

a x (y − z) + ay (z − x) + az (x − y) ≥ 0. Source: Use the fact that f (x) = a x is convex. See [1860, pp. 19, 111]. Fact 2.3.28. Let x, y, z be real numbers, let α be a real number, and assume that at least one of

the following statements holds: i) α is a positive even integer. ii) x, y, z are nonnegative, and either xyz > 0 or α ≥ 0.

154

CHAPTER 2

Then,

xα (x − y)(x − z) + yα (y − z)(y − x) + zα (z − x)(z − y) ≥ 0.

Equality holds if and only if either x = y = z or two of the numbers x, y, z are equal and the third number is zero. Source: Case i) is given in [140]. Case ii) is given in [806, p. 121], where α can be an arbitrary real number, whereas α > 0 is assumed in [140]. Remark: This is Schur’s inequality. Remark: Setting α = 0 yields xy + yz + zx ≤ x2 + y2 + z2 . See Fact 2.3.36. Setting α = 1 yields 4(x + y + z)(xy + yz + zx) ≤ (x + y + z)3 + 9xyz, x(y2 + z2 ) + y(z2 + x2 ) + z(x2 + y2 ) ≤ 3xyz + x3 + y3 + z3 , 0 ≤ (x − y)2 (x + y − z) + (y − z)2 (y + z − x) + (z − x)2 (z + x − y). Equivalently, in the case where x, y, z are nonnegative, 2(xy + yz + zx) ≤ x2 + y2 + z2 +

9xyz , x+y+z

2≤

x y z 4xyz + + + . y + z z + x x + y (x + y)(y + z)(z + x)

See [752, p. 329] and Fact 2.3.60. Remark: If x, y, z are nonnegative, xyz = 0, and α < 0, then, △ defining 1/0 = ∞, the inequality has the form ∞ ≥ 0. With this convention, the second condition in ii) can be deleted. Related: Fact 2.3.16, Fact 2.3.67, and Fact 2.3.70. Fact 2.3.29. Let w, x, y, z be nonnegative numbers. Then, (w6 − 3w4 + 2w3 − 3w2 + 1)xyz(x + y + z) + w2 (3 − w4 )(x3 y + y3 z + z3 x) + (3w4 − 1)(xy3 + yz3 + zx3 ) ≤ 2w3 (x4 + y4 + z4 ). Source: [98]. Fact 2.3.30. Let x, y, z, a, b, c be nonnegative numbers, and assume that either a ≤ b ≤ c or

c ≤ b ≤ a. Then,

a(x − y)(x − z) + b(y − z)(y − x) + c(z − x)(z − y) ≥ 0.

Source: [806, p. 123]. Remark: This is a consequence of Schur’s inequality. See Fact 2.3.28. Fact 2.3.31. For all r ∈ R and x, y, z ∈ (0, ∞), define △

αr = xr (x − y)(x − z) + yr (y − z)(y − x) + zr (z − x)(z − y). Now, let p, r ∈ R, assume that pr ≥ 0, and let x, y, z ∈ (0, ∞). Then, α p αr ≤ α0 α p+r . Source: [748, p. 68]. Fact 2.3.32. Let x, y, z ∈ [0, ∞), let r0 ≈ 1.5583 be the unique positive solution of (1 + s)1+s = (3s) s , and let r ∈ [0, r0 ]. Then, ( x + y + z )r+1 1 r . (x y + yr z + zr x) ≤ 3 3 Source: [748, p. 113]. Fact 2.3.33. Let x, y, z ∈ [0, ∞). If r ∈ [−1, 2], then 0 ≤ x2 (x − y)(x − ry) + y2 (y − z)(y − rz) + z2 (z − x)(z − rx). If r ∈ [−2, 2], then 0 ≤ x(x − y)(x2 − ry2 ) + y(y − z)(y2 − rz2 ) + z(z − x)(z2 − rx2 ). Source: [748, p. 109]. Fact 2.3.34. Let x, y, z ∈ (−1, ∞). Then,

2≤

x2 + 1 y2 + 1 z2 + 1 + + . z2 + y + 1 x2 + z + 1 y2 + x + 1

155

EQUALITIES AND INEQUALITIES

Source: [748, p. 10]. Fact 2.3.35. Let x, y, z ∈ [0, 1]. Then,

1 ≤ (1 − x)(1 − y)(1 − z) + x + y + z,

x(1 − y2 ) + y(1 − z2 ) + z(1 − x2 ) ≤ 45 .

Source: [752, p. 39] and [112, p. 37]. Remark: The first inequality is equivalent to xyz ≤

xy + yz + zx.

Fact 2.3.36. Let x, y, z be nonnegative numbers. Then,

3xyz ≤ x2 y + y2 x + z3 , 3xyz ≤ x2 y + y2 z + z2 x, 8xyz ≤ (xy + 1)(yz + 1)(zx + 1), 6(x2 + 2y2 + 3z2 ) ≤ (x + 2y + 3z)2 , 5xy + 3yz + 7zx ≤ 6x2 + 4y2 + 5z2, 81xyz ≤ (x + 1)(x + y)(y + z)(z + 16), 5(x + y + z) ≤ 2(x2 + y2 + z2 ) + xyz + 8, 9(xy + yz + zx) ≤ (x2 + 2)(y2 + 2)(z2 + 2), (x + 1)(y + 1)(z + 1) ≤ 2xyz + x2 + y2 + z2 + 3, 6(xy + yz + zx) ≤ (x + y + z)(xy + yz + zx + 3), 5(x + y + z + 1)2 ≤ 16(x2 + 1)(y2 + 1)(z2 + 1), 8(x2 y2 z2 + xyz + 1)2 ≤ 9(x4 + 1)(y4 + 1)(z4 + 1), xy + yz + zx + x + y + z ≤ x2 + y2 + z2 + xyz + 2, 8(xy + yz + zx) ≤ x3 + y3 + z3 + 4(x + y + z) + 9xyz, (x + y + z)3 ≤ (x5 − x2 + 3)(y5 − y2 + 3)(z5 − z2 + 3), √ √ √ xy + yz + zx ≤ ( xy + yz + zx)2 ≤ 3(xy + yz + zx), x2 y2 z2 + xyz + 1 ≤ 3(x2 − x + 1)(y2 − y + 1)(z2 − z + 1), √ √ √ √ 2(xy + yz + zx) ≤ x2 + y2 + z2 + xyz( x + y + z), (x + 1)(y + 1)(z + 1)(xyz + 1) ≤ 2(x2 + 1)(y2 + 1)(z2 + 1), 2(xy + yz + zx) ≤ x2 + y2 + z2 + 3(xyz)2/3 ≤ x2 + y2 + z2 + 2xyz + 1, 3(x + y)2 (y + z)2 (z + x)2 (x2 + y2 + z2 ) ≤ 8(x2 + y2 )(y2 + z2 )(z2 + x2 )(x + y + z)2 , 3xyz ≤ x(xy − y + 1)(yz − z + 1) + y(yz − z + 1)(zx − x + 1) + z(zx − x + 1)(xy − y + 1), 1 1 1 2(xy + yz + zx + 1)2 1 3 ≤ + + , ≤ x+y+z+ , xyz + 1 x(y + 1) y(z + 1) z(x + 1) (x + y)(y + z)(z + x) xyz √ √ √ x yz + y zx + z xy ≤ xy + yz + zx ≤ 31 (x + y + z)2 ( 2 ) y2 z2 2 2 2 2 3 x ≤ x + y + z ≤ (x + y + z) ≤ 6 + + . 33 43 53 Source: [107, pp. 475, 477, 478], [748, pp. 372, 373, 381], [752, pp. 27, 28], [806, pp. 12, 39,

92, 101, 112, 113, 196, 206, 212, 213], [1371, pp. 117, 126], [1938, pp. 8, 32, 111], and [2380, p. 101]. Remark: xy + yz + zx ≤ x2 + y2 + z2 follows from Fact 2.11.50 as well as from Fact 2.3.28 with α = 0. Remark: 31 (x + y + z)2 ≤ x2 + y2 + z2 relates the mean to the quadratic mean. See Fact

156

CHAPTER 2

2.11.89. Remark: Note that 1≤

x2 + y2 + z2 , xy + yz + zx

1≤

(x + y + z)2 ≤ 3, x2 + y2 + z2

3≤

(x + y + z)2 , xy + yz + zx

which can be contrasted with the corresponding bounds for the sides of a triangle given by Fact 5.2.25. Fact 2.3.37. Let x, y, z be nonnegative numbers, and let r be a real number. If r ∈ [1, 3], then xr y4−r + yr z4−r + zr x4−r ≤ 31 (x2 + y2 + z2 )2 . If x + y + z ≥ 3 and r = 21 (x + y + z − 1), then x1+r yr + y1+r zr + z1+r xr ≤ rr (r + 1)r+1 . If x + y + z ≥ 3 and r = x + y + z − 1, then xr y + yr z + zr x ≤ rr . Source: [748, pp. 68, 69]. Fact 2.3.38. Let x, y, z be nonnegative numbers, and let α ∈ (0, 3]. Then,

2(xy + yz + zx) + α ≤ x2 + y2 + z2 + α(xyz)2/α + 3. In particular, 2(xy + yz + zx) ≤ x2 + y2 + z2 + 3(xyz)2/3 , 2(xy + yz + zx) ≤ x2 + y2 + z2 + 2xyz + 1, 2(xy + yz + zx) ≤ x2 + y2 + z2 + (xyz)2 + 2. If, in addition, xyz ≤ 1, then 2(xy + yz + zx) ≤ x2 + y2 + z2 + 3. Source: [107, p. 475] and [806, p. 124]. Related: Fact 2.2.76. Fact 2.3.39. Let x, y, z be real numbers. Then,

√ √ x2 + y2 − y2 + z2 ≤ |x − z|.

Source: [1566, p. 13]. Fact 2.3.40. Let x, y, z be nonnegative numbers. Then,

√

√ √ x + z ≤ x + y + y + z, √ √ √ 3 x2 + yz + y2 + zx + z2 + xy ≤ (x + y + z), 2 √ √ √ 4x2 + yz + 4y2 + zx + 4z2 + xy ≤ 52 (x + y + z), √ √ √ √ x2 + 2yz + y2 + 2zx + z2 + 2xy ≤ 3(x + y + z), √ √ √ 0 ≤ (x2 − yz) y + z + (y2 − zx) z + x + (z2 − xy) x + y, √ √ √ 2(xy + yz + zx) ≤ x x2 + 3yz + y y2 + 3zx + z z2 + 3xy, √ 4 2(x + yz)(y + zx)(z + xy) ≤ (xy + yz + zx + 1)(x + y + z + 1), √ √ √ √ 2(x + y + z) ≤ x2 + y2 + y2 + z2 + z2 + x2 ≤ 2(x + y + z),

EQUALITIES AND INEQUALITIES

157

√ √ √ √ 3 xy + yz + zx ≤ x2 + xy + y2 + y2 + yz + z2 + z2 + zx + x2 , √ √ √ 0 ≤ (x2 − yz) x2 + 4yz + (y2 − zx) y2 + 4zx + (z2 − xy) z2 + 4xy, √ 3(x3 y3 + y3 z3 + z3 x3 ) ≤ xy(x + y − z) + yz(y + z − x) + zx(z + x − y), √ √ √ xy 2(x2 + y2 ) + yz 2(y2 + z2 ) + zx 2(z2 + x2 ) ≤ x3 + y3 + z3 + 3xyz, [√ ] √ √ √ √ (x + y)(y + z)(z + x) ≤ x + y + z x(y + z) + y(z + x) + z(x + y) , √ √ √ √ √ √ √ 2 x + y + y + z + z + x ≤ 6(x + y + z) ≤ x + 1 + y2 + 1 + z2 + 1, √ √ √ √3 3 3 3 x3 + 3(x + z)y2 + 2xyz + y3 + 3(y + x)z2 + 2yzx + z3 + 3(z + y)x2 + 2zxy ≤ 9(x + y + z), √ √ x2 + y2 + z2 ≤ (x2 − xy + y2 )(y2 − yz + z2 ) + (y2 − yz + z2 )(z2 − zx + x2 ) √ + (z2 − zx + x2 )(x2 − xy + y2 ), √ √ √ √ x + y + z + 3(xy + yz + zx) ≤ (x + y)(y + z) + (y + z)(z + x) + (z + x)(x + y) √ ≤ 13 [5(x + y + z) + 3(xy + yz + zx)]. √ If r ∈ [0, 2], then √ √ √ (1 + r) x3 y + y3 z + z3 x ≤ x4 + y4 + z4 + r x2 y2 + y2 z2 + z2 x2 . If r ≥ 2, then √ √ √ √ x2 + rxy + y2 + y2 + ryz + z2 + z2 + rzx + x2 ≤ 4(x2 + y2 + z2 ) + (3r + 2)(xy + yz + zx). If a, b, c > 0, then √ 3(xy + yz + zx) ≤

b c a (x + y) + (y + z) + (z + x). b+c c+a a+b Source: [47, p. 3], [377], [748, pp. 6, 109, 375], [752, pp. 43, 243], [806, pp. 45, 89, 104, 201], and [1956]. Fact 2.3.41. Let x, y, z be positive numbers. Then,  √   3     16(x + y + z)     √  √  √    √   x+y y+z z+x  3(x + y)(y + z)(z + x)   √ 3 2≤ ≤ + + ,      z x y   6(x + y + z)       √3       xyz √ √ √ √ √ √ √ y z x x y z 3 3 ≤ + + , 2< + + , √ y+z z+x x+y 2 x+y+z y+z z+x x+y √ √ √ √ √ √ 2x 2y 2z x y z + + ≤ 3, + + ≤ 1, x+y y+z z+x 4x + 5y 4y + 5z 4z + 5x √ √ √ x2 + yz y2 + zx z2 + xy x2 − yz y2 − zx z2 − xy + + , 0≤ √ 3≤ + √ + √ , (z + x)y (x + y)z (y + z)x x2 + yz y2 + zx z2 + xy

158

CHAPTER 2

√ √ √ √ √ x(y + z) y(z + x) z(x + y) x3 y3 z3 + + , 1 ≤ + + , 2≤ x2 + yz y2 + zx z2 + xy x3 + (y + z)3 y3 + (z + x)3 z3 + (x + y)3 √ √ √ √ √ √ √ x2 + yz y2 + zx z2 + xy 3 2 48x 48y 48z ≤ + + , 15 ≤ 1 + + 1+ + 1+ , 2 y+z z+x x+y y+z z+x x+y √ √ √ x x2 + 3yz y y2 + 3zx z z2 + 3xy 1 1 3 1 + √ + √ + + , ≤ √ , x+y+z ≤ √ y+z z+x x+y 2xyz x x + y y y + z z z + x √ 2 √ 2 √ 2 x 2x 2y 2z y z x+y+z x+y+z ≤ √ , √ , +√ +√ +√ +√ ≤ √ xy + yz + zx z2 + x 2 x 2 + y2 y2 + z2 x2 + 2yz y2 + 2zx z2 + 2xy √ )3/5 ( )3/5 ( )3/5 ( x−1 + y−1 + z−1 2y 2z x 2x y z + + , , 3≤ + 2 + 2 ≤ 2 y+z z+x x+y x+y+z 2x + yz 2y + zx 2z + xy √ √ √ √ √ √ √ 2 + yz 2 2 1 x xy + yz + zx 2x 2y 2z 3 y + zx 3 3 z + xy 2 + √3 ≤ + + , + + , 3 ≤ 2 2 2 2 2 2 2 2 2 x+y y+z z+x y +z z +x x +y x +y +z 2 √ √ √ xy yz zx √ 9(x3 + y3 + z3 ) xy(x + z) yz(y + x) zx(z + y) ≤ + + , 3xyz(x + y + z) ≤ + + , 4 2 2 2 y+z z+x x+y 2(x + y + z) xy + z yz + x zx + y √ 1 x2 + y2 y2 + z2 z2 + x2 2 1 1 ≤ 3(x2 + y2 + z2 ) ≤ + + , √3 + + , x+y y+z z+x (x + y)(y + z)(z + x) 2x + y 2y + z 2z + x √ √ √ √ 1 1 1 1 1 1 2 2 2 2 2 ≤ √ + √ + √ , 2 3≤ x +y +z + + 2 + 2, √ 2 x y z xy + yz + zx x2 + yz y2 + zx z2 + xy √ xy(x + z) yz(y + x) zx(z + y) 3xyz(x + y + z) ≤ + + , y+z z+x x+y √ x2 y2 z2 y2 z2 x2 √ √ + + + + + , 2( xy + yz + zx) ≤ y z x x y z √ √ √ √ x3 + y3 + z3 x3 y3 z3 ≤ + + , 2 2 2 2 2 2 2 2 x +y +z x + 8y y + 8z z + 8x2 √

x2 − yz y2 − zx z2 − xy 0≤ √ + √ + √ , 2x2 + y2 + z2 2y2 + z2 + x2 2z2 + x2 + y2 x2 − yz

y2 − zx z2 − xy + √ + √ , 8x2 + (y + z)2 8y2 + (z + x)2 8z2 + (x + y)2 √ √ √ x(x + y + z) y(x + y + z) z(x + y + z) + + , 2≤ (x + y)(x + z) (y + z)(y + x) (z + x)(z + y) √ √ √ x2 (y2 + z2 ) y2 (z2 + x2 ) z2 (x2 + y2 ) + + ≤ x + y + z, 2 2 x + yz y + zx z2 + xy √ √ √ √ √ 2 + yz 2 + zx 2 3 3 xyz x y 4 4 4 z + xy 10 − ≤ + + , x+y+z x(y + z) y(z + x) z(x + y)

0≤ √

159

EQUALITIES AND INEQUALITIES

√

(

) x y z x2 y2 z2 + + ≤ x+y+z+ + + , y z x y z x √ √ √ √ x2 y2 z2 3 5 + + ≤ , 5 y2 + (z + x)2 z2 + (x + y)2 x2 + (y + z)2 √ √ √ 4x(y + z) + yz 4y(z + x) + zx 4z(x + y) + xy 9 ≤ + + , 2 y+z z+x x+y √ √ √ x3 y3 z3 1≤ + + , 3 3 3 3 3 x + 7xyz + y y + 7xyz + z z + 7xyz + x3 √ xy(x3 + y3 ) yz(y3 + z3 ) zx(z3 + x3 ) 3xyz(x3 + y3 + z3 ) ≤ + + , x2 + y2 y 2 + z2 z2 + x2 √ √ √ √ √ y z x 3 3 (x + y)(y + z)(z + x) +y +z ≤ , x z+x x+y y+z 4 x+y+z √ √ √ 2x(y + z) 2y(z + x) 2z(x + y) 2≤ + + , (2y + z)(y + 2z) (2z + x)(z + 2x) (2x + y)(x + 2y) √   x2 + y2 3  (x + y + z)(x2 + y2 + z2 ) y2 + z2 z2 + x2 − 1 ≤ + + ,  2 xyz (x + y)z (y + z)x (z + x)y √ √ √ x y z 9 xy yz zx ≤ + + + + + , 4 (x + y)2 (y + z)2 (z + x)2 y+z z+x x+y x y z + + ≤ 1, √ √ √ x + (x + y)(x + z) y + (y + z)(y + x) z + (z + x)(z + y) x y z 2 + ≤ 1, + √ ≤ √ √ √ 2 2 2 2 1 + 2 x + 2(y + z ) y + 2(z + x ) z + 2(x2 + y2 ) √ √ √ x3 + y3 y 3 + z3 6(xy + yz + zx) z3 + x 3 + + , ≤ 2 √ x + y2 y2 + z2 z2 + x 2 (x + y + z) (x + y)(y + z)(z + x) v u √ t ) √ ) ( ( 1 1 1 1 1 1 2 2 2 1 + 1 + (x + y + z ) 2 + 2 + 2 ≤ (x + y + z) + + , y z x x y z √ √ √ 2(xy + yz + zx) x2 − xy + y2 y2 − yz + z2 z2 − zx + x2 4≤ + + + , x2 + y2 + z2 xy + z2 yz + x2 zx + y2 √ √ √ ) √ (√ √ √ x y z y+z z+x x+y + + ≤ + + + 3 2 − 3, 2 y+z z+x x+y 2x 2y 2z x y z y z x + + . + √ + √ ≤ √ 2 2 2 x + 2y y + 2z z + 2x 3(x + xy + yz) 3(y + yz + zx) 3(z + zx + xy) 2

If r ∈ R, then

√

(x2 + y2 + z2 )

8(x + y + z)(xr yr + yr zr + zr xr ) xr + yr yr + zr zr + xr ≤ + + . (x + y)(y + z)(z + x) x+y y+z z+x

160

CHAPTER 2

If n ≥ 2, then

√

3(xn−1 + yn−1 + zn−1 )(xn + yn + zn ) xn + yn yn + zn zn + xn ≤ + + . x+y+z x+y y+z z+x

Source: [107, pp. 45, 58, 59, 157, 251, 252, 310, 314, 489, 490, 484–486, 586, 590, 591], [108,

pp. 27, 171, 172, 356], [748, pp. 6, 8, 12, 151, 343, 345, 372, 375, 376], [752, pp. 31–36, 41, 42, 43, 242–245, 310–312, 318], [806, pp. 13, 91, 104, 185, 193, 202, 207], [960], [1771], [1956], [2294, pp. 271, 272], and [2527, p. 187]. Fact 2.3.42. Let x, y, z be positive numbers, and let p ≤ 1/2. Then, )p ( )p ( ( z )p 3 y x + + ≤ p. x+y y+z z+x 2 In particular,

√

√

x + x+y

y + y+z

√

√ z 3 2 ≤ . z+x 2

Source: [2850]. Fact 2.3.43. Let x, y, z be positive numbers. Then,

√ 4

√

√

x2 + x2 − xy + y2

4

y2 + y2 − yz + z2

4

z2 ≤ z2 − zx + x2

√

√ 2x + x+y

2y + y+z

√

2z ≤ 3. z+x

Source: [162]. Fact 2.3.44. Let x, y, z be positive numbers, and let p ≥ (log 3)/(log 2) − 1 ≈ 0.585. Then,

( ( )p ( )p z 3 x y )p + ≤ + . 2p y+z z+x y+z

Source: [748, p. 151]. Fact 2.3.45. Let x, y, z be positive numbers, and let p ≥ 1. Then,

9 xp yp zp ≤ + + . p 2·3 y+z z+x y+z Source: [2527, p. 131]. Fact 2.3.46. Let x, y, and z be positive numbers, and let α ≥ 8. Then,

√

3 1+α

≤ √

In particular, 1≤ √

x

y z + √ + √ . x2 + αyz y2 + αzx z2 + αxy

y z + √ + √ . 2 2 + 8yz y + 8zx z + 8xy

x x2

Source: [108, pp. 442, 446]. Related: This result is a special case of Fact 2.11.80. Fact 2.3.47. Let x, y, z ∈ (0, 1). Then,

√

xyz +

√ (1 − x)(1 − y)(1 − z) < 1.

Source: [1158, p. 239] and [1938, p. 42]. Fact 2.3.48. Let x, y, z ∈ [0, 1]. Then,

3 ≤ (1 − x + xy)2 + (1 − y + yz)2 + (1 − z + zx)2 . 2

161

EQUALITIES AND INEQUALITIES

Source: [752, p. 241]. Fact 2.3.49. Let x, y, z be nonnegative numbers. Then,

√3

√

xyz ≤ 13 ( xy +

√

yz +

√

  √     1   (xy + yz + zx)  1  3 ≤ (x + y + z). zx) ≤    √   1 1  3  (x + y + z) + 3 xyz  6

2

Source: The first inequality is given by Fact 2.11.97; the second upper inequality is given in [2527, p. 179]; the second lower inequality is given in [1507, 2129, 2925]. Remark: See [50]. Fact 2.3.50. Let x, y, z be nonnegative numbers, let α, β, γ be real numbers, and assume that

α + β + γ = π. Then,

√ √ √ 2(cos α) xy + 2(cos β) yz + 2(cos γ) zx ≤ x + y + z.

Source: [1893]. Fact 2.3.51. Let x, y, z be nonnegative numbers. Then,

√3

xyz ≤ 31 (x + y + z) ≤

√3

√ √ √ √ √ √ xyz + max{( x − y)2 , ( y − z)2 , ( z − x)2 }.

Source: [107, p. 478] and [1158, p. 146]. Fact 2.3.52. Let x, y be nonnegative numbers, and let z be a positive number. Then,

x + y ≤ zy x + z−xy. Source: [1371, p. 163]. Fact 2.3.53. Let x, y, z be nonnegative numbers. Then,

    xy yz z x ( x + y + z ) x+y+z       z x y x y z ≤ min {xy yz z x , xz y x zy } ≤  ≤ x x y y zz .     3   (xyz)(x+y+z)/3 

Source: [644, p. 107], [806, p. 115], and [2294, p. 267]. Fact 2.3.54. Let x, y, z be positive numbers. Then,

xy+z 3 yz+x zz+y ≤ + + . 2 2 4 (y + z) (z + x) (x + y)2 Source: [752, p. 245]. Fact 2.3.55. Let x, y, z be positive numbers, assume that x ≤ y ≤ z, and let α ∈ [0, e). Then,

xαy + yαz + zαx ≤ xαx + yαy + zαz . Source: [783]. Related: Fact 2.2.73. Fact 2.3.56. Let x, y, z ∈ (0, 1), and assume that x + y + z = 1. Then,

3 x y z ≤ + + . 2 1−x 1−y 1−z Source: Use Nesbitt’s inequality. See Fact 2.3.60 and [2088]. Fact 2.3.57. Let x, y, z ∈ (−1, 1). Then,

1 1 1 1 1 1 + + ≤ + + , 2 2 1 − xy 1 − yz 1 − zx 1 − x 1−y 1 − z2 1 1 1 1 1 1 + + ≤ + + , 2 2 2 4 4 1 − x yz 1 − y zx 1 − z xy 1 − x 1−y 1 − z4 32 32 17 17 17 45 32 + + ≤ + + + , 2 2 2 3 3 3 1 − xyz 1−x y 1−y z 1−z x 1−x 1−y 1−z

162

CHAPTER 2

( 2 )2 ( 2 )2 ( 2 )2 x2 yz y2 zx z2 xy x y z + + ≤ + + . 2 2 2 2 2 2 4 4 (1 − x yz) (1 − y zx) (1 − z xy) 1−x 1−y 1 − z4 Source: [2088]. Fact 2.3.58. Let x, y, z be distinct positive numbers. Then,

√

x + y y + z z + x , 3 ≤ + + x − y y − z z − x

1 1 4 1 + + . ≤ 2 2 xy + yz + zx (x − y) (y − z) (z − x)2

Source: [1938, p. 113]. Fact 2.3.59. Let n be a positive integer, let k be a nonnegative integer, and let x, y, z be positive

numbers. Then, xk + yk + zk ≤ In particular, 3≤

xn+1+k yn+1+k zn+1+k xn+k yn+k zn+k + n+1 + n+1 . + + ≤ yn zn xn yn+1 z x

x y z x 2 y 2 z2 x3 y3 z3 + + ≤ 2 + 2 + 2 ≤ 3 + 3 + 3, y z x y z x y z x

x+y+z≤

x2 y2 z2 x4 y4 z4 x3 y3 z3 + + ≤ 2 + 2 + 2 ≤ 3 + 3 + 3, y z x y z x y z x

x2 + y2 + z2 ≤

x3 y3 z3 x 5 y 5 z5 x4 y4 z4 + + ≤ 2 + 2 + 2 ≤ 3 + 3 + 3. y z x y z x y z x

Source: [806, pp. 131, 184, 193, 194], Fact 2.11.67, and Fact 2.11.70. Fact 2.3.60. Let x, y, z be nonnegative numbers. Then,

6(x + y − z)(y + z − x)(z + x − y) ≤ 2[x2 (z + y − x) + y2 (z + x − y) + z2 (x + y − z)] = 2[x(y2 + z2 ) + y(z2 + x2 ) + z(x2 + y2 ) − (x3 + y3 + z3 )] ≤ 2(x + y + z)(xy + yz + zx) − 94 (x + y + z)3 − 49 [(x + y + z)2 − 3(xy + yz + zx)]3/2 ≤ 6xyz   2(x + y + z)(xy + yz + zx) − 49 (x + y + z)3       + 94 [(x + y + z)2 − 3(xy + yz + zx)]3/2 ≤  ≤ ( )    3 x+y+z    xyz 1 + √  3 2 xyz

√ 2 3 3 (xy

     3/2   + yz + zx)          

≤ 32 (x + y + z)(xy + yz + zx) { 2 }   x(y + z2 ) + y(z2 + x2 ) + z(x2 + y2 )    3     (x + y)(y + z)(z + x) ≤  4 2 3 (x + y + z) ≤  9         3xyz + 31 (x + y + z)(x2 + y2 + z2 ) {2 3 } (x + y3 + z3 ) + 54 [x(y2 + z2 ) + y(z2 + x2 ) + z(x2 + y2 )] ≤ 23 (x + y + z)(x2 + y2 + z2 ) ≤ 5 3xyz + x3 + y3 + z3 ≤ 2xyz + 34 (x3 + y3 + z3 ) ≤ 32 (xyz + x3 + y3 + z3 ) ≤ 2(x3 + y3 + z3 ),     2(xy2 + yz2 + zx2 )         3 (x + 1)(y + 1)(x + z)(y + z) 6xyz ≤  ≤ x2y2 + y2z2 + z2 x2 + x2 + y2 + z2 ,  8        x(y2 + z2 ) + y(z2 + x2 ) + z(x2 + y2 ) 

163

EQUALITIES AND INEQUALITIES

    2(xy2 + yz2 + zx2 )       2 2 2 2 2 2     x(y + z ) + y(z + x ) + z(x + y )     √ 6xyz ≤  ≤ 2(x3 + y3 + z3 ), √        6xyz + 2 9 + 6 3|(x − y)(y − z)(z − x)|       6xyz + 1 max {(x + y − 2z)3 , (y + z − 2x)3 , (z + x − 2y)3 }   2 6xyz ≤ 43 (x + y)(y + z)(z + x) ≤ 49 (x + y + z)3 − [x(y2 + z2 ) + y(z2 + x2 ) + z(x2 + y2 )] ≤ 3xyz + x3 + y3 + z3 , 6xyz ≤ 23 (x2y2z2 + xy + yz + zx), 6xyz + 3(x2 + y2 )z ≤ 49 (x + y + z)3 , 6xyz + 12xy ≤ 6x2 + y2 (z + 2)(2z + 3), 108xyz ≤ (x + 2y)(x + 5y)(3x + 2y + z), 27(xyz + xy2 + yz2 + zx2 ) ≤ 4(x + y + z)3 , 100xyz ≤ (x + y + z)[(x + y)2 + (x + y + 4z)2 ], (x + y)3 + (y + z)3 + (z + x)3 ≤ 8(x3 + y3 + z3 ), 4(x + y + z)(xy + yz + zx) ≤ (x + y + z)3 + 9xyz, 7(x + y + z)(xy + yz + zx) ≤ 2(x + y + z)3 + 9xyz, √ √3 3 x2 y + y2 z + z2 x ≤ 34 (x3 + y3 + z3 ) + (3 − 4)xyz, 9xyz ≤ 3(xy2 + yz2 + zx2 ) ≤ 2(x3 + y3 + z3 ) + 2(x2 y + y2 z + z2 x), √ x3 + y3 + z3 ≤ (x2 + y2 + z2 )3/2 + 3(1 − 3)xyz ≤ (x2 + y2 + z2 )3/2 , 3xyz + 15(xy2 + yz2 + zx2 ) ≤ 12(x2 y + y2 z + z2 x) + 4(x3 + y3 + z3 ), 6(x + y + z)(x2 y + y2 z + z2 x) ≤ x4 + y4 + z4 + 17(x2 y2 + y2 z2 + z2 x2 ), 6xyz ≤ 32 (x + y + z)(xy + yz + zx) ≤ (x2 + x + 1)(y2 + y + 1)(z2 + z + 1), 6xyz ≤ 2(xy2 + yz2 + zx2 ) ≤ 25 (x3 + y3 + z3 ) + 45 [x(y2 + z2 ) + y(z2 + x2 ) + z(x2 + y2 )],

√ √ √ √ ( 4 2 + 13/4 + 12 )(xy2 + yz2 + zx2 ) ≤ ( 4 2 + 13/4 − 12 )(x2 y + y2 z + z2 x) + x3 + y3 + z3 ,

( ) 1 1 1 (x + y)(y + z)(z + x) ≤ (x + y + z) + + + log 8 − 9, 3 log 2 ≤ log xyz x y z 9 2 2 2 1 1 1 ≤ + + ≤ + + x+y+z x+y y+z z+x x y z ( ) x y z y z x 3 + + + + + ≤ 2(x + y + z) y z x x y z { } 3 x y z y z x ≤ max + + , + + , x+y+z y z x x y z

3≤

x+y y+z z+x x y z + + ≤ + + , x+z y+x z+y y z x

1≤

x y z + + < 2, x + 2y y + 2z z + 2x

x y z 3 1 x y z 1 ≤ + + ≤ , ≤ + + , 2 2x + y + z 2y + z + x 2z + x + y 4 2 x + 2y + 3z y + 2z + 3x z + 2x + 3y x y z 1 + + ≤ , 4x + 4y + z 4y + 4z + x 4z + 4x + y 3

164

CHAPTER 2

3≤

2x + y 2y + z 2z + x + + , 2x + z 2y + x 2z + y

4xyz x y z + + + , (x + y)(y + z)(z + x) y + z z + x x + y ( ) 1 1 1 1 1 1 ≤ + + , + 2(x2 + y2 + z2 ) xy + yz + zx x + y + z x + y y + z z + x 2≤

y2 + zx z2 + xy x y z x2 + yz + + ≤ + + , (x + y)(x + z) (y + z)(y + x) (z + x)(z + y) y + z z + x x + y 9(x + y + z)(xy + yz + zx) + 3(xy + yz + zx) ≤ 27xyz + 2(x + y + z)3 + 2(x + y + z)2 + [(x + y + z)2 − 3(xy + yz + zx)]2 . If x, y, z ∈ [ 21 , 2], then

(

) ( ) y z x x y z 5 + + +9≤8 + + . x y z y z x

Source: Fact 2.3.61, [107, pp. 10, 24, 92, 93, 148, 149], [108, pp. 45, 264, 265, 446], [176, 287],

[748, pp. 5, 7, 111], [752, pp. 29, 30], [806, pp. 138, 181, 188, 198, 204, 214], [993, pp. 166, 169, 179, 182], [1158, pp. 36, 45], [1371, pp. 117, 120, 152], [1757, pp. 247, 257], [1772], [1938, pp. 11, 34], [1962, 2002, 2381], and [2527, p. 204]. 3xyz ≤ xy2 + yz2 + zx2 is equivalent to 3 ≤ yx + yz + xz , which follows from Fact 2.3.66 and Fact 2.11.50. x(y2 +z2 )+y(z2 + x2 )+z(x2 +y2 ) ≤ 3xyz+ x3 +y3 +z3 is given in [1938, p. 49]. This inequality follows from Schur’s inequality given by Fact 2.3.28. See [1569]. 29 (x + y + z)3 ≤ 32 (xyz + x3 + y3 + z3 ) is a slight improvement of the inequality 17 (x + y + z)3 ≤ xyz + x3 + y3 + z3 given in [1938, p. 48]. 6xyz ≤ 94 (x + y + z)3 − 3(x2 + y2 )z follows from Fact 2.11.104. 23 (x + y + z)(x2 + y2 + z2 ) ≤ 2xyz + 34 (x3 + y3 + z3 ) follows from Suranyi’s inequality given by Fact 2.11.22. Remark: x3 + y3 + z3 − 3xyz = 21 (x + y + z)[(x − y)2 + (y − z)2 + (z − x)]2 yields 6xyz ≤ 2(x3 + y3 + z3 ). Remark: For x, y, z > 0, 9xyz ≤ (x + y + z)(xy + yz + zx) is given by Fact 2.11.43. Remark: For x, y, z > 0, 3xyz ≤ xy2 + yz2 + zx2 is given by Fact 2.11.48. Remark: x(y2 + z2 ) + y(z2 + x2 ) + z(x2 + y2 ) ≤ 3xyz + x3 + y3 + z3 is a special case of Schur’s inequality. See Fact 2.3.28. Remark: x(y2 + z2 ) + y(z2 + x2 ) + z(x2 + y2 ) ≤ 2(x3 + y3 + z3 ) can be written as 9 2 2 2 ≤ + + . x+y+z x+y y+z z+x Remark: For x, y, z > 0, x(y2 + z2 ) + y(z2 + x2 ) + z(x2 + y2 ) ≤ 2(x3 + y3 + z3 ) is equivalent to

x y z 3 ≤ + + , 2 y+z z+x x+y which is Nesbitt’s inequality. See [1757, p. 267]. Nesbitt’s inequality is interpolated by   √3  xyz 3 (x + y + z)2 1  ≤ +  1 − 1 2 2(xy + yz + zx) 2 3 (x + y + z)   (x + y + z)2             2(xy + yz + zx)   x y z     √ ≤ + + , ≤  3       xyz y+z z+x x+y x y z 1         + + + − 1       y + z z + x x + y 2  1 (x + y + z) 3 {

} x 2(y + z) y 2(z + x) z 2(x + y) + , + , + y + z 2x + y + z z + x 2y + z + x x + y 2z + x + y { } x 2(y + z) y 2(z + x) z 2(x + y) ≤ max + , + , + y + z 2x + y + z z + x 2y + z + x x + y 2z + x + y

3 ≤ min 2

EQUALITIES AND INEQUALITIES

≤

165

y z x + + . y+z z+x x+y

y z x See [397, 1770]. An upper bound for y+z + z+x + x+y is given by Fact 2.3.61. In the case where y z x x, y, z are the sides of a triangle, an upper bound for y+z + z+x + x+y is given by Fact 5.2.25. A generalization is given by Fact 2.11.53. Remark:

2[x(y2 + z2 ) + y(z2 + x2 ) + z(x2 + y2 ) − (x3 + y3 + z3 )] ≤ 6xyz is equivalent to

4(x + y + z)(xy + yz + zx) ≤ (x + y + z)3 + 9xyz,

which in turn is equivalent to x(x − y)(x − z) + y(y − z)(y − x) + z(z − x)(z − y) ≥ 0, which is Schur’s inequality. See Fact 2.3.28. An equivalent form is given by 2(xy + yz + zx) − (x2 + y2 + z2 ) ≤ 9xyz(x + y + z). Remark: 3xyz + 3(xy2 + yz2 + zx2 ) ≤ 2(x3 + y3 + z3 ) + 2(x2 y + y2 z + z2 x) can be written as y z x + + . 1≤ y + 2z z + 2x x + 2y See [806, p. 40] and [1938, p. 104]. Remark: 6xyz ≤ x(y2 + z2 ) + y(z2 + x2 ) + z(x2 + y2 ) follows from Fact 2.11.78. Remark: The left-hand inequality in the penultimate string is 6xyz ≤ x(y2 + z2 ) + y(z2 + x2 ) + z(x2 + y2 ). In the case where x, y, z represent the sides of a triangle, this string 4 (x + y + z)3 ≤ x3 + y3 + z3 + xyz is sharp and is interpolated by Fact 5.2.25. Remark: The bound 27 thus tighter than the bound given in [806, p. 21], where 4/27 is replaced by 1/7. Fact 2.3.61. Let x, y, z be positive numbers. Then, ( ) 3 x y z 1 x+y y+z z+x x+y y+z z+x 9 ≤ + + ≤ + + ≤ + + − . 2 y+z z+x x+y 4 z x y z x y 2 Source: The second inequality is given in [1938, p. 32]. This inequality is equivalent to

6x2 y2 z2 + 2xyz(x3 + y3 + z3 ) ≤ 2(x3 y3 + y3 z3 + z3 x3 ) + x4 (y2 + z2 ) + y4 (z2 + x2 ) + z4 (x2 + y2 ), which can be written as 2(x3 y3 + y3 z3 + z3 x3 ) − 6x2 y2 z2 + x4 (y2 + z2 − 2yz) + y4 (z2 + x2 − 2zx) + z4 (x2 + y2 − 2xy) ≥ 0. Remark: The left-most inequality is Nesbitt’s) inequality. See Fact 2.3.60. Remark: The third ( z+x 3 1 x+y inequality follows from 2 ≤ 4 z + y+z x + y . Remark: The third term interpolates

x y z x+y y+z z+x 9 3 ≤ + + ≤ + + − , 2 y+z z+x x+y z x y 2 which is given in [209, pp. 33, 34]. This string is equivalent to 6≤

x y z x+y y+z z+x 9 + + + ≤ + + , 2 y+z z+x x+y z x y

which in turn is equivalent to 6xyz ≤

9 x2 yz xy2 z xyz2 xyz + + + ≤ x(y2 + z2 ) + y(z2 + x2 ) + z(x2 + y2 ). 2 y+z z+x x+y

The left-hand inequality and the second inequality yield ( 2 ) x yz xy2 z xyz2 6xyz ≤ 4 + + ≤ x(y2 + z2 ) + y(z2 + x2 ) + z(x2 + y2 ), y+z z+x x+y

166

CHAPTER 2

which interpolates 6xyz ≤ x(y2 + z2 ) + y(z2 + x2 ) + z(x2 + y2 ) in Fact 2.3.60. This interpolating term is not comparable with the three interpolating terms in Fact 2.3.60. Fact 2.3.62. Let x, y, z be positive numbers. Then,  9 x y z        + + +       2 y + z z + x x + y    x+y y+z z+x  ≤ 6≤ + + . ( )  √     z x y   xy + yz + zx        6 + 11 1 − x2 + y2 + z2  Hence,

x+y y+z z+x 17 ≤ + + + 11 z x y

√

xy + yz + zx . x 2 + y 2 + z2

Source: [107, pp. 483, 484]. Fact 2.3.63. Let x, y, z, p be positive numbers. Then,

3 x y z ≤ + + . 1+ p py + z pz + x px + y Source: [2850]. Remark: Setting p = 1 yields Nesbitt’s inequality. See Fact 2.3.60. Fact 2.3.64. Let x, y, z be positive numbers, and let r ≥ 1. Then, 3 1 2 [ 3 (x

+ y + z)]r−1 ≤

xr yr zr + + . y+z z+x x+y

Source: [2850]. Remark: Setting r = 1 yields Nesbitt’s inequality. See Fact 2.3.60. Fact 2.3.65. Let x, y, z be positive numbers, let p and q be nonnegative numbers, and assume

that p + q > 0. Then, 3 (x + y + z)2 x y z ≤ ≤ + + . p + q (p + q)(xy + yz + zx) py + qz pz + qx px + qy Source: [2743]. Remark: Setting p = q = 1 yields Nesbitt’s inequality. See Fact 2.3.60. Fact 2.3.66. Let x, y, z be nonnegative numbers. Then,

 xyz(x + y + z)     2 2   2 2 2 2    2xyz|x + y − z|   x y +y z +z x    ≤ (xy + yz + zx)2 ≤ 3(x2y2 + y2z2 + z2 x2 ) ≤        2xyz|x − y + z|   3xyz(x + y + z)   2xyz| − x + y + z|  ≤ 32 (x3 y + y3 z + z3 x + xy3 + yz3 + zx3 ) ≤ (x2 + y2 + z2 )2 { } 3(x4 + y4 + z4 ) 3 3 3 ≤ (x + y + z)(x + y + z ) ≤ ≤ 27(x4 + y4 + z4 ), (x + y + z)4 x2y2 + y2z2 + z2 x2 ≤ 21 [x4 + y4 + z4 + xyz(x + y + z)] ≤ x4 + y4 + z4 ≤ (x2 + y2 + z2 )2 ,  2xyz|x + y − z|     2xyz|x − y + z|  ≤ 3(x3y + y3z + z3 x) ≤ (x2 + y2 + z2 )2 ,    2xyz| − x + y + z|  3  3 3    x y + y z + z x  xyz(x + y + z) ≤  ≤ x4 + y4 + z4 ,    xy3 + yz3 + zx3   xyz(x + y + z) ≤ 14 (x + y)2 (x + z)2 ,

x[x3 + (y + z)3 )] ≤ (x2 + y2 + z2 )2 ,

167

EQUALITIES AND INEQUALITIES 4 27 (x 2 2

+ y + z)4 ≤ x4 + y4 + z4 + 3(x2 y2 + y2 z2 + z2 x2 ),

6(x y + y2 z2 + z2 x2 ) ≤ x4 + y4 + z4 + 5(x3 y + y3 z + z3 x), |(x3 y + y3 z + z3 x − (xy3 + yz3 + zx3 )| ≤

√ 9 2 2 32 (x

+ y2 + z2 )2

4(x + y + z)2 (xy + yz + zx) ≤ (x + y + z)4 + 3(xy + yz + zx)2 , 0 ≤ (x − y)(3x + 2y)3 + (y − z)(3y + 2z)3 + (z − x)(3z + 2x)3 ,

√ 4 4 3 3 3 (x y

√ 4

+ y3 z + z3 x) ≤ ( 4 3 3 − 1)xyz(x + y + z) + x4 + y4 + z4 ,

4(xy + yz + zx)2 ≤ (x + y + z)2 (xy + yz + zx) + 3xyz(x + y + z), x2 (y + z)2 + y2 (z + x)2 + z2 (x + y)2 ≤ 34 (x + y + z)(x3 + y3 + z3 ), x2 y2 + y2 z2 + z2 x2 + 2|x3 y + y3 z + z3 x − xy3 − yz3 − zx3 | ≤ x4 + y4 + z4 , √ 6[xy(z − x)2 + yz(x − y)2 + zx(y − z)2 ] ≤ (x2 − yz)2 + (y2 − zx)2 + (z2 − xy)2 , 5(x + y + z)2 (xy + yz + zx) ≤ (x + y + z)4 + 4(xy + yz + zx)2 + 6xyz(x + y + z), 3(x + y + z) x y z ≤ + + . 3+x+y+z y+1 z+1 x+1

√ √ √ √ If x, y, z ∈ [1/ 4 + 3 2, 4 + 3 2], then

(x + y + z)4 ≤ 9(x2 + y2 + z2 )(xy + yz + zx). Now, assume that x, y, z are positive numbers. Then, 3≤

x2 + 1 y2 + 1 z2 + 1 + + , y+z z+x x+y

z2 − y2 x2 − z2 y2 − x2 + + , x+y y+z z+x 1 1 1 1 1 1 + + ≤ + + , x + y y + z z + x 2x 2y 2z 0≤

1≤

x y z + + , x + 2y y + 2z z + 2x

x2 − yz y2 − zx z2 − xy + + , x+y y+z z+x x+y z+x y+z x y z 3≤ + + ≤ + + , z+x y+z x+y y z x 0≤

3 x2 + 2yz y2 + 2zx z2 + 2xy 4xy 4yz 4zx (x + y + z) ≤ + + , + + ≤ x + y + z, 2 y+z z+x x+y 2x + y + z 2y + z + x 2z + x + y y2 − zx x2 − xy x2 − yz + + , 3x + y + z 3y + z + x 3z + x + y 9xy 9yz 9zx + + ≤ x + y + z, 3x + 4y + 2z 3y + 4z + 2x 3z + 4x + 2y  2 x + y2 y2 + z2 z2 + x2    + +    y+z x+y  2xy 2yz 2zx  x+y + + ≤ x+y+z≤   2 2 2 2 2 x+y y+z z+x  x +y y +z z + x2    + + ,  2z 2x 2y 0≤

4xy 4yz 4zx 2x2 2y2 2z2 + + ≤ x+y+z≤ + + . x + y + 2z y + z + 2x z + x + 2y y+z z+x x+y Source: [108, pp. 11, 100], [287], [748, pp. 111, 112], [752, p. 29], [806, pp. 11, 40, 41, 64, 65, 138, 186, 252, 324], [993, pp. 170, 180], [1158, pp. 172, 173], [1371, pp. 106, 122, 147, 149], [1569], [1757, pp. 247, 257], [1938, pp. 12, 40, 41, 49, 112], Fact 2.11.15, and Fact

168

CHAPTER 2

2.11.98. Remark: 3xyz(x + y + z) ≤ 32 (x3 y + y3 z + z3 x + xy3 + yz3 + zx3 ) is given in [1371, p. 147]. Remark: 2xyz(x + y − z) ≤ x2y2 + y2z2 + z2 x2 follows from (xy − yz − zx)2, and thus is valid for all real x, y, z. See [993, p. 194]. Remark: 3xyz(x + y + z) ≤ (xy + yz + zx)2 follows from 2 2 2 2 2 2 Newton’s inequality.√See Fact 2.11.35. √ Remark: 2xyz(z + x − y) ≤ x y + y z + z x is equivalent √ to x2 + xz + z2 ≤ x2 − xy + y2 + y2 − yz + z2 . See [47, p. 17], [993, p. 184], [1158, p. 36], and [1938, p. 52]. Fact 2.3.67. Let xyz be nonnegative numbers, let α and β be real numbers, and assume that, if β ≤ 0, then α ≥ 2β, whereas, if β ≥ 0, then α ≥ β2 + 2β. Then, (1+β)[xy(x2 +y2 )+yz(y2 +z2 )+zx(z2 +x2 )] ≤ (1+2β−α)xyz(x+y+z) ≤ α(x2 y2 +y2 z2 +z2 x2 )+x4 +y4 +z4 . Source: [751]. Remark: Setting α = β = 0 yields Schur’s inequality, which is given by Fact

2.3.16, Fact 2.3.28, and Fact 2.3.70, and which holds for real x, y, z. Fact 2.3.68. Let x, y, z be nonnegative numbers. Then, x4 (y + z) + y4 (z + x) + z4 (x + y) ≤

1 12 (x + 4 5

y + z)5 ,

xyz(xy + yz + zx) ≤ xy4 + yz4 + zx ≤ x + y5 + z5 , xyz(x + y + z)2 ≤ 83 [xy(x + y)3 + yz(y + z)3 + zx(z + x)3 ], (x2 y2 + y2 z2 + z2 x2 )(x + y + z) ≤ (x2 + y2 + z2 )(x3 + y3 + z3 ), 2xyz(x + y + z)2 + 3xyz(xy + yz + zx) ≤ (x + y + z)(xy + yz + zx)2 , 3xyz(x2 + y2 + z2 ) ≤ (x2 + y2 + z2 )(x3 + y3 + z3 ) ≤ 3(x5 + y5 + z5 ), 33xyz(x2 + y2 + z2 ) ≤ 2(x2 + y2 + z2 )(x3 + y3 + z3 ) + 9xyz(x + y + z)2 , x2 y3 + y2 z3 + z2 x3 + x3 y2 + y3 z2 + z3 x2 ≤ xy4 + yz4 + zx4 + x4 y + y4 z + z4 x, 6xyz[(y + z)(z + x) + (z + x)(x + y) + (x + y)(y + z)] ≤ (x + y)(y + z)(z + x)(x + y + z)2 , x2 (z + x)(x + y)(y + z − x) + y2 (x + y)(y + z)(z + x − y) + z2 (y + z)(z + x)(x + y − z) ≤ 21 (x + y)(y + z)(z + x)(xy + yz + zx). Now, assume that x, y, z are positive. Then, x+y y+z x y z + +1≤ + + , y+z x+y y z x xy + yz + zx xyz 2 ≤ 3 + , x2 + y2 + z2 x + y3 + z3 3 4≤

(x − z)2 x y z +3≤ + + , xy + yz + zx y z x

2≤

x + y y + z z + x 3(xy + yz + zx) , + + + y+z z+x x+y (x + y + z)2

x2 + y2 + z2 8xyz + , xy + yz + zx (x + y)(y + z)(z + x)

x2 + y2 + z2 ≤

x3 + xyz y3 + xyz z3 + xyz + + , y+z z+x x+y

x y z 2(xy + yz + zx) 2 2 x3 + 3y3 y3 + 3z3 z3 + 3x3 13 2 2 ≤ + + + , (x + y + z ) ≤ + + , 6 y + z z + x x + y 3(x2 + y2 + z2 ) 3 5x + y 5y + z 5z + x 6(x2 + y2 + z2 ) x2 y2 z2 ≤ x+y+z+ + + , x+y+z y z x 6(x − z)2 x y z +3≤ + + , 2 y z x (x + y + z)

1 x3 + 2xyz y3 + 2xyz z3 + 2xyz (x + y + z)2 ≤ + + , 2 y+z z+x x+y ( ) 27(x − z)2 1 1 1 + 9 ≤ (x + y + z) + + , x y z 2(x + y + z)2

7(x − z)2 3 x y z + ≤ + + , 16(xy + yz + zx) 2 y + z z + x x + y

17 x y z 7(xy + yz + zx) ≤ + + + 2 , 2 y z x x + y2 + z2

169

EQUALITIES AND INEQUALITIES

2(xy + yz + zx) ≤

x3 + 3xyz y3 + 3xyz z3 + 3xyz + + , y+z z+x x+y

5 1 1 1 3xyz , ≤ + + + x + y + z x + y y + z z + x 2(xy + yz + zx)2 x+y y+z z+x 2(xy + yz + zx) 13 ≤ + + + , x + y + 2z y + z + 2x z + x + 2y 3(x2 + y2 + z2 ) 6 x3 − y3 y3 − z3 z3 − x3 1 + + ≤ [(x − y)2 + (y − z)2 + (z − x)2 ], x+y y+z z+x 8 x2 + y2 + z2 − xy − yz − zx 2x2 − xy − y2 2y2 − yz − z2 2z2 − zx − x2 ≤ + + . x+y+z x+y y+z z+x If x, y, z are positive and distinct, then 4 1 1 1 ≤ + + . 2 2 xy + yz + zx (x − y) (y − z) (z − x)2 Source: [107, pp. 9, 51, 88, 274, 578, 579], [108, pp. 25, 162], [109, pp. 72, 113], [287], [748,

pp. 13, 216, 377, 380], [752, pp. 34, 242, 244, 317], [806, pp. 63, 138, 157, 165, 186, 212], [1006, 1731], [1938, pp. 11, 49], and [2527, p. 184]. Fact 2.3.69. Let x, y, z be nonnegative numbers. Then, 0 ≤ 27x2 y2 z2 ≤ 3xyz(x + y + z)(xy + yz + zx) ≤ 27 8 xyz(x + y)(y + z)(z + x)  27  3     64 xyz(2x + y + z)(2y + z + x)(2z + x + y) ≤ xyz(x + y + z)   27 ≤ ≤ (x + y)2 (y + z)2 (z + x)2    3   64 (xy + yz + zx)   9 2 2 2 2 2 2      16 [x (x + y)(y + z) (z + x) + y (x + y)(y + z)(z + x) + z (x + y) (y + z)(z + x)]  ≤    2 2 2 2   (x + y + z )(xy + yz + zx) ≤ ≤

6 2 2 1 3 27 (x + y + z) ≤ 16 (x + y + z)[x(x + y) (x + z) + y(y 6 6 6 6 6 6 9 64 [(x + y) + (y + z) + (z + x) ] ≤ 9(x + y + z ), 2 9 16 [x (x

+ z)2 (y + x)2 + z(z + x)2 (z + y)2 ]

+ y)(y + z)2 (z + x) + y2 (x + y)(y + z)(z + x)2 + z2 (x + y)2 (y + z)(z + x)]

≤ (xy + yz + zx)3 ≤ (x2 + xy + y2 )(y2 + yz + z2 )(z2 + zx + x2 ) ≤ 3(x2 + y2 + z2 )(x2 y2 + y2 z2 + z2 x2 ) ≤

27 2 8 (x

+ y2 )(y2 + z2 )(z2 + x2 )

≤ (x2 + y2 + z2 )3 ≤ 3(x3 + y3 + z3 )2 ,  3 2   2 4  3 2 3 2 2 4 2 4       x yz + y zx + z xy    x y +y z +z x   3x y z ≤  ≤ 4 2 ,     xy3 z2 + yz3 x2 + zx3 y2     x y + y4z2 + z4 x2   2 2 2

12x2 y2 z2 ≤ 6x2 y2 z2 + 2xyz(x3 + y3 + z3 ) ≤ 2(x3 y3 + y3 z3 + z3 x3 ) + x4 (y2 + z2 ) + y4 (z2 + x2 ) + z4 (x2 + y2 ), 3 4 (x

+ y)(y + z)(z + x)(x + y + z)(x2 + y2 + z2 )   2 2 2 2 2 2 2 2 2      (x + y + z)(x + y + z )[x(y + z ) + y(z + x ) + z(x + y )]  ≤    2 2 2 2 3   (x + y + z)(x + y + z )(x + y + z) 9

≤ 9x2 y2 z2 + (x + y + z)(x2 + y2 + z2 )(x3 + y3 + z3 + 2xyz),

170

CHAPTER 2

432xy2z3 ≤ (x + y + z)6, (x4 + y4 + z4 )(xy + yz + zx) ≤ (x3 + y3 + z3 )2 , 9(x2 + yz)(y2 + zx)(z2 + xy) ≤ 8(x3 + y3 + z3 )2, 3xyz(x3 + y3 + z3 ) ≤ (xy + yz + zx)(x4 + y4 + z4 ), x2 y2 z2 + xyz + 1 ≤ 3(x2 − x + 1)(y2 − y + 1)(z2 − z + 1), (xy + yz + zx)3 ≤ (x2 + xy + yz)(y2 + yz + zx)(z2 + zx + xy), 4xyz(xy2 + yz2 + zx2 ) ≤ (x2 + yz)(y2 + zx)(z2 + xy) + 4x2 y2 z2 , 2xyz(xy2 + yz2 + zx2 ) + 2x2 y2 z2 ≤ (x2 + yz)(y2 + zx)(z2 + xy), xyz(x3 + y3 + z3 ) ≤ 3(x2 − xy + y2 )(y2 − yz + z2 )(z2 − zx + x2 ), 9x2 y2z2 ≤ 31 (xy + yz + zx)3 ≤ (x2 y + y2z + z2 x)(xy2 + yz2 + zx2 ), (x3 + y3 + z3 )2 + 36x2 y2 z2 ≤ (x2 + y2 + z2 )3 + 6xyz(x3 + y3 + z3 ), 3(x3 y3 + y3 z3 + z3 x3 ) ≤ [xy(x + y − z) + yz(y + z − x) + zx(z + x − y)]2 , (x + y + z)2 (xy + yz + zx)2 ≤ 3(x2 + xy + y2 )(y2 + yz + z2 )(z2 + zx + x2 ), xyz(x + y + z)3 ≤ 9(x4 yz + y4 zx + z4 xy) ≤

9 64 [(x 3 2

+ y)6 + (y + z)6 + (z + x)6 ],

6x2 y2 z2 ≤ 2(x3y3 + y3z3 + z3 x3 ) ≤ 21 [(x3 + y3 + z ) + 3x2 y2 z2 ] ≤ x6 + y6 + z6 + 3x2y2 z2, [27xyz + (x + y + z)(2x2 + 2y2 + 2z2 − 5xy − 5yz − 5zx)]2 ≤ 4(x2 + y2 + z2 − xy − yz − zx)3 , (x2 + xy + y2 )(y2 + yz + z2 )(z2 + zx + x2 ) ≤ (x3 + y3 + z3 )(xy + yz + zx)(x + y + z) ≤ (x2 + y2 + z2 )3 . If α ∈ [3/7, 7/3], then (α + 1)6 (xy + yz + zx)3 ≤ 27(αx + y)2 (αy + z)2 (αz + x)2 . In particular, 64(xy + yz + zx)3 ≤ 27(x + y)2 (y + z)2 (z + x)2 ,

27(xy + yz + zx)3 ≤ (2x + y)2 (2y + z)2 (2z + x)2 .

Now, assume that x, y, z are positive. Then, 9 x2 + 2yz y2 + 2zx z2 + 2xy ≤ + + , 4 (y + z)2 (z + x)2 (x + y)2

10 ≤

x2 + 16yz y2 + 16zx z2 + 16xy + 2 + 2 , y 2 + z2 z + x2 x + y2

9 2x2 + yz 2y2 + zx 2z2 + xy xy − 2yz + zx yz − 2zx + xy zx − 2xy + yz + + , ≤ 2 + 2 + 2 , 2 y2 − yz + z2 z2 − zx + x2 x2 − xy + y2 y + z2 z + x2 x + y2 x2 + yz y2 + zx z2 + xy 2x2 − yz 2y2 − zx 2z2 − xy 2≤ 2 + + , 3 ≤ + + , y + yz + z2 z2 + zx + x2 x2 + xy + y2 y2 − yz + z2 z2 − zx + x2 x2 − xy + y2 xy + 4yz + zx yz + 4zx + xy zx + 4xy + yz 4≤ + + , y2 + z2 z2 + x 2 x2 + y2 0≤

1≤ 0≤

2y2

x2 y2 z2 + 2 + 2 , 2 2 − yz + 2z 2z − zx + 2x 2x − xy + 2y2

x2 − yz y2 − zx z2 − xy + + , 2y2 − 3yz + 2z2 2z2 − 3zx + 2x2 2x2 − 3xy + 2y2

x2 y2 z2 1 + + ≤ , (2x + y)(2x + z) (2y + z)(2y + x) (2z + x)(2z + y) 3

171

EQUALITIES AND INEQUALITIES

5≤

x y z 6xyz , + + + 2 y z x x y + y2 z + z2 x

x2 y2 (x + y)2 ≤ + , x2 + y2 + 2z2 x2 + z2 y2 + z2 x2 y 6≤ 2≤

3≤

3xyz 6xyz ≤ 2 + 1, 2 2 + y z + z x xy + yz2 + zx2

(x + y)2 (y + z)2 (z + x)2 + 2 + 2 , z2 + xy x + yz y + zx

x(y + z) y(z + x) z(x + y) + 2 + 2 , x2 + yz y + zx z + xy

x2 + y2 + z2 3(x3 y + y3 z + z3 x) + , xy + yz + zx x2 y2 + y2 z2 + z2 x2

xy yz zx + 2 + 2 ≤ 1, 2 2 + 2y y + 2z z + 2x2 x+y y+z z+x x y z + + ≤ + + , y+z z+x x+y y z x

x+y+z≤

x2 + yz y2 + zx z2 + xy + + , y+z z+x x+y

8 1 1 1 + + , ≤ (x + y + z)2 2x2 + yz 2y2 + zx 2z2 + xy ( )2 1 1 1 1 1 1 1 + + , + + ≤ 2x2 + yz 2y2 + zx 2z2 + zx 9 x y z

3(x3 + y3 + z3 ) y2 z2 x2 + + , ≤ 2 2 2 2(x + y + z ) y + z z + x x + y 4≤

x2

0≤

x2 − yz y2 − zx z2 − xy + 2 + 2 , 2 2 2 2 +y +z 2y + z + x 2z + x2 + y2

2x2

3x2 − yz 3y2 − zx 3z2 − xy 3 + + ≤ , 2 2 2 2 2 2 2 2 2 2 2x + y + z 2y + z + x 2z + x + y

1 1 1 x2 y2 z2 12 ≤ + + , 1 ≤ + + , (x + y + z)2 x2 + yz y2 + zx z2 + xy x2 + xy + y2 y2 + yz + z2 z2 + zx + x2 xyz 27 2 2 xy yz zx 2 + + + , + + , 2(x + y + z) ≤ ≤ z x y xy + yz + zx (x + y + z)2 (x + y)y (y + z)z (z + x)x 2 1 1 1 + 2 + 2 , ≤ 2 xy + yz + zx x + 2yz y + 2zx z + 2xy

3 1 1 1 + 2 + 2 , ≤ 2 xy + yz + zx x + yz y + zx z + xy

6 x+y y+z z+x yz zx xy 3 ≤ 2 + 2 + 2 , + 2 + 2 ≤ , 2 2 2 2 2 2 2 x + y + z 2z + xy 2x + yz 2y + zx 3x + y + z 5 3y + z + x 3z + x + y y(z + x) z(x + y) 3 xy − yz + zx yz + xy zx − xy + yz x(y + z) + + , ≤ + 2 + , 2≤ 2 y + yz + z2 z2 + zx + x2 x2 + xy + y2 2 y2 + z2 z − zx + xy x2 + y2 9 1 1 1 9 z x y ≤ + + , ≤ + + , 4(xy + yz + zx) (x + y)2 (y + z)2 (z + x)2 4(x + y + z) (x + y)2 (y + z)2 (z + x)2 1 1 1 xy + 4yz + zx yz + 4zx + xy zx + 4xy + yz 10 ≤ 2 + + , 6≤ + + , (x + y + z)2 x + y2 y2 + z2 z2 + x2 x2 + yz y2 + zx z2 + xy 1 1 1 1 ≤ + + , xy + yz + zx (x + 2y)2 (y + 2z)2 (z + 2x)2 4x2 − y2 − z2 4y2 − z2 − x2 4z2 − x2 − y2 + + ≤ 3, x(y + z) y(z + x) z(x + y) x y z x2 y2 z2 + + ≤ 2 + + , y + z z + x x + y y + z2 z2 + x2 x2 + y2 x 1 1 y z 1 + + , + 2 + 2 ≤ + yz y + zx z + xy x + y y + z z + x 1 1 1 1 ≤ + + , 2 2 2 2 (x + y + z) 22x + 5yz 22y + 5zx 22z + 5xy x2

(2x + y)2 (2y + z)2 (2z + x)2 + + ≤ 3, 4x2 + y2 + 4z2 4y2 + z2 + 4x2 4z2 + x2 + 4y2

172

CHAPTER 2

0≤

4x2

y2 − zx z2 − xy x2 − yz + 2 + 2 , 2 2 2 2 + 4y + z 4y + 4z + x 4z + 4x2 + y2

9xyz xy2 yz2 zx2 ≤ + + ≤ 1 (x2 + y2 + z2 ), 2(x + y + z) x + y y + z z + x 2 2(xy + yz + zx) ≤

x2 (y + z)2 y2 (z + x)2 z2 (x + y)2 + 2 + 2 , y2 + z2 z + x2 x + y2

1 x2 y2 z2 1 ≤ 2 + + ≤ , 2 2 2 2 2 3 3x + (y + z) 2 3y + (z + x) 3z + (x + y) 1 1 1 9 ≤ 2 + + , (x + y + z)2 x + xy + y2 y2 + yz + z2 z2 + zx + x2 ( ) 4 y z 1 1 x 1 + + , + + ≤ 2 5 x+y y+z z+x y + z2 z2 + x2 x2 + y2 3 1 1 1 ≤ 2 + 2 + 2 , 2 2 xy + yz + zx x − xy + y y − yz + z z − zx + x2 1≤

z2 x2 2xyz y2 + + + , (x + y)2 (y + z)2 (z + x)2 (x + y)(y + z)(z + x)

y2 z2 1 x2 + + ≤ , (2x + y)(2x + z) (2y + z)(2y + x) (2z + x)(2z + y) 3 6 1 1 1 ≤ + + , x2 + y2 + z2 + xy + yz + zx 2x2 + yz 2y2 + zx 2z2 + xy 3x(y + z) − 2yz 3y(z + x) − 2zx 3z(x + y) − 2xy 3 ≤ + + , 2 (2x + y + z)(y + z) (2y + z + x)(z + x) (2z + x + y)(x + y) 1 1 1 1 ≤ + + , x2 + y2 + z2 5(x2 + y2 ) − xy 5(y2 + z2 ) − yz 5(z2 + x2 ) − zx 1≤ If r ≤ 52 , then

2(x2

y 2 + z2 z2 + x 2 x2 + y2 + + . 2 2 2 2 + y ) + (x + y)z 2(y + z ) + (y + z)x 2(z + x2 ) + (z + x)y 3(1 + r) x2 + ryz y2 + rzx z2 + rxy + + . ≤ 4 (y + z)2 (z + x)2 (x + y)2

In particular, 3 x2 − 2yz y2 − 2zx z2 − 2xy x2 − yz y2 − zx z2 − xy ≤ + + , 0≤ + + , 2 2 2 2 2 4 (y + z) (z + x) (x + y) (y + z) (z + x) (x + y)2 ( )2 ( ( )2 x y )2 z 3 x2 + yz 3 y2 + zx z2 + xy ≤ + + , ≤ + + . 4 y+z z+x x+y 2 (y + z)2 (z + x)2 (x + y)2

−

If r > −2, then 3(3 + 2r) 2x2 + (1 + 2r)yz 2y2 + (1 + 2r)zx 2z2 + (1 + 2r)xy + 2 + 2 . ≤ 2+r y2 + ryz + z2 z + αzx + x2 x + rxy + y2 In particular, 9 2x2 + yz 2y2 + zx 2z2 + xy ≤ 2 + 2 + 2 , 2 y + z2 z + x2 x + y2

21 2x2 + 5yz 2y2 + 5zx 2z2 + 5xy ≤ + + , 4 (y + z)2 (z + x)2 (x + y)2

173

EQUALITIES AND INEQUALITIES

5≤

2y2 + 3zx 2z2 + 3xy 2x2 + 3yz + 2 + 2 , 2 2 2 y + yz + z z + zx + x x + xy + y2 1≤

3≤

2x2 − yz 2y2 − zx 2z2 − xy + 2 + 2 , 2 2 − yz + z z − zx + x x − xy + y2

y2

x2 y2 z2 + + , 2y2 − yz + 2z2 2z2 − zx + 2x2 2x2 − xy + 2y2

x2 − yz y2 − zx z2 − xy + + , 2y2 − 3yz + 2z2 2z2 − 3zx + 2x2 2x2 − 3xy + 2y2 9 1 1 1 ≤ 2 + 2 + 2 . 2 2 2 2 2 7(x + y + z ) 4y − yz + 4z 4z − zx + 4x 4x − xy + 4y2 0≤

If r ≥ 0, then

x y z rx + y ry + z rz + x + + ≤ + + . ry + z rz + x rx + y y z x

If r ∈ [0, 25 ], then 1 1 1 3(r + 1) ≤ + + . x2 + y2 + z2 + r(xy + yz + zx) x2 + xy + y2 y2 + yz + z2 z2 + zx + x2 √ If r ∈ [ 23 , 3 + 7], then

If r ≥ 3 +

√ 7, then

1 1 r+2 1 + 2 + 2 . ≤ 2 r(xy + yz + zx) rx + yz ry + zx rz + xy 1 9 1 1 ≤ + + . (r + 1)(xy + yz + zx) rx2 + yz ry2 + zx rz2 + xy

Source: [107, pp. 30, 54, 168, 289, 290], [108, pp. 12, 104, 353], [109, pp. 179, 180, 388, 389],

[135], [287], [289, pp. 20, 120, 121], [371], [603, p. 244], [725, p. 114], [748, pp. 6, 9, 11, 12, 217, 218, 301, 302, 318, 342, 344, 375, 377, 378, 381], [749], [752, pp. 27, 29, 31, 35, 37–40, 242], [806, pp. 15, 100, 101, 111, 114, 127, 138, 139, 186, 188, 189, 191, 196, 201, 204, 211, 212, 213, 214], [961], [993, pp. 179, 182], [1025], [1371, pp. 105, 134, 149, 150, 155, 169], [1507], [1757, pp. 247, 252, 257], [1903], [1938, pp. 31, 107, 110, 111], [2128, p. 14], [2380, p. 72], [2791]. Remark: The inequalities 2(x3y3 + y3z3 + z3 x3 ) ≤ 21 [(x3 + y3 + z3 )2 + 3(x2 y2 z2 ] ≤ x6 + y6 + z6 + 3x2y2 z2 hold for all real numbers. The left-hand inequality is given in [1860, pp. 20, 114, 115]. Related: Fact 2.3.61, Fact 2.11.15, Fact 2.11.25, and Fact 5.2.25. Credit: (xy + yz + zx)2 (xyz2 + x2 yz + xy2z) ≤ 3(y2z2 + z2 x2 + x2 y2 )2 is due to M. Klamkin. See Fact 5.2.7 and [2791]. Fact 2.3.70. Let x, y, z be nonnegative numbers. Then, x2 y2z2 (xy + yz + zx) ≤ x8 + y8 + z8, (xyz + 1)3 ≤ (x3 + 1)(y3 + 1)(z3 + 1), x3 y3 z3 (x + y + z) ≤ x5 y5 + y5 z5 + z5 x5 , x2 y2 z2 (x + y + z) ≤ x5 yz + y5 zx + z5 xy, x2 y2 z2 (x3 + y3 + z3 ) ≤ x3 y6 + y3 z6 + z3 x6 , (x5 + y5 + z5 )(x + y + z)3 ≤ 9(x4 + y4 + z4 )2 , (x3 + y3 + z3 )(x5 + y5 + z5 ) ≤ 3(x8 + y8 + z8 ), xyz(x + y + z)(x3 + y3 + z3 ) ≤ (xy + yz + zx)(x5 + y5 + z5 ), (xy + yz + zx)2 (xyz2 + yzx2 + zxy2 ) ≤ 3(x2 y2 + y2z2 + z2 x2 )2, (x3 + y3 + z3 + xyz)(x + y − z)(y + z − x)(z + x − y) ≤ 4x2 y2 z2 ,

174

CHAPTER 2

x2y2z2 (x + y + z)2 ≤ 3x2y2z2 (x2 + y2 + z2 ) ≤ (x2 y2 + y2z2 + z2 x2 )2 , (x3 y3 + y3 z3 + z3 x3 )(x + y)(y + z)(z + x) ≤ 3(x3 + y3 )(y3 + z3 )(z3 + x3 ), [2(xy + yz + zx) − (x2 + y2 + z2 )][2(x3 y3 + y3 z3 + z3 x3 ) − (x6 + y6 + z6 )] ≤ [2(x2 y2 + y2 z2 + z2 x2 ) − (x4 + y4 + z4 )]2 . Now, assume that x, y, z are positive. Then, 6≤

3x3 + xyz 3y3 + yzx 3z3 + zxy + 3 + 3 , y3 + z3 z + x3 x + y3

x5 y5 z5 (xy2 + yz2 + zx2 )5 ≤ 2 + 2 + 2, 3 3 3 4 (x + y + z ) y z x

1≤

x3

xy2 yz2 zx2 3 + + ≤ , 3 3 3 3 3 3 2 x +y y +z z +x

y3 z3 x3 + 3 + 3 , 3 3 + y + xyz y + z + xyz z + x3 + xyz

2y3 2z3 3 y3 z z3 x 2x3 x3 y + + , + + , ≤ 2 z2 (y2 + zx) x2 (z2 + xy) y2 (x2 + yz) y2 + z2 z2 + x2 x2 + y2 x2 yz y3 z3 y2 zx z2 xy (x + y + z)6 x3 2 2 1 2 + + , + + ≤ (x + y + z ), ≤ x + 2y y + 2z z + 2x x2 + yz y2 + zx z2 + xy 2 81(x2 + y2 + z2 )2 x+y+z≤

0≤

x2 y2 (y − z) y2 z2 (z − x) z2 x2 (x − y) + + , x+y y+z z+x

3(x3 + y3 + z3 ) x3 + xyz y3 + yzx z3 + zxy ≤ + + , 2(x2 + y2 + z2 ) (y + z)2 (z + x)2 (x + y)2

y z 3 y3 z z3 x 9 x x3 y + + , + + , ≤ ≤ 4(xy + yz + zx) y(y + z)2 z(z + x)2 x(x + y)2 2 z2 (y2 + xz) x2 (z2 + yx) y2 (x2 + yz) 2x4 2y4 2z4 x3 + 3xyz y3 + 3xyz z3 + 3xyz + 3 + 3 , x+y+z≤ + + , 3 3 3 +y y +z z +x (y + z)2 (z + x)2 (x + y)2 ( 5 )3 x y2 z2 y5 z5 x2 3(x6 + y6 + z6 )2 ≤ + 2 + 2 , + + , 1≤ 2 2 2 y z x x + xy + y y + yz + z z + zx + x2 ( 2 )2 x + y2 + z2 x3 + y3 + z3 1 3 x3 + 3xyz y3 + 3xyz z3 + 3xyz + + , ≤ + , ≤ xy + yz + zx 4xyz 4 2 (y + z)3 (z + x)3 (x + y)3 x+y+z x y z ≤ + + , x2 + y2 + z2 2x2 + yz 2y2 + zx 2z2 + xy

x+y+z≤

x3

y3 z3 x3 + 3 + 3 , 3 3 + y + xyz y + z + xyz z + x3 + xyz x(y + z) y(z + x) z(x + y) xy + yz + zx +1≤ 2 + + , x2 + y2 + z2 x + 2yz y2 + 2zx z2 + 2xy ( )2 ( )2 ( )2 9(x + y + z)2 2x 2y 2z ≤ +1 + +1 + +1 , xy + yz + zx y z x xy yz zx x y z + 2 + 2 ≤ + + , 2 z + xy x + yz y + zx y + z z + x x + y 1≤

x3

9 1 1 1 + + , ≤ 3 3 + y + z ) xy(x + y) yz(y + z) zx(z + x) 1 1 1 1 + + ≤ , x3 + y3 + xyz y3 + z3 + xyz z3 + x3 + xyz xyz 2(x3

y z x2 x y2 z2 + + ≤ 2 + + , y + z z + x x + y y + z2 z2 + x2 x2 + y2

175

EQUALITIES AND INEQUALITIES

x2

y z 1 x 1 1 + 2 + 2 ≤ + + , + yz y + zx z + xy x + y y + z z + x

x+y+z≤

x3 y3 z3 + + , y2 − yz + z2 z2 − zx + x2 x2 − xy + y2

2y 2z 1 1 1 2x + + ≤ + + , 3x2 + yz 3y2 + zx 3z2 + xy x + y y + z z + x x4 x3 + y3 + z3 y4 z4 ≤ 2 + 2 + 2 , 2 2 x+y+z x + xy + y y + yz + z z + zx + x2 y2 z2 x2 + y2 + z2 x2 , + + ≤ x2 + xy + yz y2 + yz + zx z2 + zx + xy xy + yz + zx 9 1 1 1 + 2 + 2 , ≤ 2 2 (x + y + z) x + xy + yz y + yz + zx z + zx + xy yz zx x2 + y2 + z2 xy , + + ≤ xy + yz + zx y2 + yz + z2 z2 + zx + x2 x2 + xy + y2 x2

xy yz zx x2 + y2 + z2 , + 2 + 2 ≤ + yz + zx y + zx + xy z + xy + yz xy + yz + zx

x3 + y3 + z3 x4 y4 z4 ≤ + + , x2 + y2 + z2 x3 + xyz + y3 y3 + xyz + z3 z3 + xyz + x3 9(xy + yz + zx) x y z ≤ 2 + 2 + 2 , 3 (x + y + z) x + xy + yz y + yz + zx z + zx + xy 1 1 1 1 2 ≤ 2 + 2 + 2 , + 2 2 2 xy + yz + zx x + y + z 2x + yz 2y + zx 2z + xy 4xyz(xy + yz + zx) xy4 + yz4 + zx4 + x4 y + y4 z + z4 x + , xy + yz + zx x 2 + y 2 + z2 12 12 12 1 1 1 4 4 4 + + ≤ + + + + + , 3x + y 3y + z 3z + x x y z x + y y + z z + x 6xyz ≤

3(x + 1)(y + 1)(z + 1) 1 1 1 x y z ≤3+x+y+z+ + + + + + , xyz + 1 x y z y z x √   y2 + z2 3  (x + y + z)(x2 + y2 + z2 ) z2 + x2 x2 + y2 − 1 ≤ + + ,  2 xyz x(y + z) y(z + x) z(x + y) 1≤

x3

x3 + 4xyz y3 + 4xyz z3 + 4xyz + 3 + 3 , 3 3 + (y + z) + 6xyz y + (z + x) + 6xyz z + (x + y)3 + 6xyz

[(x + y + z)(x2 + y2 + z2 ) − (x3 + y3 + z3 )]4 (x + y)4 (y + z)4 (z + x)4 ≤ + + , z x y (x3 + y3 + z3 )3 y z xy yz zx x + + ≤ 1, + + ≤ x2 + xy + yz y2 + yz + zx z2 + zx + xy 2x + z 2y + x 2z + y ) ( 1 1 1 (x + y + z)2 1 1 1 + + ≤ + + , x2 + yz y2 + zx z2 + xy 3(xy + yz + zx) x2 + y2 y2 + z2 z2 + x2 x3 y3 z3 1 + + ≤ , (2x2 + y2 )(2x2 + z2 ) (2y2 + z2 )(2y2 + x2 ) (2z2 + x2 )(2z2 + y2 ) x + y + z

176

CHAPTER 2

4xyz(x2 − y2 )2 ≤ x(x − y)(x − z) + y(y − z)(y − x) + z(z − x)(z − y), (x + y + z)(x + y)(y + z)(z + x) )5 ( )5 ( )5 ( 6 6 1 2 1 1 + + ≤ 5 + 5 + 5, x+y 3x + y + 2z 3x + 3y + z x y z 2 2 8 8 8 1 1 1 8 8 8 2 + + + + + ≤ + + + + + , x + y y + z z + x 3x + y 3y + z 3z + x x y z x + 3y y + 3z z + 3x ( ) 1 1 1 x2 + y2 + z2 2 xyz + + ≤ 31 (x + y + z) ≤ 3 2 2 2 x+y+z x + yz y + zx z + xy 3 y3 z3 x + + . ≤ 2 x + xy + y2 y2 + yz + z2 z2 + zx + x2 If r > −2, then 4r + 10 ≤

x2 + 4(r + 2)2 yz y2 + 4(r + 2)2 zx z2 + 4(r + 2)2 xy + 2 + 2 , y2 + ryz + z2 z + rzx + x2 x + rxy + y2

3(2r + 3) 2x2 + (2r + 1)yz 2y2 + (2r + 1)zx 2z2 + (2r + 1)xy ≤ + 2 + 2 , r+2 y2 + ryz + z2 z + rzx + x2 x + rxy + y2 3(r + 1) xy + (r − 1)yz + zx yz + (r − 1)zx + xy zx + (r − 1)xy + yz ≤ + + , r+2 y2 + ryz + z2 z2 + rzx + x2 x2 + rxy + y2 xy + (r + 2)2 yz + zx yz + (r + 2)2 zx + xy zx + (r + 2)2 xy + yz + + . y2 + ryz + z2 z2 + rzx + x2 x2 + rxy + y2

r+4≤

If r > −2 and p ≤ (2r + 1)/2, then yz + pzx zx + pxy xy + pyz 3(p + 2) + + . ≤ 2 r+2 y + ryz + z2 z2 + rzx + x2 x2 + rxy + y2 If r > −2 and p ∈ [(2r + 1)/2, 4(r + 2)2 ], then yz + pzx zx + pxy p xy + pyz + 2 + 2 . +2≤ 2 2 2 r+2 y + ryz + z z + rzx + x x + rxy + y2 If r > −2 and p ≥ 1, then 4pr + 12p2 − 2 ≤

xy + 4p(r + 2p)2 yz yz + 4p(r + 2p)2 zx zx + 4p(r + 2p)2 xy + + . y2 + ryz + z2 z2 + rzx + x2 x2 + rxy + y2

If r > −2 and p ≤ r − 1, then 3(p + 2) xy + pyz + zx yz + pzx + xy zx + pxy + yz ≤ 2 + 2 + 2 . r+2 y + ryz + z2 z + rzx + x2 x + rxy + y2 If r > −2 and p ∈ [r − 1, (r + 2)2 ], then p xy + pyz + zx yz + pzx + xy zx + pxy + yz +2≤ 2 + 2 + 2 . r+2 y + ryz + z2 z + rzx + x2 x + rxy + y2 If r > −2 and p ≥ (r + 2)2 , then xy + pyz + zx yz + pzx + xy zx + pxy + yz √ 2 p−r ≤ 2 + 2 + 2 . y + ryz + z2 z + rzx + x2 x + rxy + y2 If n ≥ 0, then 1 n 2 (x

+ yn + zn ) ≤

xn+1 yn+1 zn+1 + + , y+z z+x x+y

(

x + 2y 3

)n

( +

y + 2z 3

)n

( +

z + 2x 3

)n ≤ xn + yn + zn ,

177

EQUALITIES AND INEQUALITIES

(xyz)n+2 (xn + yn + zn ) ≤ (xy)2n+3 + (yz)2n+3 + (zx)2n+3 . If n ≥ 2, then

√

3(xn−1 + yn−1 + zn−1 )(xn + yn + zn ) xn + yn yn + zn zn + xn ≤ + + . x+y+z x+y y+z z+x

If n ≥ 3, then xn−1 y + yn−1 z + zn−1 x + xyn−1 + yzn−1 + zxn−1 ≤ xn + yn + zn + xyz(xn−3 + yn−3 + zn−3 ). If n ≥ 1, then

2xn − yn − zn 2yn − zn − xn 2zn − xn − yn + 2 + 2 , y2 − yz + z2 z − zx + x2 x − xy + y2 ( n ) √ x yn zn xn+1 yn+1 zn+1 n 1 n n n 3 (x + y + z ) y + z + z + x + x + y ≤ y + z + z + x + x + y . 0≤

Source: [55, 99], [107, pp. 12, 15, 54, 99, 100, 108, 109, 290, 487, 488], [108, pp. 15, 55, 115,

116, 318, 354–356], [109, p. 156], [288, p. 229], [318, 751], [748, pp. 12, 13, 111, 319, 373, 376, 382], [752, pp. 29, 30, 32, 33, 35–38, 42, 244], [806, pp. 45, 116, 129, 157, 187, 198, 199, 213, 215], [993, p. 184], [1371, pp. 108, 117, 134, 150], [1569], [1860, pp. 19, 112, 113, 134, 228], [1772], [1938, pp. 12, 43, 49, 107, 108], [2527, p. 204], [2601], and [2791]. Remark: Setting n = 0 in the third to last inequality yields Nesbitt’s inequality. See Fact 2.3.60. Remark: The last inequality is due to Schur. See [99], Fact 2.3.16, Fact 2.3.28, and Fact 2.3.67. Fact 2.3.71. Let x, y, z be positive numbers. Then, ( ) 3 3 1 1 1 1 1 1 1 3 + + ≤ + + + + + , 3x + y 3y + z 3z + x x + y y + z z + x 4 x y z ( ) 2 2 2 1 1 1 1 1 1 1 + + ≤ + + + + + , 3x + y 3y + z 3z + x x + 3y y + 3z z + 3x 4 x y z ( ) 3 15 15 15 12 12 12 4 1 1 1 + + + ≤ + + + + + . x + y + z x + 2y y + 2z z + 2x 2x + y 2y + z 2z + x 3 x y z If r ∈ [1, 3], then

( ) 3 3 1 1 1 1 1 1 1 3 + + ≤ + + + + + . rx + (4 − r)y ry + (4 − r)z rz + (4 − r)x x + y y + z z + x 4 x y z

If r ∈ [−1, 2], then

( ) ( ) r+1 r+1 r+1 r 1 1 1 1 1 1 1 + + ≤ + + + + + . 3x + y 3y + z 3z + x 2 x + y y + z z + x 4 x y z

Source: [381].

2.4 Facts on Equalities and Inequalities in Four Variables Fact 2.4.1. Let a, b be positive numbers, let x, y be real numbers, and assume that x/a ≤ y/b.

Then,

x+y y x ≤ ≤ . a a+b b Source: [47, p. 32]. Remark: The center term is the mediant of x/a and y/b. See Fact 2.12.6. Fact 2.4.2. Let a, b, x, y be positive numbers, and assume that a ≤ x and b ≤ y. Then, x+y x+b ≤ . a+y a+b

178

CHAPTER 2

Fact 2.4.3. Let a, b be positive numbers, and let x, y be real numbers. Then,

x2 y2 (x + y)2 ≤ + . a+b a b Source: Fact 2.12.19. Related: Fact 2.6.7. Fact 2.4.4. Let x1 , x2 , x3 , x4 be complex numbers. Then, ∑ ∑ 6(x12 + x22 + x32 + x42 ) = (xi + x j )2 + (xi − x j )2 , 1≤i< j≤4

6x1 (x12

+

x22

+

x32

+

x42 )

=

1≤i< j≤4

∑

(x1 + xi ) + 3

2≤i≤4

6(x12

+

x22

+

x32

+

x42 )2

=

∑

∑

(x1 − xi )3 ,

2≤i≤4

(xi + x j ) + 4

1≤i< j≤4

∑

(xi − x j )4 .

1≤i< j≤4

Remark: The second equality is Maillet’s identity, while the second and third equalities are Liou-

ville’s identity. See [309, p. 14] and [2232]. Fact 2.4.5. Let w, x, y, z be complex numbers. Then, (w2 + x2 + y2 + z2 )2 = (w2 + x2 − y2 − z2 )2 + 4(wy + xz)2 + 4(wz − xy)2 . Source: [2316, p. 58]. Remark: This is Lebesgue’s identity. Fact 2.4.6. Let w, x, y, z be complex numbers. Then,

(x − y)(w − z)2 + (y − z)(w − x)2 + (z − x)(w − y)2 + (x − y)(y − z)(z − x) = 0, z2 [(3wxy − 2x3 − w2 z)2 + 4(wy − x2 )3 ] = w2 [(3xyz − 2y3 − z2 w)2 + 4(xz − y2 )3 ], 2(w4 + x4 +y4 +z4 )+8wxyz = (w2 + x2 +y2 +z2 )2 +(w+ x−y−z)(w− x−y+z)(w− x+y−z)(w+ x+y+z). △

Now, let s = w + x + y + z. Then, 16(s − w)(s − x)(s − y)(s − z) + (w2 − x2 − y2 + z2 )2 = 4(wz + xy)2 , s(s − w − z)(s − x − z)(s − y − z) + wxyz = (s − w)(s − x)(s − y)(s − z). Source: [732, pp. 57, 71] and [289, pp. 27, 154]. Fact 2.4.7. Let a, w, x, y, z be complex numbers. Then,

(w2 − ax2 )(y2 − az2 ) = (wy + axz)2 − a(wz + xy)2 = (wy − axz)2 − a(wz − xy)2 . In particular, (w2 + x2 )(y2 + z2 ) = (wz + xy)2 + (wy − xz)2 = (wy + xz)2 + (wz − xy)2 . If, in addition, w, x, y, z are real, then max {(wz + xy)2 , (wy − xz)2 , (wy + xz)2 , (wz − xy)2 } ≤ (w2 + x2 )(y2 + z2 ). Remark: The first equality is Brahmagupta’s identity. See [1016]. Remark: The case a = −1 is

Diophantus’s identity, which is a special case of Lagrange’s identity given by Fact 2.12.13. This equality is a statement of the fact that |w + x ȷ|2 |y + z ȷ|2 = |(w + x ȷ)(y + z ȷ)|2 . See [773, p. 77], [1171], [1258, pp. 13, 14], [2107, pp. 25, 26], [2380, p. 6], and [2527, p. 47]. Related: Fact 1.11.31. Fact 2.4.8. Let w, x, y, z be complex numbers. Then, (4w + 1)2 + (4x + 1)2 + (4y + 1)2 + (4z + 1)2 = 4(w + x + y + z + 1)2 + 4(w + x − y − z)2 + 4(w − x + y − z)2 + 4(−w + x + y − z)2 . Source: [2116]. Related: Fact 1.11.28.

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Fact 2.4.9. Let w, x, y, z be complex numbers. Then,

(w + x + y + z)3 = w3 + x3 + y3 + z3 + 3[w2 (x + y + z) + w(x + y + z)2 + (x + y)(x + z)(y + z)]. Fact 2.4.10. Let w, x, y, z be distinct, nonzero complex numbers, and let a, b, c be complex

numbers. Then, (w + a)(w + b)(w + c) (x + a)(x + b)(x + c) (y + a)(y + b)(y + c) + + w(w − x)(w − y)(w − z) x(x − y)(x − z)(x − w) y(y − z)(y − x)(y − x) abc (y + a)(y + b)(y + c) =− . + y(y − z)(y − x)(y − x) wxyz Source: [732, p. 161]. Fact 2.4.11. Let w, x, y, z be complex numbers such that division by zero does not occur in the

expressions below. Then, (wx − yz)(w2 − x2 + y2 − z2 ) + (wy − xz)(w2 + x2 − y2 − z2 ) (w + z)(x + y) = . (w2 − x2 + y2 − z2 )(w2 + x2 − y2 − z2 ) + 4(wx − yz)(wy − xz) (w + z)2 + (x + y)2 Source: [732, p. 161]. Fact 2.4.12. Let w, x, y, z be complex numbers. Then,

(w2 − x2 )2 + (y2 − z2 )2 + 2(wx − yz)2 + 4wxyz = w4 + x4 + y4 + z4 . If w, x, y, z are real, then

4wxyz ≤ w4 + x4 + y4 + z4 .

Furthermore, if w, x, y, z are nonnegative, then (w2 − x2 )2 + (y2 − z2 )2 + 2(wx − yz)2 ≤ w4 + x4 + y4 + z4 . Remark: This result yields the arithmetic-mean–geometric-mean inequality for four variables. See

[288, pp. 226, 367]. Fact 2.4.13. Let w, x, y, z be complex numbers. Then, (w2 + wx + x2 )[wy3 + xz3 + (w + x)(y + z)3 ] = (y2 + yz + z2 )[yw3 + zx3 + (y + z)(w + x)3 ]. Fact 2.4.14. Let x1 , x2 , x3 , x4 be complex numbers. Then,

60(x12 + x22 + x32 + x42 )3 =

∑

(xi ± x j ± xk )6 + 2

4 ∑

(xi ± x j )6 + 36

1≤i< j≤4

1≤i< j 0. Then, 3(a + b + c) ax + by + cz ay + bz + cx az + bx + cy ≤ + + , 2 x+y y+z z+x 3(a + b + c)2 4 (ax + by + cz)(ay + bz + cx) (ay + bz + cx)(z + x + 4y) (az + bx + cy)(ax + by + cz) ≤ + + , (x + y)(y + z) (y + z)(z + x) (z + x)(x + y) (a + b + c)3 (ax + by + cz)(ay + bz + cx)(az + bx + cy) ≤ . 8 (x + y)(y + z)(z + x) Source: [1772]. △ △ Fact 2.6.17. Let x1 , x2 , x3 , x4 , x5 , x6 be nonnegative numbers, and define x7 = x1 and x8 = x2 .

Then, 216

6 ∑

xi xi+1 xi+2 xi+3

 6 4 ∑  xi  , ≤  i=1

i=1

1296

6 ∑

xi xi+1 xi+2 xi+3 xi+4

i=1

 6 6 ∑  xi  . 46656x1 x2 x3 x4 x5 x6 ≤ 

 6 5 ∑  xi  , ≤  i=1

i=1

Source: Fact 2.11.16. The first inequality is given in [108, p. 462]. Related: Fact 2.5.5. √ √ Fact 2.6.18. Let x1 , x2 , x3 , x4 , x5 , x6 ∈ [ 3/3, 3]. Then,

0≤

x1 − x2 x2 − x3 x3 − x4 x4 − x5 x5 − x6 x6 − x1 + + + + + . x2 + x3 x3 + x4 x4 + x5 x5 + x6 x6 + x5 x1 + x2

Source: [748, p. 379].

2.7 Facts on Equalities and Inequalities in Seven Variables Fact 2.7.1. Let x, y, z be positive numbers, let p, q, r, s be nonnegative numbers, and assume that r ≤ s and p ≤ q. Then,

(x s + y s + z s )(xq + yq + zq ) x s+q + y s+q + z s+q ≤ r+p . r r r p p p (x + y + z )(x + y + z ) x + yr+p + zr+p Source: [53].

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EQUALITIES AND INEQUALITIES

2.8 Facts on Equalities and Inequalities in Eight Variables Fact 2.8.1. Let x1 , x2 , x3 , x4 , y1 , y2 , y3 , y4 be complex numbers. Then,

(x12

+ x22 + x32 + x42 )(y21 + y22 + y23 + y24 ) = (x1 y1 − x2 y2 − x3 y3 − x4 y4 )2 + (x1 y2 + x2 y1 + x3 y4 − x4 y3 )2 + (x1 y3 − x2 y4 + x3 y1 + x4 y2 )2 + (x1 y4 + x2 y3 − x3 y2 + x4 y1 )2.

Hence,

 (x1 y1 − x2 y2 − x3 y3 − x4 y4 )2 + (x1 y2 + x2 y1 + x3 y4 − x4 y3 )2       + (x1 y3 − x2 y4 + x3 y1 + x4 y2 )2      2 2  (x1 y1 − x2 y2 − x3 y3 − x4 y4 ) + (x1 y2 + x2 y1 + x3 y4 − x4 y3 )     2  + (x1 y4 + x2 y3 − x3 y2 + x4 y1 )       (x1 y1 − x2 y2 − x3 y3 − x4 y4 )2 + (x1 y3 − x2 y4 + x3 y1 + x4 y2 )2      2  + (x1 y4 + x2 y3 − x3 y2 + x4 y1 )     2 2  (x1 y2 + x2 y1 + x3 y4 − x4 y3 ) + (x1 y3 − x2 y4 + x3 y1 + x4 y2 )     2 + (x1 y4 + x2 y3 − x3 y2 + x4 y1 )  ≤ (x12 + x22 + x32 + x42 )(y21 + y22 + y23 + y24 ).

Credit: L. Euler. See [2228, p. 8]. Remark: Replacing x2 , x3 , x4 by −x2 , −x3 , −x4 yields

(x12 + x22 + x32 + x42 )(y21 + y22 + y23 + y24 ) = (x1 y1 + x2 y2 + x3 y3 + x4 y4 )2 + (x1 y2 − x2 y1 − x3 y4 + x4 y3 )2 + (x1 y3 + x2 y4 − x3 y1 − x4 y2 )2 + (x1 y4 − x2 y3 + x3 y2 − x4 y1 )2. Remark: This equality represents a relationship between a pair of quaternions. An analogous

equality holds for two sets of eight variables representing a pair of octonions. See [773, p. 77]. Fact 2.8.2. Let a, b, c, x, y, z, p, q be positive numbers. Then, 18 (p + 1)(q + 1)[(b + c)x + (c + a)y + (a + b)z] 1 1 1 ≤ + + . (a + pb)(x + qy) (b + pc)(y + qz) (c + pa)(z + qx) Source: [752, p. 320].

2.9 Facts on Equalities and Inequalities in Nine Variables Fact 2.9.1. Let x1 , x2 , x3 , y1 , y2 , y3 , z1 , z2 , z3 be nonnegative numbers. Then,

(x1 y1 z1 + x2 y2 z2 + x3 y3 z3 )3 ≤ (x13 + x23 + x33 )(y31 + y32 + y33 )(z31 + z32 + z33 ). Source: [806, p. 98].

2.10 Facts on Equalities and Inequalities in Sixteen Variables Fact 2.10.1. Let x1 , . . . , x8 , y1 , . . . , yn be complex numbers. Then,

 8  8 ∑  ∑ 2  y2i = (x1 y1 − x2 y2 − x3 y3 − x4 y4 − x5 y5 − x6 y6 − x7 y7 − x8 y8 )2 xi  i=1 i=1 + (x1 y2 + x2 y1 + x3 y4 − x4 y3 + x5 y6 − x6 y5 − x7 y8 + x8 y7 )2 + (x1 y3 − x2 y4 + x3 y1 + x4 y2 + x5 y7 + x6 y8 − x7 y5 − x8 y6 )2 + (x1 y4 + x2 y3 − x3 y2 + x4 y1 + x5 y8 − x6 y7 + x7 y6 − x8 y5 )2 + (x1 y5 − x2 y6 − x3 y7 − x4 y8 + x5 y1 + x6 y2 + x7 y3 + x8 y4 )2

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+ (x1 y6 + x2 y5 − x3 y8 + x4 y7 − x5 y2 + x6 y1 − x7 y4 + x8 y3 )2 + (x1 y7 + x2 y8 + x3 y5 − x4 y6 − x5 y3 + x6 y4 + x7 y1 − x8 y2 )2 + (x1 y8 − x2 y7 + x3 y6 + x4 y5 − x5 y4 − x6 y3 + x7 y2 + x8 y1 )2 . Source: [127] and [2316, p. 94]. Remark: This is Degen’s eight-square identity.

2.11 Facts on Equalities and Inequalities in n Variables Fact 2.11.1. Let z1, . . . , zn be complex numbers. Then,

  n 2 n 1 ∑  ∑ 2  zi z j =  zi  + zi  . 2 i=1 j=1 i=1

n ∑ i ∑ i=1

Source: [1219, p. 37]. ∑ Fact 2.11.2. Let z1, . . . , zn ∈ C, let α1 , . . . , αn ∈ [0, 1], and assume that ni=1 αi = 1. Then,

∑

αi αj (zi − z j )2 =

n ∑

1≤i< j≤n

j=1

 n 2 2  n n ∑ ∑ ∑    αi z2i −  αi zi  . αi zi  = αj z j − i=1

i=1

i=1

Source: [1115]. Fact 2.11.3. Let n ≥ 2, and let z1, . . . , zn be distinct complex numbers. Then, n ∑ i=1

  zi + z j  0, n even, =  1, n odd. z − z i j j∈{1,...,n}, j,i ∏

Source: [1042]. Fact 2.11.4. Let n ≥ 2, let z1, . . . , zn be distinct complex numbers, and let 0 ≤ k ≤ n. Then, n ∑ i=1

   0, 0 ≤ k < n − 1,     ∏n = 1, k = n − 1,    j=1, j,i (zi − z j )  ∑ni=1 zi , k = n. zki

In particular, z21 z22 1 1 z1 z2 + = 0, + = 1, + = z1 + z2 , z1 − z2 z2 − z1 z1 − z2 z2 − z1 z1 − z2 z2 − z1 1 1 1 + + = 0, (z1 − z2 )(z1 − z3 ) (z2 − z1 )(z2 − z3 ) (z3 − z1 )(z3 − z2 ) z1 z2 z3 + + = 0, (z1 − z2 )(z1 − z3 ) (z2 − z1 )(z2 − z3 ) (z3 − z1 )(z3 − z2 ) z23 z21 z22 + + = 1, (z1 − z2 )(z1 − z3 ) (z2 − z1 )(z2 − z3 ) (z3 − z1 )(z3 − z2 ) z33 z31 z32 + + = z1 + z2 + z3 . (z1 − z2 )(z1 − z3 ) (z2 − z1 )(z2 − z3 ) (z3 − z1 )(z3 − z2 ) Source: [1650, pp. 36–38]. Related: Fact 2.11.5. The first equality is a special case of Abel’s

theorem given by Fact 12.16.6. Fact 2.11.5. Let n ≥ 2, let z1, . . . , zn be distinct complex numbers, and let k ≥ 0. Then, n ∑ i=1

n ∑∏ zn+k i i z jj , = (z − z ) j j=1, j,i i j=1

∏n

189

EQUALITIES AND INEQUALITIES

where the second sum is taken over all n-tuples (i1 , . . . , in ) of nonnegative integers whose sum is k + 1. In particular, z53 z52 z51 + + (z1 − z2 )(z1 − z3 )(z1 − z4 ) (z2 − z1 )(z2 − z3 )(z2 − z4 ) (z3 − z1 )(z3 − z2 )(z3 − z4 ) z54 = z21 + z22 + z23 + z24 + z1 z2 + z1 z3 + z1 z4 + z2 z3 + z2 z4 + z3 z4 . + (z4 − z1 )(z4 − z2 )(z4 − z3 ) Source: [763]. Remark: Setting k = 0 yields the third case in Fact 2.11.4. Fact 2.11.6. Let n ≥ 2, and let z1, . . . , zn be distinct, nonzero complex numbers. Then,

 n  n n ∑ ∏  ∑ 1 1 n−1  = (−1)  zi  . ∏n 2 z z j=1, j,i (zi − z j ) i=1 i=1 i i=1 i

Source: [1672]. Related: Fact 2.11.4 and Fact 2.11.5. Fact 2.11.7. Let z, z1, . . . , zn ∈ C, and assume that −z, z1, . . . , zn are distinct. Then, n ∏ i=1

n n ∑ 1 ∏ 1 1 = . z + zi z + zi j=1 z j − zi i=1 j,i

Fact 2.11.8. Let z1 , . . . , zn be complex numbers, and let k ≥ 1. Then,

 n k ∑ ( k! ) ∑  zi11 · · · zinn ,  zi  = i , . . . , i 1 n i=1 (n+k−1) ∑ where the sum is taken over all n n-tuples (i1 , . . . , in ) of nonnegative integers such that nj=1 i j = k. Remark: This is the multinomial theorem. Remark: Probabilistic interpretations are discussed in [1587]. Related: Fact 2.3.1. Fact 2.11.9. Let n ≥ 2, and let x1 . . . , xn be complex numbers. Then, n  n n ∏ ∑  ∑ ∑n i n−1 i j    j j=1  xi . (−1)  (−1) xi j  = 2 n! (i1 ,...,in )∈{1,2}n

j=1

i=1

In particular, let w, x, y, z be complex numbers. Then, (x + y)2 − (x − y)2 = 4xy,

(x + y + z)3 − (x − y + z)3 − (x + y − z)3 + (x − y − z)3 = 24xyz,

(w + x + y + z)4 − (w − x + y + z)4 − (w + x − y + z)4 − (w + x + y − z)4 + (w − x − y + z)4 + (w − x + y − z)4 + (w + x − y − z)4 − (w − x − y − z)4 = 192wxyz. Remark: This is Boutin’s identity. See [2232]. Fact 2.11.10. Let x1 . . . , xn be real numbers, and assume that x1 ≤ · · · ≤ xn . Then, n ∑ i, j=1

|(i − j)(xi − x j )| =

n n ∑ n∑ |xi − x j | = n (2i − 1 − n)xi . 2 i, j=1 i=1

Source: [394]. Related: Fact 1.12.8. Fact 2.11.11. Let x1 , . . . , xn ∈ [−1, ∞), and let α1 , . . . , αn be real numbers. Then, the following

statements hold:

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CHAPTER 2

i) Assume that either α1 , . . . , αn ∈ (−∞, 0] or α1 , . . . , αn ∈ [1, ∞), and assume that either x1 , . . . , xn ∈ [−1, 0] or x1 , . . . , xn ∈ [0, ∞). Then, n n ∑ ∏ 1+ αi x i ≤ (1 + xi )αi . i=1

ii) Assume that α1 , . . . , αn ∈ [0, 1] and n ∏

i=1

∑n i=1

αi ≤ 1. Then,

(1 + xi )αi ≤ 1 +

i=1

n ∑

αi xi .

i=1

Source: [2443]. Remark: This is a multivariable extension of Bernoulli’s inequality given by Fact 2.1.21. Related: Fact 2.11.12. Fact 2.11.12. Let x1 , . . . , xn be nonnegative numbers, and assume that either x1 , . . . , xn ∈ [0, 1]

or x1 , . . . , xn ∈ [1, ∞). Then,

1+

n ∑

xi ≤ n +

n ∏

i=1

xi .

i=1

Source: [112, p. 37], [806, p. 27], and Fact 2.11.11. △ Fact 2.11.13. Let x1 , . . . , xn ∈ [0, 1], and define xn+1 = x1 . Then, n ∑

xi ≤

⌊n⌋ 2

i=1

+

n ∑

xi xi+1 .

i=1

Source: [112, p. 37]. Fact 2.11.14. Let x1 , . . . , xn be nonnegative numbers. Then, n ∑ i=1

( ) ∑ n n ixi ≤ + xii . 2 i=1

Source: [806, p. 197]. Fact 2.11.15. Let x1 , . . . , xn be nonnegative numbers, and let k ≥ 1. Then, n ∑

xik

 n k n ∑ ∑  xik. xi  ≤ nk−1 ≤  i=1

i=1

i=1

Equality holds in the second inequality if and only if x1 = · · · = xn . Remark: The case n = 4, k = 3 is given by (w + x + y + z)3 ≤ 16(w3 + x3 + y3 + z3 ) of Fact 2.4.23. Fact 2.11.16. Let n ≥ 2, let x1 , . . . , xn be nonnegative numbers, let k be a positive integer such △ △ that n − 2 ≤ k ≤ n, and define xn+1 = x1 , . . . , xn+k−1 = xk−1 . Then,  n k n i+k−1 ∑ ∏ ∑  k−1 n x j ≤  xi  . i=1

j=i

i=1

If k = n = 2, then 4x1 x2 ≤ (x1 + x2 )2. If n = 3 and k = 2, 3, then 3(x1 x2 + x2 x3 + x3 x1 ) ≤ (x1 + x2 + x3 )2 ,

27x1 x2 x3 ≤ (x1 + x2 + x3 )3 .

If n = 4 and k = 2, 3, 4, then 4(x1 x2 + x2 x3 + x3 x4 + x4 x1 ) ≤ (x1 + x2 + x3 + x4 )2 , 16(x1 x2 x3 + x2 x3 x4 + x3 x4 x1 + x4 x1 x2 ) ≤ (x1 + x2 + x3 + x4 )3 , 264x1 x2 x3 x4 ≤ (x1 + x2 + x3 + x4 )4 .

191

EQUALITIES AND INEQUALITIES

If k = n, then nn

n ∏

 n n ∑  xi ≤  xi  .

i=i

i=1

Source: The case n = 4 and k = 2 is given in [1371, p. 144]. The case n = 6 and k = 4 is given in [108, pp. 462, 463]. See Fact 2.11.61. Related: The case n = 5 and k = 3, 4, 5 is given by Fact 2.5.5. The case n = 6 and k = 4, 5, 6 is given by Fact 2.6.17. Conjecture: Let n ≥ 4, let x1 , . . . , xn △ △ be nonnegative numbers, let 1 ≤ k ≤ n − 3, and define xn+1 = x1 , . . . , xn+k−1 = xk−1 . Then,  n k n i+k−1 ∑ ∏ ∑  k k x j ≤  xi  . j=i

i=1

If n = 6 and k = 3, then 27

6 ∑

i=1

xi xi+1 xi+2

 6 3 ∑  xi  . ≤  i=1

i=1

The case n ≥ 4 and k = 2 is given in [1371, p. 144]. Fact 2.11.17. Let x1 , . . . , xn be nonnegative numbers. Then,  n 2 n ∑ ∑   xi2. xi ≤ n i=1

i=1

Equality holds if and only if x1 = · · · = xn . Remark: This result is equivalent to i) of Fact 11.9.23 with m = 1. Fact 2.11.18. Let x1 , . . . , xn be real numbers. Then,    n 2 n ∑   ∑  2 xi  . xi ≤ (n − 1) 2x1 x2 +  i=1

i=1

Source: [1566, p. 12]. △ △ Fact 2.11.19. Let x1 , . . . , xn be real numbers, let 1 ≤ k ≤ n − 1, and define xn+1 = x1 , . . . , xn+k =

xk . Then,

n ∑

xi xi+k ≤

i=1

In particular, let w, x, y, z be real numbers. Then, wx + xy + yz + zw 2(wy + xz)

n ∑

xi2 .

i=1

} ≤ w2 + x2 + y2 + z2 .

Source: [176]. Related: Fact 2.4.16. Fact 2.11.20. Let x0 , x1 , . . . , xn+1 be real numbers. Then, the following statements hold:

i) If x1 = 0, then ( 4 sin2

)∑ n n−1 n ( ∑ π π )∑ 2 xi2 ≤ (xi+1 − xi )2 ≤ 4 cos2 x . 2(2n − 1) i=2 2n − 1 i=2 i i=1

ii) If x0 = xn+1 = 0, then

( 4 sin2

)∑ n n ∑ π xi2 ≤ (xi+1 − xi )2 . 2(n + 1) i=1 i=0

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CHAPTER 2

iii) If xn+1 = x1 and

∑n i=1

xi = 0, then n n ( π) ∑ 2 ∑ 4 sin2 xi ≤ (xi+1 − xi )2 . n i=1 i=1

iv) If xn = 0 and xn+1 = x1 , then

v) If

∑n i=1

n n ( π )∑ 2 ∑ xi ≤ (xi+1 − xi )2 . 4 sin2 2n i=1 i=1

xi = 0, then

n n−1 ( π )∑ 2 ∑ xi ≤ (xi+1 − xi )2 . 4 sin2 2n i=1 i=1

vi) If n is even and xn+1 = x1 , then ) n n ( )∑ ( π ) ( 2π ∑ π 2 π 2 4 sin sin − sin (xn/2 − xn )2 ≤ (xi+1 − xi )2 . xi + n sin n i=1 n n n i=1 ∑n vii) If i=1 xi = 0, then n n−1 ( ( π) ( π ∑ π )∑ 2 π) 4 sin2 xi + 2n sin sin − sin (x1 + xn )2 ≤ (xi+1 − xi )2 . 2n i=1 n n 2n i=1

viii) If x0 = xn+1 = 0, then )∑ ( )∑ ( n n n ∑ π π xi2 ≤ (xi+1 − xi )2 ≤ 4 cos2 xi2 . 4 sin2 2(n + 1) i=1 2(n + 1) i=0 i=1 ix) If a ≥ 0 and x0 = xn+1 = 0, then )∑ ( n n ∑ π (i + a)2 xi2 ≤ (i + a)(i + a + 1)(xi+1 − xi )2 4 sin2 2(n + 1) i=1 i=0 ( )∑ n π ≤ 4 cos2 (i + a)2 xi2 . 2(n + 1) i=1 ∑ x) If x0 = x1 , xn+1 = xn , and ni=1 xi = 0, then n n−1 ( π )∑ 2 ∑ 16 sin4 xi ≤ (xi+2 − 2xi+1 + xi )2 . 2n i=1 i=0

xi) If x0 = x1 and xn+1 = xn , then n n−1 ( ∑ π )∑ 2 x . (xi+2 − 2xi+1 + xi )2 ≤ 16 cos4 2n i=1 i i=0

Source: [2045]. Remark: These are inequalities of Wirtinger’s type. Fact 2.11.21. Let x1 , . . . , xn be nonnegative numbers, and let k be a positive integer. Then, n ∑ i=1

Source: [1757, pp. 257, 258].

xik

 n  n  n ∑ ∑  ∑ k−1     ≤  xi   xi  ≤ n xik. i=1

i=1

i=1

193

EQUALITIES AND INEQUALITIES

Fact 2.11.22. Let x1 , . . . , xn be nonnegative numbers. Then, n ∑ i=1

xin

  n  n n n n ∑ ∏ ∑ ∑  ∑ n−1  n     xi  ≤ (n − 1) xi + n ≤  xi   xi ≤ n xin. i=1

i=1

i=1

i=1

i=1

Source: [806, pp. 35, 36]. Remark: This result interpolates the right-hand inequality in Fact 2.11.21 in the case k = n. Remark: The second inequality is Suranyi’s inequality. Fact 2.11.23. Let x1 , . . . , xn be nonnegative numbers, and let α ∈ [0, 1]. Then,

  n  n   n  ∑ 1−α  ∑ √ 2 ∑ α xi  . xi   xi  ≤   i=1

i=1

i=1

Equality holds if and only if either α = 1/2 or x1 = · · · = xn . Source: [2991, p. 250]. Fact 2.11.24. Let x1 , . . . , xn be nonnegative numbers. Then,   n 3  n   n ∑ 3 2  ∑ 3 2 ∑ 2 xi  . xi  ≤ n  xi  ≤   i=1

i=1

i=1

Source: Set p = 2 and q = 3 in Fact 2.11.90 and square all terms. Fact 2.11.25. Let x1 , . . . , xn be positive numbers. Then,

 n  n   n  n   n   n  n  ∑  ∑ 1  ∑  ∑ 1  n ∑ 2  ∑ 3  ∑ 1  2         xi   x  x  x ≤ , ≤  , x   i=1 i   i=1 xi2  2  i=1 i   i=1 i   i=1 xi  i=1 i i=1   n   n  n  n  ∑ 1  ∑  ∑ 3  ∑ 5  . xi   xi  ≤  xi    x i=1 i i=1 i=1 i=1

Source: [1371, p. 150] and [1809, 2750]. Fact 2.11.26. Let x1 , . . . , xn be positive numbers. Then,

  n   n  n   n  n  n   n n n ∏ ∑ n  ∏ ∑  ∑ n−1   ∏  ∑  ∑ 1  ∑ n  ≤  xi , xi  +n xi  ≤ (n−1)  xi   xi   xi ,  xi   xi  +n(n−1)  x i=1 i=1 i=1 i i=1 i=1 i=1 i=1 i=1 i=1   n   n     n  n   n n n n ∑ ∏  ∑ 1  ∑   ∑  ∑ n  ∏  ∏ 1 n+1              xi  − n  xi , 2 ≤  xi  ≤ (n − 1) xi  − xi   xi + .  − n + x x i=1 i i=1 i=1 i=1 i=1 i=1 i=1 i=1 i Source: [748, pp. 219, 220]. Fact 2.11.27. Let x1 , . . . , xn be positive numbers, assume that x1 ≤ 1, and assume that, for all

i ∈ {1, . . . , n − 1}, xi ≤ xi+1 ≤ xi + 1. Then,

n ∑

xi3

 n 2 ∑  ≤  xi  . i=1

i=1

Source: [993, p. 183]. Remark: Equality holds in the case where xi = i, as shown in Fact 1.12.1. Fact 2.11.28. Let x1 , . . . , xn be nonnegative numbers. Then,

∑

xi x j (xi2 1≤i< j≤n Source: [752, p. 215] and [1938, p. 106].

+

x2j )

 n 4 1 ∑  ≤  xi  . 8 i=1 

194

CHAPTER 2

Fact 2.11.29. Let x1 , . . . , xn be positive numbers. Then,

 n  n n ∑ ∏  ∑   xi  xi ≤ xin+1 . i=1

i=1

i=1

Source: [993, p. 170]. △ Fact 2.11.30. Let x1 , . . . , xn be real numbers, and define xn+1 = x1 . Then,

0≤

n ∑ (xi − xi+1 )(3xi + xi+1 )3 . i=1

√

Now, let r ∈ [0, ( 3 − 1)/2]. Then, n ∑

(r + 1)

xi3 xi+1 ≤

n ∑ 3 (xi4 + rxi xi+1 ).

i=1

i=1

Source: [748, p. 110]. △ Fact 2.11.31. Let x1 , . . . , xn be nonnegative numbers, and define xn+1 = x1 . Then,

∑ 1 3 3∑ 3 xi xi+1 ≤ (xi4 + xi xi+1 ). 2 i=1 2 i=1 n

n

√3 Now, let r ∈ [1/( 4 − 1), ∞). Then,

0≤

n ∑ (xi − xi+1 )(rxi + xi+1 )3 . i=1

Source: [748, p. 110]. △ Fact 2.11.32. Let x1 , . . . , xn be positive numbers, and define xn+1 = x1 . Then,

1 + (n − 2) min {x1x2 , . . . , xnxn+1 }
0 satisfy )n−1 (∏ n i=1 xi , p(λ) = ∏ (xi − x j )2 where the product in the denominator is taken over all i, j ∈ {1, . . . , n} such that i < j. Then, v v t n t n n ∏ ∏ √ 1∑ n n xi . xi < q(λ) n xi ≤ n i=1 i=1 i=1 Source: [116, pp. 419–421]. Remark: This is Siegel’s inequality. △ 1

Fact 2.11.84. Let x1 , . . . , xn be nonnegative numbers, define A = △

n

∑n i=1

△

xi and G =

(∏

n i=1

xi

)1/n

,

assume that ε = (A−G)/A < 1, and let r1 and r2 be the unique solutions of xe = (1−ε) satisfying 0 < r1 ≤ 1 ≤ r2 . Then, for all i ∈ {1, . . . , n}, Ar1 ≤ xi ≤ Ar2 . Source: [2527, p. 35]. Fact 2.11.85. Let x1 , . . . , xn be positive numbers. Then,  n 1/n  n 1/n n ∏  ∏  1∑ 1 + log  xi  ≤  xi  ≤ xi . n i=1 i=1 i=1 Source: For all x > 0, 1 + log x ≤ x.

1−x

n

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CHAPTER 2

Fact 2.11.86. Let x1 , . . . , xn be positive numbers, let r be a real number, and define

 n 1/n  ∏       r = 0, xi  ,      △  i=1 Mr =   n 1/r      1 ∑ r     xi  , r , 0.   n  i=1

Then, M0 = limr→0 Mr . Now, let p and q be real numbers such that p ≤ q. Then, M p ≤ Mq . Therefore, lim Mr = min {x1 , . . . , xn } ≤ M−1 ≤ M0 ≤ M1 ≤ lim Mr = max {x1 , . . . , xn }.

r→−∞

r→∞

Finally, p < q and at least two of the numbers x1 , . . . , xn are distinct if and only if M p < Mq . Source: [603, p. 210] and [1952, p. 105]. To verify the limit for M0 , take the log and use l’Hˆopital’s rule given by Fact 12.17.13. Remark: This is the power-mean inequality. M0 ≤ M1 is the arithmetic-mean–geometric-mean inequality given by Fact 2.11.81. M−1 ≤ M0 is the harmonicmean–geometric-mean inequality. Related: Fact 2.2.58, Fact 2.11.87, Fact 2.11.89, Fact 2.11.135, and Fact 10.13.8. Fact 2.11.87. Let x1 , . . . , xn be positive numbers, let α1 , . . . , αn be nonnegative numbers, and ∑ assume that ni=1 αi = 1. Then, n n ∏ ∑ 1 αi ≤ x ≤ αi x i . i n ∑ αi i=1 i=1 x i=1 i Now, let r be a real number, define

 n   ∏ α 1/n    i   xi  , r = 0,      △  i=1 Mr =    n    ∑ r 1/r     αi xi  , r , 0,     i=1

and let p and q be real numbers such that p ≤ q. Then, M p ≤ Mq . If, in addition, α1 , . . . , αn are positive, then lim Mr = min {x1 , . . . , xn } ≤ lim Mr = M0 ≤ lim Mr = max {x1 , . . . , xn }.

r→−∞

r→0

r→∞

Equality holds if and only if x1 = x2 = · · · = xn . Source: Since f (x) = −log x is convex, it follows that n n n ∏ ∑ ∑ log xiαi = αi log xi ≤ log αi xi . i=1

i=1

i=1

∏ △ ∑ To prove the last statement, define f : [0, ∞)n 7→ [0, ∞) by f (µ1 , . . . , µn ) = ni=1 αi µi − ni=1 µαi i. Note that f (µ, . . . , µ) = 0 for all µ ≥ 0. If x1 , . . . , xn minimizes f, then ∂f/∂µi (x1 , . . . , xn ) = 0 for all i ∈ {1, . . . , n}, which implies that x1 = x2 = · · · = xn . Source: [1074] and [2128, p. 11]. Remark: This is the weighted arithmetic-mean–geometric-mean inequality. Setting α1 = · · · = αn = 1/n yields Fact 2.11.81. Remark: The second inequality generalizes Young’s inequality. See Fact 2.2.50 and Fact 2.2.53. Fact 2.11.88. Let x1 , . . . , xn be positive numbers, let α1 , . . . , αn ∈ [1, ∞), and let p and q be

205

EQUALITIES AND INEQUALITIES

real numbers. If either p < q < 0 or 0 < p < q, then  n 1/q  n 1/p  ∑ ∑  q p  αi x  ≤  αi x  . i

i

i=1

i=1

If p < 0 < q, then

1/q 1/p  n  n  ∑  ∑ q p  αi xi  ≤  αi xi  . i=1

i=1

Source: [1621]. Fact 2.11.89. Let x1 , . . . , xn be positive numbers, and let p and q be nonzero real numbers such

that p ≤ q. Then,

 n  1/q  n   1 ∑ p 1/p  1 ∑ q xi  ≤  xi  .  n i=1 n i=1

Furthermore, min {x1 , . . . , xn } ≤

1 x1

n + ··· +

1 xn

≤

√n

x1 + · · · + xn x1 · · · xn ≤ ≤ n

v t

1∑ 2 x ≤ max {x1 , . . . , xn }. n i=1 i n

Equality holds in each inequality if and only if x1 = x2 = · · · = xn . Source: [2060, pp. 28–30]. Remark: The lower bound for the geometric mean is the harmonic mean, while the second and third terms are the harmonic-mean–geometric-mean inequality. See Fact 2.11.135. Remark: The upper bound for the arithmetic mean is the quadratic mean. See [1283]. Related: Fact 2.2.58 and Fact 2.11.86. Fact 2.11.90. Let x1 , . . . , xn be nonnegative numbers, and let p, q ∈ [1, ∞), where p ≤ q. Then,   n 1/p  n   n ∑ q 1/q  ∑ q 1/q ∑ p 1/p−1/q       x  . x  ≤ n x  ≤   Equivalently,

i

i

i

i=1

i=1

i=1 n ∑

xiq

  n n ∑ ∑ p q/p xi  ≤ nq/p−1 xiq . ≤  i=1

i=1

i=1

Furthermore, the first inequality is strict if and only if p < q and at least two of the numbers x1 , . . . , xn are positive. Source: Fact 11.8.7. Remark: This is the power-sum inequality. See [603, p. 213]. This result implies that the H¨older norm is a monotonic function of the exponent. Remark: Setting p = 1 and q = k yields Fact 2.11.15. Remark: The power-mean inequality is given by Fact 2.11.86. Related: Fact 11.8.7. Fact 2.11.91. Let x1 , . . . , xn be nonnegative numbers, and let p, q ∈ (0, 1], where p ≤ q. Then,  n   n 1/p ∑ q 1/q ∑  p x  ≤  x  .  i

i=1

i

i=1

Furthermore, this inequality is strict if and only if p < q and at least two of the numbers x1 , . . . , xn are positive. Remark: This is the reverse power-sum inequality. Related: Fact 2.11.90 and Fact 11.8.21. Fact 2.11.92. Let 0 ≤ a < b, and let x1 , . . . , xn ∈ [a, b]. Then,  n 1/n  n 1/n  n 1/n n ∏   ∏  √ 2 ∏ n−1 √ 1 ∑ 1∑     xi ≤  xi  + ( b − a) ≤  xi  + |xi − x j |. xi  ≤ n i=1 n n 1≤i< j≤n i=1 i=1 i=1

206

CHAPTER 2

Source: [748, p. 373] and [993, p. 186]. Fact 2.11.93. Let x1 , . . . , xn be nonnegative numbers. Then,

(n − 1)

v t n ∏ n

i=1

v t

xi +

1∑ 2 ∑ x ≤ xi . n i=1 i i=1 n

n

Source: [748, p. 382]. Fact 2.11.94. Let 0 < a < b, and let x1 , . . . , xn ∈ [a, b]. Then,

√ 2−2/n  n 1/n  √ n ∏  b  1∑  1  a  xi  . + xi ≤    n i=1 2 b a i=1

Source: [752, p. 217]. Fact 2.11.95. Let 0 < a < b, and let x1 , . . . , xn ∈ [a, b]. Then,

 n 1/n  n 1/n n ∏  ∏  1∑   xi  ≤ xi  , xi ≤ S(b/a)  n i=1 i=1 i=1

where S is Specht’s ratio given by Fact 12.17.5. Remark: The right-hand inequality is a reverse arithmetic-mean–geometric mean inequality. See [1087, 1092, 2501, 2938]. Related: Fact 2.2.54, Fact 2.2.66, and Fact 15.15.23. Credit: W. Specht. Fact 2.11.96. Let x1 , . . . , xn be positive numbers, and let k and l satisfy 1 ≤ k ≤ l ≤ n. Then, 1/k 1/l ( )−1  n 1/n ( )−1 n k l ∑ ∏ ∑ ∏   n   n ∏  1∑      xi . xi j  ≤ xi j  ≤  xi  ≤  n i=1 k i 0,

    x − 21 x2    x < log(x + 1) <    1 2 1 3 1 4  x − 1 x2 + 1 x3 . x − 2x + 3x − 4x  2 3

Source: [806, p. 136] and [1567, p. 55]. Fact 2.15.5. Let x ∈ (0, ∞). Then,

|log x| ≤

|x − 1| √ , x

log(x + 1) ≤ √

x

. x+1 Equality in the first inequality holds if and only if x = 1. Source: [2294, p. 309]. Fact 2.15.6.

lim

x→1

Furthermore, for all x > 0, x2 where

log x 1 = . x2 − 1 2

1 log x 1 , ≤ 2 ≤ + 1 x − 1 2x

log x △ 1 = for x = 1. Finally, for all x ∈ (0, 1), x2 − 1 2 x2 − 1 x2 − 1 , < log x < 2 2x x +1

whereas, for all x > 1,

x2 − 1 x2 − 1 . < log x < 2 2x x +1

Fact 2.15.7. If x ∈ (0, 1], then

x − 1 x2 − 1 x − 1 (x − 1)(1 + x1/3 ) 2(x − 1) x2 − 1 ≤ ≤ √ ≤ ≤ log x ≤ ≤ 2 ≤ x − 1. 1/3 x 2x x+1 x +1 x+x x If x ≥ 1, then x − 1 x2 − 1 2(x − 1) (x − 1)(1 + x1/3 ) x − 1 x2 − 1 ≤ √ ≤ ≤ 2 ≤ ≤ log x ≤ ≤ x − 1. x x+1 2x x +1 x + x1/3 x If x > 0, then

2|x − 1| |x − 1|(1 + x1/3 ) |x − 1| ≤ |log x| ≤ ≤ √ . x+1 x + x1/3 x

228

CHAPTER 2

Equality holds in each inequality if and only if x = 1. Source: [604, p. 8], [946], and [1300]. Fact 2.15.8. Let x ∈ (−1, 1). Then, |x|(|x| + 1) |x| ≤ | log(x + 1)| ≤ . |x| + 1 |x + 1| Source: [2294, p. 309]. Fact 2.15.9. Let x ≥ 1. Then,

(x3 − 1)(x + 1) 3(x2 − 1) ≤ log x ≤ . x2 + 4x + 1 3x(x2 + 1) Source: [374]. Fact 2.15.10. Let x ≥ 1. Then,

√

log x ≤ (x − 1)

3

√   2x2 + 5x + 2  x − 1  log x ≤ 1 +  . 2(x + 1)  x

2 , x(x + 1)

Source: [374]. Fact 2.15.11. Let x ≥ 1. Then,

(x − 1)2 +

x2 − 1 √ 4 ≤ 2(x + 1). log x

Source: [375]. Fact 2.15.12. Let x > 1. Then,

( 1/2 )2 ( 1/3 )3 x−1 x + x1/4 + 1 x +1 < < . log x 3 2

Source: [1530]. Fact 2.15.13. Let x ∈ (0, 1). Then,

0 < log

1+x 2x3 − 2x < . 1−x 3(1 − x2 )

Source: [2294, p. 414]. Fact 2.15.14. Let x ∈ (0, ∞). Then,

) ( 1 1 1 < + , log 1 + x x + 1 x(2x + 1)

( ) ( ) 1 1 1 1 1< x+ log 1 + 0. Then,

ex

x2 < log(x + 1). −1

Source: [2294, p. 308]. Fact 2.15.16. Let x be a positive number. Then,

x−1≤

3(x − 1)2 + x − 1 ≤ x log x. 2(x + 2)

Equality holds in each inequality if and only if x = 1. Source: [523, p. 63].

229

EQUALITIES AND INEQUALITIES

Fact 2.15.17. Let x ∈ [0, 1). Then,

e−1 x ≤ . e − ex 1 − x

(e − 1) log Source: [2294, p. 176]. Fact 2.15.18. Let α > 1. If x ∈ [0, 1), then

log

( )α 1 1 ≤ log . 1 − xα 1−x

If x ∈ [0, ∞), then log

1 ≤ xα . 1 − (1 − e−x )α

Source: [2294, p. 175]. Fact 2.15.19. Let n ≥ 1. Then,

∑ 1 1 1 1 n(n + 1)(n + 2) < ≤ n + n(n + 1)(n + 2). 2 3 4 3 i=1 log (1 + 1/i) n

Source: [386]. Fact 2.15.20. Let x ∈ (−1, ∞) and y ∈ R. Then,

xy + x + y + 1 ≤ (x + 1) log(x + 1) + ey . Equality holds if and only if y = 1 + log x. Source: Fact 12.13.16. Related: Fact 2.2.39. Fact 2.15.21. Let x and y be positive numbers. Then, (x + y) log[ 21 (x + y)] ≤ x log x + y log y. Source: Use the fact that f (x) = x log x is convex on (0, ∞). See [1567, p. 62]. Fact 2.15.22. Let x, y ∈ (0, 1]. Then, |x log x − y log y| ≤ |x − y|1−1/e . Source: [2550]. Fact 2.15.23. Let a and b be positive numbers, assume that a < b, and let x ≥ 0. Then, 1 2 (a

1 x b(x + a) x ≤ log ≤ √ √ . 1 b − a a(x + b) + b)[x + 2 (a + b)] ab(x + ab)

Source: [1965]. ∑ Fact 2.15.24. Let x1 , . . . , xn be positive numbers, and assume that ni=1 xi = 1. Then, the fol-

lowing statements hold: ∑ i) 0 ≤ ni=1 xi log x1i ≤ log n. ∑ ii) ni=1 xi log x1i = 0 if and only if n = 1. iii)  n 2  n  n n 1 ∑   ∏ x  3 ∑ (xi − n )2 1 ∑ 1 1  i   xi  . ≤ log n − xi log = log n  |xi − n | ≤ 2  i=1 2 i=1 xi + 2n xi i=1 i=1 ∑ iv) ni=1 xi log x1i = log n if and only if x1 = · · · = xn = 1/n. ∑ Source: [2061, Chapter XXIII]. iii) is given in [523, p. 63]. Remark: ni=1 xi log x1i is the entropy. ∑n Fact 2.15.25. Let x1 , . . . , xn be positive numbers, and assume that i=1 xi = 1. Then, log ∑n

1

2 i=1 xi

≤

n ∑ i=1

xi log

1 ≤ log n. xi

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CHAPTER 2

Furthermore, each inequality is an equality if and only if x1 = · · · = xn = 1/n. Source: Theorem 8 of [1991] and [2991, p. 348]. ∑ Fact 2.15.26. Let x1 , . . . , xn be positive numbers, and assume that ni=1 xi = 1. Then, 1 2   n 2  n n  ∑  ∑   1∑ 1   2 (n − n) maxi, j∈{1,...,n} (xi − x j )  2 2   xi  − 1 ≤  (xi − x j ) = n 0 ≤ log n − xi log ≤ (∑ )1/2 [(∑ ) ]1/2  n 1  2 xi 2 i, j=1  ni=1 xi3 . i=1 i=1 i=1 xi − n Equality holds in the first and second inequalities if and only if x1 = · · · = xn = 1/n. Source: ∑ [944, 945]. Remark: Fact 2.11.135 implies that n2 ≤ ni=1 x1i . ∑ △ Fact 2.15.27. Let x1 , . . . , xn be positive numbers, assume that ni=1 xi = 1, and define a = △ mini∈{1,...,n} xi and b = maxi∈{1,...,n} xi . Then, ⌊ ⌋ ⌊ ⌋ n ∑ 1 n2 b 1 n2 (b − a)2 1 . (b − a) log ≤ 0 ≤ log n − xi log ≤ √ xi n 4 a n 4 ab i=1 Equality holds in each inequality if and only if x1 = · · · = xn = 1/n. Source: [946]. Related: Fact 2.12.46 and Fact 3.25.7. Fact 2.15.28. Let x1 , . . . , xn , y1 , . . . , yn be positive numbers. Then, the following statements hold: i)  n  ∑n n ∑ ∑  xi j=1 x j   xi  log ∑n ≤ xi log . y y i j=1 j i=1 i=1 ∑n ii) Assume that i=1 xi = 1. Then, n ∑

iii) Assume that

∑n

i=1 yi


0, (n − i)! i=0 n ∑ 4n−i (i + 1) sin (i + 1)x > 0, (n − i)! i=0

If x ∈ [0, π], then

i=1

) n ( ∑ n−i+k k

i=1

In particular,

2

2

sin ix ≥ 0.

n ∑ (n + 1 − i) sin ix ≥ 0,

n ∑ (n + 1 − i)(n + 2 − i) sin ix ≥ 0.

i=1

i=1

If x ∈ [0, π], then ∑ sin ix 9 ≤ , 200 i=1 i + 1 n

−

1 ∑ cos ix ≤ , 2 i=1 i + 1 n

−

1 ∑ sin ix + cos ix ≤ . 2 i=1 i+1 n

−

If n ≥ 2 and x ∈ [0, π], then 5 ∑ cos ix ≤ , 6 i=1 i n

−

If x ∈ [0, π] and p ∈ [0, 9/2], then

41 ∑ cos ix ≤ . 96 i=1 i + 1 n

−

∑ cos ix 1 ≤ . 1 + p i=1 i + p n

− If x ∈ [0, 2π], then n ∑ i=0

sin (i + 12 )x =

2 1 1 − cos (n + 1)x sin 2 (n + 1)x = ≥ 0, 2 sin 12 x sin 12 x

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CHAPTER 2

1+2

n ( ∑ i=1 n ∑

1+2

i=1

1−

 2 1 1  sin 2 (n + 1)x  i )   ≥ 0, cos ix = n+1 n+1 sin 12 x

(n!)2 (n!)2 cos ix = (2 cos 12 x)2n ≥ 0. (n − i)!(n + i)! (2n)!

Source: [63, 64, 65], [116, pp. 371, 372, 382], [178, pp. 1–6], and [1691]. Related: Fact 2.16.9.

2.18 Facts on Equalities and Inequalities for Inverse Trigonometric Functions Fact 2.18.1. The following statements hold:

i) asin : [−1, 1] 7→ [− π2 , π2 ] is given by

√ π + ȷ log(x + 1 − x2 ȷ). 2 asin : (−∞, −1] ∪ [1, ∞) 7→ [− π2 , π2 ] + IA is given by (π ) √ √ π − ȷ log( x2 − 1 + |x|) . asin x = + ȷ log(x − x2 − 1) = (sign x) 2 2 asin : C 7→ [− π2 , π2 ] + IA is given by √ √ π asin z = + ȷ log(z + 1 − z2 ȷ) = − ȷ log( 1 − z2 + z ȷ). 2 acos : [−1, 1] 7→ [0, π] is given by √ √ π acos x = + ȷ log( 1 − x2 + x ȷ)] = − ȷ log(x + 1 − x2 ȷ). 2 acos : (−∞, −1] ∪ [1, ∞) 7→ C is given by √ π acos x = + ȷ log(x ȷ + x2 − 1 ȷ). 2 acos : C 7→ C is given by √ √ π acos z = + ȷ log( 1 − z2 + z ȷ)] = − ȷ log(z + 1 − z2 ȷ). 2 π π atan : R 7→ (− 2 , 2 ) is given by ȷ atan x = [log(1 − x ȷ) − log(1 + x ȷ)]. 2 atan : C\{− ȷ, ȷ} 7→ (− π2 , π2 ) + IA is given by ȷ atan z = [log(1 − z ȷ) − log(1 + z ȷ)]. 2 acsc : (−∞, −1] ∪ [1, ∞) 7→ [− π2 , 0) ∪ (0, π2 ] is given by √   1 1   acsc x = − ȷ log  1 − 2 + ȷ . x x asin x =

ii)

iii)

iv)

v)

vi)

vii)

viii)

ix)

x) acsc : C\{0} 7→ [− π2 , π2 ]\{0} + IA is given by

√   1 1   acsc z = − ȷ log  1 − 2 + ȷ . z z

255

EQUALITIES AND INEQUALITIES

xi) asec : (−∞, −1] ∪ [1, ∞) 7→ [0, π2 ) ∪ ( π2 , π] is given by √    1 1 π asec x = + ȷ log  1 − 2 + ȷ . 2 x x xii) asec : C\{0} 7→ [0, π]\{ π2 } + IA is given by

√  1 1  π  asec z = + ȷ log  1 − 2 + ȷ . 2 z z

xiii) acot : R 7→ (− π2 , 0) ∪ (0, π2 ] is given by     π2 , acot x =    ȷ [log(−1 − x ȷ) − log(−1 + x ȷ) + log x ȷ − log −x ȷ], 2

x = 0, x , 0.

xiv) acot : C\{− ȷ, ȷ} 7→ ((− π2 , 0) ∪ (0, π2 ]) + IA is given by     π2 , acot z =    ȷ [log(−1 − z ȷ) − log(−1 + z ȷ) + log z ȷ − log −z ȷ], 2

z = 0, z , 0.

Source: To prove i), note that

sin[ π2 + ȷ log(x +

√ √ √ 1 − x2 ȷ)] = cos[ ȷ log(x + 1 − x2 ȷ)] = cosh log(x + 1 − x2 ȷ) ) √ 1 ( log(x+ √1−x2 ȷ) 2 = e + e− log(x+ 1−x ȷ) 2  √  1 1  2 =  x + 1 − x ȷ + √  = x. 2 x + 1 − x2 ȷ

Remark: The logarithm and square root are the principal inverses. Remark: The range of acot and its value at zero are consistent with Matlab and Mathematica. The expression for acot for nonzero z ∈ ȷ(−1, 1) is consistent with Mathematica but differs from Matlab by the sign of the real part. The expression for atan for nonzero z ∈ ȷ[(−∞, −1) ∪ (1, ∞)] is consistent with Mathematica but differs from Matlab by the sign of the real part. Fact 2.18.2. Let z be a complex number. Then, the following statements hold: i) asin(−z) = − asin z, acos(−z) = π − acos z. ii) acsc(−z) = − acsc z, asec(−z) = π − asec z. iii) asin z + acos z = π2 , acsc z + asec z = π2 .

iv) If z , 0, then acsc z = asin 1z and asec z = acos 1z . v) If z < {− ȷ, ȷ}, then atan(−z) = − atan z. vi) If z < {− ȷ, ȷ, 0}, then acot(−z) = − acot z and acot z = atan 1z . If Re z , 0, then the following statement holds: vii) atan z + atan 1z = atan z + acot z = acot z + acot 1z = (sign Re z) π2 . Fact 2.18.3. Let x be a real number. Then, the following statements hold: i) asin x + acos x = π2 . ii) acos2 x − asin2 x + π asin x = π4 . iii) atan 0 = 0 and acot 0 = π2 . iv) If x , 0, then atan x + acot x = (sign x) π2 . 2

v) If |x| ≥ 1, then acsc x = asin 1x and asec x = acos 1x .

256

CHAPTER 2

vi) If x , 0, then acot x = atan 1x . vii) If x ∈ [−1, 0) ∪ (0, 1], then asin x + asec 1x = π2 . viii) If x , 0, then atan x + atan 1x = (sign x) π2 . √ x ix) If x ∈ [0, 1], then asin x = acos 1 − x2 = atan √ . 1 − x2 √ x) If x ∈ [0, 1], then acos x = asin 1 − x2 = 12 acos(2x2 − 1). π 1−x 1 xi) If |x| < 1, then atan √ + asin x = . 2 2 4 1−x √ √ xii) If x ∈ [−1, 0], then asin x + acos 1 − x2 = 0 and acos x + asin 1 − x2 = π. √ √ 1 − x2 1 − x2 and acos x = atan . xiii) If x ∈ (0, 1], then asin x = acot x √ x √ 1 − x2 1 − x2 xiv) If x ∈ [−1, 0), then asin x + π = acot and acos x = atan + π. x x 2 xv) If x ∈ [−1, 0], then acos(2x − 1) + 2 acos x = 2π. √ 1−x x . = 2 atan xvi) If |x| < 1, then acos x = acot √ 1+x 1 − x2 x 1 xvii) sin atan x = √ and cos atan x = √ . 2 2 x +1 x +1 x x 1 π xviii) atan x = asin √ = (sign x) acos √ = − acos √ . 2 2 2 2 x +1 x +1 x +1 sign x |x| xix) sin acot x = √ and cos acot x = √ . x2 + 1 x2 + 1 1 |x| x xx) acot x = (sign x) asin √ = acos √ = acos √ − truth(x < 0)π. 2 2 2 x +1 x +1 x +1 x . xxi) atan x = 2 atan √ 1 + 1 + x2 1 xxii) If x , 0, then atan 1x = 2 atan 2x − atan 4x31+3x .

1 1 x2 − 1 π 1 and acot x = asin √ . asin 2 + = acos √ 2 x +1 4 1 + x2 1 + x2 1 1 If x ≤ 0, then atan x + acos √ = 0 and acot x + asin √ = π. 2 1+x 1 + x2 2x x . and 2 atan x = asin 2 If |x| ≤ 1, then asin x = 2 atan √ 2 x +1 1+ 1−x 2x If |x| ≥ 1, then 2 atan x + asin 2 = (sign x)π. x +1 2x . If |x| < 1, then 2 atan x = atan 1 − x2 2x If |x| > 1, then 2 atan x = atan + (sign x)π. 2 1 − x√ 1 − x2 If x ∈ (−1, 1], then acos x = 2 atan . 1+x √ √ atan x = 2 atan(x − x2 + 1) + π2 and 2 atan(x + x2 + 1) = atan x + π2 . 1−x 3π If x < −1, then atan x + atan =− . 1+x 4

xxiii) If x ≥ 0, then atan x = xxiv) xxv) xxvi) xxvii) xxviii) xxix) xxx) xxxi)

257

EQUALITIES AND INEQUALITIES

xxxii) If x > −1, then atan x + atan

1−x π = . 1+x 4

1 − x2 = 2(sign x) atan x. 1 + x2 √ √ √ If |x| ≤ 2/2, then 2 acos x + asin(2x 1 − x2 ) = π and 2 asin x = asin(2x 1 − x2 ). √ √ If |x| ≥ 2/2, then 2 asin x + asin(2x 1 − x2 ) = (sign x)π. x−3 2π x+3 = asin √ + . If |x| ≤ 1, then asin √ 2 2 3 12 + 4x 12 + 4x 1 1 1 . If x , 0, then atan = 2 atan − atan 3 x 2x( ) 4x + 3x ( )

xxxiii) acos xxxiv) xxxv) xxxvi) xxxvii)

√

√

xxxviii) If |x| < 1, then atan 1−x5x2 = atan 1+2 5 x − atan √ 2 ) xxxix) If |x| < 1 + 2, then 4 atan x = atan x4x(1−x 4 −6x2 +1 . xl) If x > 0, then 2 atan x =

√ 1− 5 2

x.

π 2

+ atan 12 (x − 1x ). √ 2 x+1 1 = asin = asec(1 + 2x ). xli) If x > 0, then 2 asin √ x+2 x+2 xlii) (sign x) atan sinh x + asin sech x = π2 . x ȷ − 1 x2 − 1 + 2x ȷ . = xȷ+ 1 x2 + 1 Remark: See [650, 1514, 1619, 2458] and [1217, pp. 57–59]. Remark: Some of these results can be extended to complex numbers. Fact 2.18.4. The following equalities hold: √ √ i) atan 33 = π6 , atan 1 = π4 , atan 3 = π3 . √ √ π ii) atan(2 − 3) = 12 , atan(2 + 3) = 5π 12 . √ √ √ √ iii) atan 5 − 2 5 = π5 , atan 5 + 2 5 = 2π 5 . √ iv) atan 43 = 2 atan 31 , 2 atan 12 ( 5 − 1) = atan 2. xliii) e2(acot x) ȷ =

v) asin 53 + asin 45 = π2 . vi) atan 35 + atan 14 = π4 . vii) atan 12 + atan 2 = π2 . viii) atan 43 + atan 43 = π2 . ix) atan 12 + atan 13 = π4 . x) atan 34 + atan 17 = π4 . xi) 2 atan 12 − atan 17 = π4 . 5 xii) 3 atan 14 + atan 99 = π4 . 1 = π4 . xiii) 4 atan 15 − atan 239 √ √ xiv) atan 7 + acot 7 = π2 . 3 xv) 5 atan 71 + 2 atan 79 = π4 .

xvi) atan 12 = atan 13 + atan 17 . 2 . xvii) atan 31 = atan 17 + atan 11 29 3 xviii) 5 atan 278 + 7 atan 79 = π4 .

258

CHAPTER 2

2 3 xix) atan 11 = atan 17 + atan 79 . xx) atan 1 + atan 2 + atan 3 = π. xxi) atan 12 + atan 51 + atan 18 = π4 .

xxii) 2 atan 15 + atan 17 + 2 atan 18 = π4 . 1 1 + atan 1985 = π4 . xxiii) 3 atan 41 + atan 20 1 1 + atan 239 = π4 . xxiv) 6 atan 18 + 2 atan 57 1 1 1 − atan 239 − 4 atan 515 = π4 . xxv) 8 atan 10 1 1 1 xxvi) 12 atan 18 + 8 atan 57 − 5 atan 239 = π4 . 1 = π2 . xxvii) atan 21 + atan 31 + 4 atan 51 − atan 239 xxviii) atan 1 + atan 3 + atan 5 + atan 7 + atan 8 = 2π. 1 1 1 1 xxix) 22 atan 28 + 2 atan 443 − 5 atan 1393 − 10 atan 11018 = π4 . 1 1 1 1 + 7 atan 239 − 12 atan 682 + 24 atan 12943 = π4 . xxx) 44 atan 57 1 1 1 1 xxxi) 48 atan 49 + 128 atan 57 − 20 atan 239 + 48 atan 110443 = π. 1 1 1 1 + 28 atan 239 − 48 atan 682 + 96 atan 12943 = π. xxxii) 176 atan 57 xxxiii) atan 2 + atan 3 + atan 4 + atan 5 + atan 7 + atan 8 + atan 13 = 3π. xxxiv) atan 1 + atan 5 + atan 7 + atan 8 + atan 12 + atan 13 + atan 17 + atan 18 + atan 21 = 4π. Source: xxviii) and the last two equalities are given in [1180]. Fact 2.18.5. Let n ≥ 1. Then, n n ∑ ∑ n π 1 2 atan 2 = atan , (−1)i atan 2 = acot(n + 1) − acot n − , n+1 4 2i i i=1 i=1 n ∑

atan

i=0

1 π 1 = atan(n + 1). = − atan n+1 i2 + i + 1 2

Source: [112, pp. 27, 28], [1311, p. 277], and [2180]. Related: Fact 13.6.13. Fact 2.18.6. The following statements hold:

i) Let x, y ∈ [−1, 1]. Then,

 √   acos( (1 − y2 )(1 − x2 ) − xy),     √    2 2   acos( (1 − y )(1 − x ) − xy), asin x + asin y =  √    − acos( (1 − y2 )(1 − x2 ) − xy),      √   − acos( (1 − y2 )(1 − x2 ) − xy),

ii) Let x, y ∈ [−1, 1]. Then, √ √   asin(x 1 − y2 + y 1 − x2 ),     √ √   asin x + asin y =  − asin(x 1 − y2 + y 1 − x2 ) + π,      − asin(x √1 − y2 + y √1 − x2 ) − π,

x ≥ 0 and y > 0, xy < 0 and x + y > 0, x < 0 and y < 0, xy < 0 and x + y < 0.

x2 + y2 ≤ 1 or xy ≤ 0, x2 + y2 > 1, x > 0, and y > 0, x2 + y2 > 1, x < 0, and y < 0.

259

EQUALITIES AND INEQUALITIES

iii) Let x, y ∈ [−1, 1]. Then,      atan       asin x + asin y =          atan

√ √ x 1 − y2 + y 1 − x2 + (sign x)π, x2 + y2 > 1 and xy > 0, √ (1 − x2 )(1 − y2 ) − xy √ √ x 1 − y2 + y 1 − x2 , else. √ (1 − x2 )(1 − y2 ) − xy

iv) Let x, y ∈ [−1, 1]. Then,

 √   − acos[xy − (1 − y2 )(1 − x2 )] + 2π, x + y < 0,    acos x + acos y =  √    acos[xy − (1 − y2 )(1 − x2 )], x + y > 0, √ acos x − acos y = sign(y − x) acos[xy + (1 − y2 )(1 − x2 )].

v) Let x, y ∈ R. Then,

 x+y   atan ,    1 − xy  atan x + atan y =   x+y    + (sign x)π, atan 1 − xy

xy < 1, xy > 1.

vi) Let x, y ∈ R, and assume that x , 0, y , 0, and x + y , 0. Then,  xy − 1    acot , xy < 0 or xy ≥ 1,    x+y  acot x + acot y =    xy − 1    + (sign x)π, 0 < xy < 1.  acot x+y vii) Let p, q, r be positive numbers, and assume that qr = 1 + p2 . Then, atan

1 1 1 = atan + atan . p p+q p+r

1 1 1 In particular, atan 13 = atan 51 + atan 18 and atan 70 = atan 99 + atan 239 . 2 viii) Let x, y ∈ C, and assume that x , 0, x + y , 0, and x + xy + 1 , 0. Then,

atan

1 y 1 = atan − atan 2 . x+y x x + xy + 1

ix) Let x, y ∈ C, and assume that x , 0, x − y , 0, and x2 − xy + 1 , 0. Then, atan

1 1 y . = atan + atan 2 x−y x x − xy + 1

x) Let x, y ∈ R, and assume that x2 < y2 . Then, x 1 x x+y − atan √ = acos . atan √ 2 2 2 2 2 y y −x y −x Remark: See [649], [1217, pp. 57, 58], and [2013, p. 166]. Fact 2.18.7. Let x and y be real numbers, and define the four-quadrant inverse tangent function

atan2 by

△

atan2(y, x) = arg(x + ȷy).

If x , 0, then atan

y = atan2[(sign x)y, |x|]. x

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CHAPTER 2

Furthermore,

  0,        tan−1 yx ,       − π2 , atan2(y, x) =   π    2,      −π + tan−1 yx ,     π + tan−1 y , x

y = x = 0, x > 0, y < 0, x = 0, y > 0, x = 0, y < 0, x < 0, y ≥ 0, x < 0.

Equivalently,   0, y = x = 0,     √   atan2(y, x) =  2 atan √ 2 y 2 , x2 + y2 + x > 0,   x +y +x     π, y = 0, x < 0. Finally, if x1 , y1 , x2 , y2 are real numbers, then there exists k ∈ {−2, 0, 2} such that atan2(y1 , x1 ) − atan2(y2 , x2 ) = atan2(y1 x2 − y2 x1 , y1 y2 + x1 x2 ) + kπ. Remark: The range of atan is (− π2 , π2 ), whereas the range of atan2 is (−π, π]. Fact 2.18.8. The following statements hold:

i) If x ∈ [0, 1), then

2 ( πx ) sin asin x ≤ x2 ≤ (sin x) asin x, π 2 √ √  6( 1 + x − 1 − x)  3x    ≤ √ √ √    πx 2   4+ 1+x+ 1−x  2+ 1−x   √     πx   2 + 1 − x2 ≤ ≤ asin x ≤ √  (π2 /4)x   x   2   2 + (π − 2) 1 − x √   .  π  2 1 − x2 1−x    2 +  x √

ii) If x ∈ (0, 1), then acos x < 1−x x . iii) If x > 0, then  x log(x2 + 1)     ≤    x x2 + 1   3x πx πx  < < atan x < √ √    2  1+2 x +1 π − 2 + 2 x2 + 1 2x + π       πx x − 31 x3    √     1 + 2 x2 + 1  0, then

8 3 45 x tanh

x+2≤

sinh x x

)2 +

tanh x . x

    1     √    x < sinh x < 2 sinh 2x 2 2  < tanh x <  sinh x + cosh x       1.    (sin x) cos x sinh x

vii) Let x > 0, let p > 0, and let q ≥ p + 1. Then,

( )q p cosh x (2 − p) sinh x sinh x + < . 2 2x x

In particular, if q ≥ 3, then

( cosh x
0 satisfy α = coth α, and define β = (cosh α)/α ≈ 1.5088. Then, the following statements hold: v) For all x ∈ (0, ∞), β ≤ (cosh x)/x. vi) f : (0, α) 7→ (β, ∞) defined by f (x) = (cosh x)/x is decreasing. vii) f : (α, ∞) 7→ (β, ∞) defined by f (x) = (cosh x)/x is increasing. Remark: (sinh 0)/0 = 1. Fact 2.19.6. Let z ∈ C be such that all terms below are defined. Then, d sinh z = cosh z, dz d csch z = −(csch z) coth z, dz

d d cosh z = sinh z, tanh z = sech2 z, dz dz d d sech z = −(sech z) tanh z, coth z = − csch2 z. dz dz

2.20 Facts on Equalities and Inequalities for Inverse Hyperbolic Functions Fact 2.20.1. The following statements hold: i) asinh : R 7→ R is given by √ asinh x = log(x + x2 + 1).

ii) asinh : C 7→ R + ȷ[− π2 , π2 ] is given by asinh z = log(z + iii) acosh : [1, ∞) 7→ [0, ∞) is given by acosh x = log(x +

√ z2 + 1). √

x2 − 1).

265

EQUALITIES AND INEQUALITIES

iv) acosh : C 7→ R + ȷ[0, π] is given by acosh z = log(z +

√ √ z − 1 z + 1).

v) atanh : (−1, 1) 7→ (−∞, ∞) is given by 1+x 1 log . 2 1−x vi) atanh : C\{−1, 1} 7→ R + ȷ(− π2 , π2 ) is given by atanh x =

atanh z = 12 [log(1 + z) − log(1 − z)]. vii) acsch : R\{0} 7→ R\{0} is given by

√  1 acsch x = log  1 + 2 + x

viii) acsch : C\{0} 7→ R + ȷ(− π2 , π2 ] is given by

ix) asech : (0, 1) 7→ R is given by

 1  . x

√  1 acsch z = log  1 + 2 + z

 1  . z

√  1 −1+ asech x = log  x2

 1  . x

x) asech : C\{0} 7→ R + ȷ(−π, π] is given by √ √  1 1 asech z = log  −1 +1+ z z

 1  . z

xi) acoth : (−∞, −1] ∪ [1, ∞) 7→ R\{0} is given by acoth x =

x+1 1 log . 2 x−1

xii) acoth : C\{0} 7→ R + ȷ(− π2 , π2 ] is given by ( ) z+1 z−1 1 log − log . acoth z = 2 z z Fact 2.20.2. If x ∈ (−1, 0) ∪ (0, 1), then

(

) ( atan x ) atanh x asin x asinh x , 1< , x x x x asin x asinh x atan x atanh x 2< + , 2< + , x x x x ( ( )2 )2 asinh x atan x asinh x asin x asin x atanh x < < 0.

267

EQUALITIES AND INEQUALITIES

vi) y3 + py + q = 0 has three real roots if and only if 27q2 + 4p3 < 0. Source: [1485, pp. 673–676]. Remark: This is Cardano’s formula. Remark: In the case where a, b, c are not real and p is nonzero, the six roots of u6 + qu3 − p3 /27 may be distinct. However, computing y = u − p/(3u) yields at most three distinct roots of y3 + py + q. Related: Fact 2.21.2, Fact 6.10.7, Fact 15.18.4, Fact 15.21.27, and Fact 15.21.29. Fact 2.21.3. Let z be a complex number with complex conjugate z, real part Re z, and imaginary part Im z. Then, the following statements hold: i) ii) iii) iv) v) vi) vii) viii) ix) x) xi) xii) xiii) xiv) xv) xvi) xvii) xviii)

z = z, 0 ≤ |z| = | − z| = |z|, Re ȷz = − Im z, Im ȷz = Re z. −|z| ≤ Re z ≤ | Re z| ≤ |z|, −|z| ≤ Im z ≤ | Im z| ≤ |z|. Re z = | Re z| = |z| if and only if Re z ≥ 0 and Im z = 0. Im z = | Im z| = |z| if and only if Im z ≥ 0 and Re z = 0. If z , 0, then z−1 = z −1 , z−1 = z/|z|2, and |z−1 | = 1/|z|. If |z| = 1, then z−1 = z. If z , 0, then Re z−1 = (Re z)/|z|2. Re z , 0 if and only if Re z−1 , 0. √ If Re z , 0, then |z| = (Re z)/(Re z−1 ). |z2 | = |z|2 = |z2 | = |z|2 = zz = (Re z)2 + (Im z)2 = (Re z)2 + (Im z)2 . z2 ∈ [0, ∞) if and only if Im z = 0. z2 ∈ (−∞, 0] if and only if Re z = 0. z2 + z2 + 4(Im z)2 = 2|z|2 , z2 + z2 + 2|z|2 = 4(Re z)2 , z2 + z2 + 2(Im z)2 = 2(Re z)2. { 2 } |z + z2 | 2 2 z +z ≤ ≤ 2|z|2. 2(Re z)2 If at least two of the quantities z2 + z2 , 2(Re z)2 , and 2|z|2 are equal, then Im z = 0. Either Re z = 0 or Im z = 0 if and only if z2 = z2 . Both Re z = 0 and Im z = 0 if and only if z = 0. Let n be a positive integer. Then, n−1 ∑

zi = 1 + z + · · · + zn−1 .

i=0

If z , 1, then

1 − zn ∑ i = z = 1 + z + · · · + zn−1 . 1−z i=0 n−1

Furthermore, lim z→1

xix) Let n, k, m ≥ 1, and define △

S =

1 − zn = n. 1−z n−1 ∑

e(mπi/k) ȷ .

i=0

If 2k - m, then S =

1 − e(mnπ/k) ȷ . 1 − e(mπ/k) ȷ

268

CHAPTER 2

Furthermore, S =

n−1 ∑ i=0

   n, 2k|m,   mπi   cos = 1, 2k - m and 2k|m(n − 1),   k   0, 2k - m and 2k|mn.

In all other cases, Im S , 0. xx) Let n, m ≥ 1, and define △

S =

n−1 ∑

e(2mπi/n) ȷ .

i=0

Then,

   n, S =  0,

xxi) Let n ≥ 1, and let k be an integer. Then, k+n−1 ∑

(2πi/n) ȷ

e

i=k

n|m, n - m.

   1, n = 1, =  0, n ≥ 2.

Source: xx) is given in [672]; xx) implies xxi). Remark: A matrix version of i) is given in [2613]. Fact 2.21.4. Let n ≥ 1, and let z be a complex number. Then, n−1 ∏

[z − e(2πi/n) ȷ ] =

i=0

n ∏

n−1 ∏

[z − e(2πi/n) ȷ ] = zn − 1,

i=1

[1 − e(2πi/n) ȷ z] =

i=0

Now, define

△

pn (z) =

∏

n ∏

[1 − e(2πi/n) ȷ z] = 1 − zn .

i=1

[z − e(2πi/n) ȷ ],

where the product is taken over all i ∈ {1, . . . , n} such that gcd {i, n} = 1. Then, pn is monic, all of its coefficients are integers, and ∏ zn − 1 = pi (z), where the product is taken over all i ∈ {1, . . . , n} such that i|n. In particular, p1 (z) = z − 1,

p2 (z) = z + 1,

p5 (z) = z4 + z3 + z2 + z + 1, p8 (z) = z4 + 1,

z3 (z) = z2 + z + 1,

p6 (z) = z2 − z + 1, p9 (z) = z6 + z3 + 1,

p11 (z) = z10 +z9 +z8 +z7 +z6 +z5 +z4 +z3 +z2 +z+1, z2 − 1 = p1 (z)p2 (z),

z3 − 1 = p1 (z)p3 (z),

p4 (z) = z2 + 1,

p7 (z) = z6 + z5 + z4 + z3 + z2 + z + 1, p10 (z) = z4 − z3 + z2 − z + 1,

p12 (z) = z4 −z2 +1, z4 − 1 = p1 (z)p4 (z),

z6 − 1 = p1 (z)p2 (z)p3 (z)p6 (z),

z7 − 1 = p1 (z)p7 (z),

z9 − 1 = p1 (z)p3 (z)p9 (z),

z10 − 1 = p1 (z)p2 (z)p5 (z),

z12 − 1 = p1 (z)p2 (z)p3 (z)p4 (z)p6 (z)p12 (z),

p13 (z) = z12 +z11 + p11 (z), z5 − 1 = p1 (z)p5 (z),

z8 − 1 = p1 (z)p2 (z)p4 (z)p8 (z), z11 − 1 = p1 (z)p11 (z),

z13 − 1 = p1 (z)p13 (z).

Furthermore, the following statements hold: ∑ i) If n is prime, then pn (z) = ni=0 zi . ii) If n is odd, then p2n (z) = pn (−z). iii) If m and n are distinct primes, then all of the coefficients of pmn are either −1, 0, or 1. △ iv) Let n ≥ 2, and define m = deg pn . Then, pn (z) = zm pn (1/z).

269

EQUALITIES AND INEQUALITIES

v) If m and n are distinct primes, then pn (zm ) = pmn (z)pn (z). vi) If m and n are distinct primes, then (zmn − 1)pmn (z) = pm (zn )pn (zm )(z − 1). Source: [126, 577]. Remark: pn is the nth cyclotomic polynomial. Remark: p105 is the first cyclotomic polynomial that has a coefficient that is neither 1, −1, nor 0. In particular, the coefficient of z7 in p105 is 2. Fact 2.21.5. Let n, m ≥ 1, and let z be a complex number. Then, n ∏ m ∏

(z − e

2πi n

ȷ

e

2π j m

ȷ

) = (zlcm {n,m} − 1)gcd {n,m} .

i=1 j=1

Source: [107, pp. 60, 319, 321]. Fact 2.21.6. Let z1 and z2 be complex numbers. If n ≥ 0, then 2n ∏

(z1 + e[2iπ/(2n+1)] ȷ z2 ) = z2n+1 + z2n+1 . 1 2

i=0

If n ≥ 1, then

2n−1 ∏

(z1 + e(2iπ/n) ȷ z2 ) = (zn1 − (−1)n zn2 )2 .

i=0

Source: [109, pp. 298–300]. Fact 2.21.7. Let z be a complex number, and assume that |z| ≤ 1. Then, 4 5 |z| 3 4 |z|

≤ (sin 1)|z| ≤ |sin z| ≤ 12 (e − 1e )|z| ≤ 65 |z|,

≤

e2 −1 |z| e2 +1

≤ |tan z| ≤ (tan 1)|z| ≤ 85 |z|,

3 5 |z|

1 2 |z|

≤ (cos 1) ≤ |cos z| ≤ 12 (e + 1e ) ≤ 85 ,

≤ (1 − 1e )|z| ≤ |ez − 1| ≤ (e − 1)|z| ≤ 47 |z|.

Source: [2346, p. 116] and [2994]. Fact 2.21.8. Let z1 and z2 be complex numbers, and let α be a real number. Then, the following

statements hold: i) z1 z2 = 0 if and only if either z1 = 0 or z2 = 0. ii) z1 z2 = z1 z2 . iii) |z1z2 | = |z1 | |z2 |. iv) If z2 , 0, then |z1 /z2 | = |z1 |/|z2 |. v) |z1 | − |z2 | ≤ |z1 + z2 | ≤ |z1 | + |z2 |. vi) |z1 + z2 | = |z1 | + |z2 | if and only if Re(z1 z2 ) = |z1 ||z2 |. vii) |z1 + z2 | = |z1 | + |z2 | if and only if there exists α ≥ 0 such that either z1 = αz2 or z2 = αz1 . viii) |z1 | − |z2 | ≤ |z1 − z2 |. ix) |z | − |z | = |z − z | if and only if there exists α ≥ 0 such that either z = αz or z = αz . 1

2

1

2

1

2 2

2

1

x) |1 + z1 z2 | = (1 − |z1 | )(1 − |z2 | ) + |z1 + z2 | = (1 + |z1 | )(1 + |z2 | ) − |z1 − z2 | . xi) |z1 − z2 |2 ≤ (1 + |z1 |2 )(1 + |z2 |2 ). xii) If z1 and z2 are nonzero, then 21 z1 − z2 + zz21 z1 − zz21 z2 = 12 (|z1 | + |z2 |) |zz11 | − |zz22 | ≤ |z1 − z2 |. 2

xiii) xiv) xv) xvi) xvii)

2

2

2

2 Re(z1z2 ) ≤ 2| Re(z1z2 )| ≤ 2|z1z2 | ≤ |z1 |2 + |z2 |2. 2|z1z2 | = |z1 |2 + |z2 |2 if and only if |z1 | = |z2 |. 2 Re(z1z2 ) = |z1 |2 + |z2 |2 if and only if z1 = z2 . |z1 + z2 |2 = |z1 |2 + |z2 |2 + 2 Re z1 z2 . |z1 + z2 |2 = |z1 |2 + |z2 |2 if and only if Re z1 z2 = 0.

2

2

270

xviii) xix) xx) xxi) xxii)

CHAPTER 2

If α , 0, then α1 (αz1 + z2 )2 + (z1 − z2 )2 = (1 + α)z21 + (1 + α1 )z22 . (z1 + z2 )2 + (z1 − z2 )2 = 2z21 + 2z22 . If α , 0, then α1 |αz1 + z2 |2 + |z1 − z2 |2 = (1 + α)|z1 |2 + (1 + α1 )|z2 |2 . |z1 + z2 |2 + |z1 − z2 |2 = 2|z1 |2 + 2|z2 |2 . Let a1 and a2 be real numbers, and assume that a1 , a2 , and a1 + a2 are nonzero. Then, |z1 |2 |z2 |2 |z1 + z2 |2 |a1 z2 − a2 z1 |2 + = + . a1 a2 a1 + a2 a1 a2 (a1 + a2 )

xxiii) (z1 + z2 )2 + (z1 + z2 ȷ)2 ȷ = (z1 − z2 )2 + (z1 − z2 ȷ)2 ȷ. xxiv) 4z1 z2 = |z1 + z2 |2 − |z1 − z2 |2 + (|z1 − z2 ȷ|2 − |z1 + z2 ȷ|2 ) ȷ. xxv) Let n ≥ 3. Then, n−1 1 ∑ −(2iπ/n) ȷ e |z1 + e(2iπ/n) ȷ z2 |2 . z1 z2 = n i=0 xxvi) 2z1 z2 = |z1 + z2 |2 − |z1 |2 − |z2 |2 + (|z1 |2 + |z2 |2 − |z1 + z2 ȷ|2 ) ȷ. xxvii) If z1 z2 , 0 and arg z1 + arg z2 ∈ (−π, π], then arg z1 z2 = arg z1 + arg z2 . In particular, if z1 z2 , 0, Re z1 > 0, and Re z2 > 0, then arg z1 z2 = arg z1 + arg z2 . xxviii) |z1 | + |z2 | ≤ |1 + z1 | + |1 + z2 | + |1 + z1 z2 |. xxix) (|z1 − z2 | − |z1 z2 − 1|)2 ≤ |(z21 − 1)(z22 − 1)| ≤ (|z1 − z2 | + |z1 z2 − 1|)2 . |z1 + z2 | |z1 | |z2 | xxx) ≤ + . 1 + |z1 + z2 | 1 + |z1 | 1 + |z2 | xxxi) If z1 and z2 are nonzero, then z1 − |z |z = z2 − |z |z . 1 2 2 1 |z1 | |z2 | xxxii) If p ∈ [1, 2], then |z1 + z2 | p ≤ 2 p−1 (|z1 | p + |z2 | p ) ≤ |z1 + z2 | p + |z1 − z2 | p ≤ 2(|z1 | p + |z2 | p ). xxxiii) If p ≥ 2, then 2(|z1 | p + |z2 | p ) ≤ |z1 + z2 | p + |z1 − z2 | p ≤ 2 p−1 (|z1 | p + |z2 | p ). xxxiv) If p ∈ (1, 2], q ≥ 2, and 1/p + 1/q = 1, then |z1 + z2 |q + |z1 − z2 |q ≤ 2(|z1 | p + |z2 | p )q−1. xxxv) If p ≥ 2, q ∈ (1, 2], and 1/p + 1/q = 1, then 2(|z1 | p + |z2 | p )q−1 ≤ |z1 + z2 |q + |z1 − z2 |q. xxxvi) If p, q > 1 and 1/p + 1/q = 1, then |z1 + z2 |2 ≤ p|z1 |2 + q|z2 |2 . Equality holds if and only if z2 = (p − 1)z1 . xxxvii) Let n ≥ 1. If z1 , z2 , then zn1 − zn2 n−2 + · · · + zn−1 = zn−1 1 + z2 z1 2 , z1 − z2

lim

z2 →z1

zn1 − zn2 = nzn−1 1 . z1 − z2

xxxviii) Let p ∈ [0, 1]. Then, ||z1 | p − |z2 | p | ≤ ||z1 | − |z2 || p ≤ |z1 − z2 | p .

271

EQUALITIES AND INEQUALITIES

xxxix) Let p ∈ [1, ∞). Then,

 p p    ||z1 | − |z2 | | ||z1 | − |z2 || ≤    |z1 − z2 | p . p

xl) Let n ≥ 1. Then,

 n n n n    ||z1 | − |z2 | | ≤ |z1 − z2 | ||z1 | − |z2 ||n ≤    |z1 − z2 |n .

xli) If α ∈ [0, 1], then |z1 − z2 |2 + |αz1 + z2 |2 ≤ (1 + α)|z1 |2 + (1 + α1 )|z2 |2 . xlii) If either α < 0 or α ≥ 1, then (1 + α)|z1 |2 + (1 + α1 )|z2 |2 ≤ |z1 − z2 |2 + |αz1 + z2 |2 . xliii) If r ≥ 1, then

|ez1 − ez2 | ≤ |z1 − z2 |[ 12 (|ez1 |r + |ez2 |r )]1/r .

xliv) If x ∈ R, then |e x ȷ − 1| = 2| sin 2x | and |e x ȷ − 1 − x ȷ| ≤ 21 x2 . xlv) If p, q ∈ (1, ∞) satisfy 1/p + 1/q = 1, then |z1 z2 | ≤

|z1 | p |z2 |q + . p q

Equality holds if and only if |z1 | p = |z2 |q. Remark: Matrix versions of iii), v), vii)–ix) are given in [2613]; x) is given in [110, p. 19] and [2933]; xii) is the Dunkl-Williams inequality, see [935, p. 43], [936, p. 52], and ii) of Fact 11.8.3; xvii) is the Pythagorean theorem; xx) is the generalized parallelogram law, see [1095]; xxi) is the parallelogram law, see [978] and Fact 11.8.3; xxii) is given in [2060, p. 315] and [2252] and follows from Fact 2.12.22; xxiv) is the polarization identity, see [828, p. 54], [2112, p. 276], and Fact 11.8.3; xxv) is given in [828, p. 54]; xxvi) is given in [2238, p. 261]; xxvii) is given in [582, p. 23]; xxviii) is given in [2682]; xxix) is given in [365]; xxx) is given in [993, p. 183]; xxxii)–xxxv) are due to J. A. Clarkson; see [1419], [2061, p. 536], [2294, p. 253], Fact 11.8.3, and Fact 11.10.54; xxxvi) is given in [1984]; xli) and xlii) are given in [1095]; xliii) is given in [942]; xliv) is given in [968, p. 274]. Remark: The absolute value |z| = |x + y ȷ|, where x and y are real, is equal to the [ ] Euclidean norm ∥ yx ∥2 . Hence, results involving the Euclidean norm on R2 can be recast in terms of complex numbers. Remark: xxxvi) is Bohr’s inequality. Extensions are given in Fact 2.21.23 and Fact 10.11.83. Remark: The lower bounds for |z1 − z2 | given by viii) and xii) cannot be ordered. Remark: xlv) is Young’s inequality. See Fact 2.2.50. Fact 2.21.9. Let z1 and z2 be nonzero complex numbers. Then,   |z1 | − |z2 |   |z − z | + 1 2          |z1 − z2 | − |z1 | − |z2 | z1   max {|z1 |, |z2 |} z2    ≤  ≤ −      min {|z1 |, |z2 |} |z1 | |z2 |     2|z − z | 1 2       |z1 | + |z2 |   2|z1 − z2 |             max {|z |, |z |} 1 2    4|z1 − z2 |  ≤ . ≤     |z1 | + |z2 |   2(|z1 − z2 | + |z1 | − |z2 | )          |z1 | + |z2 |

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1 −z2 | Source: Fact 11.7.11 and Fact 11.8.3. Remark: |zz11 | − |zz22 | ≤ 2|z |z1 |+|z2 | is given by xii) in Fact 2.21.8. Fact 2.21.10. Let w, z ∈ C, and assume that |w| < 1 and |z| < 1. Then,

 1   |w − z|[(1 − 21 |w + z|)−2 + 12 (1 − |w|)−2 + 12 (1 − |z|)−2 ],    2 −1 −1 |(1 − w) − (1 − z) | ≤  |w − z|(1 − |w|)−1 (1 − |z|)−1 , |w| , |z|,     −2 |w − z|(1 − |w|) , |w| = |z|,

 1 1 −1   + 21 (1 − |w|)−1 + 21 (1 − |z|)−1 ],  2 |w − z|[(1 − 2 |w + z|)      |w − z| | log (1 − w)−1 − log (1 − z)−1 | ≤  [log (1 − |w|)−1 − log (1 − |z|)−1 ], |w| , |z|,    |w| − |z|    |w − z|(1 − |w|)−1 , |w| = |z|. Source: [942]. Fact 2.21.11. Let z1 , z2 , z3 be complex numbers. Then, the following statements hold:

i) ii) iii) iv)

(z1 − z3 )2 + (z3 − z2 )2 = 21 (z1 − z2 )2 + 2[z3 − 12 (z1 + z2 )]2 . (z1 + z2 )2 + (z2 + z3 )2 + (z3 + z1 )2 = z21 + z22 + z23 + (z1 + z2 + z3 )2 . (z1 − z2 )2 + (z2 − z3 )2 + (z3 − z1 )2 + (z1 + z2 + z3 )2 = 3(z21 + z22 + z23 ). (z1 + z2 − z3 )2 + (z2 + z3 − z1 )2 + (z3 + z1 − z2 )2 + (z1 + z2 + z3 )2 = 4(z21 + z22 + z23 ).

|z1 − z3 |2 + |z3 − z2 |2 = 21 |z1 − z2 |2 + 2|z3 − 21 (z1 + z2 )|2 . |z1 + z2 |2 + |z2 + z3 |2 + |z3 + z1 |2 = |z1 |2 + |z2 |2 + |z3 |2 + |z1 + z2 + z3 |2 . |z1 − z2 |2 + |z2 − z3 |2 + |z3 − z1 |2 + |z1 + z2 + z3 |2 = 3(|z1 |2 + |z2 |2 + |z3 |2 ). |z1 + z2 − z3 |2 + |z2 + z3 − z1 |2 + |z3 + z1 − z2 |2 + |z1 + z2 + z3 |2 = 4(|z1 |2 + |z2 |2 + |z3 |2 ). |z1 + z2 | + |z2 + z3 | + |z3 + z1 | ≤ |z1 | + |z2 | + |z3 | + |z1 + z2 + z3 |. |z1 | + |z2 | + |z3 | ≤ |z1 + z2 − z3 | + |z2 + z3 − z1 | + |z3 + z1 − z2 |. If z1, z2 , z3 are nonzero and z71 + z72 + z73 = z1 + z2 + z3 = 0, then |z1 | = |z2 | = |z3 |. Source: The first four equalities are given by Fact 2.3.2; v) is the Appolonius identity, see [2238, p. 260]; vi) is given in [110, p. 19]; vii) is given in [2238, p. 244]; viii) is given in [110, p. 19] and [978]; ix) is Hlawka’s inequality, see Fact 1.21.9 and Fact 11.8.3; x) is given in [993, p. 181]; xi) is given in [110, pp. 186, 187]. Remark: If z1 , z2 , z3 are positive numbers that represent the lengths of the sides of a triangle, then equality holds in x). See Fact 5.2.14. Fact 2.21.12. Let z1 , z2 , z3 , z4 be complex numbers. Then, v) vi) vii) viii) ix) x) xi)

(z1 − z3 )(z2 − z4 ) = (z1 − z2 )(z3 − z4 ) + (z1 − z4 )(z2 − z3 ). Consequently,

|z1 − z3 | |z2 − z4 | ≤ |z1 − z2 | |z3 − z4 | + |z1 − z4 | |z2 − z3 |.

Source: [2238, p. 473]. Remark: This is Ptolemy’s inequality. △ Fact 2.21.13. Let n ≥ 1, define p ∈ R[s] by p(s) = sn + 1, define {λ1 , . . . , λn } = roots(p), and

let a ∈ (0, ∞). Then,

n n−1 ∑ 1 1 1∑ = a2i . n i=1 |λi − a|2 (an + 1)2 i=0

Source: [2547]. Fact 2.21.14. Let z1 , . . . , zn ∈ C, let α1 , . . . , αn be nonzero real numbers, and assume that

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EQUALITIES AND INEQUALITIES

∑n i=1

αi = 1. Then,

2 n n ∑ ∑ αi 1 2 ∑ |zi | − zi = α α i=1 i=1 i 1≤i≤ j≤n j

2 α j z − z . j αi i

Source: [1095, 2989]. Related: Fact 10.11.85 and Fact 11.8.4. Fact 2.21.15. Let z, z1, . . . , zn be complex numbers. Then,

2 n n ∑ 1 1 ∑ 1∑ |zi − z j |2, |z − zi |2 = z − zi + 2 n i=1 n i=1 n 1≤i< j≤n

n ∑ 1 ∑ |zi − z j |2 ≤ |zi |2 . n 1≤i< j≤n i=1

Source: [110, p. 146] and [2252]. Fact 2.21.16. Let n ≥ 3, and let z1, . . . , zn be distinct complex numbers. Then, the following

statements are equivalent: i) In the ordering listed, z1 , . . . , zn lie on the same line or circle in the complex plane. ii) n−1 ∑ |zn − z2 | |zi+1 − zi | = . |z2 − z1 ||zn − z1 | i=2 |zi+1 − z1 ||zi − z1 | Fact 2.21.17. Let z1, . . . , zn be complex numbers. Then,

2   n 2 n 2 n n ∑  ∑  ∑ ∑  |Re zi | − Re zi  ≤  |zi | − zi . i=1 i=1 i=1 i=1

Source: [2548]. Fact 2.21.18. Let z1, . . . , zn be complex numbers, and assume that there exists ϕ ∈ [0, π2 ] such

that, for all i ∈ {1, . . . , n}, | arg zi | ≤ ϕ. Then, 1/n n n ∏ 1 ∑ (cos ϕ) zi ≤ zi . n i=1 i=1

Remark: This is an extension of the arithmetic-mean–geometric-mean inequality to complex numbers. Source: [2527, p. 36]. △ Fact 2.21.19. Let z1, . . . , zn be complex numbers, and assume that r = max {|z1 |, . . . , |zn |} < 1. ∏ Then, there exists z ∈ C such that |z| ≤ r and (z + 1)n = ni=1 (zi + 1). Source: [2527, p. 103]. Fact 2.21.20. Let n ≥ 2, let z1 , . . . , zn ∈ C, let a1 , . . . , an be nonzero real numbers, and assume

that

∑n

i=1

ai , 0. Then,

∑n 2 ∑ |ai z j − a j zi |2 i=1 zi 1 = ∑n . + ∑n ai ai a j i=1 ai i=1 ai 1≤i< j≤n

n ∑ |zi |2 i=1

If, in addition, a1 , . . . , an are positive numbers, then n ∑ |ai z j − a j zi |2 ∑ |zi |2 ≤ . ai a j ai i=1 ai 1≤i< j≤n i=1

1 ∑n

Source: [2252]. Related: Fact 2.12.19. The case n = 2 is given by Fact 2.21.8. Fact 2.21.21. Let z1 , . . . , zn be complex numbers and, for all i ∈ {1, . . . , n}, let zi = ri eϕi ȷ , where

ri ≥ 0 and ϕi ∈ R. Furthermore, assume that there exist θ1 , θ2 ∈ R such that 0 < θ2 − θ1 < π and such

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that, for all i ∈ {1, . . . , n}, θ1 ≤ ϕi ≤ θ2 . Then,

n ∑ [cos 21 (θ2 − θ1 )] |zi | ≤ zi . i=1 i=1 n ∑

Source: [938]. Credit: M. Petrovich. Fact 2.21.22. Let z1 , . . . , zn be complex numbers. Then,

∑ |zi | ≤ π max zi , i∈S i=1

n ∑

where the maximum is taken over all subsets S of {1, . . . , n}. Source: [2527, p. 224]. Fact 2.21.23. Let z1 , . . . , zn be complex numbers, let p1 , . . . , pn be positive numbers, and let r > 1. Then, r  r−1 n n n ∑ ∑ 1   ∑ 1−r zi ≤  pi |zi |r . pi  i=1 i=1 i=1 Source: [1984]. Related: The special case n = r = 2 is Bohr’s inequality. See Fact 2.21.8. A matrix version of this result is given in Fact 11.10.56. Fact 2.21.24. Let n, m ≥ 1, and let z1 , . . . , zm be complex numbers. Then,  m (m+n−1 m ∏∏ ∏  m ) ij , zi  z =  i

i=1

j=1

where the first product is taken over all m-tuples (i1 , . . . , im ) of nonnegative integers such that ∑m j=1 i j = n. Source: [771, p. 35]. Fact 2.21.25. Let n ≥ 3, let z1, . . . , zn be complex numbers, and, for all k ∈ {1, . . . , n}, define △ ∑ S k = |zi1 + · · · + zik |, where the sum is taken over all k-tuples (i1 , . . . , ik ) such that 1 ≤ i1 < · · · < ik ≤ n. Then, for all k ∈ {2, . . . , n − 1}, ( ) (n−2) S k ≤ n−2 k−1 S 1 + k−2 S n . In addition,

n−1 ∑

S i ≤ (2n−2 − 1)(S 1 + S n ).

i=2

Source: [2470]. Example: If n = 3, then |z1 + z2 | + |z2 + z3 | + |z3 + z1 | ≤ |z1 | + |z2 | + |z3 | + |z1 + z2 + z3 |. △

If n = 4, then, with z = z1 + z2 + z3 + z4 ,

|z1 + z2 | + |z1 + z3 | + |z1 + z4 | + |z2 + z3 | + |z2 + z4 | + |z3 + z4 | ≤ 2(|z1 | + |z2 | + |z3 | + |z4 |) + |z|, |z1 + z2 + z3 | + |z1 + z2 + z4 | + |z1 + z3 + z4 | + |z2 + z3 + z4 | ≤ |z1 | + |z2 | + |z3 | + |z4 | + 2|z|. Remark: These inequalities concern the diagonals of a polygon. Related: Fact 11.8.8. Fact 2.21.26. Let z1, . . . , zn , w1, . . . , wn be complex numbers. Then,

2 n n ∑ ∑ |zi − z j | + |wi − w j | + 2 (zi − wi ) = 2 |zi − w j |2 . i=1 i, j=1 i, j=1 i, j=1 n ∑

Equivalently,

n ∑

2

2

2 n n ∑ ∑ |zi − z j | + |wi − w j | + (zi − wi ) = |zi − w j |2 . i, j=1 i, j=1,i< j i, j=1,i< j i=1 n ∑

2

n ∑

2

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EQUALITIES AND INEQUALITIES

Source: [2093]. Remark: This is a generalized parallelogram law. Setting z1 = −z2 = z, z3 =

−z4 = w, and w1 = w2 = w3 = w4 = 0 yields the parallelogram law |z + w|2 + |z − w|2 = 2|z|2 + 2|w|2 given by xix) of Fact 2.21.8. Related: Fact 11.8.17. Fact 2.21.27. Let z be a complex number. Then, the following statements hold: i) 0 < |ez| ≤ e|z|. ii) |ez| = e|z| if and only if Im z = 0 and Re z ≥ 0. iii) |ez | = 1 if and only if Re z = 0. iv) |ez | − 1 ≤ |ez − 1| ≤ e|z| − 1. v) ez = eRe z [cos Im z + (sin Im z) ȷ]. vi) Re ez = 0 if and only if Im z is an odd integer multiple of ± π2 . vii) Im ez = 0 if and only if Im z is an integer multiple of ±π. viii) If z is nonzero, then |z ȷ | < eπ. ix) ȷ ȷ = e−π/2 . ∑ x) If θ is a real number and θ/(2π) is not an integer, then limk→∞ 1k ki=1 eiθ ȷ = 0. Furthermore, let θ1 and θ2 be real numbers. Then, the following statements hold: xi) |eθ1 ȷ − eθ2 ȷ | = 2| sin 12 (θ1 − θ2 )|, |eθ1 ȷ + eθ2 ȷ | = 2| cos 12 (θ1 − θ2 )|. xii) |eθ1 ȷ − eθ2 ȷ | ≤ |θ1 − θ2 |, and |eθ1 ȷ − eθ2 ȷ | = |θ1 − θ2 | if and only if θ1 = θ2 . θ ȷ θ ȷ xiii) |ee 1 − ee 2 | ≤ 2e| sin 21 (θ1 − θ2 )|.

xiv) | sinh eθ1 ȷ − sinh eθ2 ȷ | ≤ e e+1 | sin 12 (θ1 − θ2 )| ≤ e 2e+1 |θ1 − θ2 |. Finally, let r1 and r2 be nonnegative numbers, at least one of which is positive. Then, the following statement holds: 2|r1eθ1 ȷ − r2 eθ2 ȷ | xv) |eθ1 ȷ − eθ2 ȷ | ≤ . r1 + r2 Source: ix) is discussed in [1351, p. 48]; x) is given in [970, p. 503]; xi)–xiv) are given in [942]; xv) is given in [1391, p. 218]. Remark: A matrix version of x) is given by Fact 15.17.15. Fact 2.21.28. Let p, q ∈ (1, ∞), assume that 1/p + 1/q = 1, and let x and y be complex numbers such that |x| < 1 and |y| < 1. Then, 2

2

1 1 1 ≤ + , p p q 2 p(1 − |x| )(1 − |y| ) q(1 − |x| )(1 − |y|q ) |1 − xy| 1 1 1 ≤ + , p q q q−1 p−1 p(1 − |x| )(1 − |y| ) q(1 − |x| )(1 − |y| p ) |1 − x|y| ||1 − x|y| | 1 p p 1 q q 1 p q 1 q p q−1 p−1 |e xy |2 ≤ e|x| +|y| + e|x| +|y| , |e x|y| +x|y| | ≤ e|x| +|y| + e|x| +|y| , p q p q 1 1 | log(1 − xy)|2 ≤ log(1 − |x| p ) log(1 − |y| p ) + log(1 − |x|q ) log(1 − |y|q ), p q 1 1 | log(1 − x|y|q−1 ) log(1 − x|y| p−1 )| ≤ log(1 − |x| p ) log(1 − |y|q ) + log(1 − |x|q ) log(1 − |y| p ). p q Source: [1496]. △ Fact 2.21.29. Let z be a complex number. Define D = {z ∈ C: z , 0 and |z − 1| ≤ 1}. Then, the

following statements hold:

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i) For all z ∈ D, log z is given by the convergent series log z =

∞ ∑ (−1)i+1 i=1

i

(z − 1)i .

ii) If z ∈ C and −π < Im z ≤ π, then log e = z. iii) If z1, z2 ∈ D, then log z1 z2 = log z1 + log z2 . iv) If |z| < 1, then z

| log(1 + z)| ≤ − log(1 − |z|),

|z| |z|(1 + |z|) ≤ | log(1 + z)| ≤ . 1 + |z| |1 + z|

Remark: Let z = reθ ȷ ∈ C satisfy |z − 1| < 1. Then, − π2 < θ
0}, and, for all z1 , z2 ∈ OUHP, define △

f (z1 , z2 ) = 2 atanh

|z1 − z2 | + |z1 − z2 | |z1 − z2 | = log . |z1 − z2 | |z1 − z2 | − |z1 − z2 |

Then, the following statements hold: i) For all z1 , z2 ∈ OUHP, f (z1 , z2 ) ≥ 0. ii) Let z1 , z2 ∈ OUHP. Then, f (z1 , z2 ) = 0 if and only if z1 = z2 . iii) For all z1 , z2 , z3 ∈ OUHP, f (z1 , z3 ) ≤ f (z1 , z2 ) + f (z2 , z3 ). Remark: f is a Poincar´e metric. See [137]. Related: Fact 2.21.30.

2.22 Notes Reference works on inequalities include [340, 603, 604, 605, 764, 1316, 1952, 1969, 1971, 2061]. Texts on complex variables include [217, 582, 645, 1136, 1369, 1472, 1697, 2113, 2191]. A collection of identities involving binomial coefficients is given in [1208].

Chapter Three Basic Matrix Properties In this chapter we provide a detailed treatment of the basic properties of matrices, such as range, null space, rank, and invertibility. We also consider properties of convex sets, cones, and subspaces.

3.1 Vectors The set Fn consists of vectors x of the form

   x(1)    x =  ...  ,   x(n)

(3.1.1)

where x(1) , . . . , x(n) ∈ F are the components of x, and F represents either R or C. Hence, the elements of Fn are column vectors. Since F1 = F, it follows that every scalar is also a vector. If x ∈ Rn and every component of x is nonnegative, then x is nonnegative, and, if every component of x is positive, then x is positive. If α ∈ F and x ∈ Fn, then αx ∈ Fn is given by    αx(1)   .  αx =  ..  . (3.1.2)   αx(n) If x, y ∈ Fn, then x and y are linearly dependent if there exists α ∈ F such that either x = αy or y = αx. Furthermore, vectors add component by component; that is, if x, y ∈ Fn, then    x(1) + y(1)    .. (3.1.3) x + y =   . .   x(n) + y(n) Thus, if α, β ∈ F, then the linear combination αx + βy is given by    αx(1) + βy(1)    ..  . αx + βy =  .   αx(n) + βy(n)

(3.1.4)

If x ∈ Rn and x is nonnegative, then we write x ≥≥ 0, and, if x is positive, then we write x >> 0. If x, y ∈ Rn, then x ≥≥ y means that x − y ≥≥ 0, and x >> y means that x − y >> 0. △ Let S ⊆ Fn. For α ∈ F, define αS = {αx: x ∈ S}. We write −S for (−1)S. The set S is symmetric if S = −S; that is, x ∈ S if and only if −x ∈ S. For S1 , S2 ⊆ Fn, define the Minkowski sum △

S1 + S2 = {x + y: x ∈ S1 and y ∈ S2 }.

(3.1.5)

If S1 and S2 are multisets, then △

S1 + S2 = {x + y: x ∈ S1 and y ∈ S2 }ms .

(3.1.6)

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Note that S1 + S2 = S2 + S1 and, if S3 ⊆ Fn , then (S1 + S2 ) + S3 = S1 + (S2 + S3 ). Furthermore, △ for all y ∈ Fn and S ⊆ Fn, define y + S = {y} + S = {y + x: x ∈ S}. Note that, for all α, β ∈ F, (α + β)S ⊆ αS + βS. Finally, S + {0} = {0} + S = S and S + ∅ = ∅ + S = ∅. Let S1 ⊆ Fn and S2 ⊆ Fm. Then, the Cartesian product S1 × S2 ⊆ Fn × Fm is the subset of Fn+m given by [ ] [ ] [ ] S1 △ In 0n×m = S + S2 (3.1.7) S2 0m×n 1 Im and likewise for multisets. Let S ⊆ Fn. Then, S is a cone if, for all x ∈ S and α > 0, the vector αx is an element of S. Now, assume that S is a cone. Then, S is pointed if 0 ∈ S, while S is blunt if 0 < S. Furthermore, S is a one-sided cone if x, −x ∈ S implies that x = 0. Hence, S is one-sided if and only if S ∩ −S ⊆ {0}. The empty set is a cone. For x, y ∈ Fn and α ∈ [0, 1], the vector αx + (1 − α)y is a convex combination of x and y with barycentric coordinates α and 1 − α. The set S ⊆ Fn is a convex set if, for all x, y ∈ S, every convex combination of x and y is an element of S. The empty set is a convex set. Furthermore, S is a convex cone if it is a convex set and a cone. The empty set is a convex cone. For x, y ∈ Fn and α ∈ F, the vector αx + (1 − α)y is an affine combination of x and y. The set S ⊆ Fn is an affine subspace of Fn if, for all x, y ∈ S, every affine combination of x and y is an element of S. The empty set is an affine subspace. For x, y ∈ Fn and α, β ∈ F, the vector αx + βy is a linear combination of x and y. The set S ⊆ Fn is a subspace of Fn if, for all x, y ∈ S, every linear combination of x and y is an element of S. The empty set is a subspace. Note that R is a convex cone and a subspace of R2 . However, R is not a subspace of C. The following result shows how affine subspaces and subspaces are related. Proposition 3.1.1. Let S ⊆ Fn . Then, the following statements hold: i) S is an affine subspace if and only if there exist z, x1 , . . . , xr ∈ Fn such that  r      ∑  S=z+ αi xi : α1 , . . . , αr ∈ F . (3.1.8)     i=1

ii) S is a subspace if and only if there exist x1 , . . . , xr ∈ Fn such that  r      ∑  S= αi xi : α1 , . . . , αr ∈ F .    

(3.1.9)

i=1

iii) S is an affine subspace if and only if there exists z ∈ Fn such that S + z is a subspace. iv) S is a subspace if and only if there exists z ∈ Fn such that S + z is an affine subspace. Proof. The third statement is given in [1267, Theorem 4.4].  n n The affine subspaces S1 , S2 ⊆ F are parallel if there exists z ∈ F such that S1 + z = S2 . If S is an affine subspace, then there exists a unique subspace parallel to S. Proposition 1.3.15 implies that the greatest lower bound of a collection of sets is given by their intersection. Proposition 3.1.2. Let S be a collection of subsets of Fn, each of which is a (cone, convex set, convex cone, affine subspace, subspace). Then, the intersection glb(S) of all elements of S is a (cone, convex set, convex cone, affine subspace, subspace). Proof. The case of convex sets is given in [2487, p. 75].  Let S ⊆ Fn , and let S denote the collection of all (cones, convex sets, convex cones, affine

279

BASIC MATRIX PROPERTIES

subspaces, subspaces) that contain S. Then, Proposition 1.3.16 implies that the intersection S0 = glb(S) of all elements of S is the smallest (cone, convex set, convex cone, affine subspace, subspace) that contains S in the sense that, if S1 is a (cone, convex set, convex cone, affine subspace, subspace) that contains S, then S0 ⊆ S1 . Let S ⊆ Fn. Then, the conical hull of S, denoted by cone S, is the smallest cone in Fn containing S. The convex hull of S, denoted by conv S, is the smallest convex set containing S. The convex conical hull of S, denoted by coco S, is the smallest convex cone in Fn containing S. The affine hull of S, denoted by affin S, is the smallest affine subspace in Fn containing S, while the span of S, denoted by span S, is the smallest subspace in Fn containing S. If x, y ∈ R3 are distinct, then affin {x, y} is the line passing through x and y, while, if x, y, z ∈ R3 are not contained in a single line, then affin {x, y, z} is the plane passing through x, y, and z. Note that cone ∅ = conv ∅ = coco ∅ = affin ∅ = span ∅ = ∅. Let x1 , . . . , xr ∈ Fn. Then, x1 , . . . , xr are linearly independent if α1 , . . . , αr ∈ F and r ∑

αi x i = 0

(3.1.10)

i=1

imply that α1 = α2 = · · · = αr = 0. Note that x1 , . . . , xr are linearly independent if and only if x1, . . . , xr are linearly independent. If x1 , . . . , xr are not linearly independent, then x1 , . . . , xr are linearly dependent. Note that 0n×1 is linearly dependent. Let S ⊆ Fn, assume that S is nonempty, and assume that S is a subspace. If S is not equal to {0n×1 }, then there exist x1 , . . . , xr ∈ Fn such that x1 , . . . , xr are linearly independent over F and such that span {x1 , . . . , xr } = S. The set {x1 , . . . , xr } is a basis for S. The positive integer r is the dimension dim S of S. For n ≥ 0, we define dim {0n×1 } = 0. The subspace S is a hyperplane if dim S = n − 1. △ We define dim ∅ = −∞. Let S ⊆ Fn. If S is an affine subspace, then the dimension dim S of S is the dimension of the subspace parallel to affin S. If, however, S is not an affine subspace, then dim S is defined to be the dimension of the affine hull of S; that is, △

dim S = dim affin S.

(3.1.11)

The affine subspace S is an affine hyperplane if dim S = n − 1. The following result is the subspace dimension theorem. Theorem 3.1.3. Let S1 , S2 ⊆ Fn be subspaces. Then, dim(S1 + S2 ) + dim(S1 ∩ S2 ) = dim S1 + dim S2 .

(3.1.12)

 For the next result, note that “⊂” indicates proper inclusion. Proposition 3.1.4. Let S1 , S2 ⊆ Fn be affine subspaces such that S1 ⊆ S2 . Then, S1 ⊂ S2 if and only if dim S1 < dim S2 . Equivalently, S1 = S2 if and only if dim S1 = dim S2 . Corollary 3.1.5. Let S1 , S2 ⊆ Fn be subspaces such that S1 ⊆ S2 . Then, S1 ⊂ S2 if and only if dim S1 < dim S2 . Equivalently, S1 = S2 if and only if dim S1 = dim S2 . Let S1 , S2 ⊆ Fn be subspaces. Then, S1 and S2 are complementary subspaces if S1 + S2 = Fn and S1 ∩ S2 = {0}. In this case, we say that S1 is complementary to S2 , and vice versa. Corollary 3.1.6. Let S1 , S2 ⊆ Fn be subspaces, and consider the following statements: i) dim(S1 + S2 ) = n. ii) S1 ∩ S2 = {0}. iii) dim S1 + dim S2 = n. iv) S1 and S2 are complementary subspaces. Proof. See [1305, p. 227].

280

Then,

CHAPTER 3

[i), ii)] ⇐⇒ [i), iii)] ⇐⇒ [ii), iii)] ⇐⇒ [i), ii), iii)] ⇐⇒ [iv)].

The following result shows that cones can be used to induce relations on Fn. Proposition 3.1.7. Let S ⊆ Fn be a cone and, for x, y ∈ Fn, let x ≼ y denote the relation y − x ∈ S. Then, the following statements hold: i) “≼” is reflexive if and only if S is a pointed cone. ii) “≼” is antisymmetric if and only if S is a one-sided cone. iii) “≼” is symmetric if and only if S is a symmetric cone. iv) “≼” is transitive if and only if S is a convex cone. Proof. The proofs of i), ii), and iii) are immediate. To prove necessity in iv), let x, y ∈ S so that, for all α ∈ [0, 1], 0 ≼ αx ≼ αx + (1 − α)y. Since “≼” is transitive, it follows that αx + (1 − α)y ∈ S for all α ∈ [0, 1], and thus S is a convex set. Conversely, suppose that S is a convex cone, and assume ] [ that x ≼ y and y ≼ z. Then, y − x ∈ S and z − y ∈ S imply that z − x = 2 21 (y − x) + 12 (z − y) ∈ S. Hence, x ≼ z, and thus “≼” is transitive. 

3.2 Matrices The vectors x1 , . . . , xm ∈ Fn placed side by side form the matrix △

A = [x1 · · · xm ],

(3.2.1)

which has n rows and m columns. The components of the vectors x1 , . . . , xm are the entries of A. We write A ∈ Fn×m and say that A has size n × m. Since Fn = Fn×1, it follows that every vector is also a matrix. Note that F1×1 = F1 = F. If n = m, then A is square and has size n. The ith row of A and the jth column of A are denoted by rowi (A) and col j (A), respectively. Hence,    row1 (A)    .. (3.2.2) A =   = [col1 (A) · · · colm (A)]. .   rown (A) The entry x j(i) of A in both the ith row of A and the jth column of A is denoted by A(i, j) . Therefore, x ∈ Fn can be written as      x(1)   x(1,1)      (3.2.3) x =  ...  =  ...  .     x(n,1) x(n) Let A ∈ Fn×m. For b ∈ Fn, the matrix obtained from A by replacing coli (A) with b is denoted by i

A ← b.

(3.2.4)

Likewise, for b ∈ F , (3.2.4) denotes the matrix obtained from A by replacing rowi (A) with b. △ Let A ∈ Fn×m, and let l = min {n, m}. Then, the entries A(i,i) for all i ∈ {1, . . . , l} and A(i, j) for all i , j are the diagonal entries and off-diagonal entries of A, respectively. Moreover, for all i ∈ {1, . . . , l −1}, the entries A(i,i+1) and A(i+1,i) are the superdiagonal entries and subdiagonal entries of A, respectively. In addition, the entries A(i,l+1−i) for all i ∈ {1, . . . , l} are the reverse-diagonal entries of A. If the diagonal entries A(1,1) , . . . , A(l,l) of A are real, then the diagonal entries of A are labeled as 1×m

dl (A) ≤ · · · ≤ d1 (A),

(3.2.5)

281

BASIC MATRIX PROPERTIES

and we define

△

△

dmax (A) = d1(A),

dmin (A) = dl (A).

With this notation, the vector of diagonal entries of A is defined by    d1 (A)    △  d(A) =  ...  .   dl (A) Partitioned matrices are of the form

  A11  ..  .  Ak1

··· . · .· · ···

 A1l   ..  .  ,  Akl

(3.2.6)

(3.2.7)

(3.2.8)

where, for all i ∈ {1, . . . , k} and j ∈ {1, . . . , l}, the block Ai j of A is a matrix of size ni × m j . If ni = m j and the diagonal entries of Ai j lie on the diagonal of A, then the square matrix Ai j is a diagonally located block; otherwise, Ai j is an off-diagonally located block. Let A ∈ Fn×m. Then, a submatrix of A is formed by deleting rows and columns of A. In particular, A is a submatrix of A, as is the matrix obtained by deleting rows and columns of A. Alternatively, a submatrix can be specified in terms of the rows and columns that are retained. If A is a partitioned matrix, then every block of A is a submatrix of A. A block of A is thus a submatrix of A whose entries are entries of adjacent rows and adjacent columns of A. If like-numbered rows and columns of A are retained, then the resulting square submatrix of A is a principal submatrix of A. Every diagonally located block is a principal submatrix. Finally, if rows and columns 1, . . . , j of A are retained, then the resulting j × j submatrix of A is a leading principal submatrix of A. Let A ∈ Fn×m, and let S1 and S2 be subsets of {1, . . . , n} and {1, . . . , m}, respectively. Then, A(S1 ,S2 ) is the card(S1 ) × card(S2 ) submatrix of A formed by retaining the rows of A listed in S1 and the columns of A listed in S2 . Hence, A(i, j) denotes A({i},{ j}) . If S ⊆ {1, . . . , min{n, m}}, then we define △ A(S) = A(S,S) , which is a principal submatrix of A. Furthermore, A(S1 ,·) denotes the submatrix of A formed by retaining the rows listed in S1 , and A(·,S2 ) denotes the submatrix of A formed by retaining the columns listed in S2 . Likewise, A[S1 ,S2 ] is the [n − card(S1 )] × [m − card(S2 )] submatrix of A formed by deleting the rows of A listed in S1 and the columns of A listed in S2 . In particular, A[i, j] denotes the (n − 1) × (m − 1) submatrix of A formed by deleting the ith row and jth column of A. Furthermore, A[S1 ,·] denotes the submatrix of A formed by deleting the rows listed in S1 , and A[·,S2 ] denotes the submatrix of A formed by deleting the columns listed in S2 . If S ⊆ {1, . . . , min{n, m}}, △ then we define A[S] = A[S,S] . Let A, B ∈ Fn×m. Then, A and B add entry by entry; that is, for all i ∈ {1, . . . , n} and j ∈ {1, . . . , m}, (A + B)(i, j) = A(i, j) + B(i, j) . Furthermore, for all i ∈ {1, . . . , n} and j ∈ {1, . . . , m}, it follows that, for all α ∈ F, (αA)(i, j) = αA(i, j) . Hence, for all α, β ∈ F, (αA + βB)(i, j) = αA(i, j) + βB(i, j) . If A, B ∈ Fn×m, then A and B are linearly dependent if there exists α ∈ F such that either A = αB or B = αA. Let A ∈ Rn×m. If every entry of A is nonnegative, then A is nonnegative, which is written as A ≥≥ 0. If every entry of A is positive, then A is positive, which is written as A >> 0. If A, B ∈ Rn×m, then A ≥≥ B means that A − B ≥≥ 0, while A >> B means that A − B >> 0. Let z ∈ F1×n and y ∈ Fn = Fn×1 . Then, the scalar zy ∈ F is defined by △

zy =

n ∑ i=1

z(1,i) y(i) .

(3.2.9)

282

CHAPTER 3

Now, let A ∈ Fn×m and x ∈ Fm. Then, the matrix-vector product Ax is defined by    row1(A)x    △  ..  . Ax =  .   rown (A)x It can be seen that Ax is a linear combination of the columns of A, that is, m ∑ Ax = x(i) coli (A).

(3.2.10)

(3.2.11)

i=1 △

Let A ∈ Fn×m. Then, A can be associated with the function f : Fm 7→ Fn defined by f (x) = Ax for all x ∈ Fm. For all α, β ∈ F and x, y ∈ Fm, it follows that f (αx + βy) = α f (x) + β f (y) = αAx + βAy.

(3.2.12)

Therefore, f is linear. For all S ⊆ F , we define m

△

AS = f (S) = {Ax : x ∈ S},

(3.2.13)

and, for all S ⊆ F , we define n

△

Ainv (S) = f inv (S) = {x ∈ Fm : Ax ∈ S}.

(3.2.14)

Note that A∅ = A (∅) = ∅. Now, let b ∈ F . Then, the function f : F 7→ F defined by inv

n

m

△

f (x) = Ax + b

n

(3.2.15)

is affine. Theorem 3.2.1. Let A ∈ Fn×m and B ∈ Fm×l, and define f : Fm 7→ Fn and g: Fl 7→ Fm by △

△

△

f (x) = Ax and g(y) = By. Furthermore, define the composition h = f ◦ g : Fl 7→ Fn. Then, for all y ∈ Rl, h(y) = f [g(y)] = A(By) = (AB)y, (3.2.16) where, for all i ∈ {1, . . . , n} and j ∈ {1, . . . , l}, AB ∈ Fn×l is defined by △

(AB)(i, j) =

m ∑

A(i,k) B(k, j) .

(3.2.17)

k=1

Hence, we write ABy for (AB)y and A(By). Let A ∈ Fn×m and B ∈ Fm×l. Then, AB ∈ Fn×l is the product of A and B. The matrices A and B are conformable, and the product (3.2.17) defines matrix multiplication. Let A ∈ Fn×m and B ∈ Fm×l. Then, AB can be written as    row1 (A)B    .. AB = [Acol1 (B) · · · Acoll (B)] =  (3.2.18)  . .   rown (A)B Thus, for all i ∈ {1, . . . , n} and j ∈ {1, . . . , l}, (AB)(i, j) = rowi (A)col j (B), col j (AB) = Acol j (B),

(3.2.19) (3.2.20)

rowi (AB) = rowi (A)B.

(3.2.21)

283

BASIC MATRIX PROPERTIES

For conformable matrices A, B, C, the associative and distributive equalities (AB)C = A(BC), A(B + C) = AB + AC,

(3.2.22) (3.2.23)

(A + B)C = AC + BC

(3.2.24)

are valid. Hence, we write ABC for (AB)C and A(BC). Note that (3.2.22) is a special case of (1.6.3). For S1 ⊆ Fn×m and S2 ⊆ Fm×l, define the Minkowski product △

S1 S2 = {AB : A ∈ S1 and B ∈ S2 }.

(3.2.25)

If S1 and S2 are multisets, then △

S1 S2 = {AB : A ∈ S1 and B ∈ S2 }ms .

(3.2.26)

In particular, for A ∈ Fn×m and S ⊆ Fm×l, AS = {A}S = {AB : B ∈ S}, which generalizes (3.2.13). Let A, B ∈ Fn×n. Then, the commutator [A, B] ∈ Fn×n of A and B is the matrix △

[A, B] = AB − BA. The adjoint operator adA : F

n×n

7→ F

n×n

(3.2.27)

is defined by △

adA(X) = [A, X].

(3.2.28)

Let x, y ∈ R3. Then, the cross product x × y ∈ R3 of x and y is defined by    x(2) y(3) − x(3) y(2)   △  x × y =  x(3) y(1) − x(1) y(3)  .   x(1) y(2) − x(2) y(1) Furthermore, the 3 × 3 cross-product matrix is defined by    0 −x(3) x(2)   △  0 −x(1)  . K(x) =  x(3)   −x(2) x(1) 0 Note that

x × y = K(x)y.

(3.2.29)

(3.2.30)

(3.2.31)

Multiplication of partitioned matrices is analogous to matrix multiplication with scalar entries. For example, for matrices with conformable blocks, [ ] [ ] C AC [A B] = AC + BD, [A B]C = , (3.2.32) D BC [ ] [ ] [ ][ ] [ ] A AC AD A B E F AE + BG AF + BH [C D] = , = . (3.2.33) B BC BD C D G H CE + DG CF + DH The n × m zero matrix, all of whose entries are zero, is written as 0n×m. If the dimensions are unambiguous, then we write just 0. Let x ∈ Fm and A ∈ Fn×m. Then, the zero matrix satisfies 0k×m x = 0k×1 ,

A0m×l = 0n×l ,

0k×n A = 0k×m .

(3.2.34)

Another special matrix is the empty matrix. For n ∈ N, the 0 × n empty matrix, which is written as 00×n, has zero rows and n columns, while the n × 0 empty matrix, which is written as 0n×0 , has n rows and zero columns. For A ∈ Fn×m, where n, m ∈ N, the empty matrix satisfies the multiplication rules 00×n A = 00×m , A0m×0 = 0n×0 . (3.2.35)

284

CHAPTER 3

Although empty matrices have no entries, it is useful to define the product △

0n×0 00×m = 0n×m . Also, we define For n, m ∈ N, we define F

(3.2.36)

△ △ I0 = Iˆ0 = 00×0 .

0×m △

n×0 △

(3.2.37) 0 △

= {00×m }, F = {0n×0 }, and F = F [ ] 0n×0 0n×m = 0n×m . 00×0 00×m

. Note that

0×1

(3.2.38)

The empty matrix is analogous to 0 for real numbers and ∅ for sets. The n × n identity matrix, which has 1’s on the diagonal and 0’s elsewhere, is denoted by In or just I. Let x ∈ Fn and A ∈ Fn×m. Then, the identity matrix satisfies In x = x, 2 △

AIm = In A = A. k △

(3.2.39) △

Let A ∈ Fn×n. Then, A = AA and, for all k ≥ 1, A = AAk−1. We use the convention A0 = I. In particular, 00n×n = In . The n×n reverse permutation matrix, which has 1’s on the reverse diagonal and 0’s elsewhere, is △ denoted by Iˆn or just I.ˆ In particular, Iˆ1 = 1. Multiplication of x ∈ Fn by Iˆn reverses the components of x. Likewise, left multiplication of A ∈ Fn×m by Iˆn reverses the rows of A, while right multiplication of A by Iˆm reverses the columns of A. Note that Iˆn2 = In . △ The n × n cyclic permutation matrix Pn is defined by P1 = 1 and, for all n ≥ 2,   0 · · · 0 0   0 1    ..  . 0 0  1  0 0   ..  0 0  . 0 0 0 △  . Pn =   .. . . . . ..  .. ..  . . . . . .    . . . 0 1   0 0 0   1 0 0 ··· 0 0 Note that P1 = Iˆ1, P2 = Iˆ2 , and, for all n ≥ 1, Pnn = In . △ △ The n × n standard nilpotent matrix Nn , or just N, is defined by N0 = 00×0 , N1 = 0, and, for all n ≥ 2,   0 · · · 0 0   0 1    ..  . 0 0  1  0 0   ..  0 0  . 0 0 0 △  . Nn =   .. . . . . ..  .. ..  . . . . . .    ..   0 0 . 0 0 1   0 0 0 ··· 0 0 Note that, for all n ≥ 0, Nnn = 0.

285

BASIC MATRIX PROPERTIES

3.3 Transpose and Inner Product A fundamental vector and matrix operation is the transpose. If x ∈ Fn, then the transpose xT of x is defined to be the row vector △

xT = [x(1) · · · x(n) ] ∈ F1×n. Similarly, if x = [x(1,1) · · · x(1,n) ] ∈ F1×n, then    x(1,1)     xT =  ...  ∈ Fn×1.   x(1,n)

(3.3.1)

(3.3.2)

Let x, y ∈ Fn. Then, xTy ∈ F is a scalar, and xTy = yTx =

n ∑

x(i) y(i) .

(3.3.3)

i=1

Note that xTx =

n ∑

2 x(i) .

(3.3.4)

i=1

The vector ei,n ∈ Rn, or just ei , has 1 as its ith component and 0’s elsewhere. Thus, ei,n = coli (In ).

(3.3.5)

Let A ∈ Fn×m. Then, eTi A = rowi (A) and Aei = coli (A). Furthermore, the (i, j) entry of A can be written as A(i, j) = eTi Ae j .

(3.3.6)

The n × m matrix Ei, j,n×m ∈ Rn×m, or just Ei, j , has 1 as its (i, j) entry and 0’s elsewhere. Thus, Ei, j,n×m = ei,n eTj,m . Note that Ei,1,n×1 = ei,n and In = E1,1 + · · · + En,n =

(3.3.7)

n ∑

ei eTi .

(3.3.8)

i=1

Finally, the n × m ones matrix, all of whose entries are 1, is written as 1n×m or just 1. Thus, 1n×m =

n,m ∑

Ei, j,n×m .

(3.3.9)

i, j=1

Note that 1n×1 =

n ∑

ei,n

i=1

   1  . =  ..  ,   1

1n×m = 1n×1 11×m .

(3.3.10)

Lemma 3.3.1. Let x ∈ R. Then, xTx = 0 if and only if x = 0.

Let x, y ∈ Rn. Then, xTy ∈ R is the inner product of x and y. Furthermore, x and y are mutually orthogonal if xTy = 0. If x and y are nonzero, then the angle θ ∈ [0, π] between x and y is defined by △

θ = acos √

xTy xTxyTy

.

(3.3.11)

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CHAPTER 3

Note that x and y are mutually orthogonal if and only if θ = π2 . Let x ∈ Cn. Then, x = y + ȷz, where y, z ∈ Rn. Therefore, the transpose xT of x is given by xT = yT + ȷzT.

(3.3.12)

The complex conjugate x of x is defined by △

x = y − ȷz,

(3.3.13)

while the complex conjugate transpose x∗ of x is defined by △

x∗ = xT = yT − ȷzT.

(3.3.14)

The vectors y and z are the real and imaginary parts Re x and Im x of x, respectively, which are defined by △ △ (3.3.15) Re x = 21 (x + x) = y, Im x = 21ȷ (x − x) = z. Note that x∗x =

n ∑

x(i) x(i) =

n ∑

|x(i) |2 =

n ( ∑

i=1

i=1

) y2(i) + z2(i) .

(3.3.16)

i=1

If w, x ∈ Cn, then wTx = xTw. Let x ∈ Fn. Then, the Euclidean norm of x is defined by 1/2  n  ∑ √ △  2  ∥x∥2 =  |x(i) |  = x∗x.

(3.3.17)

i=1

If x ∈ Rn, then

  n ∑ 2 1/2 √  x(i)  = xTx. ∥x∥2 = 

(3.3.18)

i=1

Lemma 3.3.2. Let x ∈ Cn. Then, x∗x = 0 if and only if x = 0.

Let x, y ∈ Cn. Then, x∗y ∈ C is the inner product of x and y, which is given by x∗y =

n ∑

x(i) y(i) .

(3.3.19)

i=1

Furthermore, x and y are mutually orthogonal if x∗y = 0. If x and y are nonzero, then the angle θ ∈ [0, π] between x and y is defined by Re x∗y △ θ = acos √ ∗ ∗ . x xy y

(3.3.20)

Note that x and y are mutually orthogonal if and only if θ = π2 . It follows from the Cauchy-Schwarz inequality given by Corollary 11.1.7 that the arguments of acos in (3.3.11) and (3.3.20) are elements of the interval [−1, 1]. Furthermore, θ ∈ R(acos) = [0, π]. Let A ∈ Fn×m. Then, the transpose AT ∈ Fm×n of A is defined by    [col1 (A)]T    △ .. AT = [[row1(A)]T · · · [rown (A)]T ] =  (3.3.21)  . .   T [colm (A)] Note that (AT )T = A. Furthermore, for all i ∈ {1, . . . , n}, coli (AT ) = [rowi (A)]T ; for all j ∈ {1, . . . , m}, row j (AT ) = [col j (A)]T ; and, for all i ∈ {1, . . . , n} and j ∈ {1, . . . , m}, (AT )( j,i) = A(i, j) . If B ∈ Fm×l, then (AB)T = BTAT.

(3.3.22)

287

BASIC MATRIX PROPERTIES

In particular, if x ∈ Fm, then (Ax)T = xTAT,

(3.3.23)

and if, in addition, y ∈ F , then y Ax is a scalar and n

If B ∈ F

T

yTAx = (yTAx)T = xTATy.

(3.3.24)

(αA + βB)T = αAT + βBT.

(3.3.25)

, then, for all α, β ∈ F,

n×m

Let S ⊆ Cn×m. Then,

△

ST = {AT : A ∈ S},

(3.3.26)

and likewise for multisets. Let x ∈ Fn and y ∈ Fm. Then, the matrix xyT ∈ Fn×m is the outer product of x and y. The outer product xyT is nonzero if and only if both x and y are nonzero. Let A ∈ Fn×m and B ∈ Fm×l . Then, m ∑ AB = coli (A)rowi B. (3.3.27) i=1

Therefore, the product of two matrices can be written as the sum of outer-product matrices. The trace of a square matrix A ∈ Fn×n, denoted by tr A, is defined to be the sum of its diagonal entries; that is, n ∑ △ tr A = A(i,i) . (3.3.28) i=1

Note that

tr A = tr AT.

(3.3.29)

Let A ∈ Fn×m and B ∈ Fm×n. Then, AB and BA are square, tr AB = tr BA = tr ATBT = tr BTAT =

n,m ∑

A(i, j) B( j,i) =

i, j=1

and

m ∑

rowi (A)coli B,

(3.3.30)

i=1 n,m ∑

A2(i, j) .

(3.3.31)

tr(αA + βB) = α tr A + β tr B.

(3.3.32)

tr AAT = tr ATA =

i, j=1

Furthermore, if n = m, then, for all α, β ∈ F, Lemma 3.3.3. Let A ∈ Rn×m. Then, tr ATA = 0 if and only if A = 0.

Let A, B ∈ Rn×m. Then, the inner product of A and B is tr ATB. Furthermore, A and B are mutually orthogonal if tr ATB = 0. Let C ∈ Cn×m. Then, C = A + ȷB, where A, B ∈ Rn×m. Then, the transpose CT of C is given by CT = AT + ȷBT.

(3.3.33)

The complex conjugate C of C is △

C = A − ȷB,

(3.3.34)

288

CHAPTER 3

while the complex conjugate transpose C ∗ of C is T

△

C ∗ = C = AT − ȷBT.

(3.3.35)

Note that C = C if and only if B = 0, and that (CT )T = C = (C ∗ )∗ = C.

(3.3.36)

The matrices A and B are the real and imaginary parts Re C and Im C of C, respectively, which are denoted by △ △ Re C = 12 (C + C) = A, Im C = 21ȷ (C − C) = B. (3.3.37) If C is square, then tr C = tr A + (tr B) ȷ = tr C T = tr C = tr C ∗. Let S ⊆ Cn×m. Then,

△

(3.3.38)

△

S∗ = {A∗ : A ∈ S},

S = {A: A ∈ S},

(3.3.39)

and likewise for multisets. Let A ∈ Fn×n. Then, for all k ∈ N, k

△

AkT = (Ak )T = (AT )k,

Ak = A ,

△

Ak∗ = (Ak )∗ = (A∗ )k.

(3.3.40)

Lemma 3.3.4. Let A ∈ Cn×m. Then, tr A∗A = 0 if and only if A = 0.

Let A, B ∈ Cn×m. Then, the inner product of A and B is tr A∗B. Furthermore, A and B are mutually orthogonal if tr A∗B = 0. If A, B ∈ Cn×m, then, for all α, β ∈ C, (αA + βB)∗ = αA∗ + βB∗, while, if A ∈ C

n×m

and B ∈ C

(3.3.41)

, then

m×l

(AB)∗ = B∗A∗.

AB = A B, In particular, if A ∈ C

n×m

(3.3.42)

and x ∈ C , then m

(Ax)∗ = x∗A∗,

(3.3.43)

y∗Ax = (y∗Ax)T = xTAT y

(3.3.44)

(y∗Ax)∗ = (y∗Ax)T = (yT Ax)T = x∗A∗y.

(3.3.45)

while, if, in addition, y ∈ C , then n

and

For A ∈ Fn×m, define the reverse transpose of A by ˆ △ AT = Iˆm ATIˆn

(3.3.46)

and the reverse complex conjugate transpose of A by △ A∗ˆ = Iˆm A∗Iˆn .

For example,

Furthermore,

[

1 2 4 5

3 6

]Tˆ

  6  =  5  4

(3.3.47)  3   2  .  1

ˆ ˆ (A∗ )∗ˆ = (A∗ˆ )∗ = (AT )T = (AT )T = Iˆn AIˆm ,

(3.3.48)

(A∗ˆ )∗ˆ = (AT )T = A. ˆ ˆ

(3.3.49)

289

BASIC MATRIX PROPERTIES

Finally, if B ∈ Fm×l, then (AB)∗ˆ = B∗ˆA∗ˆ , Let S ⊆ Cn×m. Then,

ˆ △

ˆ

ˆ ˆ

(AB)T = BTAT.

(3.3.50)

△

S∗ˆ = {A∗ˆ : A ∈ S},

ˆ

ST = {AT : A ∈ S},

(3.3.51)

and likewise for multisets. For x ∈ Fm and A ∈ Fn×m, every component of x can be replaced by its absolute value to obtain |x| ∈ Rm , where, for all i ∈ {1, . . . , n}, △

|x|(i) = |x(i) |, and every entry of A can be replaced by its absolute value to obtain |A| ∈ R i ∈ {1, . . . , n} and j ∈ {1, . . . , m}, △

(3.3.52) , where, for all

n×m

|A|(i, j) = |A(i, j) |.

(3.3.53)

|Ax| ≤≤ |A||x|,

(3.3.54)

|AB| ≤≤ |A||B|.

(3.3.55)

Note that and, if B ∈ F

, then

m×l

For x ∈ Rn and A ∈ Rn×m, every component of x can be replaced by its sign to obtain sign x ∈ Rn , where, for all i ∈ {1, . . . , n}, △

(sign x)(i) = sign x(i) ,

(3.3.56)

and every entry of A can be replaced by its sign to obtain sign A ∈ Rn×m , where, for all i ∈ {1, . . . , n} and j ∈ {1, . . . , m}, △ (sign A)(i, j) = sign A(i, j) . (3.3.57) Let S1 , S2 ⊆ Fn be sets. Then, S1 and S2 are mutually orthogonal if x∗y = 0 for all x ∈ S1 and y ∈ S2 . Let S ⊆ Fn be nonempty. Then, the orthogonal complement S⊥ of S is defined by △

S⊥ = {x ∈ Fn : x∗y = 0 for all y ∈ S}.

(3.3.58)

The orthogonal complement S⊥ of S is a subspace. Furthermore, S and S⊥ are mutually orthogonal. Proposition 3.3.5. Let S1 , S2 ⊆ Fn be mutually orthogonal sets. Then, S1 ∩ S2 ⊆ {0}.

(3.3.59)

S1 ∩ S2 = {0}.

(3.3.60)

Furthermore, 0 ∈ S1 ∩ S2 if and only if Proposition 3.3.5 shows that, if S1 and S2 are mutually orthogonal subspaces, then (3.3.60) holds. Let S1 , S2 ⊆ Fn be subspaces. Then, S1 and S2 are orthogonally complementary if S1 and S2 are complementary and mutually orthogonal. Proposition 3.3.6. Let S1 , S2 ⊆ Fn be subspaces. Then, S1 and S2 are orthogonally complementary if and only if S1 = S⊥2 . If these conditions hold, then dim S1 + dim S2 = n. Corollary 3.3.7. Let S ⊆ Fn be a subspace. Then, S and S⊥ are orthogonally complementary.

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CHAPTER 3

3.4 Geometrically Defined Sets Let S ⊆ Fn . Then, S is a hyperplane if and only if there exists a nonzero vector y ∈ Fn such that S = {y}⊥. Furthermore, S is an affine hyperplane if and only if there exists z ∈ Fn such that S + z is a hyperplane. Let S ⊆ Fn. Then, the dual cone of S is defined by △

dcone S = {x ∈ Fn : Re x∗y ≤ 0 for all y ∈ S}.

(3.4.1)

Note that dcone S is a pointed convex cone. Furthermore, dcone S = dcone conv S = dcone cone S = dcone coco S.

(3.4.2)

The set {x ∈ Fn : Re x∗y ≤ 0} = dcone {y} is a closed half space, while the set {x ∈ Fn : Re x∗y < 0} is an open half space. Furthermore, S is an affine (closed, open) half space if there exists z ∈ Fn such that S + z is a (closed, open) half space. Let S ⊆ Fn . Then, S is a polytope if S is the union of a finite number of sets, each of which is the intersection of a finite number of closed half spaces. If F = R and dim S = n = 3, then S is a polyhedron, whereas, if F = R and dim S = n = 2, then S is a polygon. Note that every polytope, and thus every polyhedron and polygon, is a closed set (see Definition 12.1.6) that is not necessarily convex, connected, or bounded (see Definition 12.1.13). If S ⊂ Fn , card S = n + 1, and dim S = n, then the convex polytope conv S is a simplex. S is a polyhedral cone if S is a cone and a polytope. S is a zonotope if there exist z, x1 , . . . , xm ∈ Fn such that   m      ∑ . αi xi : 0 ≤ αi ≤ 1 for all i ∈ {1, . . . , m} S=z+     i=1

S is a zonotope if and only if S is the Minkowski sum of a finite number of line segments. S is a parallelotope if S is a zonotope and dim S = m = n. If F = R and dim S = n = 3, then the parallelotope S is a parallelepiped, while, if F = R and dim S = n = 2, then the parallelotope S is a parallelogram. Let S ⊆ Fn. Then, △ polar S = {x ∈ Fn : Re x∗y ≤ 1 for all y ∈ S} (3.4.3) is the polar of S. Note that polar S is a convex set and polar S = polar conv S.

(3.4.4)

3.5 Range and Null Space Two key features of a matrix A ∈ Fn×m are its range and null space, denoted by R(A) and N(A), respectively. The range of A is defined by △

R(A) = {Ax: x ∈ Fm } = AFm.

(3.5.1)

Note that, for all nonnegative n and m, it follows that R(0n×0 ) = {0n×1 } and R(00×m ) = {00×1 }. Letting αi denote x(i) , it can be seen that  m      ∑  R(A) =  α col (A): α , . . . , α ∈ F , (3.5.2)  i i 1 m     i=1

which shows that R(A) is a subspace of Fn. It follows from Fact 3.11.5 that R(A) = span {col1 (A), . . . , colm(A)}.

(3.5.3)

291

BASIC MATRIX PROPERTIES

The null space of A ∈ Fn×m is defined by △

N(A) = {x ∈ Fm : Ax = 0}.

(3.5.4)

Note that N(0n×0 ) = F = {00×1 } and N(00×m ) = F . Equivalently, { } N(A) = x ∈ Fm : xT [rowi (A)]T = 0 for all i ∈ {1, . . . , n} { }⊥ = [row1(A)]T, . . . , [rown (A)]T , 0

m

(3.5.5) (3.5.6)

which shows that N(A) is a subspace of F . Note that, if α ∈ F is nonzero, then R(αA) = R(A) and N(αA) = N(A). Finally, if F = C, then R(A) and R(A) are not necessarily identical. For example, △ [ ] let A = 1ȷ . Let A ∈ Fn×n, and let S ⊆ Fn be a subspace. Then, S is an invariant subspace of A if AS ⊆ S. Note that AR(A) ⊆ AFn = R(A) and AN(A) = {0n } ⊆ N(A). Hence, R(A) and N(A) are invariant subspaces of A. If A ∈ Fn×m and B ∈ Fm×l, then m

R(AB) = AR(B). Lemma 3.5.1. Let A ∈ F

,B∈F

n×m

, and C ∈ F

m×l

(3.5.7)

. Then,

k×n

R(AB) ⊆ R(A),

(3.5.8)

N(A) ⊆ N(CA).

(3.5.9)

Proof. Since R(B) ⊆ F , it follows that R(AB) = AR(B) ⊆ AF = R(A). Furthermore, y ∈ N(A) implies that Ay = 0, and thus CAy = 0.  n×n Corollary 3.5.2. Let A ∈ F , and let k ≥ 1. Then, m

m

R(Ak ) ⊆ R(A),

(3.5.10)

N(A) ⊆ N(Ak ).

(3.5.11)

Although R(AB) ⊆ R(A) for arbitrary conformable matrices A, B, we now show that equality holds in the special case B = A∗. This result, along with others, is the subject of the following basic theorem. Theorem 3.5.3. Let A ∈ Fn×m. Then, the following statements hold: i) R(A)⊥ = N(A∗ ). ii) R(A) = R(AA∗ ). iii) N(A) = N(A∗A). Proof. To prove i), we first show that R(A)⊥ ⊆ N(A∗ ). Let x ∈ R(A)⊥. Then, x∗z = 0 for all z ∈ R(A). Hence, x∗Ay = 0 for all y ∈ Rm. Equivalently, y∗A∗x = 0 for all y ∈ Rm. Letting y = A∗x, it follows that x∗AA∗x = 0. Now, Lemma 3.3.2 implies that A∗x = 0. Thus, x ∈ N(A∗ ). Conversely, let us show that N(A∗ ) ⊆ R(A)⊥. Letting x ∈ N(A∗ ), it follows that A∗x = 0, and, hence, y∗A∗x = 0 for all y ∈ Rm. Equivalently, x∗Ay = 0 for all y ∈ Rm. Hence, x∗z = 0 for all z ∈ R(A). Thus, x ∈ R(A)⊥, which proves i). To prove ii), note that Lemma 3.5.1 with B = A∗ implies that R(AA∗ ) ⊆ R(A). To show that R(A) ⊆ R(AA∗ ), let x ∈ R(A), and suppose that x < R(AA∗ ). Then, it follows from Proposition 3.3.6 that x = x1 + x2 , where x1 ∈ R(AA∗ ) and x2 ∈ R(AA∗ )⊥ with x2 , 0. Thus, x2∗ AA∗y = 0 for all y ∈ Rn, and setting y = x2 yields x2∗ AA∗x2 = 0. Hence, Lemma 3.3.2 implies that A∗x2 = 0, so that, by i), x2 ∈ N(A∗ ) = R(A)⊥. Since x ∈ R(A), it follows that 0 = x2∗ x = x2∗ x1 + x2∗ x2 . However, x2∗ x1 = 0 so that x2∗ x2 = 0 and x2 = 0, which is a contradiction. This proves ii).

292

CHAPTER 3

To prove iii), note that ii) with A replaced by A∗ implies that R(A∗A)⊥ = R(A∗ )⊥ . Furthermore, replacing A by A∗ in i) yields R(A∗ )⊥ = N(A). Hence, N(A) = R(A∗A)⊥. Now, i) with A replaced by A∗A implies that R(A∗A)⊥ = N(A∗A). Hence, N(A) = N(A∗A), which proves iii).  i) of Theorem 3.5.3 can be written equivalently as N(A)⊥ = R(A∗ ), N(A) = R(A∗ )⊥ ,

(3.5.12) (3.5.13)

N(A∗ )⊥ = R(A),

(3.5.14)

∗

while replacing A by A in ii) and iii) of Theorem 3.5.3 yields R(A∗ ) = R(A∗A), N(A∗ ) = N(AA∗ ).

(3.5.15) (3.5.16)

Using ii) of Theorem 3.5.3 and (3.5.15), it follows that R(AA∗A) = AR(A∗A) = AR(A∗ ) = R(AA∗ ) = R(A).

(3.5.17)

△

Letting A = [1 ȷ] shows that R(A) and R(AAT ) may be different.

3.6 Rank and Defect Let A ∈ Fn×m. Then, the rank of A is defined by △

rank A = dim R(A).

(3.6.1)

It can be seen that the rank of A is equal to the number of linearly independent columns of A over F. For example, if F = C, then rank [1 ȷ] = 1, and, if either F = R or F = C, then rank [1 1] = 1. If all of the entries of A are real, then rank A is the same whether F in (3.5.2) is chosen to be either R or C. Furthermore, rank A = rank A, rank AT = rank A∗, rank A ≤ m, and rank AT ≤ n. If rank A = m, then A has full column rank, and, if rank AT = n, then A has full row rank. If A has either full column rank or full row rank, then A has full rank; otherwise, A is rank deficient. For all nonnegative n and m, rank(0n×m ) = rank(0n×0 ) = rank(00×m ) = 0. Finally, the defect of A is △

def A = dim N(A).

(3.6.2)

The following result follows from Theorem 3.5.3. Corollary 3.6.1. Let A ∈ Fn×m. Then, the following statements hold: i) rank A∗ + def A = m. ii) rank A = rank AA∗. iii) def A = def A∗A. Proof. (3.5.12) and Proposition 3.1.6 imply that rank A∗ = dim R(A∗ ) = dim N(A)⊥ = m − dim N(A) = m − def A, which proves i). ii) and iii) follow from ii) and iii) of Theorem 3.5.3.  ∗ Replacing A by A in Corollary 3.6.1 yields rank A + def A∗ = n, ∗

∗

rank A = rank A A, def A∗ = def AA∗. Furthermore, note that Lemma 3.6.2. Let A ∈ F

def A = def A, n×m

and B ∈ F

def AT = def A∗.

(3.6.3) (3.6.4) (3.6.5) (3.6.6)

. Then,

m×l

rank AB ≤ min {rank A, rank B}.

(3.6.7)

293

BASIC MATRIX PROPERTIES

Proof. Since, by Lemma 3.5.1, R(AB) ⊆ R(A), it follows that rank AB ≤ rank A. Next, suppose △ that rank B < rank AB. Let {y1 , . . . , yr } ⊂ Fn be a basis for R(AB), where r = rank AB, and, since yi ∈ AR(B) for all i ∈ {1, . . . , r}, let xi ∈ R(B) be such that yi = Axi for all i ∈ {1, . . . , r}. Since rank B < r, it follows that x1 , . . . , xr are linearly dependent. Hence, there exist α1 , . . . , αr ∈ F, not ∑ ∑ ∑ all zero, such that ri=1 αi xi = 0, which implies that ri=1 αi Axi = ri=1 αi yi = 0. Thus, y1 , . . . , yr are linearly dependent, which is a contradiction.  n×m Corollary 3.6.3. Let A ∈ F . Then,

rank A = rank A∗ , ∗

Therefore, If, in addition, n = m, then

(3.6.8)

def A = def A + m − n.

(3.6.9)

rank A = rank A∗A.

(3.6.10)

def A = def A∗.

(3.6.11)

Proof. It follows from (3.6.7) with B = A∗ that rank AA∗ ≤ rank A∗. Furthermore, ii) of Corollary

3.6.1 implies that rank A = rank AA∗. Hence, rank A ≤ rank A∗. Interchanging A and A∗ and repeating this argument yields rank A∗ ≤ rank A. Hence, rank A = rank A∗. Next, i) of Corollary 3.6.1, (3.6.8), and (3.6.3) imply that def A = m −rank A∗ = m −rank A = m − (n −def A∗ ), which proves (3.6.9).  Corollary 3.6.4. Let A ∈ Fn×m. Then, rank A ≤ min {m, n}.

(3.6.12)

Proof. By definition, rank A ≤ m, while it follows from (3.6.8) that rank A = rank A∗ ≤ n.



The dimension theorem is given by (3.6.13) in the following result. Corollary 3.6.5. Let A ∈ Fn×m. Then, rank A + def A = m, ∗

rank A = rank A A.

(3.6.13) (3.6.14)

Proof. (3.6.13) follows from i) of Corollary 3.6.1 and (3.6.8), while (3.6.14) follows from (3.6.4) and (3.6.8).  Corollary 3.6.6. Let A ∈ Fn×n and l ≥ k ≥ 1. Then,

rank Al ≤ rank Ak ,

(3.6.15)

def A ≤ def A .

(3.6.16)

k

l

Proposition 3.6.7. Let A ∈ Fn×n. If rank A2 = rank A, then, for all k ≥ 1, rank Ak = rank A.

Equivalently, if def A2 = def A, then, for all k ≥ 1, def Ak = def A. Proof. Since rank A2 = rank A and R(A2 ) ⊆ R(A), it follows from Lemma 3.1.5 that R(A2 ) = R(A). Hence, R(A3 ) = AR(A2 ) = AR(A) = R(A2 ). Thus, rank A3 = rank A. For all k ≥ 1, similar arguments yield rank Ak = rank A.  The following results follow from the subspace dimension theorem (3.1.12) and the dimension theorem (3.6.13). Corollary 3.6.8. Let A ∈ Fn×n. Then, dim[R(A) + N(A)] + dim[R(A) ∩ N(A)] = n.

(3.6.17)

Corollary 3.6.9. Let A ∈ Fn×m and B ∈ Fm×l. Then,

dim[N(A) + R(B)] + dim[N(A) ∩ R(B)] = def A + rank B.

(3.6.18)

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CHAPTER 3

The following result is Sylvester’s rank formula. Proposition 3.6.10. Let A ∈ Fn×m and B ∈ Fm×l. Then, rank AB + dim[N(A) ∩ R(B)] = rank B.

(3.6.19)

Proof. Using Proposition 8.1.7 and Fact 8.9.15 it follows that

dim[N(A) + R(B)] = dim[R(I − A+A) + R(B)] = dim R([B I − A+A]) = rank [B I − A+A] = m + rank AB − rank A. Therefore, using (3.6.18) it follows that rank AB + dim[N(A) ∩ R(B)] = rank AB + def A + rank B − dim[N(A) + R(B)] = rank AB + def A + rank B − (m + rank AB − rank A) = def A + rank B − m + rank A = rank B.



The next result is Sylvester’s inequality. Proposition 3.6.11. Let A ∈ Fn×m and B ∈ Fm×l. Then, rank A + rank B ≤ m + rank AB.

(3.6.20)

Furthermore, equality holds if and only if N(A) ⊆ R(B). Proof. It follows from (3.6.13) and (3.6.19) that rank B − rank AB = dim[N(A) ∩ R(B)] ≤ dim N(A) = def A = m − rank A, which implies (3.6.20). Alternatively, using (3.6.7) to obtain the second inequality below, it follows that [ ] [ ] [ ][ ] 0 A 0 A −I A AB 0 rank A + rank B = rank ≤ rank = rank B 0 B I 0 I B I [ ] AB 0 = rank ≤ rank [AB 0] + rank [B I] = rank AB + m.  B I Combining (3.6.7) with (3.6.20) yields lower and upper bounds for rank AB. Corollary 3.6.12. Let A ∈ Fn×m and B ∈ Fm×l. Then, max {0, rank A + rank B − m} ≤ rank AB ≤ min {rank A, rank B} ≤ min {n, m, l}. Corollary 3.6.13. Let A ∈ F

(3.6.21)

and B ∈ F . Then, the following statements hold: i) If rank AB = n, then rank A = n. ii) If rank AB = l, then rank B = l. iii) rank AB = m if and only if rank A = rank B = m. n×m

m×l

3.7 Invertibility Let A ∈ Fn×m. Then, A is left invertible if there exists AL ∈ Fm×n such that ALA = Im , while A is right invertible if there exists AR ∈ Fm×n such that AAR = In . These definitions are consistent with the definitions of left and right invertibility given in Chapter 1 applied to the function f : Fm 7→ Fn given by f (x) = Ax. Note that AL (if it exists) and A∗ are the same size, and likewise for AR. Theorem 3.7.1. Let A ∈ Fn×m. Then, the following statements are equivalent: i) A is left invertible. ii) A is one-to-one.

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iii) def A = 0. iv) rank A = m. v) A has full column rank. The following statements are equivalent: vi) A is right invertible. vii) A is onto. viii) def A = m − n. ix) rank A = n. x) A has full row rank. Proposition 3.7.2. Let A ∈ Fn×m. Then, the following statements are equivalent: i) A has a unique left inverse. ii) A has a unique right inverse. iii) rank A = n = m. Proof. To prove i) =⇒ iii), suppose that rank A = m < n so that A is left invertible but nonsquare. Then, it follows from the dimension theorem Corollary 3.6.5 that def AT = n − m > 0. Hence, there exist infinitely many matrices AL ∈ Fm×n such that ALA = Im . Conversely, suppose that B ∈ Fn×n and C ∈ Fn×n are left inverses of A. Then, (B − C)A = 0, and it follows from Sylvester’s inequality Proposition 3.6.11 that B = C.  The following result shows that the rank and defect of a matrix are not affected by either left multiplication by a left invertible matrix or right multiplication by a right invertible matrix. This result is an extension of Lemma 3.5.1. Proposition 3.7.3. Let A ∈ Fn×m, let B ∈ Fm×l be right invertible, and let C ∈ Fk×n be left invertible. Then, R(A) = R(AB), N(A) = N(CA), rank A = rank CA = rank AB,

(3.7.1) (3.7.2) (3.7.3)

def A = def CA = def AB + m − l.

(3.7.4)

Proof. Let C L be a left inverse of C. Using both inequalities in (3.6.21) and the fact that rank A ≤

n, it follows that rank A = rank A + rank C LC − n ≤ rank C LCA ≤ rank CA ≤ rank A, which implies that rank A = rank CA. Finally, (3.6.13) and (3.7.3) imply that def A = m − rank A = m − rank AB = m − (l − def AB) = def AB + m − l.



Proposition 3.7.4. Let A ∈ Fn×m, assume that A is left invertible, let B ∈ Fm×n and C ∈ Fm×n be

left inverses of A, and let y ∈ R(A). Then, By = Cy. Proof. Let x ∈ Fm satisfy y = Ax. Then, By = BAx = x = CAx = Cy.  As shown in Proposition [ ] 3.7.2, left and right inverses of nonsquare matrices are not unique. For example, the matrix A = 01 is left invertible and has left inverses [0 1] and [1 1]. In spite of this nonuniqueness, however, left inverses are useful for solving equations of the form Ax = b, where A ∈ Fn×m, x ∈ Fm, and b ∈ Fn. If A is left invertible, it follows from Ax = b that x = ALAx = ALb, where AL ∈ Rm×n is a left inverse of A. However, it is necessary beforehand whether [ ] to determine [ ] or not there exists x satisfying Ax = b. For example, if A = 01 and b = 10 , then A is left invertible but there does not exist a vector x satisfying Ax = b. The following result addresses the various

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possibilities that can arise. A consequence of Proposition 3.7.4 is the fact that, if there exists a solution of Ax = b and A is left invertible, then Ax = b has a unique solution in the case where A does not have a unique left inverse, which holds if and only if m < n. The notation [A b] denotes the n × (m + 1) partitioned matrix formed from A ∈ Fn×m and b ∈ Fn. Note that rank A ≤ rank [A b] ≤ m + 1, while rank A = rank [A b] is equivalent to b ∈ R(A). Theorem 3.7.5. Let A ∈ Fn×m and b ∈ Fn. Then, the following statements hold: i) The following statements are equivalent: a) Ax = b has no solution. b) rank A < rank [A b]. ii) The following statements are equivalent: a) Ax = b has at least one solution. b) rank A = rank [A b]. c) b ∈ R(A). If these statements hold and xˆ ∈ Fm satisfies A xˆ = b, then the set of solutions of Ax = b is given by xˆ + N(A). iii) The following statements are equivalent: a) Ax = b has a unique solution. b) rank A = rank [A b] = m. c) Ax = b has at least one solution, and A is left invertible. iv) The following statements are equivalent: a) Ax = b has infinitely many solutions. b) rank A = rank [A b] < m. c) Ax = b has at least one solution, and A is not left invertible. v) Assume that A is left invertible. Then, the following statements hold: a) Ax = b has at most one solution. b) Ax = b has no solution if and only if rank [A b] = m + 1. c) Ax = b has a unique solution if and only if rank [A b] = m. If these conditions hold and AL ∈ Fm×n is a left inverse of A, then the unique solution of Ax = b is given by x = ALb. vi) Assume that A is right invertible. Then, the following statements hold: a) Ax = b has at least one solution. b) Let AR ∈ Fm×n be a right inverse of A. Then, x = AR b is a solution of Ax = b. c) Ax = b has a unique solution if and only if n = m. If these conditions hold and AR ∈ Fm×n is a right inverse of A, then AR = A−1. Proof. To prove i), note that rank A < rank [A b] is equivalent to the statement that b is not a linear combination of columns of A; equivalently, Ax = b does not have a solution x ∈ Fm. To prove ii), note that the equivalence of a) and b) follows from i). To prove b) =⇒ c), note that, if b < R(A), then [A b] has more linearly independent columns than A, and thus rank A < rank [A b]. To prove c) =⇒ b), let x ∈ Fn satisfy Ax = b. Then, rank A ≤ rank [A b] = rank [A Ax] = rank A[I x] ≤ rank A. Hence, rank A = rank [A b]. Finally, assume that b ∈ R(A), and let xˆ ∈ Fn satisfy A xˆ = b. Then, x ∈ Fn satisfies Ax = b if and only if A(x − xˆ) = 0, which is equivalent to x ∈ xˆ + N(A). To prove a) =⇒ b) of iii), note that it follows from ii) that N(A) = {0}. Theorem 3.7.1 thus implies that rank A = m, which, along with ii), implies b). Conversely, it follows from b) that

BASIC MATRIX PROPERTIES

297

rank A = m, and thus it follows from the last statement of ii) that Ax = b has a unique solution. Finally, c) is a restatement of b). iv) and v) are consequences of i)–iii). To prove vi), note that, since rank A = n, it follows that rank A = rank[A b], and thus it follows from ii) that Ax = b has at least one solution. In fact, AAR b = b. The last statement follows from Proposition 3.7.2.  If A is right invertible, then it does not follow that, for every solution x of Ax = b, there exists [ ] a right inverse AR of A such that x = AR b. For example, let A = [1 0] and b = 0. Then, x = 01 satisfies Ax = b, but AR b = 0 for every right inverse of A. See Fact 8.3.16. The set of solutions of Ax = b is further characterized by Proposition 8.1.8, while connections to least squares solutions are discussed in Fact 11.17.10. Let A ∈ Fn×m. Proposition 3.7.2 considers the uniqueness of left and right inverses of A, but does not consider the case where a matrix is both a left inverse and a right inverse of A. Consequently, we say that A is nonsingular if there exists B ∈ Fm×n, the inverse of A, such that BA = Im and AB = In; that is, B is both a left and right inverse of A. Proposition 3.7.6. Let A ∈ Fn×m. Then, the following statements are equivalent: i) A is nonsingular. ii) rank A = n = m. If these statements hold, then A has a unique inverse. Proof. If A is nonsingular, then, since B is both left and right invertible, it follows from Theorem 3.7.1 that rank A = m and rank A = n. Hence, ii) holds. Conversely, it follows from Theorem 3.7.1 that A has both a left inverse B and a right inverse C. Then, B = BIn = BAC = InC = C. Hence, B is also a right inverse of A. Thus, A is nonsingular. In fact, the same argument shows that A has a unique inverse.  n The following result can be viewed as a specialization of Theorem 1.6.5 to the function f : F 7→ Fn, where f (x) = Ax. Corollary 3.7.7. Let A ∈ Fn×n. Then, the following statements are equivalent: i) A is nonsingular. ii) A has a unique inverse. iii) A is one-to-one. iv) A is onto. v) A is left invertible. vi) A is right invertible. vii) A has a unique left inverse. viii) A has a unique right inverse. ix) rank A = n. x) def A = 0. Let A ∈ Fn×n be nonsingular. Then, the inverse of A, denoted by A−1 , is a unique n × n matrix with entries in F. If A is not nonsingular, then A is singular. The following results follow from Theorem 3.7.5 in the case n = m. Corollary 3.7.8. Let A ∈ Fn×n and b ∈ Fn. Then, the following statements hold: i) A is nonsingular if and only if there exists a unique vector x ∈ Fn satisfying Ax = b. If these conditions hold, then x = A−1b. ii) A is singular and rank A = rank [A b] if and only if there exist infinitely many x ∈ Rn

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satisfying Ax = b. Assume that these conditions hold, and let xˆ ∈ Fn satisfy A xˆ = b. Then, the set of solutions of Ax = b is given by xˆ + N(A). Corollary 3.7.9. Let A ∈ Fn×n . Then, the following statements are equivalent: i) A is nonsingular. ii) For all b ∈ Fn, there exists a unique solution x ∈ Fm of Ax = b. iii) For all b ∈ Fn, there exists a solution x ∈ Fm of Ax = b. iv) There exists b ∈ Fn such that Ax = b has a unique solution x ∈ Fm . Proposition 3.7.10. Let S ⊆ Fm, let Sˆ ⊆ Fn, and let A ∈ Fn×m. Then, S ⊆ Ainv (AS), ˆ = Sˆ ∩ R(A) ⊆ S. ˆ AAinv (S)

(3.7.5) (3.7.6)

Furthermore, if A is left invertible and AL is a left inverse of A, then S = Ainv (AS), ˆ = AL [Sˆ ∩ R(A)] ⊆ AL S. ˆ Ainv(S)

(3.7.7) (3.7.8)

In addition, if A is right invertible and AR is a right inverse of A, then ˆ = S, ˆ AAinv (S) Rˆ inv ˆ A S ⊆ A (S).

(3.7.10)

Ainv(S) = A−1S.

(3.7.11)

(3.7.9)

Finally, if A is nonsingular, then

Proof. The inclusions (3.7.5) and (3.7.6) follow from Fact 1.10.3 and Fact 1.10.4, respectively. To prove (3.7.8), multiply (3.7.6) by AL . To prove (3.7.7), replace Sˆ in (3.7.8) with AS and use ˆ To prove (3.7.9), multiply (3.7.10) by A (3.7.5). To prove (3.7.10), replace S in (3.7.5) with AR S. and use (3.7.6).  inv A more complete characterization of A (S) is given by Proposition 8.1.10. Proposition 3.7.11. Let A ∈ Fn×n. Then, the following statements are equivalent: i) A is nonsingular. ii) A is nonsingular. iii) AT is nonsingular. iv) A∗ is nonsingular. If these statements hold, then

(A)−1 = A−1 , T −1

(A )

∗ −1

(A )

−1 T

= (A ) , −1 ∗

= (A ) .

Proof. Since AA−1 = I, it follows that (A−1 )∗A∗ = I. Hence, (A−1 )∗ = (A∗ )−1.

(3.7.12) (3.7.13) (3.7.14) 

We thus use A−T to denote (AT )−1 and (A−1 )T and A−∗ to denote (A∗ )−1 and (A−1 )∗. Proposition 3.7.12. Let A, B ∈ Fn×n be nonsingular. Then, (AB)−1 = B−1A−1 , −T

−T −T

(AB) = A B , (AB)−∗ = A−∗B−∗.

(3.7.15) (3.7.16) (3.7.17)

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BASIC MATRIX PROPERTIES

Proof. Note that ABB−1A−1 = AIA−1 = I, which shows that B−1A−1 is the inverse of AB. Similarly,

(AB)∗A−∗B−∗ = B∗A∗A−∗B−∗ = B∗IB−∗ = I, which shows that A−∗B−∗ is the inverse of (AB)∗. For a nonsingular matrix A ∈ Fn×n and r ∈ Z we write △

A−r = (Ar )−1 = (A−1 )r, −rT △

r −T

−r∗ △

r −∗

= (A )

A

A

= (A )



(3.7.18)

−T r

−r T

T −r

−∗ r

−r ∗

∗ −r

= (A ) = (A ) = (A ) , = (A ) = (A ) = (A ) .

(3.7.19) (3.7.20)

For example, A−2∗ = (A−∗ )2.

3.8 The Determinant One of the most useful quantities associated with a square matrix is its determinant. In this section we develop some basic results pertaining to the determinant of a matrix. The determinant of A ∈ Fn×n is defined by △

det A =

∑

sign(σ)

n ∏

A(i,σ(i)) ,

(3.8.1)

i=1

where the sum is taken over all n! permutations σ of the column indices (1, . . . , n), and where ∏ σ( j) − σ(i) △ . (3.8.2) sign(σ) = j−i 1≤i< j≤n It can be seen that sign(σ) = (−1)k , where k is the smallest number of transpositions needed to transform (σ(1), . . . , σ(n)) to (1, . . . , n). The following result is an immediate consequence of this definition. Proposition 3.8.1. Let A ∈ Fn×n. Then, det AT = det A,

(3.8.3)

det A = det A,

(3.8.4)

∗

det A = det A,

(3.8.5)

det αA = αn det A.

(3.8.6)

and, for all α ∈ F, If, in addition, B ∈ Fm×n and C ∈ Fm×m, then [ ] A 0 det = (det A)(det C). B C

(3.8.7)

The following observations are immediate consequences of the definition of the determinant. Proposition 3.8.2. Let A, B ∈ Fn×n. Then, the following statements hold: i) If every off-diagonal entry of A is zero, then det A =

n ∏

A(i,i) .

i=1

In particular, det In = 1. ii) If A has a row or column consisting entirely of 0’s, then det A = 0. iii) If A has two identical rows or two identical columns, then det A = 0.

(3.8.8)

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iv) If x ∈ Fn and i ∈ {1, . . . , n}, then i

det(A + xeTi ) = det A + det(A ← x).

(3.8.9)

v) If x ∈ F1×n and i ∈ {1, . . . , n}, then i

det(A + ei x) = det A + det(A ← x).

(3.8.10)

vi) If B is equal to A except that, for some i ∈ {1, . . . , n} and α ∈ F, either coli (B) = αcoli (A) or rowi (B) = αrowi (A), then det B = α det A. vii) If B is formed from A by interchanging two rows or two columns of A, then det B = − det A. viii) If B is formed from A by adding a multiple of a (row, column) of A to another (row, column) of A, then det B = det A. vi)–viii) correspond, respectively, to multiplying the matrix A on the left or right by matrices of the form    Ii−1 0 0    (3.8.11) In + (α − 1)Ei,i =  0 α 0  ,   0 0 In−i

In + Ei, j + E j,i − Ei,i − E j, j

where i , j, and

In + βEi, j

    =   

Ii−1 0 0 0 0

    =   

0 1 0 0 0

Ii−1 0 0 0 0

0 0 0 1 0

0 0 I j−i−1 0 0

0 0 I j−i−1 0 0

0 β 0 1 0

0 0 0 0 In− j

0 1 0 0 0

0 0 0 0 In− j

     ,  

     ,  

(3.8.12)

(3.8.13)

where β ∈ F and i , j. The matrices in (3.8.12) and (3.8.13) illustrate the case i < j. Since I +(α−1)Ei,i = I +(α−1)ei eTi , I + Ei, j + E j,i − Ei,i − E j, j = I −(ei −e j )(ei −e j )T, and I +βEi, j = I +βei eTj , it follows that all of these matrices are of the form I − xyT. In terms of Definition 4.1.1, (3.8.11) is an elementary matrix if and only if α , 0, (3.8.12) is an elementary matrix, and (3.8.13) is an elementary matrix if and only if either i , j or β , −1. Proposition 3.8.3. Let A, B ∈ Fn×n. Then, det AB = det BA = (det A)(det B). Proof. First note the equality

[

A I

] [ 0 I = B 0

A I

][

−AB 0 0 I

][

I 0 B I

][

(3.8.14)

] 0 I . I 0

The[ first and ] third matrices on the right-hand [ side] of this equality add multiples of rows and columns 0 to other rows and columns of −AB 0 . These operations do not affect the determinant of −AB 0 I [ 0 I] 0 . In addition, the fourth matrix on the right-hand side of this equality interchanges n of −AB 0 I [ ] pairs of columns of B0 AI . Using (3.8.6), (3.8.7), and the fact that every interchange of a pair of

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BASIC MATRIX PROPERTIES

[ ] [ ] columns of B0 AI multiplies the determinant by −1, it thus follows that (det A)(det B) = det AI B0 = [ ] n 0 (−1)n det −AB  0 I = (−1) det(−AB) = det AB. Corollary 3.8.4. Let A ∈ Fn×n be nonsingular. Then, det A , 0 and det A−1 = (det A)−1.

(3.8.15)

Proof. Since AA−1 = In, it follows that det AA−1 = (det A) det A−1 = 1. Hence, det A , 0. In

addition, det A−1 = 1/det A.  Let A ∈ Fn×m. The determinant of a square submatrix of A is a subdeterminant of A. By convention, the determinant of A is a subdeterminant of A. The determinant of a j × j (principal, leading principal) submatrix of A is a j × j (principal, leading principal) subdeterminant of A. Let A ∈ Fn×n. Then, the cofactor of A(i, j) is the (n − 1) × (n − 1) submatrix A[i, j] of A obtained by deleting the ith row and jth column of A. The following result provides a cofactor expansion of det A. Proposition 3.8.5. Let A ∈ Fn×n. Then, for all i ∈ {1, . . . , n}, n ∑ (−1)i+kA(i,k) det A[i,k] = det A.

(3.8.16)

k=1

Furthermore, for all i, j ∈ {1, . . . , n} such that j , i, n ∑ (−1)i+kA( j,k) det A[i,k] = 0.

(3.8.17)

k=1

Proof. Equality (3.8.16) is an equivalent recursive form of the definition det A, while the righthand side of (3.8.17) is equal to det B, where B is obtained from A by replacing rowi (A) by rowj (A). Hence, det B = 0.  n×n Let A ∈ F , where n ≥ 2. To simplify (3.8.16) and (3.8.17) it is useful to define the adjugate of A, denoted by AA ∈ Fn×n, where, for all i, j ∈ {1, . . . , n}, i

△

(AA )(i, j) = (−1)i+ j det A[ j,i] = det(A ← e j ).

(3.8.18)

Then, (3.8.16) implies that, for all i ∈ {1, . . . , n}, n ∑

A(i,k) (AA )(k,i) = (AAA )(i,i) = (AAA)(i,i) = det A,

(3.8.19)

k=1

while (3.8.17) implies that, for all i, j ∈ {1, . . . , n} such that j , i, n ∑

A(i,k) (AA )(k, j) = (AAA )(i, j) = (AAA)(i, j) = 0.

(3.8.20)

k=1

Thus,

AAA = AAA = (det A)I.

(3.8.21)

Consequently, if det A , 0, then 1 AA , det A

(3.8.22)

AAA = AAA = 0.

(3.8.23)

A−1 = whereas, if det A = 0, then A △

For a scalar A ∈ F, we define A = 1. In particular, 01×1 = 1. A

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The following result provides the converse of Corollary 3.8.4 by using (3.8.22) to construct A−1 in terms of (n − 1) × (n − 1) subdeterminants of A. Corollary 3.8.6. Let A ∈ Fn×n. Then, A is nonsingular if and only if det A , 0. If these conditions hold, then, for all i, j ∈ {1, . . . , n}, the (i, j) entry of A−1 is given by (A−1 )(i, j) = (−1)i+ j

det A[ j,i] . det A

(3.8.24)

Finally, the following result characterizes the rank of a matrix in terms of the nonsingularity of its submatrices. Proposition 3.8.7. Let A ∈ Fn×m. Then, rank A is the largest size of all nonsingular submatrices of A.

3.9 Partitioned Matrices Partitioned matrices were used to state or prove several results in this chapter including Proposition 3.6.11, Theorem 3.7.5, Proposition 3.8.1, and Proposition 3.8.3. In this section we give several useful equalities for partitioned matrices. Proposition 3.9.1. Let Ai j ∈ Fni ×m j for all i ∈ {1, . . . , k} and j ∈ {1, . . . , l}. Then, T  AT · · · AT    11  A11 · · · A1l   k1   .   .  . . . . . .  .   .. (3.9.1) = · · ·  · .· · . .  ,  . .      T T Ak1 · · · Akl A1l · · · Akl ∗  A∗ · · · A∗    11  A11 · · · A1l   k1   .   .  . . . . . .  .   .. (3.9.2) = · · ·  · .· · . .  .  . .      ∗ ∗ Ak1 · · · Akl A1l · · · Akl If, in addition, k = l and ni = mi for all i ∈ {1, . . . , m}, then     A11 A12  A11 · · · A1k  ∑  0 A k 22   .  . ..  = tr A , det tr  ..  .. · · ·  ii . . .  ..  .  i=1   Ak1 · · · Akk 0 0 Lemma 3.9.2. Let B ∈ F

[

I 0

··· ··· .. . ···

 A1k   k A2k  ∏  = det Aii . ..  .  i=1 Akk

and C ∈ F . Then, ]−1 [ ] [ ]−1 [ B I −B I 0 I = , = I 0 I C I −C

n×m

(3.9.3)

m×n

Let A ∈ Fn×n and D ∈ Fm×m be nonsingular. Then, [ ]−1 [ −1 A 0 A = 0 0 D

] 0 . I

] 0 . D−1

(3.9.4)

(3.9.5)

Proposition 3.9.3. Let A ∈ Fn×n, B ∈ Fn×m, C ∈ Fl×n, and D ∈ Fl×m, and assume that A is

nonsingular. Then,

[

A C

] [ ][ ][ ] B I 0 A 0 I A−1B = , D CA−1 I 0 D − CA−1B 0 I [ ] A B rank = n + rank(D − CA−1B). C D

(3.9.6) (3.9.7)

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BASIC MATRIX PROPERTIES

If, furthermore, l = m, then

[

A det C

] B = (det A) det(D − CA−1B). D

(3.9.8)

Proposition 3.9.4. Let A ∈ Fn×m, B ∈ Fn×l, C ∈ Fl×m, and D ∈ Fl×l, and assume that D is

nonsingular. Then,

[

] [ ][ ][ B I BD−1 A − BD−1C 0 I = D 0 I 0 D D−1C [ ] A B rank = l + rank(A − BD−1C). C D

A C

If, furthermore, n = m, then

[

A det C

] B = (det D) det(A − BD−1C). D

Corollary 3.9.5. Let A ∈ Fn×m and B ∈ Fm×n. Then,

[

Hence,

[ rank

In B

] 0 , I

] [ A I = n Im B [ I = n 0

0 Im A Im

][ ][

In 0

0 Im − BA

In − AB 0

0 Im

][ ][

(3.9.9) (3.9.10)

(3.9.11) ]

In 0

A Im

In B

] 0 . Im

] A = n + rank(Im − BA) = m + rank(In − AB), Im [ ] I A det n = det(Im − BA) = det(In − AB). B Im

In B

(3.9.12) (3.9.13)

(3.9.14) (3.9.15)

Hence, In + AB is nonsingular if and only if Im + BA is nonsingular. Lemma 3.9.6. Let A ∈ Fn×n, B ∈ Fn×m, C ∈ Fm×n, and D ∈ Fm×m, and assume that A and D are nonsingular. Then, (det A) det(D − CA−1B) = (det D) det(A − BD−1C). −1

(3.9.16)

−1

Furthermore, D − CA B is nonsingular if and only if A − BD C is nonsingular. Proposition 3.9.7. Let A ∈ Fn×n, B ∈ Fn×m, C ∈ Fm×n, and D ∈ Fm×m. If A and D − CA−1B are nonsingular, then ]−1  A−1 + A−1B(D − CA−1B)−1CA−1 −A−1B(D − CA−1B)−1  [   A B  . =  (3.9.17) −1 −1 −1 −1 −1 C D −(D − CA B) CA (D − CA B) If D and A − BD−1C are nonsingular, then ]−1  [ (A − BD−1C)−1  A B =  C D −D−1C(A − BD−1C)−1

   .  D−1 + D−1C(A − BD−1C)−1BD−1 −(A − BD−1C)−1BD−1

If A, D, and D − CA−1B are nonsingular, then A − BD−1C is nonsingular, and  [ ]−1  (A − BD−1C)−1 −(A − BD−1C)−1BD−1   A B  . =   C D −(D − CA−1B)−1CA−1 (D − CA−1B)−1

(3.9.18)

(3.9.19)

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The following result is the matrix inversion lemma. A special case is the Sherman-MorrisonWoodbury formula given by Fact 3.21.3. Corollary 3.9.8. Let A ∈ Fn×n, B ∈ Fn×m, C ∈ Fm×n, and D ∈ Fm×m. If A, D − CA−1B, and D are nonsingular, then A − BD−1C is nonsingular, (A − BD−1C)−1 = A−1 + A−1B(D − CA−1B)−1CA−1 , −1

−1

−1

−1

C(A − BD C) A = D(D − CA B) C.

(3.9.20) (3.9.21)

If A and I − CA−1B are nonsingular, then A − BC is nonsingular and (A − BC)−1 = A−1 + A−1B(I − CA−1B)−1CA−1.

(3.9.22)

If D − CB, and D are nonsingular, then I − BD−1C is nonsingular and (I − BD−1C)−1 = I + B(D − CB)−1C.

(3.9.23)

If I − CB is nonsingular, then I − BC is nonsingular and (I − BC)−1 = I + B(I − CB)−1C.

(3.9.24)

Corollary 3.9.9. Let A, B, C, D ∈ Fn×n. If A, B, C − DB−1A, and D −CA−1B are nonsingular, then

[

A C

B D

]−1

 (C − DB−1A)−1   . −1 −1  (D − CA B)

(3.9.25)

 −A−1B(D − CA−1B)−1   .  (D − CA−1B)−1

(3.9.26)

 −1  A − (C − DB−1A)−1CA−1 =  −(D − CA−1B)−1CA−1

If A, C, B − AC −1D, and D − CA−1B are nonsingular, then [ ]−1  A−1 − A−1B(B − AC −1D)−1  A B =  C D (B − AC −1D)−1

If A, B, C, B − AC −1D, and D − CA−1B are nonsingular, then C − DB−1A is nonsingular and [ ]−1  A−1 − A−1B(B − AC −1D)−1 (C − DB−1A)−1    A B  . (3.9.27) =  −1 −1 −1 −1  C D (B − AC D) (D − CA B) If B, D, A − BD−1C, and C − DB−1A are nonsingular, then  [ ]−1   (A − BD−1C)−1 (C − DB−1A)−1  A B  . =   C D −D−1C(A − BD−1C)−1 D−1 − D−1C(C − DB−1A)−1 If C, D, A − BD−1C, and B − AC −1D are nonsingular, then  [ ]−1  (A − BD−1C)−1 −(A − BD−1C)−1BD−1   A B  . =  −1 −1 −1 −1 −1 −1  C D (B − AC D) D − (B − AC D) BD

(3.9.28)

(3.9.29)

If B, C, D, A − BD−1C, and C − DB−1A are nonsingular, then B − AC −1D is nonsingular and  [ ]−1  (A − BD−1C)−1  (C − DB−1A)  A B  . =  (3.9.30) −1 −1 −1 −1 −1 −1  C D (B − AC D) D − D C(C − DB A) Finally, if A, B, C, D, A − BD−1C, and B − AC −1D are nonsingular, then C − DB−1A and D − CA−1B are nonsingular and [ ]−1  (A − BD−1C)−1 (C − DB−1A)−1    A B  . =  (3.9.31) −1 −1 −1 −1  C D (B − AC D) (D − CA B)

305

BASIC MATRIX PROPERTIES

Corollary 3.9.10. Let A, B ∈ Fn×n, and assume that A and I − A−1B are nonsingular. Then, A − B

is nonsingular, and (A − B)−1 = A−1 + A−1B(I − A−1B)−1A−1.

(3.9.32)

If, in addition, B is nonsingular, then (A − B)−1 = A−1 + A−1 (B−1 − A−1 )−1A−1.

(3.9.33)

3.10 Majorization Let x ∈ Rn . Then, the components of x↓ ∈ Rn are the same as the components of x and satisfy (x )(n) ≤ · · · ≤ (x↓ )(1) . An analogous definition is used for x ∈ R1×n . Furthermore, the components of x↑ ∈ Rn are the same as the components of x and satisfy (x↑ )(1) ≤ · · · ≤ (x↑ )(n) . An analogous definition is used for x ∈ R1×n . Definition 3.10.1. Let x, y ∈ Rn or x, y ∈ R1×n. Then, the following terminology is defined: ↓

w

i) y weakly majorizes x (x ≺ y) if, for all k ∈ {1, . . . , n}, k ∑

k ∑ (y↓ )(i) .

(x↓ )(i) ≤

i=1

(3.10.1)

i=1

s

ii) y strongly majorizes (x ≺ y) x if y weakly majorizes x and n ∑

n ∑ (y↓ )(i) .

(x↓ )(i) =

i=1

(3.10.2)

i=1

Now, assume that x and y are nonnegative. Then, the following terminology is defined: wlog

iii) y weakly log majorizes x (x ≺ y) if, for all k ∈ {1, . . . , n}, k ∏

k ∏

(x↓ )(i) ≤

i=1

(y↓ )(i) .

(3.10.3)

i=1

slog

iv) y strongly log majorizes x (x ≺ y) if y weakly log majorizes x and n ∏

n ∏

(x↓ )(i) =

i=1

(y↓ )(i) .

(3.10.4)

i=1

An equivalent formulation of (3.10.1) is the condition max

k ∑

x(i j ) ≤

k ∑ (y↓ )(i) .

j=1

(3.10.5)

i=1

where the maximum is taken over all 1 ≤ i1 < · · · < ik ≤ n. Therefore, (3.10.1) implies that k ∑ i=1

x(i) ≤

k ∑ (y↓ )(i) ,

(3.10.6)

i=1

but the converse is not true. If y strongly majorizes x, then y weakly majorizes x, and, if y strongly log majorizes x, then y weakly log majorizes x. Fact 3.25.15 states that, if y weakly log majorizes x, then y weakly majorizes x. Furthermore, Fact 3.25.3 states that y strongly log majorizes x and y strongly majorizes x if and only if x and y have the same components.

306

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Note that, if x weakly majorizes y and y weakly majorizes x, then x and y have the same compow nents but are not necessarily equal. Therefore, “≺” is not an antisymmetric relation and thus is not a partial ordering. w

s

Proposition 3.10.2. Weak majorization “≺” and strong majorization “≺” are reflexive and wlog

transitive relations on Rn . Furthermore, weak log majorization “ ≺ ” and strong log majorization slog

“ ≺ ” are reflexive and transitive relations on [0, ∞)n . Definition 3.10.3. Let S ⊆ Rn, and let f : S 7→ R. Then, f is Schur-convex if, for all x, y ∈ S s such that x ≺ y, it follows that f (x) ≤ f (y). Furthermore, f is Schur-concave if −f is Schur-convex.

3.11 Facts on One Set Fact 3.11.1. Let α ∈ F, and let S ⊆ Fn be a (cone, pointed cone, convex set, polytope, convex

cone, polyhedral cone, affine subspace, subspace). Then, so is αS. Fact 3.11.2. Let S ⊆ Fn, let α, β ∈ F, and assume that at least one of the following statements holds: i) card(S) ≤ 1. ii) S is a convex set, and α and β are nonnegative numbers. iii) S is an affine subspace, and either α = β = 0 or α + β , 0. Then, (α + β)S = αS + βS. Source: For ii), let x, y ∈ S and let α and β be nonnegative numbers such △ that α + β > 0. Then, αx + βy ∈ αS + βS. Define θ = α/(α + β) ∈ (0, 1). Then, θx + (1 − θ)y ∈ S, and thus αx + βy = (α + β)[θx + (1 − θ)y] ∈ (α + β)S. See [2606, p. 6]. For iii), let x, y ∈ S and let α, β ∈ F △ satisfy α + β , 0. Then, αx + βy ∈ αS + βS. Define θ = α/(α + β) ∈ F. Then, θx + (1 − θ)y ∈ S, and thus αx + βy = (α + β)[θx + (1 − θ)y] ∈ (α + β)S. Fact 3.11.3. Let S ⊆ Fn. Then, the following statements are equivalent: i) S is convex. ii) For all α, β > 0, αS + βS = (α + β)S. ∑ ∑ iii) For all k ≥ 2 and α1 , . . . , αk ∈ [0, ∞) such that ki=1 αi = 1, ki=1 αi S = S. Source: [1267, p. 107] and [2487, p. 73]. Fact 3.11.4. Let S ⊆ Fn. Then, the following statements hold: i) S = cone S if and only if S is a cone. ii) S = conv S if and only if S is a convex set. iii) S = coco S if and only if S is a convex cone. iv) S = affin S if and only if S is an affine subspace. v) S = span S if and only if S is a subspace. Fact 3.11.5. Let S ⊆ Rn be nonempty. Then, cone S = {αx: x ∈ S and α > 0},  k  k  ∪ ∑   ∑  conv S = αi xi : αi > 0, xi ∈ S, i = 1, . . . , k, and αi = 1      i=1 k∈P i=1  k  k  ∪  ∑   ∑  = α x : α > 0, x ∈ S, i = 1, . . . , k, and α = 1 ,   i i i i i     1≤k≤n+1

i=1

i=1

307

BASIC MATRIX PROPERTIES

  k  ∪    ∑ αi xi : αi > 0, xi ∈ S, i = 1, . . . , k coco S =      k∈P i=1  k   ∪    ∑  α x : α > 0, x ∈ S, i = 1, . . . , k , =   i i i i     i=1

1≤k≤n

 k  k  ∪ ∑   ∑  αi xi : αi ∈ R, xi ∈ S, i = 1, . . . , k, and affin S = αi = 1      i=1 k∈P i=1  k  k  ∑ ∪    ∑  αi xi : αi ∈ R, xi ∈ S, i = 1, . . . , k, and αi = 1  , =      1≤k≤n+1

i=1

i=1

 k   ∪   ∑  αi xi : αi ∈ R and xi ∈ S, i = 1, . . . , k span S =      k∈P i=1   k  ∪     ∑ . α x : α ∈ R and x ∈ S, i = 1, . . . , k =   i i i i     i=1

1≤k≤n

Source: The second expression for conv S is Caratheodory’s theorem. See [309, p. 10], [1260, p.

43], and [1769, Theorem 2.23]. The first expression for conv S is given in [1769, Theorem 2.15], while the second expression for conv S is given in [1769, Theorem 2.23]. The second expression for coco S is given in [1267, Theorem 4.21]. Fact 3.11.6. Let S ⊆ Cn be nonempty. Then, cone S = {αx: x ∈ S and α > 0},   k k  ∑ ∪    ∑ αi xi : αi > 0, xi ∈ S, i = 1, . . . , k, and αi = 1 conv S =      i=1 k∈P i=1   k k  ∑ ∪     ∑ , α x : α > 0, x ∈ S, i = 1, . . . , k, and α = 1 =   i i i i i     1≤k≤2n+1

i=1

i=1

  k  ∪    ∑ coco S = α x : α > 0, x ∈ S, i = 1, . . . , k   i i i i     k∈P i=1  k   ∪    ∑  = αi xi : αi > 0, xi ∈ S, i = 1, . . . , k ,      1≤k≤2n

i=1

 k  k  ∪ ∑   ∑  affin S = αi xi : αi ∈ C, xi ∈ S, i = 1, . . . , k, and αi = 1      i=1 k∈P i=1  k  k  ∪  ∑   ∑  = α x : α ∈ C, x ∈ S, i = 1, . . . , k, and α = 1 ,   i i i i i     1≤k≤n+1

i=1

i=1

 k   ∪   ∑  span S = α x : α ∈ C and x ∈ S, i = 1, . . . , k   i i i i     k∈P

i=1

308

CHAPTER 3

  k   ∪    ∑ . αi xi : αi ∈ C and xi ∈ S, i = 1, . . . , k =      1≤k≤n

i=1

Fact 3.11.7. Let S ⊂ F . Then, the following statements hold: n

If S is a closed half space, then S is a pointed, polyhedral, convex cone that is not one-sided. If S is an open half space, then S is a blunt, one-sided, convex cone. If S is a convex cone, then S is contained in a closed half space. If S is an open convex cone, then S is a blunt, one-sided cone and is contained in an open half space. Fact 3.11.8. Let S ⊆ Fn. Then, the following statements hold: i) coco S = conv cone S = cone conv S. ii) coco(S ∪ −S) = affin(S ∪ −S) = affin(S ∪ {0}) = span S = S⊥⊥ . { } affin S ⊆ span S. iii) S ⊆ conv S ⊆ coco S iv) S ⊆ cone S ⊆ coco S ⊆ span S. Remark: See [362, p. 52]. “Pointed” in [362] means one-sided. Fact 3.11.9. Let S ⊆ Rn, and assume that S is a convex cone. Then, i) ii) iii) iv)

affin S = span S = S − S = conv[S ∪ (−S)]. Hence, S − S is a subspace. Source: [2487, p. 152]. Remark: R is not a subspace of C, and thus the result does not hold for S ⊆ Cn . Related: Fact 3.11.12. Fact 3.11.10. Let S ⊆ Fn, and consider the following statements: i) S is a cone. ii) S is a convex set. iii) S is a convex cone. iv) S is an affine subspace. v) S is a subspace. vi) 0 ∈ S. vii) 0 ∈ affin S. viii) affin S is a subspace. ix) affin S = span S. x) dim S = n. xi) affin S = Fn. xii) span S = Fn. Then, v) ⇐⇒ {i), iv)} ⇐⇒ {iii), iv)} ⇐⇒ {iv), vi)} =⇒ iii) ⇐⇒ {i), ii)}, } x) ⇐⇒ xi) ⇐⇒ xii) =⇒ vii) ⇐⇒ viii) ⇐⇒ ix). vi) Fact 3.11.11. Let S ⊆ Fn, assume that S is an affine subspace, and let z ∈ Fn. Then, S + z is an affine subspace. If, in addition, z ∈ S, then S − z is a subspace. Source: Fact 3.11.10. Fact 3.11.12. Let S ⊆ Fn, and assume that S is an affine subspace. Then, there exist a unique vector x ∈ Fn and a unique subspace S0 ⊆ Fn such that S = x + S0 . Source: [2487, p. 12]. Remark: S − S = {y − z : y, z ∈ S} = S0 is a subspace. Related: Fact 3.11.9. Fact 3.11.13. Let S ⊂ Fn. Then, the following statements hold:

309

BASIC MATRIX PROPERTIES

i) S is an affine hyperplane if and only if there exist a nonzero vector y ∈ Fn and α ∈ R such that S = {x ∈ Fn : Re x∗y = α}. ii) S is an affine closed half space if and only if there exist a nonzero vector y ∈ Fn and α ∈ R such that S = {x ∈ Fn : Re x∗y ≤ α}. iii) S is an affine open half space if and only if there exist a nonzero vector y ∈ Fn and α ∈ R such that S = {x ∈ Fn : Re x∗y < α}. Source: Let z ∈ Fn satisfy z∗y = α. Then, {x: x∗y = α} = {y}⊥ + z. Fact 3.11.14. Let x1 , . . . , xk ∈ Fn. Then, affin {x1 , . . . , xk } = x1 + span {x2 − x1 , . . . , xk − x1 }. Related: Fact 12.11.13. Fact 3.11.15. Let A ∈ Fn×m, and let S ⊆ Fm. Then,

cone AS = A cone S,

conv AS = A conv S,

affin AS = A affin S,

coco AS = A coco S,

span AS = A span S.

Hence, if S is a (cone, convex set, polytope, convex cone, polyhedral cone, affine subspace, subspace), then so is AS. Now, assume that A is left invertible, and let AL ∈ Fm×n be a left inverse of A. Then, cone S = AL cone AS, conv S = AL conv AS, coco S = AL coco AS, affin S = AL affin AS,

span S = AL span AS.

Hence, if AS is a (cone, convex set, polytope, convex cone, polyhedral cone, affine subspace, subspace), then so is S. Related: Fact 3.11.18. Fact 3.11.16. Let A ∈ Fn×m , let S ⊆ Fn , and assume that S is a (cone, convex set, convex cone, affine subspace, subspace). Then, so is Ainv (S). Fact 3.11.17. Let A ∈ Fn×m and S ⊆ Fn . Then, conv Ainv (S) = Ainv (conv[S ∩ R(A)]) ⊆ Ainv (conv S). Source: [2487, pp. 126–128]. Related: Fact 12.11.22. △ Fact 3.11.18. Let A ∈ Fn×m, let X = {col1(A), . . . , colm (A)} ⊂ Fm, and define △

Θm = {θ ∈ [0, ∞)m :

m ∑

θ(i) = 1},

△

Γm = {αei,m : α ∈ [0, ∞), i ∈ {1, . . . , m}},

i=1 △

Φm = {α ∈ Fm :

m ∑

α(i) = 1}.

i=1

Then, the following statements hold: i) Γm = cone {e1,m , . . . , em,m } is a cone. ii) Θm = conv {e1,m , . . . , em,m } is a convex polytope. iii) [0, ∞)m = coco {e1,m , . . . , em,m } is a polyhedral cone. iv) Φm = affin {e1,m , . . . , em,m } = N(11×m ) + e1,m is an affine subspace. v) Fm = span {e1,m , . . . , em,m } is a subspace. vi) cone X = AΓm . vii) conv X = AΘm . viii) coco X = A[0, ∞)m . ix) affin X = AΦm .

310

CHAPTER 3

x) span X = AFm = R(A). Related: Fact 3.11.15. Fact 3.11.19. Let S ⊆ Fn . Then, the following statements hold: i) S is a convex polytope if and only if S is the intersection of a finite number of closed half spaces. ii) S is a bounded, convex polytope if and only if S is the convex hull of a finite number of points. iii) S is a polyhedral cone if and only if S is the convex conical hull of a finite number of points that includes 0. iv) Let z, x1 , . . . , xm ∈ Fn , and let S be the zonotope   m      ∑ . α x : 0 ≤ α ≤ 1 for all i ∈ {1, . . . , m} S=z+  i i i     i=1

Then, S is convex. In particular,     k m      ∑ ∪     x : 1 ≤ i < · · · < i ≤ m S = z + conv {0} ∪  .   i 1 k j       k=1

j=1

Remark: The number of vertices of the zonotope in iv) does not exceed 2m . Related: Fact 5.4.3

and Fact 5.4.4. Fact 3.11.20. Let S ⊆ Fn be a subspace. Then, for all m ≥ dim S, there exists A ∈ Fn×m such

that S = R(A).

△

Fact 3.11.21. Let A ∈ Fn×n, let S ⊆ Fn, assume that S is a subspace, let k = dim S, let S ∈ Fn×k,

and assume that R(S ) = S. Then, S is an invariant subspace of A if and only if there exists M ∈ Fk×k such that AS = SM. Source: Use Fact 7.14.1 with B = I. To prove sufficiency, note that S ⊆ AS + S. It then follows from dim S = dim(AS + S) and Corollary 3.1.5 that S = AS + S. Finally, Fact 3.12.12 implies that AS ⊆ S. See [1762, pp. 89, 90]. Fact 3.11.22. Let S ⊆ Fn. Then, S⊥ = (span S)⊥. Now, assume that S is a subspace. Then, S and S⊥ are orthogonally complementary. Fact 3.11.23. Define the convex pointed cone S ⊂ R2 by △

S = {(x1 , x2 ) ∈ [0, ∞) × R : if x1 = 0, then x2 ≥ 0} = ([0, ∞) × R)\[{0} × (−∞, 0)]. d

d

Furthermore, for all x, y ∈ R2, define x ≤ y if and only if y − x ∈ S. Then, “≤” is a total ordering on d

R2. Remark: “≤” is the lexicographic (dictionary) ordering. See Fact 1.8.16 and [309, p. 161].

3.12 Facts on Two or More Sets Fact 3.12.1. Let S1 , S2 ⊆ Fn, and assume that S1 ⊆ S2 . Then,

cone S1 ⊆ cone S2 , conv S1 ⊆ conv S2 , affin S1 ⊆ affin S2 , span S1 ⊆ span S2 ,

coco S1 ⊆ coco S2 , dim S1 ≤ dim S2 .

Furthermore, dim S1 = dim S2 if and only if affin S1 = affin S2 . Remark: The last statement follows from Proposition 3.1.4. Fact 3.12.2. Let S1 , S2 ⊆ Fn. Then, cone(S1 + S2 ) ⊆ cone S1 + cone S2 , conv(S1 + S2 ) = conv S1 + conv S2 , coco(S1 + S2 ) = coco S1 + coco S2 , affin(S1 + S2 ) = affin S1 + affin S2 ,

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span(S1 + S2 ) ⊆ span S1 + span S2 . Source: [1267, p. 138] and [2398, pp. 2, 3]. To prove conv S1 + conv S2 ⊆ conv(S1 + S2 ), let

∑k

∑ ∑ ∈ conv S1 and y = ki=1 ϕi yi ∈ conv S2 , where θ1 , . . . , θk , ϕ1 , . . . , ϕk ≥ 0 and ki=1 θi = ∑k i=1 ϕi = 1. Then, x + y = i, j=1 θi ϕ j (xi + y j ) ∈ conv(S1 + S2 ). Fact 3.12.3. Let S1 , S2 ⊆ Fm, and let A ∈ Fn×m . Then, A(S1 + S2 ) = AS1 + AS2 . Fact 3.12.4. Let S1 , S2 ⊆ Fn be (cones, pointed cones, convex sets, polytopes, convex polytopes, convex cones, polyhedral cones, affine subspaces, subspaces), and let α, β ∈ F. Then, so are αS1 ∩ βS2 and αS1 + βS2 . Source: Fact 3.14.7. See [1267, p. 90] for the case where S1 and S2 are convex. Fact 3.12.5. Let S1 , . . . , Sk ⊆ Fn , and let α1 , . . . , αk ∈ R. Then, x= ∑k

i=1 θi xi

conv

k ∑ i=1

αi Si =

k ∑

αi conv Si .

i=1

Source: [2487, p. 124]. Related: Fact 12.12.12. Fact 3.12.6. Let S1 , S2 ⊆ Fn . Then, (S∼ − S2 )∼ = {x ∈ Fn : x + S2 ⊆ S1 }. Remark: These

expressions define the Minkowski difference. Fact 3.12.7. Let A ∈ Fn×m. Then, for all S1 , S2 ⊆ Fm , the following statements hold: i) S1 ⊆ Ainv (AS1 ). ii) AS1 = AAinv (AS1 ). iii) If S1 ⊆ S2 , then AS1 ⊆ AS2 . iv) A(S1 ∩ S2 ) ⊆ AS1 ∩ AS2 . v) A(S1 ∪ S2 ) = AS1 ∪ AS2 . vi) A(S1 + S2 ) = AS1 + AS2 . vii) (AS1 )\(AS2 ) ⊆ A(S1\S2 ). Furthermore, the following statements are equivalent: viii) A is left invertible. ix) For all S ⊆ Fm , Ainv (AS) = S. iv) For all S1 , S2 ⊆ Fm , A(S1 ∩ S2 ) = AS1 ∩ AS2 . x) For all disjoint S1 , S2 ⊆ Fm , AS1 and AS2 are disjoint. xi) For all S1 , S2 ⊆ Fm , (AS1 )\(AS2 ) = A(S1\S2 ). Source: Fact 1.10.3 and [688, p. 12]. Fact 3.12.8. Let A ∈ Fn×m. Then, for all S1 , S2 ⊆ Fn, the following statements hold: i) AAinv (S1 ) ⊆ S1 . ii) If S1 ⊆ S2 , then Ainv (S1 ) ⊆ Ainv (S2 ). iii) Ainv (S1 ∩ S2 ) = Ainv (S1 ) ∩ Ainv (S2 ). iv) Ainv (S1 ∪ S2 ) = Ainv (S1 ) ∪ Ainv (S2 ). v) Ainv (S1 ) + Ainv (S2 ) ⊆ Ainv (S1 + S2 ). vi) (Ainv S1 )\(Ainv S2 ) = Ainv (S1\S2 ). In addition, the following statements are equivalent: vii) A is right invertible. viii) For all S ⊆ Fn , AAinv (S) = S. Source: Fact 1.10.4 and [688, p. 12].

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Fact 3.12.9. Let A1 ∈ Fn×m, A2 ∈ Fn×l, S1 ⊆ Fm , and S2 ⊆ Fl . Then,

[

] S1 [A1 A2 ] = A1 S1 + A2 S2 . S2

Fact 3.12.10. Let A ∈ Fn×m, and let S1 ⊆ Fm and S2 ⊆ Fn be subspaces. Then, the following

statements are equivalent: i) AS1 ⊆ S2 . ii) A∗ S⊥2 ⊆ S⊥1 . ⊥ ∗inv ⊥ Source: It follows from i) and Fact 3.18.19 that S⊥ (S1 ). It now follows from 2 ⊆ (AS1 ) = A Proposition 3.7.10 that A∗ S⊥2 ⊆ A∗A∗inv (S⊥1 ) ⊆ S⊥1 . See [688, p. 12]. Fact 3.12.11. Let S1 , S2 ⊆ Fn. Then, } (span S1 ) ∪ span S2 ⊆ span(S1 ∪ S2 ) = (span S1 ) + span S2 , span(S1 + S2 ) span(S1 ∩ S2 ) ⊆ (span S1 ) ∩ span S2 . If, in addition, 0 ∈ S1 ∩ S2 , then span(S1 + S2 ) = span(S1 ∪ S2 ) = (span S1 ) + span S2 . Source: [2238, p. 532], [2418, p. 11], and [2487, p. 4]. Fact 3.12.12. Let S1 , S2 ⊆ Fn. Then, the following statements hold: i) If 0 ∈ S2 and S1 + S2 ⊆ S2 , then S1 ⊆ S2 . ii) If S2 is a convex cone and S1 ⊆ S2 , then S1 + S2 ⊆ S2 . Now, assume that S2 is a subspace. Then, the following statements are equivalent: iii) S1 ⊆ S2 . iv) S1 + S2 ⊆ S2 . v) S1 + S2 = S2 . Source: To prove i), note that S1 = S1 + {0} ⊆ S1 + S2 ⊆ S2 . To prove ii), note that S1 + S2 ⊆ S2 + S2 = S2 . To prove iv) =⇒ v), let x ∈ S1 and z ∈ S2 . Then, x ∈ S2 , and thus z − x ∈ S2 . Finally, z = x + (z − x) ∈ S1 + S2 . Fact 3.12.13. Let S1 , S2 ⊆ Fn be cones. Then, the following statements hold: i) S1 ∪ S2 is a cone. ii) If either S1 or S2 is a pointed cone, then S1 ∪ S2 is a pointed cone. iii) If S1 and S2 are convex cones, then S1 + S2 ⊆ conv(S1 ∪ S2 ). iv) If S1 and S2 are pointed convex cones, then S1 + S2 = conv(S1 ∪ S2 ). Source: [1267, p. 107]. ⊥ Fact 3.12.14. Let S1 , S2 ⊆ Fn, and assume that S1 ⊆ S2 . Then, S⊥ 2 ⊆ S1 . n ⊥ ⊥ ⊥ Fact 3.12.15. Let S1 , S2 ⊆ F . Then, S1 ∩ S2 ⊆ (S1 + S2 ) . Fact 3.12.16. Let S1 , S2 ⊆ Fn be subspaces. Then, the following statements hold: i) S1 ∪ S2 ⊆ span(S1 ∪ S2 ) = S1 + S2 . ii) If S3 ⊆ Fn is a subspace and S1 ∪ S2 ⊆ S3 , then S1 + S2 ⊆ S3 . iii) S1 ∪ S2 is a subspace if and only if either S1 ⊆ S2 or S2 ⊆ S1. If these conditions hold, then    S1 , S2 ⊆ S1 , S1 ∪ S2 = S1 + S2 =   S2 , S1 ⊆ S2 .

iv) S1 + S2 is the intersection of all subspaces containing S1 ∪ S2 and thus is the smallest subspace containing S1 ∪ S2 .

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v) The following statements are equivalent: a) S1 ⊆ S2 . b) S⊥2 ⊆ S⊥1 . c) S1 + S2 = S2 . d) S1 ∩ S2 = S1 . e) S1 and S⊥2 are mutually orthogonal. f ) For each subspace S ⊆ Fn , S1 + (S2 ∩ S) = S2 ∩ (S1 + S). g) There exists a subspace S ⊆ Fn such that S1 + (S2 ∩ S) = S2 ∩ (S1 + S). vi) S1 ⊂ S2 if and only if S⊥2 ⊂ S⊥1 . vii) S1 = (S1 ∩ S2 ) + [S1 ∩ (S1 ∩ S2 )⊥ ]. Source: [2403, p. 81]. Fact 3.12.17. Let S1 , S2 ⊆ Fn be subspaces. Then, the following statements hold: i) (S1 ∩ S2 )⊥ = S⊥1 + S⊥2 . ii) (S1 + S2 )⊥ = S⊥1 ∩ S⊥2 . iii) S1 = S2 if and only if S1 ∩ (S⊥1 + S⊥2 ) = S2 ∩ (S⊥1 + S⊥2 ). iv) S1 ∩ (S⊥1 + S⊥2 ) and S2 + (S⊥1 ∩ S⊥2 ) are complementary subspaces. v) S1 + S2 = S1 + [S⊥1 ∩ (S1 + S2 )]. vi) The following statements are equivalent: a) S1 and S2 are complementary subspaces. b) S⊥1 and S⊥2 are complementary subspaces. c) S1 + S2 = S⊥1 + S⊥2 . d) S1 ∩ S2 = S⊥1 ∩ S⊥2 . vii) The following statements are equivalent: a) S1 ∩ S⊥2 = S⊥1 ∩ S2 = {0}. b) S1 + S⊥2 = S⊥1 + S2 . c) S1 ∩ S⊥2 = S⊥1 ∩ S2 . Source: [255]. Related: Fact 4.15.4. Fact 3.12.18. Let S1 ⊆ Fn and S2 ⊆ Fm be (cones, pointed cones, [convex sets, polytopes, con] vex cones, polyhedral cones, affine subspaces, subspaces). Then, so is SS12 . Furthermore, [ ] S dim 1 = dim S1 + dim S2 . S2 Fact 3.12.19. Let S1 , S2 ⊆ Fn be subspaces. Then,

dim S1 + dim S2 − n ≤ dim(S1 ∩ S2 ) ≤ min {dim S1 , dim S2 } { } dim S1 ≤ ≤ max {dim S1 , dim S2 } dim S2 ≤ dim(S1 + S2 ) = dim S1 + dim S2 − dim(S1 ∩ S2 ) ≤ min {dim S1 + dim S2 , n}. Furthermore, the following statements hold: i) If dim(S1 + S2 ) = dim(S1 ∩ S2 ) + 1, then either S1 ⊆ S2 or S2 ⊆ S1 . ii) If S1 ∩ S2 = {0}, then dim S1 + dim S2 = dim(S1 + S2 ) ≤ n.

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dim S1 + dim(S⊥1 ∩ S2 ) = dim S2 + dim(S1 ∩ S⊥2 ). dim[S1 ∩ (S⊥1 + S2 )] = dim[S2 ∩ (S1 + S⊥2 )]. dim(S1 + S2 ) = dim(S1 ∩ S2 ) + dim[(S1 + S2 ) ∩ (S⊥1 + S⊥2 )]. If affin S1 ∩ affin S2 is nonempty, then dim(S1 ∪ S2 ) = dim(S1 + S2 ). If affin S1 ∩ affin S2 is empty, then dim(S1 ∪ S2 ) = dim(S1 + S2 ) + 1. Source: Theorem 3.1.3. To prove the first inequality, note that dim S1 + dim S2 = dim(S1 + S2 ) + dim(S1 ∩ S2 ) ≤ n + dim(S1 ∩ S2 ). i) is given in [2991, p. 7]; iii)–v) are given in [255]; and vi) and vii) are given in [2487, p. 45]. Related: Fact 4.18.1. △ Fact 3.12.20. Let S1 , S2 ⊆ Fn be subspaces, and define f (S1 , S2 ) = dim(S1 + S2 ) − dim(S1 ∩ S2 ). Then, the following statements hold: i) If S1 , S2 , then f (S1 , S2 ) > 0. ii) f (S1 , S2 ) = 0 if and only if S1 = S2 . iii) f (S1 , S2 ) = f (S2 , S1 ). iv) Let S3 ⊆ Fn be a subspace. Then, f (S1 , S3 ) ≤ f (S1 , S2 ) + f (S2 , S3 ). v) f (S1 , S2 ) = f [S1 ∩ (S1 ∩ S2 )⊥ , S2 ∩ (S1 ∩ S2 )⊥ ]. Source: [249, 253]. Remark: f is a metric on the vector space of subspaces of Fn . Fact 3.12.21. Let S1 , S2 ⊆ Fn be nonzero subspaces, and define θ ∈ [0, π2 ] by iii) iv) v) vi) vii)

cos θ = max {|x∗ y| : (x, y) ∈ S1 × S2 and x∗ x = y∗ y = 1}. Then,

cos θ = max {|x∗ y| : (x, y) ∈ S⊥1 × S⊥2 and x∗ x = y∗ y = 1}.

Furthermore, θ = 0 if and only if S1 ∩ S2 = {0}, and θ = π2 if and only if S1 = S⊥2 . Remark: θ is a principal angle. See [1134, 1504]. Related: Fact 7.10.29, Fact 7.12.42, and Fact 7.13.27. Fact 3.12.22. Let S1 , S2 ⊆ Fn be subspaces, assume that S1 and S2 are complementary, assume that dim S2 ≥ 1, and let S3 ⊆ Fn . Then, the following statements hold: i) If S3 ⊂ S2 , then S1 + S3 ⊂ Fn . ii) If S3 is a subspace and S1 + S3 = Fn , then S2 ⊆ S3 . Fact 3.12.23. Let S1 , S2 ⊆ Fn be subspaces, and assume that dim S2 < dim S1 . Then, there exists a subspace S3 ⊆ Fn such that the following statements hold: i) S3 ⊆ S1 . ii) S2 and S3 are mutually orthogonal. iii) dim S1 ≤ dim S2 + dim S3 . Source: [2991, p. 34]. Fact 3.12.24. Let S1 , S2 , S3 ⊆ Fn be subspaces. Then, dim(S1 + S2 + S3 ) + max {dim(S1 ∩ S2 ), dim(S1 ∩ S3 ), dim(S2 ∩ S3 )} ≤ dim S1 + dim S2 + dim S3 ≤ dim(S1 ∩ S2 ∩ S3 ) + 2n. Source: [860, p. 124], [2238, p. 127], and [2991, p. 267]. Remark: Setting S3 = {0} yields a weaker version of Theorem 3.1.3. Fact 3.12.25. Let S1 , S2 , S3 ⊆ Fn be subspaces. Then,

S1 + (S2 ∩ S3 ) ⊆ (S1 + S2 ) ∩ (S1 + S3 ) = S1 + (S1 + S2 ) ∩ S3 , S1 ∩ (S2 + S3 ) ⊇ (S1 ∩ S2 ) + (S1 ∩ S3 ).

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If, in addition, S2 ⊆ S1 , then

S1 ∩ (S2 + S3 ) = S2 + (S1 ∩ S3 ).

Source: [688, pp. 11, 12] and [2991, p. 7]. ∑ Fact 3.12.26. Let S1 , . . . , Sk ⊆ Fn be convex sets, and let α1 , . . . , αk ∈ F. Then, ki=1 αi Si is convex. Source: [2487, pp. 75, 76]. Fact 3.12.27. Let S1 , . . . , Sk ⊆ Fn be subspaces having the same dimension. Then, there exists a subspace Sˆ ⊆ Fn such that, for all i ∈ {1, . . . , k}, Sˆ and Si are complementary. Source: [1304, pp.

78, 79, 259, 260].

3.13 Facts on Range, Null Space, Rank, and Defect Fact 3.13.1. Let x1 , . . . , xr ∈ Fn . Then, dim ∩ri=1 {xi }⊥ = n − rank [x1 · · · xr ]. Remark: This

result determines the dimension of an intersection of hyperplanes in terms of the number of linearly independent vectors that define the hyperplanes. Related: Fact 4.17.9. Fact 3.13.2. Let A ∈ Fn×n. Then, N(A) ⊆ R(I − A) and N(I − A) ⊆ R(A). Related: Fact 4.15.5. Fact 3.13.3. Let A ∈ Fn×m, and let S ∈ Fm×l. Then, S N(AS ) ⊆ N(A). If, in addition, S is right invertible, then S N(AS ) = N(A). Source: S N(AS ) = S {x ∈ Fl : ASx = 0} = {Sx ∈ Fl : ASx = 0} ⊆ {y ∈ Fl : Ay = 0} = N(A). Now, assume that S is right invertible. Then, S is onto, and thus S N(AS ) = S {x ∈ Fl : ASx = 0} = {Sx ∈ Fl : ASx = 0} = {y ∈ Fl : Ay = 0} = N(A). Related: Proposition 3.7.3. Fact 3.13.4. Let S ⊆ Fm, assume that S is an affine subspace, and let A ∈ Fn×m. Then, the following statements hold: i) rank A + dim S − m ≤ dim AS ≤ min {rank A, dim S}. ii) dim(AS) + dim[N(A) ∩ S] = dim S. iii) dim AS ≤ dim S. iv) If A is left invertible, then dim AS = dim S. Source: For ii), see [2314, p. 413]. For iv), note that dim AS ≤ dim S = dim ALAS ≤ dim AS. Remark: The proof of Proposition 3.6.10 uses the Moore-Penrose generalized inverse defined in Chapter 8. An alternative proof that avoids this technique is given in [2238, p. 111]. Similarly, △ define S = R(B). Then, it follows from ii) that dim R(B) = dim[AR(B)] + dim[N(A) ∩ R(B)] = rank AB + dim[R(B) ∩ N(A)]. Related: Fact 3.11.15 and Fact 12.11.28. Fact 3.13.5. Let A ∈ Fn×m , let B, C ∈ Fm×p , and assume that A∗AB = A∗AC. Then, AB = AC. Source: For all i ∈ {1, . . . , p}, coli (B) − coli (C) ∈ N(A∗A) = N(A). Fact 3.13.6. Let A ∈ Fn×m , let B, C ∈ Fn×p , and assume that AA∗B = AA∗C. Then, A∗B = A∗C. Source: Fact 3.13.5. Fact 3.13.7. Let A ∈ Fn×m , let B, C ∈ F p×n , and assume that BAA∗ = CAA∗ . Then, BA = CA. Fact 3.13.8. Let A ∈ Fn×m , let B, C ∈ F p×m , and assume that BA∗A = CA∗A. Then, BA∗ = CA∗ . Fact 3.13.9. Let A ∈ Fn×m , let x, y ∈ R(A∗ ), and assume that Ax = Ay. Then, x = y. Source: Fact 3.13.6. Fact 3.13.10. Let A ∈ Fn×m and B ∈ F1×m. Then, N(A) ⊆ N(B) if and only if there exists y ∈ Fn such that B = y∗A. Fact 3.13.11. Let A ∈ Fn×m and b ∈ Fn. Then, there exists x ∈ Fn satisfying Ax = b if and only if, for all y ∈ N(A∗ ), b∗y = 0. Source: Assume that A∗y = 0 implies that b∗y = 0. Then, N(A∗ ) ⊆ N(b∗ ). Hence, b ∈ R(b) ⊆ R(A). Fact 3.13.12. Let A ∈ Fn×m and B ∈ Fl×m. Then, N(B) ⊆ N(A) if and only if there exists C ∈ Fn×l such that A = CB. Now, let A ∈ Fn×m and B ∈ Fn×l. Then, R(A) ⊆ R(B) if and only if there exists C ∈ Fl×m such that A = BC.

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Fact 3.13.13. Let A, B ∈ Fn×m, and let C ∈ Fm×l be right invertible. Then, R(A) ⊆ R(B) if and only if R(AC) ⊆ R(BC). Furthermore, R(A) = R(B) if and only if R(AC) = R(BC). Source: Since

C is right invertible, it follows that R(A) = R(AC). Fact 3.13.14. Let A, B ∈ Fn×n, and assume that there exists α ∈ F[such ] that αA+B [ is] nonsingular. △ △ Then, N(A) ∩ N(B) = {0}. Remark: The converse is false. Let A = 12 00 and B = 00 12 . Fact 3.13.15. Let A, B ∈ Fn×m. Then, R(A + B) ⊆ R(A) + R(B). Furthermore, the following statements are equivalent: i) R(A) ⊆ R(A + B). ii) R(B) ⊆ R(A + B). iii) R(A + B) = R(A) + R(B). Source: To prove i) =⇒ ii), R(A) ⊆ R(A + B) implies that R(A + B) = R([A A + B]) = R([A B]) = R([B A + B]). Hence, R(B) ⊆ R(A + B). See [2991, p. 56]. Fact 3.13.16. Let A, B ∈ Fn×m. Then, N(A) ∩ N(B) ⊆ N(A + B). Furthermore, the following statements are equivalent: i) N(A + B) ⊆ N(A). ii) N(A + B) ⊆ N(B). iii) N(A + B) = N(A) ∩ N(B). Source: Fact 3.13.15. Fact 3.13.17. Let A, B ∈ Fn×m, and let α ∈ F be nonzero. Then, N(A) ∩ N(B) = N(A) ∩ N(A + αB) = N(αA + B) ∩ N(B). Related: Fact 3.14.10. Fact 3.13.18. Let x ∈ Fn and y ∈ Fm. If either x = 0 or y , 0, then R(xyT ) = R(x) = span {x}.

Furthermore, if either x , 0 or y = 0, then N(xyT ) = N(yT ) = {y}⊥. Fact 3.13.19. Let A ∈ Fn×m and B ∈ Fm×l. Then, rank AB = rank A if and only if R(AB) = R(A). Source: If R(AB) ⊂ R(A), then Lemma 3.1.5 implies that rank AB < rank A. Fact 3.13.20. Let A ∈ Fn×m, B ∈ Fm×l, and C ∈ Fl×k, and assume that rank AB = rank B. Then, rank ABC = rank BC. Source: rank BTAT = rank BT implies that R(CTBTAT ) = R(CTBT ). Fact 3.13.21. Let A ∈ Fn×m and B ∈ Fm×l. Then, the following statements hold: i) rank AB + dim[N(A) ∩ R(B)] = rank B. ii) rank AB + dim[N(B∗ ) ∩ R(A∗ )] = rank A. iii) rank AB + m = rank A + dim[N(A) + R(B)]. iv) rank AB + def A = dim[N(A) + R(B)]. v) rank AB + def A + dim[N(B∗ ) ∩ R(A∗ )] = m. vi) rank AB + def A = dim[N(A) + R(B)]. vii) rank AB + def B + dim[N(A) ∩ R(B)] = l. viii) def AB + rank B = dim[N(A) ∩ R(B)] + l. ix) def AB = def B + dim[N(A) ∩ R(B)]. x) def AB + rank A = dim[N(B∗ ) ∩ R(A∗ )] + l. xi) def AB + m = def A + dim[N(B∗ ) ∩ R(A∗ )] + l. xii) def AB + rank A + dim[N(A) + R(B)] = l + m. xiii) def AB + dim[N(A) + R(B)] = def A + l. Remark: i) is Sylvester’s rank formula given by Proposition 3.6.10. Related: Fact 8.9.2, Fact 8.9.3, and Fact 8.9.4.

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Fact 3.13.22. Let A ∈ Fn×m and B ∈ Fm×l. Then,

max {def A + l − m, def B} ≤ def AB ≤ def A + def B. If, in addition, m = l, then

max {def A, def B} ≤ def AB.

Remark: The first inequality is Sylvester’s law of nullity. Fact 3.13.23. Let A ∈ Fn×m , B ∈ Fn×l, and k ≥ 0. Then, there exists X ∈ Fm×l such that AX = B and rank X = k if and only if rank B ≤ k ≤ min {m + rank B − rank A, l}. Source: [2760]. Fact 3.13.24. The following statements hold:

i) Let A ∈ Fn×m. Then, rank A = 0 if and only if A = 0. ii) For all α ∈ F and A ∈ Fn×m , rank αA = (sign |α|) rank A. iii) For all A, B ∈ Fn×m , rank(A + B) ≤ rank A + rank B Remark: Compare these statements to the properties of a matrix norm given by Definition 11.2.1. Fact 3.13.25. Let n, m, k ∈ P. Then, rank 1n×m = 1 and 1kn×n = nk−11n×n. Fact 3.13.26. Let A ∈ Fn×m. Then, rank A = 1 if and only if there exist x ∈ Fn and y ∈ Fm such that x , 0, y , 0, and A = xyT. If these statements hold, then tr A = yTx. Related: Fact 6.10.5. Fact 3.13.27. Let A ∈ Fn×n, k ≥ 1, and l ∈ N. Then, the following statements hold: i) R[(AA∗ )k ] = R[(AA∗ )lA]. ii) N[(A∗A)k ] = N[A(A∗A)l ]. iii) rank (AA∗ )k = rank (AA∗ )lA. iv) def (A∗A)k = def A(A∗A)l. Fact 3.13.28. Let A ∈ Fn×n. Then, R(I − A) = R(A − A2 ) + R(I − A2 ),

N(I − A) = N(A − A2 ) ∩ N(I − A2 ),

R(I + A) = R(A + A2 ) + R(I − A2 ),

N(I + A) = N(A + A2 ) ∩ N(I − A2 ),

R(A) = R(A2 ) + R(A − A3 ), R(A) = R(A − A2 ) + R(A + A2 ),

N(A) = N(A2 ) ∩ N(A − A3 ), N(A) = N(A − A2 ) ∩ N(A + A2 ).

Source: [257]. Fact 3.13.29. Let A ∈ Fn×n. Then,

rank(I − A2 ) = rank(I + A) + rank(I − A) − n, rank(A − A2 ) = rank A + rank(I − A) − n, rank(A + A2 ) = rank A + rank(I + A) − n, rank(A − A3 ) = rank A + rank(I − A2 ) − n, rank(A − A3 ) + rank(I − A) = rank(A − A2 ) + rank(I − A2 ), rank(A − A3 ) + rank(I + A) = rank(A + A2 ) + rank(I − A2 ), rank A + rank(A − A3 ) = rank(A + A2 ) + rank(A − A2 ), rank A + rank(A2 − A4 ) = rank A2 + rank(A − A3 ), rank A + rank(A2 − A5 ) = rank A2 + rank(A − A4 ). Source: [257, 2659]. Related: Fact 4.9.2 and Fact 4.21.3. Fact 3.13.30. Let A ∈ Fn×n. Then, the following statements hold:

i) 2 rank A ≤ rank A2 + n.

318

ii) iii) iv) v) vi)

CHAPTER 3

rank A = rank A2 + n if and only if N(A) ⊆ R(A). 2 rank A2 ≤ rank A + rank A3 . If k ≥ 1, then k rank A ≤ rank Ak + (k − 1)n. If l ≥ k ≥ 0, then rank(Ak − Ak+l ) = rank Ak + rank(I − Al ) − n. ( ) ∑ If k ≥ 0, then rank(I − Ak+1 ) = rank(I − A) + rank ki=0 Ai − n.

vii) If k ≥ 0 and l ≥ 2, then rank(Ak − Akl ) = rank Ak + rank(I − Ak(l−1) ) − n. Furthermore, consider the following statements: viii) There exists k ≥ 2 such that Ak = A. ix) 2 rank A2 = rank A + rank A3 . x) There exist X, Y ∈ Fn×n such that A = A2X + YA2 . Then, viii) =⇒ ix) ⇐⇒ x). Source: [257] and [860, p. 126]. Fact 3.13.31. Let x, y ∈ Fn. Then, the following statements hold: i) R(xyT + yxT ) ⊆ R([x y]). ii) {x}⊥ ∩ {y}⊥ ⊆ N(xyT + yxT ). iii) rank(xyT + yxT ) ≤ 2. Furthermore, the following statements are equivalent: iv) Either x or y is zero. v) xyT + yxT = 0. vi) rank(xyT + yxT ) = 0. vii) def(xyT + yxT ) = n. In addition, the following statements are equivalent: viii) There exists α ∈ F such that x = αy , 0. ix) rank(xyT + yxT ) = 1. x) def(xyT + yxT ) = n − 1. Moreover, the following statements are equivalent: xi) x and y are linearly independent. xii) rank(xyT + yxT ) = 2. xiii) def(xyT + yxT ) = n − 2. Finally, the following statements are equivalent: xiv) x and y are nonzero. xv) x or y is nonzero and R(xyT + yxT ) = R([x y]). xvi) x or y is nonzero and rank(xyT + yxT ) = rank([x y]). xvii) x or y is nonzero and {x}⊥ ∩ {y}⊥ = N(xyT + yxT ). xviii) x or y is nonzero and dim({x}⊥ ∩ {y}⊥ ) = def(xyT + yxT ). xix) 1 ≤ rank(xyT + yxT ) ≤ 2. Remark: xyT + yxT is a doublet. See [835, pp. 539, 540]. Fact 3.13.32. Let A ∈ Fn×m, x ∈ Fn, and y ∈ Fm. Then, (rank A) − 1 ≤ rank(A + xy∗ ) ≤ (rank A) + 1. Related: Fact 8.4.10. [ ] [ ] △ △ Fact 3.13.33. Let A = 10 00 and B = 00 10 . Then, rank AB = 1 and rank BA = 0. Related: Fact

4.10.32.

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BASIC MATRIX PROPERTIES

Fact 3.13.34. Let A ∈ Fn×m and B ∈ Fn×l . Then,

R(AA∗ + BB∗ ) = R(A) + R(B),

N(AA∗ + BB∗ ) = N(A∗ ) ∩ N(B∗ ).

Source: [2239]. Fact 3.13.35. Let A ∈ Fn×m and B ∈ Fm×l. Then, rank AB = rank A∗AB = rank ABB∗. Source:

[2418, p. 37]. Fact 3.13.36. Let A, B ∈ Fn×m. Then,

} rank(A + B) |rank A − rank B| ≤ ≤ rank A + rank B. rank(A − B)

If, in addition, rank B ≤ k, then

{

{

(rank A) − k ≤

} rank(A + B) ≤ (rank A) + k. rank(A − B)

Fact 3.13.37. Let A ∈ Fn×m and B ∈ Fn×l. Then, A∗B = 0 if and only if R(A) and R(B) are mutually orthogonal. If these statements hold, then R(A) ∩ R(B) = {0}. Source: Let x = Az ∈ R(A) and y = Bw ∈ R(B). Then, x∗ y = z∗A∗Bw = 0. Hence, R(A) and R(B) are mutually orthogonal subspaces. Now, Proposition 3.3.5 implies that R(A) ∩ R(B) = {0}. See [2991, p. 34]. Related: Fact 8.4.8. Fact 3.13.38. Let A, B ∈ Fn×m, and assume that A∗B = 0 and BA∗ = 0. Then,

rank(A + B) = rank A + rank B. Source: Since A∗B = 0, it follows from Fact 3.13.37 that R(A) ∩ R(B) = {0}. Likewise, BA∗ = 0 implies that R(A∗ ) ∩ R(B∗ ) = {0}. The result now follows from [ ] Fact 3.14.11. [ ] Source: Fact 8.4.8 and [762, 1377]. Remark: The converse is false. Let A = 10 00 and B = 00 11 . Fact 3.14.11 gives

necessary and sufficient conditions for rank to be additive. Fact 3.13.39. Let A, B ∈ Fn×n, assume that A is Hermitian, and assume that [A, B] = 0. Then, rank(A + B) = rank A + rank B. Source: Fact 3.13.38. Fact 3.13.40. Let A, B ∈ Fn×m. Then,

[ ([ ]) ] rank A + rank B = rank(A + B) + dim R AB ∩ N([In In ]) + dim[R(A∗ ) ∩ R(B∗ )] [ ([ A ]) ] = rank(A + B) + dim R B ∩ N([In In ]) + dim([N(A) + N(B)]⊥ ) [ ([ ]) ] = rank(A + B) + dim R AB ∩ N([In In ]) + m − dim[N(A) + N(B)].

Remark: See [2238, pp. 114, 115] and [2991, p. 53]. Problem: Use this result to prove Fact

3.13.38. Fact 3.13.41. Let A, B ∈ Fn×n . Then,

R(A − ABA) = R(A) ∩ R(I − AB),

N(A − ABA) = N(A) + N(I − BA),

R(I − A ) = R(I − A) ∩ R(I + A),

N(I − A2 ) = N(I − A) + N(I + A).

2

Source: [257]. Fact 3.13.42. Let A, B ∈ Fn×n . Then,

rank(AB − I) ≤ rank(A − I) + rank(B − I), rank(AB − I) + rank B = rank(B − BAB) + n, rank(I − BA) + rank A = rank(A − ABA) + n.

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CHAPTER 3

Source: rank(AB−I) = rank[A(B−I)+A−I] ≤ rank[A(B−I)]+rank(A−I) ≤ rank(A−I)+rank(B−I).

See [2991, p. 55]. Fact 3.13.43. Let A ∈ Fn×m and B ∈ Fm×n . Then,

rank(A − ABA) + m = rank A + rank(Im − BA), rank(A − ABA) + n = rank A + rank(In − AB), rank(A − ABA) − rank(B − BAB) = rank A − rank B. Source: [257]. △ ∑ Fact 3.13.44. Let A1 , . . . , Ak ∈ Fn×m, and define A = ki=1 Ai . Then, the following statements

are equivalent: ∑ i) rank A = ki=1 rank Ai . ii) For all i ∈ {1, . . . , k}, rank(A − Ai ) = rank A − rank Ai . Source: [231, 1334]. Fact 3.13.45. Let a, b ∈ F be nonzero, let n, m ≥ 2, and define A ∈ Fn×m by   a a+b · · · a + (m − 1)b     a + mb a + (m + 1)b · · · a + (2m − 1)b  △   .   A =  .. .. . ..   · .· · . . .   a + (n − 1)mb a + [(n − 1)m + 1]b · · · a + (nm − 1)b Then, rank A = 2. Source: [1777]. Remark: As stated in [1777], “Given a homogeneous recurrence sequence, the rank of the associated recurrence matrix is bounded above by the order r of the recurrence. For inhomogeneous sequences, the upper bound on matrix rank is r + 1.”

3.14 Facts on the Range, Rank, Null Space, and Defect of Partitioned Matrices Fact 3.14.1. Let A ∈ Fn×m and B ∈ Fk×l, and assume that B is a submatrix of A. Then,

rank B ≤ rank A ≤ rank B + n + m − k − l. If, in particular, B = 0, then rank A ≤ n + m − k − l. Source: [284]. Related: Fact 3.13.36 and Fact 3.16.4. Fact 3.14.2. Let A ∈ Fn×m and B ∈ Fn×l. Then, R([A B]) = R([B A]),

rank [A B] = rank [B A].

Fact 3.14.3. Let A ∈ Fn×m and B ∈ Fn×l. Then, rank A∗B = rank [A B] if and only if R(A) = R(B). Source: [252]. Fact 3.14.4. Let A ∈ Fn×m , let B, C ∈ Fn×l , and assume that R(B) = R(C). Then,

R([A B]) = R([A C]),

rank [A B] = rank [A C].

Fact 3.14.5. Let A ∈ Fn×m and B ∈ Fl×m. Then,

([ ]) ([ ]) A B N =N , B A

[ ] [ ] A B def = def . B A

Fact 3.14.6. Let A ∈ Fn×m , let B, C ∈ Fl×m , and assume that N(B) = N(C). Then,

([ ]) ([ ]) A A N =N , B C

[ ] [ ] A A def = def . B C

321

BASIC MATRIX PROPERTIES

Fact 3.14.7. Let A ∈ Fn×m and B ∈ Fn×l, and let x, y ∈ Fn. Then, [R(A) + x] ∩ [R(B) + y] is

nonempty if and only if x − y ∈ R([A B]). Now, assume that these conditions hold. Then,

[R(A) + x] ∩ [R(B) + y] = [−A 0n×l ][A B]inv (x − y) + x = [0n×m B][A B]inv (x − y) + y. Finally, assume that [A B] has full column rank, and let w ∈ Fm+l satisfy x − y = [A B]w. Then, [R(A) + x] ∩ [R(B) + y] = {[−A 0n×l ]w + x} = {[0n×m B]w + y}. Remark: The intersection of two affine subspaces is an affine subspace. See Fact 3.12.4. Fact 3.14.8. Let A ∈ Fn×m and B ∈ Fn×l. Then,

R(A) + R(B) = R([A B]) = R(AA∗ + BB∗ ) = span[R(A) ∪ R(B)]. Consequently,

dim[R(A) + R(B)] = rank [A B] = rank(AA∗ + BB∗ ).

Furthermore, the following statements are equivalent: i) rank [A B] = n. [ ∗] A ii) def ∗ = 0. B iii) N(A∗ ) ∩ N(B∗ ) = {0}. Source: R(A) + R(B) = span[R(A) ∪ R(B)] follows from Fact 3.12.16. Fact 3.14.9. Let A ∈ Fn×m and B ∈ Fl×m. Then, [ ] A ∗ ∗ dim[R(A ) + R(B )] = rank . B Source: Fact 3.14.8. Fact 3.14.10. Let A ∈ Fn×m and B ∈ Fl×m. Then,

([ ]) A N(A) ∩ N(B) = N , B

[ ] A dim[N(A) ∩ N(B)] = def . B

Furthermore, the following statements are equivalent: [ ] A i) rank = m. B [ ] A ii) def = 0. B iii) N(A) ∩ N(B) = {0}. Related: Fact 3.13.17. Fact 3.14.11. Let A, B ∈ Fn×m. Then, the following statements are equivalent: i) rank(A + B) = rank A + rank B. [ ] A ii) rank [A B] = rank = rank A + rank B. B iii) dim[R(A) ∩ R(B)] = dim[R(A∗ ) ∩ R(B∗ )] = 0. iv) R(A) ∩ R(B) = {0} and R(A∗ ) ∩ R(B∗ ) = {0}. v) There exists C ∈ Fm×n such that ACA = A, CB = 0, and BC = 0. Source: [615, 762, 1966, 2666]. Remark: Equivalent statements are given by Fact 8.4.31 assuming [ ] that A + B is nonsingular. Remark: Equivalent statements for rank [A B] = rank AB are given by Fact 8.4.7. Related: Fact 3.13.38, Fact 3.13.40, and Fact 3.14.18.

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Fact 3.14.12. Let A ∈ Fn×m and B ∈ Fn×l. Then, R(A) = R(B) if and only if rank A = rank B =

rank [A B]. Fact 3.14.13. Let A ∈ Fn×m, B ∈ Fk×m, C ∈ Fm×l, and D ∈ Fm×p, and assume that

rank

[ ] A = rank A, B

rank [C D] = rank C.

[ ] A rank [C D] = rank AC. B

Then,

Source: i) of Fact 3.13.21. Fact 3.14.14. Let A ∈ Fn×m and B ∈ Fn×l. Then,

R(A) ∩ R(B) = [A 0n×l ]N([A B]) ∩ [0n×m B]N([A B]), dim([A 0n×l ]N([A B])) = dim([0n×m B]N([A B])) = def [A B] − def A − def B. Source: Let z ∈ R(A) ∩ R(B). Then, there exist x ∈ Rm and y ∈ Fl such that z = Ax = −By.

[ ] [ ] Therefore, [A B] yx = 0, that is, yx ∈ N([A B]), and thus z ∈ [A 0n×l ]N([A B]). Likewise, z ∈ [0n×m B]N([A B]). The reverse inclusion is immediate. To prove the second equality, note that ii) of Fact 3.13.4 implies that dim([A 0n×l ]N([A B])) = dim N([A B]) − dim[N([A 0n×l ]) ∩ N([A B])] [ ] N(A) = dim N([A B]) − dim = def [A B] − (def A + def B). N(B) Related: Fact 8.9.1. Fact 3.14.15. Let A ∈ Fn×m and B ∈ Fn×l. Then,

{

rank A + rank B − n ≤ dim[R(A) ∩ R(B)] ≤ min {rank A, rank B} ≤

rank A rank B

}

≤ max {rank A, rank B} ≤ rank [A B] = rank A + rank B − dim[R(A) ∩ R(B)] ≤ min {rank A + rank B, n}, max {def A + def B, m + l − n} ≤ def A + def B + dim[R(A) ∩ R(B)] = def [A B] { } l + def A ≤ min {l + def A, m + def B} ≤ m + def B ≤ max {l + def A, m + def B} ≤ m + l − dim[R(A) ∩ R(B)] ≤ def A + def B + n. Consequently, the following statements are equivalent: i) rank [A B] = rank A + rank B. ii) def [A B] = def A + def B. iii) R(A) ∩ R(B) = {0}. If, in addition, A∗B = 0, then rank [A B] = rank A + rank B,

def [A B] = def A + def B.

Source: Theorem 3.1.3, Fact 3.12.19, and Fact 3.14.8. For the case A∗B = 0, note that

[

] [ ∗ A∗ AA rank [A B] = rank ∗ [A B] = B 0

] 0 = rank A∗A + rank B∗B = rank A + rank B. B∗B

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BASIC MATRIX PROPERTIES

Remark: Note that rank [A B] + dim[R(A) ∩ R(B)] = rank A + rank B. Using Fact 3.13.37, A∗B = 0 implies that R(A) ∩ R(B) = {0}, and thus rank [A B] = rank A + rank B. Related: Fact 8.9.7. Fact 3.14.16. Let A ∈ Fn×m and B ∈ Fl×m. Then,

rank

[ ] A + dim[R(A∗ ) ∩ R(B∗ )] = rank A + rank B. B

Source: Fact 3.14.15. Fact 3.14.17. Let A ∈ Fn×m and B ∈ Fl×m. Then,

max {rank A, rank B} ≤ rank

[ ] A = rank A + rank B − dim[R(A∗ ) ∩ R(B∗ )] ≤ rank A + rank B, B

[ ] A def A − rank B ≤ def A − rank B + dim[R(A ) ∩ R(B )] = def ≤ min {def A, def B}. B ∗

If, in addition, AB∗ = 0, then [ ] A rank = rank A + rank B, B

∗

def

[ ] A = def A − rank B. B

Source: Fact 3.12.19 and Fact 3.14.15. Related: Fact 8.9.7. Fact 3.14.18. Let A, B ∈ Fn×m. Then,

max {rank A, rank B} rank(A + B)

}

   rank [A B]         [ ]  ≤ rank A + rank B, ≤  A       rank   B

[ ] A rank [A B] + rank ≤ rank A + rank B + rank(A + B) B    rank A + rank B + rank [A B]         [ ]  ≤ 2(rank A + rank B), ≤ A         rank A + rank B + rank B     def [A B] − m  {      min {def A, def B}   [ ]  ≤ def A − rank B ≤   A     def(A + B).     def B [] [ ] [ ] Source: rank(A + B) = rank [A B] II ≤ rank [A B], and rank(A + B) = rank [I I] AB ≤ rank AB . Fact 3.14.19. Let A ∈ Fn×m, B ∈ Fl×k, and C ∈ Fl×m. Then, [ ] [ ] A 0 A 0 rank A + rank B = rank ≤ rank , 0 B C B [

0 rank A + rank B = rank B Finally, let D ∈ Fk×m and E ∈ Fl×n. Then,

[

A rank A + rank B = rank BD + EA

] [ ] A 0 A ≤ rank . 0 B C ] [ 0 0 = rank B B

] A . BD + EA

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Fact 3.14.20. Let A ∈ Fn×m, B ∈ Fm×l, and C ∈ Fl×k. Then,

[

0 rank AB + rank BC ≤ rank BC

] AB = rank B + rank ABC, B

rank A + rank B + rank C − m − l ≤ rank AB + rank BC − rank B ≤ rank ABC. Furthermore, the following statements are equivalent: [ ] 0 AB i) rank = rank AB + rank BC. BC B ii) rank AB + rank BC − rank B = rank ABC. iii) There exist X ∈ Fk×l and Y ∈ Fm×n such that BCX + YAB [ = B.] [ ][ ][ ] 0 AB = I A −ABC 0 I 0 and Fact Remark: This is the Frobenius inequality. Source: Use BC 0 I B 0 B C I 3.14.19. The last statement follows from Fact 7.11.26. See [2674, 2675]. Related: Fact 8.9.16 for the case of equality. Fact 3.14.21. Let A, B ∈ Fn×m. Then,   [ ]  0 A B  A   rank [A B] + rank ≤ rank  A A 0  = rank A + rank B + rank(A + B). B   B 0 B [ ] △ △ Source: Use the Frobenius inequality with A = CT = [I I] and B replaced by A0 B0 . Fact 3.14.22. Let A ∈ Fn×m, B ∈ Fn×l, and C ∈ Fn×k. Then,

rank [A B C] ≤ rank [A B] + rank [B C] − rank B ≤ rank [A B] + rank C ≤ rank A + rank B + rank C. Source: [1896]. Fact 3.14.23. Let A ∈ Fn×m and B ∈ Fm×n. Then,

[

[ rank

] [ ][ ][ ] In − AB I A 0 In − AB In 0 = n 0 0 Im B 0 In In [ ][ ][ ] I 0 In 0 In In − AB = n , B Im 0 BAB − B 0 Im ] In − AB = rank B + rank(In − AB) = n + rank(BAB − B). 0

In B

In B

Related: Fact 3.17.7. Fact 3.14.24. Let A ∈ Fn×m and B ∈ Fm×n. Then,

[

[

A rank BA

A BA

] [ AB I = n B B [ I = n 0

0 Im A Im

][

][

A 0

0 B − BAB

A − ABA 0

0 B

][ ][

]

Im 0

B In

Im A

] 0 , In

] AB = rank A + rank(B − BAB) = rank B + rank(A − ABA). B

Related: Fact 3.17.9. ] △ [ Fact 3.14.25. Let A = CA DB ∈ F(n+m)×(n+m), and consider the following statements:

i) rank [A B] = rank

[A] C

= rank A = n.

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BASIC MATRIX PROPERTIES

ii) A is nonsingular. [ ] iii) rank [A B] = rank CA = n ≤ rank A. Then, i) =⇒ ii) =⇒ iii). Source: [1451, p. 20]. [ ] [ ] △ Fact 3.14.26. Let CA DB ∈ F(n+k)×(m+l), and define r = rank CA DB . Then, [ ] [ ] A B rank + rank + rank [A B] + rank [C D] − rank A − rank B − rank C − rank D C D [ ] A B ≤ rank ≤ rank A + rank B + rank C + rank D, C D max {0, r − k − l} ≤ rank A ≤ min{r, n, m},

max {0, r − m − k} ≤ rank B ≤ min{r, n, l},

max {0, r − n − l} ≤ rank C ≤ min{r, m, k},

max {0, r − n − m} ≤ rank D ≤ min{r, k, l}.

Source: [2238, p. 117] and [2638, 2664]. [ ] [ ] [ ] Fact 3.14.27. Let CA DB ∈ F(n1+n2 )×(m1+m2 ), assume that CA DB is nonsingular, and define GE HF ∈

F(m1+m2 )×(n1+n2 ) by

[

E G

] [ F △ A = H C

B D

]−1 .

Then, def A = def H, def B = def F, def C = def G, and def D = def E. Source: [2557, 2779] and [2780, p. 38]. Remark: The sizes of the matrix blocks differ from the sizes in Fact 3.17.32. Remark: This is the nullity theorem. See [2557] and Fact 4.24.2. Remark: A and H are complementary. The matrices U ∈ Fn×m and V ∈ Fm×n are complementary submatrices if the row numbers not used to create U are the column numbers used to create V, and the column numbers not used to create U are the row numbers used to create V. See Fact 3.14.28. Fact 3.14.28. Let A ∈ Fn×n, assume that A is nonsingular, and let S1 , S2 ⊆ {1, . . . , n}. Then, rank (A−1 )(S1 ,S2 ) = rank A(S∼2 ,S∼1 ) + card(S1 ) + card(S2 ) − n, def (A−1 )(S1 ,S2 ) = def A(S∼2 ,S∼1 ) . Source: [1451, p. 19] and [2780, p. 40]. Remark: The submatrices (A−1 )(S1 ,S2 ) and A(S∼2 ,S∼1 ) are complementary. Related: Fact 3.14.27, Fact 3.14.29, and Fact 3.17.33. Fact 3.14.29. Let A ∈ Fn×n, assume that A is nonsingular, and let S ⊆ {1, . . . , n}. Then,

rank (A−1 )(S,S∼ ) = rank A(S,S∼ ) . Source: Apply Fact 3.14.28 with S2 = S∼1 . Fact 3.14.30. Let A1 , . . . , Ak ∈ Fn×n . Then,

rank Now, assume that

∑k

k ∑

Ai ≤

i=1

Ai is nonsingular. Then,   A1 A2 A3 · · ·  0 A1 A2 A3 rank  . . .. ... ...  ..  0 0 · · · A1

k ∑

rank Ai .

i=1

i=1

Source: [2238, p. 107].

An ··· .. .

0 An .. .

··· ··· .. .

A2

A3

···

0 0 .. . An

     = kn.  

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CHAPTER 3

3.15 Facts on the Inner Product, Outer Product, Trace, and Matrix Powers Fact 3.15.1. Let x, y, z ∈ Fn, assume that x∗x = y∗y = z∗z = 1, and let p ≥ 2. Then,

√ p

1−

|x∗y|2

≤

√p

1−

|x∗z|2

√ p + 1 − |z∗y|2 ,

In particular, √ √ √ 1 − |x∗y|2 ≤ 1 − |x∗z|2 + 1 − |z∗y|2 ,

√ √ √p p p ∗ 2 ∗ 2 1 − | Re x y| ≤ 1 − | Re x z| + 1 − | Re z∗y|2 . √

1 − | Re x∗y|2 ≤

√ √ 1 − | Re x∗z|2 + 1 − | Re z∗y|2 .

Equality holds in each inequality if and only if there exists α ∈ F such that either z = αx or z = αy. Source: [1829], [2983, p. 155], and [2991, pp. 195–197]. Related: Fact 4.13.27, Fact 5.1.5, and Fact 11.8.9. Fact 3.15.2. Let x, y ∈ Fn. Then, x∗x = y∗y and Im x∗y = 0 if and only if (x − y)∗ (x + y) = 0. Fact 3.15.3. Let x, y ∈ Rn. Then, xxT = yyT if and only if either x = y or x = −y. Fact 3.15.4. Let x, y ∈ Rn. Then, xyT = yxT if and only if x and y are linearly dependent. Fact 3.15.5. Let x, y ∈ Rn. Then, xyT = −yxT if and only if either x = 0 or y = 0. Source: If x(i) , 0 and y( j) , 0, then x( j) = y(i) = 0 and 0 , x(i) y( j) , x( j) y(i) = 0. Fact 3.15.6. Let x, y ∈ Rn. Then, yxT + xyT = yTyxxT if and only if either x = 0 or y = 21 yTyx. Fact 3.15.7. Let x, y ∈ Fn. Then, (xy∗ )r = (y∗x)r−1xy∗. Fact 3.15.8. Let A ∈ Cn×n, and let x, y ∈ Cn . Then, 4y∗Ax =

3 ∑

ȷi (x + ȷi y)∗A(x + ȷi y)

i=0

= (x + y)∗A(x + y) − (x − y)∗A(x − y) + [(x + ȷy)∗A(x + ȷy) − (x − ȷy)∗A(x − ȷy)] ȷ. Source: [2238, p. 261] and [2979, p. 3]. Remark: A = In yields the polarization identity in Fact

11.8.3. Fact 3.15.9. Let x1 , . . . , xk ∈ Fn and y1 , . . . , yk ∈ Fm. Then, the following statements are equiv-

alent: i) x1 , . . . , xk are linearly independent, and y1 , . . . , yk are linearly independent. ) (∑ ii) rank ki=1 xi yTi = k. Source: [835, p. 537]. Fact 3.15.10. Let A, B, C ∈ R2×2. Then, tr(ABC + ACB) + (tr A)(tr B) tr C = (tr A) tr BC + (tr B) tr AC + (tr C) tr AB. Source: [591, p. 330]. Related: Fact 6.9.21. Fact 3.15.11. Let A ∈ Fn×m and B ∈ Fl×k. Then, AEi, j,m×l B = coli (A)rowj (B). ∑ Fact 3.15.12. Let A ∈ Fn×m, B ∈ Fm×l, and C ∈ Fl×n. Then, tr ABC = ni=1 rowi (A)Bcoli (C). Fact 3.15.13. Let A ∈ Fn×m. Then, the following statements are equivalent:

i) A = 0. ii) Ax = 0 for all x ∈ Fm. iii) tr AA∗ = 0. Fact 3.15.14. Let A ∈ Fn×n and k ≥ 1. Then, Re tr A2k ≤ tr AkAk∗ ≤ tr (AA∗ )k. Remark: To prove the left-hand inequality, consider tr (Ak − Ak∗ )(Ak∗ − Ak ). For the right-hand inequality in the case k = 2, consider tr (AA∗ − A∗A)2. Fact 3.15.15. Let A ∈ Fn×n. Then, tr Ak = 0 for all k ∈ {1, . . . , n} if and only if An = 0. Source: For sufficiency, Fact 6.10.12 implies that spec(A) = {0}, and thus the Jordan form of A is a

327

BASIC MATRIX PROPERTIES

block-diagonal matrix each of whose diagonally located blocks is a standard nilpotent matrix. For necessity, see [2983, p. 112]. Fact 3.15.16. Let A ∈ Fn×n, and assume that tr A = 0. If A2 = A, then A = 0. If Ak = A, where k ≥ 4 and 2 ≤ n < p, where p is the smallest prime divisor of k − 1, then A = 0. Source: [770]. △ Fact 3.15.17. Let A, B ∈ F2×2 , and define str A = A(1,1) − A(2,2) . Then, (tr AB)2 = tr A2B2 + tr ABAABA , str AB + str BA = (tr A) str B + (tr B) str A. Source: [1611]. Related: Fact 4.29.2. Fact 3.15.18. Let A, B ∈ Fn×m. Then,

Re tr A∗B ≤ |Re tr A∗B| ≤ | tr A∗B| ≤

√

tr(AA∗ ) tr(BB∗ ) ≤ 12 tr(AA∗ + BB∗ ).

Source: [2991, p. 32]. Related: Fact 3.15.19. Fact 3.15.19. Let A, B ∈ Fn×n. Then,

 Re tr A∗B ≤ | Re tr A∗B| ≤ | tr A∗B|    √ ≤ tr(AA∗ ) tr(BB∗ ) ≤ 12 tr(AA∗ + BB∗ ).   Re tr AB ≤ | Re tr AB| ≤ | tr AB| 

Source: Fact 3.15.18, Fact 10.14.22, [1477], and [2991, p. 32]. Fact 3.15.20. Let A, B ∈ Rn×n, and assume that tr(AAT + BBT ) = tr(AB + ATBT ). Then, A = BT . Source: [1158, p. 62]. Fact 3.15.21. Let A, B ∈ Fn×n , and let k ≥ 0. Then, tr (AB)k = tr (BA)k . Fact 3.15.22. Let A, B ∈ Fn×n, assume that AB = 0, and let k ≥ 0. Then, tr (A+B)k = tr Ak +tr Bk. Fact 3.15.23. Let A ∈ Fn×n , and assume that tr A = 0. Then, the following statements hold:

i) If n = 2, then A2 = 21 (tr A2 )I2 = −(det A)I2 and tr A2 = −2 det A. ii) If n = 3, then tr A3 = 3 det A. iii) If n = 4, then tr A4 = 21 (tr A2 )2 − 4 det A. Source: Fact 6.9.1, Fact 6.9.2, and Fact 6.9.3. Remark: These results apply to commutators A = [B, C]. Fact 3.15.24. Let A ∈ Rn×n, x, y ∈ Rn, and k ≥ 1. Then, (A + xyT )k = Ak + BIˆk C T, △

△

where B = [x Ax · · · Ak−1x] and C = [y (AT + yxT )y · · · (AT + yxT )k−1 y]. Source: [437]. Fact 3.15.25. Let A, B ∈ Fn×n. Then, the following statements hold: i) AB + BA = 21 [(A + B)2 − (A − B)2 ]. ii) (A + B)(A − B) = A2 − B2 − [A, B]. iii) (A − B)(A + B) = A2 − B2 + [A, B]. iv) A2 − B2 = 12 [(A + B)(A − B) + (A − B)(A + B)]. v) A∗A − B∗B = 21 [(A + B)∗ (A − B) + (A − B)∗ (A + B)]. Fact 3.15.26. Let A, B ∈ Fn×n and k ≥ 1. Then, Ak − Bk =

k−1 ∑ i=0

Ai (A − B)Bk−1−i =

k ∑ i=1

Ak−i (A − B)Bi−1.

328

CHAPTER 3

Fact 3.15.27. Let A, B ∈ Fn×n and k ≥ 1. Then,

1∑ (A + e(2iπ/k) ȷ B)k . k i=0 k−1

Ak + Bk =

Source: [454, p. 27]. Fact 3.15.28. Let A, B ∈ Fn×n , and assume that A, B−1 −A, and A−1 +(B−1 −A)−1 are nonsingular.

Then,

ABA = A − [A−1 + (B−1 − A)−1 ]−1 .

Source: [769, p. 345]. Remark: This is Hua’s identity. Fact 3.15.29. Let A, B ∈ Fn×m, let C ∈ Fm×n , and assume that A+ B = ACB. Then, A+ B = BCA. Source: [2937]. Fact 3.15.30. Let A ∈ Fn×n, B ∈ Fn×m, and C ∈ Fm×m, and let k ≥ 1. Then,

[

]k

[

] Ak−iBC i−1 . Ck ] △ [ △ [ B −B ] Fact 3.15.31. Let A, B ∈ Fn×n, and define A = AA AA and B = −B B . Then, AB = BA = 0. 2×2 Fact 3.15.32. Let A, B ∈ C , assume that A(2,2) = B(2,2) = 0 and A(1,2) A(2,1) = B(1,2) B(2,1) , and let n ≥ 1. Then, (An )(1,2) (An )(2,1) = (Bn )(1,2) (Bn )(2,1) . Source: [1751]. Fact 3.15.33. Let A ∈ C2×2 , assume that A is nonsingular and tr A , 0, and let n ≥ 1. Then, A(1,2) (An )(2,1) = (An )(1,2) A(2,1) . Source: [1747]. A B 0 C

Ak = 0

∑k

i=1

△ 1

Fact 3.15.34. Let A ∈ R2×2 , and define a =

2 tr A. Then, the following are equivalent: i) There exists B ∈ R2×2 such that A = B2 . √ √ ii) det A ≥ 0 and either A + det AI = 0 or tr A + 2 det A > 0. B is a square root of A. If these statements hold, then the following statements hold: iii) A has finitely many square roots if and only if A , aI2 . If these statements hold, then A has either exactly zero, two, or four square roots. iv) A has exactly√zero square roots if and only if A , aI2 and either det A < 0 or both det A ≥ 0 and tr A + 2 det A ≤ 0. √ v) A has exactly four square roots if and only if A , aI2 , det√A > 0, and tr A√− 2 det A > 0. In this √ case, the four square roots of A are ±(tr A + 2 det A)−1/2 (A + det AI2 ) and √ −1/2 ±(tr A − 2 det A) (A − det AI2 ). √ −1/2 vi) If A has √ exactly two square roots, then the two square roots of A are ±(tr A + 2 det A) (A + det AI2 ). vii) A has infinitely many square roots if and only if A = aI2 . If these statements hold, then the set B of square roots of A is given by    B1 , a < 0,     B= B1 ∪ B2 , a = 0,     B1 ∪ B3 ∪ B4 ∪ B5 , a > 0,

where △

B1 = △

B3 =

[√ a 0

{[

] } {[ ] } β 0 0 △ :α∈R , : α, β ∈ R, β , 0 , B2 = a−α2 α 0 −α β ] [ √ ] √ 0√ △ − a △ √ √0 + B2 B5 = + B2 , B4 = { aI2 , − aI2 }. − a 0 a α

329

BASIC MATRIX PROPERTIES

√

√

Source: [123]. Remark: A + det AI = 0 implies tr A + 2 det A = 0. Related: Fact 10.10.8. Fact 3.15.35. Two cube roots of I2 are given by

  − 1  √2  − 3

√ 3 3  2  

3   −1 −1   = I2 .  =  1 0 − 12

2

Fact 3.15.36. Let A, B, C, D ∈ Rn×n , assume that A, B, C, D are nonnegative, and assume that

0 ≤≤ B ≤≤ A and 0 ≤≤ D ≤≤ C. Then, 0 ≤≤ BD ≤≤ AC. Fact 3.15.37. Let A, B ∈ Fn×m . Then, ∑ AATBBT = (ABT )2 + [(coli A)T col j B − (col j A)T coli B][coli A(col j B)T − col j A(coli B)T ],

where the sum is taken over all i, j ∈ {1, . . . , m} such that i < j. Source: [2703]. Remark: This is a matrix extension of Lagrange’s identity. See Fact 2.12.13. Related: Fact 3.15.38, Fact 4.13.33, and Fact 8.3.3. Fact 3.15.38. Let A, B ∈ Rn×n . Then, the following statements are equivalent: i) AATBBT = (ABT )2 . ii) tr AATBBT = tr (ABT )2 . iii) ATB = BTA. Source: [2703]. Related: Fact 3.15.37. Fact 3.15.39. Let A ∈ Rn×m . Then, 2 2  m 2  n m  n n ∑ m m ∑ n ∑ ∑ ∑ ∑ ∑ ∑         A(i, j)  + nm A2(i, j) . A(i, j)  ≤  n A(i, j)  + m  i=1

j=1

j=1

i=1

i=1 j=1

i=1 j=1

Furthermore, equality holds if and only if there exist x ∈ Rn and y ∈ Rm such that A = x11×m +1n×1 yT . Source: [1032]. Remark: This is an extension of the Cauchy-Khinchin inequality. Credit: E. R. van Dam.

3.16 Facts on the Determinant Fact 3.16.1. Let n ≥ 1. Then, det Iˆn = (−1)⌊n/2⌋ = (−1)n(n−1)/2 = (−1)n−1 det Iˆn−1 . Consequently, Iˆn is an (even, odd) permutation matrix if and only if 21 n(n − 1) is (even, odd). Source: Since Iˆn

is a permutation matrix, its determinant reflects whether it permutes the components of a vector in an odd or even manner, which reflects the parity of pairwise component swaps that it performs on a vector. The total number of swaps performed by a permutation matrix is given by the sum over all rows of the number of 1’s in subsequent rows that are to the left of the 1 in each row. For Iˆn , this number is (n − 1) + (n − 2) + · · · + 2 + 1 = n(n − 1)/2. See [1560, pp. 29–32]. Related: Fact 4.13.18 and Fact 4.31.14. Fact 3.16.2. Let n ≥ 1. Then, det Pn = (−1)n−1. Consequently, Pn is an (even, odd) permutation matrix if and only if n is (odd, even). Related: Fact 4.13.18 and Fact 7.18.13. Fact 3.16.3. Let α be a complex number. Then, det(In + α1n×n ) = 1 + αn. Fact 3.16.4. Let A ∈ Fn×n , and assume that A has a zero submatrix of size r × s, where n + 1 ≤ r + s. Then, det A = 0. Source: [281, p. 14] and [2991, p. 158]. Related: Fact 3.14.1. Fact 3.16.5. Let A ∈ Rn×n . Then, det(I + A2 ) ≥ 0. Source: [112, p. 55] and [1158, p. 68]. Related: Fact 4.10.15 and Fact 4.13.20. Fact 3.16.6. Let A ∈ Fn×n and S ⊆ {1, . . . , n}, and assume that A(S) is nonsingular. Then, det A = (det A(S) ) det[A[S] − A(S∼ ,S) (A(S) )−1A(S,S∼ ) ]. Remark: This result generalizes (3.9.8).

330

CHAPTER 3

Fact 3.16.7. Let A ∈ Fn×m and B ∈ Fm×n, and assume that m < n. Then, det AB = 0. Fact 3.16.8. Let A ∈ Fn×m, B ∈ Fm×l, k ≤ min {n, m, l}, S1 ⊆ {1, . . . , n}, and S2 ⊆ {1, . . . , l}, and

assume that card(S1 ) = card(S2 ) = k. Then,

det (AB)(S1 ,S2 ) =

∑

det A(S1 ,S) det B(S,S2 ) ,

where the sum is taken over all subsets S of {1, . . . , m} having k elements. Source: [1882, pp. 116, 117] and [2991, p. 123]. Remark: This is the Binet-Cauchy formula. Remark: This result is equivalent to ix) of Fact 9.5.18. The case k = n = l is given by Fact 3.16.9. Related: Fact 3.16.9 and Fact 6.9.37. Fact 3.16.9. Let A ∈ Fn×m and B ∈ Fm×n, and assume that n ≤ m. Then, ∑ det AB = det A(·,S) det B(S,·) , where the sum is taken over all subsets S of {1, ( ). . . , m} having n elements. Source: [970, p. 102]. Remark: det AB is equal to the sum of all mn products of pairs of subdeterminants of A and B formed by choosing n columns of A and the corresponding n rows of B. Remark: Determinantal and minor equalities are given in [592, 1773]. Related: Fact 3.17.8. This is a special case of the Binet-Cauchy formula given by Fact 3.16.8 and Fact 9.5.18. The special case n = m is given by Proposition 3.8.3. △ Fact 3.16.10. Let A ∈ Fn×m, define r = rank A, let S1 , S2 ⊆ {1, . . . , n}, let S3 , S4 ⊆ {1, . . . , m}, and assume that card(S1 ) = card(S2 ) = card(S3 ) = card(S4 ) = r. Then, det A(S1 ,S3 ) det A(S2 ,S4 ) = det A(S1 ,S4 ) det A(S2 ,S3 ) . Source: [1451, p. 20]. Fact 3.16.11. Let A ∈ Fn×n , let S be the n × n matrix whose (i, j) entry is (−1)i+ j , let S1 , S2 ⊆

{1, . . . , n}, assume that card(S1 ) = card(S2 ), and let l be the sum of the elements in the multiset S1 ∪ S2 . Then, det (S ⊙ A)(S1 ,S2 ) = (−1)l det A(S1 ,S2 ) . Related: Fact 3.17.35. Fact 3.16.12. Let A ∈ Fn×n, assume that A is nonsingular, and let b ∈ Fn. Then, the solution

x ∈ Fn of Ax = b is given by

i

  1  det(A ← b)     det A      . x =  ..  .     n  det(A ←  b)   det A i

i

i

Source: Note that A(I ← x) = A ← b. Since det(I ← x) = x(i) , it follows that (det A)x(i) = det(A ← b). Remark: This is Cramer’s rule. Remark: See Fact 3.16.13 for nonsquare extensions. Related:

Fact 3.19.8. Fact 3.16.13. Let A ∈ Fn×m , assume that A is right invertible, and let b ∈ Fn. Then, a solution

x ∈ Fm of Ax = b is given, for all i ∈ {1, . . . , m}, by i

i

det[(A ← b)A∗ ] − det[(A ← 0)A∗ ] x(i) = . det(AA∗ ) Source: [1726]. Remark: This result extends Cramer’s rule. See Fact 3.16.12. Extensions to

generalized inverses are given in [417, 1529, 1717] and [2821, Chapter 3].

331

BASIC MATRIX PROPERTIES

Fact 3.16.14. Let A, B ∈ Rn×n, and assume that A and B are symmetric. Then, the following

statements are equivalent: i) AB = 0. ii) For all α, β ∈ R, det(I − αA − βB) = [det(I − αA)] det(I − βB). Source: [259, 1795]. Remark: This is the Craig-Sakamoto theorem. Related: Fact 4.18.8. Fact 3.16.15. Let A ∈ Fn×n, and assume that either A(i, j) = 0 for all i, j such that i + j < n + 1 or A(i, j) = 0 for all i, j such that i + j > n + 1. Then, det A = (−1)⌊n/2⌋

n ∏

A(i,n+1−i) .

i=1

Remark: A is either lower reverse triangular or upper reverse triangular. Fact 3.16.16. Let a1 , . . . , an ∈ F. Then,

  1 + a1   a  1 det  .  ..  a1

a2

···

1 + a2 .. .

··· .. .

a2

···

 an    n ∑ an   = 1 + ai . ..  .  i=1  1 + an

Fact 3.16.17. Let a1 , . . . , an ∈ F be nonzero. Then,

 1+a1  a  1  1  det  .  ..   1

1

···

1+a2 a2

.. .

··· .. .

1

···

 1   1  1 + ∑n ai i=1 ..  = ∏n a . i=1 i .    1+an  an

Source: Multiply the matrix in Fact 3.16.16 on the right by diag(1/a1 , . . . , 1/an ). Fact 3.16.18. Let a, b, c1, . . . , cn ∈ F, define A ∈ Fn×n by

  c1   b  △  A =  b  .  ..  b

a c2

a a

b .. . b

c3 .. .

△

b

··· ··· .. . ..

. ···

a a a .. . cn

      ,   

△

define p(x) = (c1 − x)(c2 − x) · · · (cn − x), and, for all i ∈ {1, . . . , n}, define pi(x) = p(x)/(ci − x). Then,  bp(a) − ap(b)    , b , a,    b−a   det A =    n−1 ∑   ′   a pi(a) + cn pn (a) = p(a) − ap (a), b = a. i=1

Source: [1757, pp. 65, 66] and [2980, p. 10].

332

CHAPTER 3

Fact 3.16.19. Let a, b ∈ F, and define A, B ∈ Fn×n by

△

A = (a − b)In + b1n×n

  a + b  b   =  b  .  ..  b

△

B = aIn + b1n×n

  a  b   =  b  .  ..  b

b a

b b

b a .. . . . . b b

··· ··· .. . ..

.

···

··· ··· . a + b .. .. .. . .

b a+b

b b

b .. . b

···

b

 b   b   b  ,  ..  .   a

     b  .  ..  .   a+b b b

Then, det A = (a − b)n−1[a + b(n − 1)] and, if det A , 0, then A−1 =

b 1 In + 1n×n. a−b (b − a)[a + b(n − 1)]

Furthermore, det B = an−1(a + nb), and, if det B , 0, then ( ) 1 b B−1 = In − 1n×n . a a + nb Remark: aIn + b1n×n arises in combinatorics. See [589, 591]. Related: Fact 3.17.30, Fact 6.10.21,

and Fact 10.10.39. Fact 3.16.20. Let n ≥ 2. Then,

det[diag(2, . . . , n) + 1(n−1)×(n−1) ] = n!Hn . Source: [1350, p. 132]. Remark: Hn is the nth harmonic number. Fact 3.16.21. Let a1 , . . . , an ∈ F, and define A ∈ Fn×n by △

A=

n ∑

diag[0(i−1)×(i−1) , ai 1(n+1−i)×(n+1−i) ].

i=1

Then, det A =

∏n

i=1 ai . Example: det

[a

a1 a1 1 a1 a1 +a2 a1 +a2 a1 a1 +a2 a1 +a2 +a3 △

Fact 3.16.22. Let a1 , . . . , an ∈ F, define s =

  −a1   s − a2  △  A =  s − a3  .  ..  s − an

Then, det A = (−1)

n−1

]

∑n

i=1

= a1 a2 a3 .

ai , and define

s − a1 −a2

s − a1 s − a2

s − a3 .. .

−a3 .. .

s − an

s − an

··· ··· .. . ..

.

···

 s − a1   s − a2   s − a3  .  ..  .   −an

(n − 2)s . In particular, for all a, b, c, d ∈ F,    −a b + c b + c    −b a + c  = (a + b + c)3 , det  a + c   a + b a + b −c n

333

BASIC MATRIX PROPERTIES

  −a  a + c + d det   a + b + d  a+b+c

b+c+d −b a+b+d a+b+c

b+c+d a+c+d −c a+b+c

 b + c + d   a + c + d   = −2(a + b + c + d)4 . a + b + d   −d

Source: [1641] and [1860, pp. 34, 219]. Fact 3.16.23. Let x, y, z ∈ C. Then,

  0  1 det   1 1

1 0 x2 y2

1 x2 0 z2

 1   y2   = (x + y + z)(x − y − z)(x − y + z)(x + y − z). z2  0

Related: Fact 5.4.7. △ Fact 3.16.24. Let A ∈ Fn×n, and define γ = maxi, j∈{1,...,n} |A(i, j) |. Then, | det A| ≤ γn nn/2. Source:

This result is a consequence of the arithmetic-mean–geometric-mean inequality Fact 2.11.81 and Schur’s inequality Fact 10.21.10. See [970, p. 200]. Related: Fact 10.15.10. Fact 3.16.25. Let A ∈ Rn×n, and, for all i ∈ {1, . . . , n}, let αi denote the sum of the positive components in rowi (A) and let βi denote the sum of the positive components in rowi (−A). Then, | det A| ≤

n ∏

max {αi , βi } −

i=1

n ∏

min {αi , βi }.

i=1

Source: [1543]. Credit: This is an extension of a result due to A. Schinzel. Fact 3.16.26. For i ∈ {1, 2, 3, 4}, let Ai , Bi ∈ F2×2, where det Ai = det Bi = 1. Furthermore, let

A, B, C, D ∈ F4×4, where, for all i, j ∈ {1, 2, 3, 4}, △

△

A(i, j) = tr Ai Aj , Then,

△

B(i, j) = tr Bi B j ,

C(i, j) = tr Ai B j ,

det C + det D = 0,

△

D(i, j) = tr Ai B−1 j .

det AB = (det C)2.

Credit: W. Magnus. See [1488]. Fact 3.16.27. Let I ⊆ R be a finite or infinite interval, and let f : I 7→ R. Then, the following

statements are equivalent: i) f is convex. ii) For all distinct x, y, z ∈ I,

   1 x f (x)    det  1 y f (y)    1 z f (z)  ≥ 0.   1 x x2    det  1 y y2    1 z z2

iii) For all x, y, z ∈ I such that x < y < z,

  1  det  1  1

x y z

 f (x)   f (y)  ≥ 0.  f (z)

Source: [2128, p. 21]. Fact 3.16.28. Let A ∈ Rn×n , where, for all i, j ∈ {1, . . . , n}, A(i, j) is defined below. Then, the

following statements hold:

334

CHAPTER 3 △

i) If A(i, j) = (i + j − 2)!, then det A = [1!2! · · · (n − 1)!]2 . △

ii) If A(i, j) = 1/ min {i, j}, then det A = (−1)n−1 /[n!(n − 1)!]. ( j−2) △ iii) If A(i, j) = i+i−1 , then det A = 1. ( ) n △ 2i−2 iv) If A(i, j) = j−1 , then det A = 2(2) . ( 2n ) ∏n △ i+ j+k−1 v) If A(i, j) = n+i− i, j,k=1 i+ j+k−2 . j , then det A = ∏ △ . vi) Let k ≥ 1. If A(i, j) = Ck+i+ j−2n−2 , then det A = i+ j≤k−2n−1 i+ j+2(n−1) i+ j ( ) n △ ki+m−2 ( ) vii) Let k, m ≥ 1. If A(i, j) = j−1 , then det A = k 2 . Source: [68], [166, p. 62], and [1699]. ii) is given in [146]. Remark: In i) and vi), A is a Hankel matrix. Remark: In vii), det A is independent of m. Related: Fact 7.18.6 and Fact 10.9.8. Fact 3.16.29. Let A ∈ R(n+1)×n , where, for all i, j ∈ {1, . . . , n}, ( i )   i ≥ j, △  j−1 , A(i, j) =   0, i < j. Then, for all i ∈ {1, . . . , n + 1}, det A[i,·]

 (n) (n+1)!   (−1)n+i+1 (n+1−i)! n−i Bn+1−i , i ∈ {1, . . . , n}, =  n!, i = n + 1.

Source: [1150]. Remark: B(i is ) the ith Bernoulli number. See Fact 13.1.6. Example: Let n = 4.

Then, det A[3,·] = (−1)8 (5!/2!)

4 1

B2 = 60(4)(1/6) = 40.

3.17 Facts on the Determinant of Partitioned Matrices Fact 3.17.1. Let A ∈ Fn×n, let k ∈ {1, . . . , n − 1}, let A0 be the k × k leading principal submatrix △

of A, and let B ∈ F(n−k)×(n−k) , where, for all i, j ∈ {1, . . . , n − k}, B(i, j) = det A({1,...,k,k+i},{1,...,k,k+ j}) . Then, det B = (det A0 )n−k−1 det A. If, in addition, A0 is nonsingular, then det A =

det B . (det A0 )n−k−1

Remark: If k = n − 1, then det B = det A. Remark: This is Sylvester’s identity. Fact 3.17.2. Let A ∈ Fn×n, x, y ∈ Fn, and a ∈ F. Then,

[

] x = a(det A) − yTAA x = (a + 1) det A − det(A + xyT ). a

A det T y Hence, [

A det T y

  (det A)(a − yTA−1x),      x  = adet(A − a−1xyT ),   a    −yTAA x, ]

[

In particular, det Furthermore,

A yTA

det A , 0, a , 0, a = 0 or det A = 0.

] Ax = 0. yTAx [

A det(A + xy ) = det A + y A x = − det T y T

T A

] x . −1

335

BASIC MATRIX PROPERTIES

If, in addition, A is nonsingular, then det(A + xyT ) = (det A)(1 + yTA−1x). Related: Fact 3.17.3, Fact 3.21.4, and Fact 3.21.5. Fact 3.17.3. Let A ∈ Fn×n, b ∈ Fn, and a ∈ F. Then, [ ] A b det ∗ = a(det A) − b∗AA b. b a In particular, [ det

A b∗

   (det A)(a − b∗A−1b), det A , 0,   ]     b  = adet(A − a−1bb∗ ), a , 0,   a      −b∗AA b, a = 0.

Remark: This is a special case of Fact 3.17.2 with x = b and y = b. Related: Fact 10.18.5. Fact 3.17.4. Let A ∈ Fn×n. Then,

[

] [ ] [ ] A A A −A A A = rank = rank A, rank = 2rank A, A A −A A −A A [ ] [ ] [ ] A A A −A A A det = det = 0, det = 2n (det A)2. A A −A A −A A

rank

Related: Fact 3.17.5. [ ] △ bA Fact 3.17.5. Let a, b, c, d ∈ F, let A ∈ Fn×n, and define A = aA cA dA . Then,

( [ ]) a b rank A = rank rank A, c d

det A = (ad − bc)n (det A)2.

Source: Proposition 9.1.11 and Fact 9.4.20. Related: Fact 3.17.4.

[

]

Fact 3.17.6. det I0m I0n = (−1)nm. In particular, det Fact 3.17.7. Let A, B, C ∈ Fn×n . Then,

] [ 0 A C det = det B C B

[

Furthermore, det

0 B

[0 I ] n

In 0

= (−1)n .

[

] A = (−1)n det AB. 0

] [ I − AB C = det C B

] I − AB = det(BAB − B). 0

Source: Fact 3.17.6. Related: Fact 3.14.23. Fact 3.17.8. Let A ∈ Fn×m, C ∈ Fm×m , and B ∈ Fm×n, and assume that C is nonsingular. Then,

[

] 0 = (−1)(m+1)n (det C) det AC −1B. B

A det C In particular,

[ det

A Im

] 0 = (−1)(m+1)n det AB, B

Source: [970]. Related: Fact 3.16.9. Fact 3.17.9. Let A, B ∈ Fn×n. Then,

[

AB 0

] [ A A = I I

0 B

][

0 I

I 0

][

−I B

] 0 , I

[ det

[

A −Im

] 0 = (−1)(n+1)m det AB. B

] [ AB 0 −I = B I 0

A I

][

A I

0 B

][

0 I

] I , 0

336

CHAPTER 3

[

] [ ][ ][ ] [ ] I I 0 I 0 I I I I = , det = det(B − A), B A I 0 B−A 0 I A B [ ] [ ][ ][ ] [ ] I A I A I − AB 0 I 0 I A = , det = det(I − BA), B I 0 I 0 I B I B I [ ] [ ][ ][ ] [ ] A+B B I I A 0 I 0 A+B B = , det = (det A) det B, B B 0 I 0 B I I B B [ ] [ ][ ][ ] [ ] A AB I A A − ABA 0 I 0 A AB = , det = (det B) det(A − ABA), BA B 0 I BA − B2A B BA I BA B [ ] [ ][ ][ ] [ ] A AB I 0 A 0 I B A AB = , det = (det A) det(B − BAB), BA B BA I BA − BA2 B − BAB 0 I BA B     [ ] [ ] 0   I I  1  I I   A + B A B  , det A B = det(A + B) det(A − B).   =    B A B A 2 I −I 0 A − B I −I I A

Fact 3.17.10. Let A, B ∈ Fn×n. Then,

[

det [

Source: Consider

I − A∗A I − B∗A

I − A∗A I − B∗A

] I − A∗B = (−1)n | det(A − B)|2 . I − B∗B

] [ I − A∗B I = I − B∗B I

A∗ B∗

][

] I I . −A −B

See [2991, p. 233]. Related: Fact 3.20.16 and Fact 10.16.28. Fact 3.17.11. Let A, B, C, D be conformable matrices with entries in F. Then,   [ ] [ ] [ ] 0   I B  A AB I 0  A  , det A AB = (det A) det(D − CB),   =  C D C D C I C − CA D − CB 0 I [ ] [ ] [ ] [ ] A B A B I 0  A B − AB  I B , det = (det A) det(D − CB), =   CA D CA D C I  0 D − CB  0 I [ [ ] [ ] ] [ ] A BD A BD I B  A − BC 0  I 0 , det = det(A − BC) det D, =   C D C D 0 I  C − DC D  C I [ ] [ ] [ ] [ ] A B A B I B  A − BC B − BD  I 0 , det = det(A − BC)det D. =   C I DC D DC D 0 I 0 D Related: Fact 8.9.34. [ ] Fact 3.17.12. Let A, B, C, D ∈ Fn×n, and assume that rank CA DB = n. Then,

  det A det  det C

 det B   = 0. det D

Fact 3.17.13. Let A, B, C, D ∈ Fn×n. Then,

[ det

A C

   (det A) det(D − CA−1B),      ]  n −1    B (−1) (det B) det(C − DB A), =   D  (−1)n (det C) det(B − AC −1D),        (det D) det(A − BD−1C),

det A , 0, det B , 0, det C , 0, det D , 0.

337

BASIC MATRIX PROPERTIES

Fact 3.17.14. Let A, B, C, D ∈ Fn×n. Then,

[ det

A C

   det(DA − CB),      ]     B det(AD − CB), =   D  det(AD − BC),        det(DA − BC),

AB = BA, AC = CA, DC = CD, DB = BD.

Source: If A is nonsingular and AB = BA, then

[

det

A C

] B = (det A) det(D − CA−1B) = det(DA − CA−1BA) = det(DA − CB). D

Alternatively, note that

[

A C

]   A B =  D C

   I   DA − CB 0 0

 BA−1   . A−1

If A is singular, then replace A with A+εI and use continuity. Remark: These are Schur’s formulas. See [302, p. 11]. Problem: Prove this result in the case where A is singular without invoking continuity. Fact 3.17.15. Let A, B, C, D ∈ Fn×n. Then,   det(ADT − BTCT ), AB = BAT ,        [ ]  T   DC = CDT , det(AD − BC), A B  det =   C D   det(ATD − CB), ATC = CA,        det(ATD − CTBT ), DTB = BD. △

Source: Define the nonsingular matrix Aε = A + εI, which satisfies Aε B = BATε . Then,

[

det

Aε C

] B T −1 T T = (det Aε )det(D − CA−1 ε B) = det(DAε − CAε BAε ) = det(DAε − CB). D

Fact 3.17.16. Let A, B, C, D ∈ Fn×n. Then,

[ det

A C

   (−1)rank C det(ATD + CTB), ATC = −CTA,          (−1)n+rank A det(ATD + CTB), ATC = −CTA,          (−1)rank B det(ATD + CTB), BTD = −DTB,        ]  n+rank D   det(ATD + CTB), BTD = −DTB,  B (−1) =   D   (−1)rank B det(ADT + BCT ), ABT = −BAT,          (−1)n+rank A det(ADT + BCT ), ABT = −BAT,          (−1)rank C det(ADT + BCT ), CDT = −DCT,         (−1)n+rank D det(ADT + BCT ), CDT = −DCT.

Source: [1949, 2839]. Remark: If ATC = −CTA and rank A + rank C + n is odd, then singular. Credit: D. Callan. See [2839].

[A B] C D

is

338

CHAPTER 3

Fact 3.17.17. Let A, B, C, D ∈ Fn×n. Then,

  det(ADT − BCT ),        ]  T T    B det(AD − BC ), =   D   det(ATD − C TB),        det(ATD − CTB),

[

A det C

ABT = BAT, DC T = CDT, ATC = C TA, DTB = BTD.

Source: [1949]. Fact 3.17.18. Let A, B, C, D ∈ Fn×n, and assume that A, B, C, D are nonsingular. Then,

[

A−1 det −1 C

[ ] [ ] ] 1 det(B − DC −1A) (−1)n A C B D B−1 det = det = . −1 = B D A C D det ABCD det ABCD det ABD

Source: [2991, p. 232]. Fact 3.17.19. Let A ∈ Fn×m, B ∈ Fn×l, C ∈ Fk×m, and D ∈ Fk×l, and assume that n + k = m + l. If

AC T + BDT = 0, then

[ det

A C

B D

]2 = det(AAT + BBT ) det(CC T + DDT ).

Alternatively, if ATB + C TD = 0, then [ ]2 A B det = det(ATA + C TC) det(BTB + DTD). C D [ ][ ]T [ ]T [ ] Source: Consider CA DB CA DB and CA DB CA DB . Fact 3.17.20. Let A ∈ Fn×m, B ∈ Fn×m, C ∈ Fk×m, and D ∈ Fk×m, and assume that n + k = 2m. If T AD + BC T = 0, then [ ]2 A B det = (−1)m det(ABT + BAT ) det(CDT + DC T ). C D Alternatively, if either ABT + BAT = 0 or CDT + DC T = 0, then [ ]2 A B det = (−1)m+n det(ADT + BC T )2. C D [ ][ T T ] [ ][ T T ] Source: Consider CA DB BAT CDT and CA DB CDT BAT . See [2839]. Fact 3.17.21. Let A ∈ Fn×m, B ∈ Fn×l, C ∈ Fn×m, and D ∈ Fn×l, and assume that m + l = 2n. If

A D + C TB = 0, then T

[ det

A C

B D

]2 = (−1)n det(C TA + ATC) det(DTB + BTD).

Alternatively, if either BTD + DTB = 0 or ATC + C TA = 0, then [ ]2 A B = (−1)n+m det(ATD + C TB)2. det C D [ T T ][ ] [ T T ][ ] Source: Consider CDT BAT CA DB and CDT BAT CA DB .

339

BASIC MATRIX PROPERTIES

Fact 3.17.22. Let A ∈ Fn×n, B ∈ Fn×k, C ∈ Fk×n, and D ∈ Fk×k. If either AB + BD = 0 or

CA + DC = 0, then

[ det

A C

B D

]2 = det(A2 + BC) det(CB + D2 ).

Alternatively, if either A2 + BC = 0 or CB + D2 = 0, then [ ]2 A B det = (−1)nk det(AB + BD) det(CA + DC). C D [ ]2 [ ][ ] Source: Consider CA DB and CA DB DB CA . Fact 3.17.23. Let A ∈ Fn×m, B ∈ Fn×n, C ∈ Fm×m, and D ∈ Fm×n. If either AD + B2 = 0 or 2 C + DA = 0, then [ ]2 A B det = (−1)nm det(AC + BA) det(CD + DB). C D Alternatively, if either AC + BA = 0 or CD + DB = 0, then [ ]2 A B det = det(AD + B2 ) det(C 2 + DA). C D [ ][ ] [ ][ ] Source: Consider CA DB CA DB and CA DB DB CA . Fact 3.17.24. Let A ∈ Fn×m, B ∈ Fn×l, C ∈ Fk×m, and D ∈ Fk×l, and assume that n + k = m + l. If ∗ AC + BD∗ = 0, then [ ] 2 A B det = det(AA∗ + BB∗ ) det(CC ∗ + DD∗ ). C D Alternatively, if A∗B + C ∗D = 0, then [ ] 2 A B ∗ ∗ ∗ ∗ det C D = det(A A + C C) det(B B + D D). [ ][ ]∗ [ ]∗ [ ] Source: Consider CA DB CA DB and CA DB CA DB . Related: Fact 10.16.30. Fact 3.17.25. Let A ∈ Fn×m, B ∈ Fn×m, C ∈ Fk×m, and D ∈ Fk×m, and assume that n + k = 2m. If ∗ AD + BC ∗ = 0, then [ ] 2 A B m ∗ ∗ ∗ ∗ det C D = (−1) det(AB + BA ) det(CD + DC ). Alternatively, if either AB∗ + BA∗ = 0 or CD∗ + DC ∗ = 0, then [ ] 2 A B = (−1)m+n |det(AD∗ + BC ∗ )|2 . det C D [ ][ ∗ ∗ ] [ ][ ∗ ∗ ] [ ] Source: Consider CA DB BA∗ CD∗ and CA DB CD∗ BA∗ . Remark: If m2 + nk is odd, then CA DB is singular. ∗

Fact 3.17.26. Let A ∈ Fn×m, B ∈ Fn×l, C ∈ Fn×m, and D ∈ Fn×l, and assume that m + l = 2n. If

A D + C ∗B = 0, then

[ A det C

B D

] 2 m ∗ ∗ ∗ ∗ = (−1) det(C A + A C) det(D B + B D).

340

CHAPTER 3

Alternatively, if either D∗B + B∗D = 0 or C ∗A + A∗C = 0, then [ ] 2 A B n+m ∗ ∗ 2 det C D = (−1) |det(A D + C B)| . [ ∗ ∗ ][ ] [ ∗ ∗ ][ ] [ ] Source: Consider CD∗ BA∗ CA DB and CD∗ BA∗ CA DB . Remark: If n + m is odd, then CA DB is singular. Fact 3.17.27. Let A ∈ Fn×m and B ∈ Fn×l. Then,

[

det

Im A

] A∗ = det BB∗ . AA∗ + BB∗

Source: [2991, p. 49]. Fact 3.17.28. Let A ∈ Fn×m and B ∈ Fn×l. Then,

   det(A∗A) det[B∗B − B∗A(A∗A)−1A∗B],     A∗B  = det(B∗B) det[A∗A − A∗B(B∗B)−1B∗A],   B∗B    0,

[

]

A∗A det ∗ BA

If, in addition, m + l = n, then

[ det

A∗A B∗A

rank A = m, rank B = l, n < m + l.

] A∗B = det(AA∗ + BB∗ ). B∗B

Related: Fact 8.9.36. ] △ [ Fact 3.17.29. Let A ∈ Fn×n , B ∈ Fn×m , C ∈ Fm×n , and D ∈ Fm×m , and define A = CA DB . If A is

singular, then the following statements are equivalent: i) A is nonsingular. ([ ]) ([ ]) ii) N(A) ∩ N(C) = {0}, N(D) ∩ N(B) = {0}, and R CA ∩ R DB = {0}. Furthermore, if A is nonsingular, then the following statements are equivalent: iii) A is nonsingular. iv) D − CA−1B is nonsingular. Source: [212]. Problem: Assume that A is nonsingular. Does i) ⇐⇒ ii) hold? Fact 3.17.30. Let A, B ∈ Fn×n, and define A ∈ Fkn×kn by    A B B · · · B     B A B · · · B    ..  △  . B  . A =  B B A   .  ..  .. .. . . . . ..   . .   B B B ··· A Then, If k = 2, then

det A = det[A + (k − 1)B][det(A − B)]k−1 . [

A det B

] B = det (A + B)(A − B) = det(A2 − B2 − [A, B]). A

Source: [1203]. For k = 2, the result follows from Fact 6.10.31. Related: Fact 3.16.19, Fact 3.22.6, and Fact 6.10.31.

341

BASIC MATRIX PROPERTIES △

F

n×m m×n m×m Fact 3.17.31. Let A ∈ F[n×n, B ] △∈ F , C ∈ F , and D ∈ F , and define M = ′ ′

. Furthermore, let

(n+m)×(n+m)

A B C ′ D′

′

= M , where A ∈ F A

det D′ = (det M)m−1 det A,

n×n

′

and D ∈ F

[A B] C D

∈

. Then,

m×m

det A′ = (det M)n−1 det D.

Source: [2418, p. 297]. Related: Fact 3.17.32. ] △ [ Fact 3.17.32. Let A ∈ Fn×n, B ∈ Fn×m, C ∈ Fm×n,[and D] ∈ Fm×m, define M = CA DB ∈ F(n+m)×(n+m), ′

′

△

and assume that M is nonsingular. Furthermore, let CA′ DB ′ = M−1, where A′ ∈ Fn×n and D′ ∈ Fm×m. Then, det D det A , det A′ = . det D′ = det M det M ′ ′ Hence, A is nonsingular if and [ only ] if[D ]is nonsingular, and D is nonsingular if and only if A is ′ I B A 0 nonsingular. Source: Use M 0 D′ = C I . See [2426]. Related: Fact 3.14.27, Fact 3.17.31, and Fact 4.13.21. This is a special case of Fact 3.17.34. Fact 3.17.33. Let A ∈ Fn×n, assume that A is nonsingular, let S1 , S2 ⊆ {1, . . . , n}, and assume that card(S1 ) = card(S2 ). Then, | det (A−1 )(S1 ,S2 ) | =

| det A(S∼2 ,S∼1 ) | | det A|

.

Source: [2780, p. 39] or use Fact 3.17.34. Remark: For card(S1 ) = card(S2 ) = 1, this result yields the absolute value of (3.8.24). Related: Fact 3.14.28. Fact 3.17.34. Let A ∈ Fn×n, let k ≤ n, let R, C ⊆ {1, . . . , n}, where card(R) = card(C) = k, and

let l be the sum of the elements of R ∪ C. Then,

det (AA )[R,C] = (−1)l (det A)n−k−1 det A(C,R) . If, in addition, A is nonsingular, then det A(C,R) . det A Source: [1448, p. 21]. Remark: This is Jacobi’s identity. Related: Fact 3.17.32 and Fact 3.19.4. Fact 3.17.35. Let A ∈ Fn×n, let k ≤ n, let R, C ⊆ {1, . . . , n}, where card(R) = card(C) = k, and △ let M ∈ Fn×n , where, for all i, j ∈ {1, . . . , n}, M(i, j) = det A[i, j] . Then, det (A−1 )[R,C] = (−1)l

det M (R,C) = (det A)k−1 det A[R,C] . In particular, (det A)n−1 = det AA and (det A)det A[{r1 ,r2 },{c1 ,c2 }] = (det A[r1 ,c1 ] ) det A[r2 ,c2 ] − (det A[r1 ,c2 ] ) det A[r2 ,c1 ] . Source: [23, 129, 1699, 2940]. Remark: The case k = 2 is the Desnanot-Jacobi identity, which is used for Dodgson condensation. See [129]. Remark: The second equality uses det 00×0 = 1. Remark: Let S be the n × n matrix whose (i, j) entry is (−1)i+ j , and let l be the sum of the 2k

elements of the multiset R ∪ C. Then, M = (S ⊙ AA )T , and Fact 3.16.11 implies that det(S ⊙ AA )(R,C) = (−1)l det (AA )(R,C) ,

which can be used to show that this result implies Fact 3.17.34. Fact 3.17.36. Let x1 , x2 , x3 , x4 ∈ F2 . Then, det([x1 x2 ][x3 x4 ]) − det([x1 x3 ][x2 x4 ]) + det([x1 x4 ][x2 x3 ]) = 0. Related: Fact 3.17.37.

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CHAPTER 3

Fact 3.17.37. Let n ≥ 2, let 1 ≤ m < n, let A ∈ Fn×(n−m), and let x1 , . . . , x2m ∈ Fn . Then,

∑

(−1)

∑m

j=1 i j

det([A xi1 · · · xim ][A xim+1 · · · xi2m ]) = 0,

where the sum is taken over all permutations (i1 , . . . , i2m ) of (1, . . . , 2m) such that i1 < · · · < im and im+1 < · · · < i2m . In particular, let n > 2, m = 2, A ∈ Fn×(n−2) , and x1 , x2 , x3 , x4 ∈ Fn . Then, det([A x1 x2 ][A x3 x4 ]) − det([A x1 x3 ][A x2 x4 ]) + det([A x1 x4 ][A x2 x3 ]) = 0. Source: [2940]. Remark: This is a Pl¨ucker relation. Related: ] 3.17.36. [ [ Fact △ △ A1 A2 n×m Fact 3.17.38. Let A1 , A2 , B1 , B2 ∈ F , and define A = A2 A1 and B = BB12

[ rank

]

B2 B1

]

. Then,

∑ A B = rank Ci , B A 4

i=1

△

△

△

△

where C1 = A1+A2 +B1+B2 , C2 = A1+A2 −B1−B2 , C3 = A1−A2 +B1−B2 , and C4 = A1−A2 −B1+B2 . If, in addition, n = m, then [ ] ∏ 4 A B det = det Ci . B A i=1

Source: [2671]. Related: Fact 4.32.8.

3.18 Facts on Left and Right Inverses Fact 3.18.1. Let A ∈ Fn×m. Then, the following statements are equivalent:

i) A is left invertible. ii) A is left invertible. iii) AT is right invertible. iv) A∗ is right invertible. Now, assume that A is left invertible, and let AL ∈ Fm×n be a left inverse of A. Then, the following statements hold: v) AL ∈ Fm×n is a left inverse of A. vi) ALT ∈ Fn×m is a right inverse of AT . vii) AL∗ ∈ Fn×m is a right inverse of A∗ . viii) AL is right invertible. ix) A is a right inverse of AL . x) R(A) = R(AAL ). xi) rank A = rank AL = rank AAL = m. xii) N(AL ) = N(AAL ). xiii) def AL = def AAL = n − m. xiv) def A = 0. Source: R(AAL ) ⊆ R(A) = R(AALA) ⊆ R(AAL ). Related: Fact 3.18.19. Fact 3.18.2. Let A ∈ Fn×m. Then, the following statements are equivalent: i) A is right invertible. ii) A is right invertible. iii) AT is left invertible. iv) A∗ is left invertible. Now, assume that A is right invertible, and let AR be a right inverse of A. Then, the following

BASIC MATRIX PROPERTIES

343

statements hold: v) AR ∈ Fm×n is a right inverse of A. vi) ART ∈ Fn×m is a left inverse of AT . vii) AR∗ ∈ Fn×m is a left inverse of A∗ . viii) AR is left invertible. ix) A is a left inverse of AR . x) R(AR ) = R(ARA). xi) rank A = rank AR = rank ARA = n. xii) N(A) = N(ARA). xiii) def A = def ARA = m − n. xiv) def AR = 0. Fact 3.18.3. Let A ∈ Fn×m, assume that A is left invertible, and let AL be a left inverse of A. Then, B ∈ Fm×n is a left inverse of A if and only if there exists S ∈ Fm×n such that B = AL + S and SA = 0. Source: For necessity, let S = B − AL . Related: Fact 8.3.14 and [2238, p. 150]. Fact 3.18.4. Let A ∈ Fn×m, assume that A is right invertible, and let AR be a right inverse of A. Then, B ∈ Fm×n is a right inverse of A if and only if there exists S ∈ Fm×n such that B = AR + S and AS = 0. Source: For necessity, let S = B − AR . Related: Fact 8.3.15. Fact 3.18.5. Let A ∈ Fn×m. If rank A = m, then (A∗A)−1A∗ is a left inverse of A. If rank A = n, then A∗ (AA∗ )−1 is a right inverse of A. Related: Fact 4.10.25, Fact 4.10.26, and Fact 4.17.8. Fact 3.18.6. Let A ∈ Fn×m, and assume that A is left invertible. Then, AL ∈ Fm×n is a left inverse of A if and only if there exists B ∈ Fm×n such that BA is nonsingular and AL = (BA)−1B. Source: For necessity, let B = AL. Fact 3.18.7. Let A ∈ Fn×m, and assume that A is right invertible. Then, AR ∈ Fm×n is a right inverse of A if and only if there exists B ∈ Fm×n such that AB is nonsingular and AR = B(AB)−1. Source: For necessity, let B = AR. Fact 3.18.8. Let A ∈ Fn×m and b ∈ Fn , and define f : Fm 7→ Fn by f (x) = Ax + b. Then, the following statements hold: i) f is invertible if and only if A is nonsingular. Now, assume that these conditions hold. Then, for all y ∈ Fn , f Inv (y) = A−1 (y − b). ii) f is left invertible if and only if A is left invertible. Now, assume that these conditions hold, and let AL be a left inverse of A. Then, for all y ∈ Fn , f inv (y) = {AL (y − b)}. iii) f is right invertible if and only if A is right invertible. Now, assume that these conditions hold, and let AR be a right inverse of A. Then, for all y ∈ Fn , AR (y − b) ∈ f inv (y). Related: Fact 1.10.5 and Proposition 3.7.10. Fact 3.18.9. Let A ∈ Fn×m and B ∈ Fm×l, and assume that A and B are left invertible. Then, AB is left invertible. If, in addition, AL is a left inverse of A and BL is a left inverse of B, then BLAL is a left inverse of AB. Source: Fact 1.10.6, Corollary 3.6.12, and Proposition 3.7.3.[ ]Remark: If [ ] 1 0 A and B have full column rank, then so does AB. The example A = 0 0 and B = 10 shows that the converse is not true. Related: Fact 8.4.20 and Fact 9.4.34. Fact 3.18.13 provides necessary and sufficient conditions for AB to be left invertible. Fact 3.18.10. Let A ∈ Fn×m and B ∈ Fm×l, and assume that A and B are right invertible. Then, AB is right invertible. If, in addition, AR is a right inverse of A and BR is a right inverse of B, then BRAR is a right inverse of AB. Remark: If A and B have full row rank, then so does AB. Related: Fact 8.4.21 and Fact 9.4.35. Fact 3.18.14 provides necessary and sufficient conditions for AB to be right invertible.

344

CHAPTER 3

Fact 3.18.11. Let A ∈ Fn×m and B ∈ Fm×l, and assume that AB is left invertible. Then, B is △

left invertible. Now, let (AB)L be a left inverse of AB, and define BL = (AB)LA. Then, BL is a left inverse of B. Finally, assume that A is right invertible, and let AR be a right inverse of A. Then, (AB)L = BLAR. Fact 3.18.12. Let A ∈ Fn×m and B ∈ Fm×l, and assume that AB is right invertible. Then, A is △ right invertible. Now, let (AB)R be a right inverse of AB, and define BR = B(AB)R . Then, AR is a right inverse of A. Finally, assume that B is left invertible, and let BL be a left inverse of B. Then, (AB)R = ARBL. Fact 3.18.13. Let A ∈ Fn×m and B ∈ Fm×l. Then, the following statements are equivalent: i) AB is left invertible. ii) B is left invertible, and N(A) ∩ R(B) = {0}. [ ] A iii) B and are left invertible. Im − B(B∗ B)−1B∗ [ ] A iv) rank B = l and rank = m. Im − B(B∗ B)−1B∗ Source: The equivalence of [ ] ii) and iii) follows from Fact 8.9.4. Remark: If A and B are left invertible, then Im −B(BA∗ B)−1B∗ is left invertible. See Fact 3.18.9. Fact 3.18.14. Let A ∈ Fn×m and B ∈ Fm×l. Then, the following statements are equivalent:

i) ii) iii) iv)

AB is right invertible. A is right invertible and N(A) + R(B) = Fm. A and [Im − A∗ (AA∗ )−1A B] are right invertible. rank A = n and rank [Im − A∗ (AA∗ )−1A B] = m. Source: Fact 3.18.13 and Fact 8.9.4. Remark: If A and B are right invertible, then [Im −A∗ (AA∗ )−1A B] is right invertible. See Fact 3.18.10. Fact 3.18.15. Let A ∈ Fn×m , assume that A is left invertible, let S ∈ Fm×m , assume that S is nonsingular, and let (AS )L ∈ Fm×n be a left inverse of AS . Then, there exists a left inverse AL of A △ such that (AS )L = S −1AL . Source: It follows from (AS )L AS = I that NA = I, where N = S (AS )L L −1 is a left inverse of A. Hence, (AS ) = S N. Fact 3.18.16. Let A ∈ Fn×m and B ∈ Fm×l, assume that A and B are left invertible, and let (AB)L be a left inverse of AB. Then, there exist a left inverse AL of A and a left inverse BL of B L such that = BLAL . Source: Let S 1 ∈ Fn×n and S 2 ∈ Fm×m be nonsingular matrices such that [ (AB) ] △ Im A = S 1 0 S 2 , let (S 2 B)L be a left inverse of S 2 B, and define BL = (S 2 B)L S 2 , which is a left inverse of B. Then, there exists C ∈ Fl×(n−m) such that ( [ ] )L ( [ ])L [ ]L Im S2B S2B L (AB) = S 1 S B = S1 = S 1−1 = [(S 2 B)L C]S 1−1 0 2 0 0 = [BL S 2−1 C]S 1−1 = BL S 2−1 [I S 2 BC]S 1−1 = BLAL , △

where AL = S 2−1 [I S 2 BC]S 1−1 is a left inverse of A. Problem: Extend this result to the composition f ◦ g of left-invertible functions f and g. Fact 3.18.17. Let A ∈ Fn×m , assume that A is right invertible, let S ∈ Fn×n , assume that S is nonsingular, and let (SA)R ∈ Fn×m be a right inverse of AS . Then, there exists a right inverse AR of A such that (SA)R = AR S −1 . Related: Fact 3.18.15. Fact 3.18.18. Let A ∈ Fn×m and B ∈ Fm×l, assume that A and B are right invertible, and let (AB)R be a right inverse of AB. Then, there exist a right inverse AR of A and a right inverse BR of B such that (AB)R = BRAR . Related: Fact 3.18.16.

345

BASIC MATRIX PROPERTIES

Fact 3.18.19. Let S ⊆ Fm, let Sˆ ⊆ Fn, and let A ∈ Fn×m. Then,

(AS)⊥ = A∗inv (S⊥ ),

AS⊥ = [A∗inv (S)]⊥ ,

ˆ ⊥ = Ainv (Sˆ ⊥ ), (A∗ S)

ˆ ⊥. A∗ Sˆ ⊥ = [Ainv (S)]

Furthermore, the following statements hold: i) If A is left invertible and AL is a left inverse of A, then AS⊥ ⊆ (AL∗ S)⊥. ii) If A is right invertible and AR is a right inverse of A, then (AS)⊥ ⊆ AR∗ S⊥. iii) If n = m and A is nonsingular, then (AS)⊥ = A−∗ S⊥. Source: Note that (AS)⊥ = {y ∈ Fn : y∗Ax = 0 for all x ∈ S} = {y ∈ Fn : (A∗ y)∗ x = 0 for all x ∈ S} = {y ∈ Fn : A∗ y ∈ S⊥ } = A∗inv (S⊥ ). The third equality is given in [688, p. 12] and [2726, p. 19]. i) follows from the first equality, (3.7.10), and Fact 3.18.1. ii) follows from the second equality, (3.7.8), and Fact 3.18.2. iii) follows from (3.7.11). △ ∑ Fact 3.18.20. For all i ∈ {1, . . . , k}, let Ai j ∈ Fni ×m j , define A ∈ Fn×m , where n = ki=1 ni and ∑ △ m = ki=1 mi , by    A11 · · · A1k   . . ..  △  · .· · A =  .. .  ,   Ak1 · · · Akk assume that A is (upper block triangular, lower block triangular), and, for all i ∈ {1, . . . , k}, assume that Aii is (left invertible, right invertible). Then, A is (left invertible, right invertible) and has (an upper block-triangular left inverse, a lower block-triangular right inverse). Fact 3.18.21. Let A ∈ Fn×m , B ∈ Fk×m , and C ∈ Fk×l , assume that A and C are [ left] invertible, let △ L A ∈ Fm×n be a left inverse of A, let C L ∈ Fl×k be a left inverse of C, define A = AB C0 ∈ F(n+k)×(m+l) , [ L ] △ and B = −CAL BAL C0L . Then, B is a left inverse of A. Related: This result provides an explicit left inverse for a special case of Fact 3.18.20.

3.19 Facts on the Adjugate Fact 3.19.1. Let A ∈ Fn×n. Then, the following statements hold:

i) ii) iii) iv) v) vi) vii) viii)

(A)A = AA. △ AAT = (AT )A = (AA )T. △ AA∗ = (A∗ )A = (AA )∗. If α ∈ F, then (αA)A = αn−1AA. det AA = (det A)n−1. (AA )A = (det A)n−2A. 2 det (AA )A = (det A)(n−1) . ∑ tr AA = ni=1 det A[i,i] . △

△

△

ix) If k ≥ 0, then AkA = (AA )k = (Ak )A , AkAT = (AAT )k = (Ak )AT , and AkA∗ = (AA∗ )k = (Ak )A∗ . x) A is nonsingular if and only if AA is nonsingular. Now, assume that A is nonsingular. Then, the following statements hold: xi) tr AA = (det A) tr A−1 . △ xii) A−A = (A−1 )A = (AA )−1. xiii) tr A−A = (tr A)/det A. △ xiv) A−AT = (A−1 )AT = (AAT )−1. △ △ △ xv) If k ∈ Z, then AkA = (AA )k = (Ak )A , AkAT = (AAT )k = (Ak )AT , and AkA∗ = (AA∗ )k = (Ak )A∗ . △ Source: [1394]. viii) follows from (6.4.20). Remark: With 0/0 = 1 and using 0A1×1 = 1 in vi), all

346

CHAPTER 3

of these results hold in the case n = 1. Related: Fact 6.9.1, Fact 6.9.2, and Fact 6.10.13. Fact 3.19.2. Let A ∈ Fn×n, and assume that A is singular. Then, R(A) ⊆ N(AA ). Hence, rank A ≤ def AA and rank A + rank AA ≤ n. Furthermore, R(A) = N(AA ) if and only if rank A = n − 1. Fact 3.19.3. Let A ∈ Fn×n. Then, the following statements hold: i) rank AA = n if and only if rank A = n. ii) rank AA = 1 if and only if rank A = n − 1. iii) AA = 0 if and only if rank A ≤ n − 2. iv) NnA = (−1)n+1 Nnn−1 . Source: [2263, p. 12] and [2980, p. 18]. Remark: Fact 8.3.21 provides an expression for AA in the case where rank AA = 1. Related: Fact 6.10.13. Fact 3.19.4. Let A ∈ Fn×n and k ≥ 1. Then, det (AA )({1,...,k}) = (det A)k−1 det A({k+1,...,n}) . Source: [2263, p. 12]. Related: This is a special case of Fact 3.17.34. Fact 3.19.5. Let A ∈ Fn×n. Then, the following statements are equivalent:

i) (AA )2 = 0. ii) For all k ∈ {1, . . . , n}, tr (AA )k = 0. Source: [1898]. Problem: For the case rank A ≤ n − 1, determine conditions on A under which AA is semisimple. See Fact 6.10.13. △ Fact 3.19.6. Let A ∈ Fn×n and A ∈ F(n−1)×(n−1) , where, for all i, j ∈ {1, . . . , n − 1}, A(i, j) = T (ei − ei+1 ) A(e j − e j+1 ). Then, det(A + 1n×n ) − det A = 11×n AA1n×1 =

n ∑

i

det(A ← 1n×1 ) = det A.

i=1

Source: [484]. Related: Fact 3.17.2, Fact 3.19.8, and Fact 12.16.19. Fact 3.19.7. Let n ≥ 2, and let A ∈ Fn×n. Then,

[(AA )[i,·] + (A[i,i] )A A[i,·] ]ei,n = 0. If, in addition, A and A[i,i] are nonsingular, then [(det A)(A−1 )[i,·] + (det A[i,i] )(A[i,i] )−1 A[i,·] ]ei,n = 0. Source: [40]. Fact 3.19.8. Let A ∈ Fn×n and b ∈ Fn . Then, for all i ∈ {1, . . . , n}, i

(AAb)(i) = det(A ← b). Now, let B ∈ Fn×m. Then, for all i ∈ {1, . . . , n} and j ∈ {1, . . . , m}, i

(AAB)(i, j) = det[A ← col j (B)]. In particular,

i

(AA )(i, j) = det(A ← e j ). Remark: See Fact 12.16.19 and [1451, p. 24]. The first equality implies Cramer’s rule. See Fact

3.16.12. Fact 3.19.9. Let A, B ∈ Fn×n. Then, the following statements hold:

i) (AB)A = BAAA. ii) A(A + B)AB = B(A + B)AA.

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BASIC MATRIX PROPERTIES

iii) If B is nonsingular, then (BAB−1 )A = BAAB−1. iv) If AB = BA, then AAB = BAA, ABA = BAA, and AABA = BAAA. Source: [1451, p. 23], [2263, p. 11], and [2980, p. 18]. Related: Fact 4.10.9. Fact 3.19.10. Let A ∈ Fn×n , B ∈ Fn×m , and C ∈ Fm×n , where B = [B1 · · · Bm ] and C =

[ C1 ] .. . . Cm

Then, [

A det C

   −C1 AA B1 ,    B  = −C2 [AA − (A + B1C1 )A ]B2 ,   0   −C3 [AA − (A + B1C1 )A − (A + B2C2 )A + (A + B1C1 + B2C2 )A ]B3 , ]

△

m = 1, m = 2, m = 3.

△

Furthermore, define f1 (A) = AA and, for all i ∈ {1, . . . , m − 1}, define fi+1 (A) = fi (A) − fi (A + BiCi ). Then, [ ] A B det = −Cm fm (A)Bm . C 0 Fact 3.19.11. Let A ∈ Fn×n , B ∈ Fn×m , C ∈ F p×n , and K ∈ Fm×p , and assume that either m = 1

or p = 1. Then,

Source: Note that

CAAB = C(A + BKC)AB. [

A + BKC C

] [ B A = 0 C

B 0

][

I KC

] 0 . I

In the case m [ =] p = 1, the result follows from Fact 3.19.10. Now, assume that p = 2 and m = 1, and let C = CC12 . Then, C1 AAB = C1 (A + BK1C1 )A B,

C2 AA B = C2 (A + BK2C2 )AB.

Therefore, C1 (A + BK2C2 )AB = C1 (A + BK2C2 + BK1C1 )AB = C1 (A + BKC)A B, C2 (A + BK1C1 )AB = C2 (A + BK1C1 + BK2C2 )AB = C2 (A + BKC)A B. [ ] Furthermore, since det CA [B0B] = 0, it follows from Fact 3.19.10 that C2 AAB = C2 (A+BK1C1 )AB = C2 (A+BK2C2 )A B,

C1 AA B = C1 (A+BK2C2 )A B = C1 (A+BK1C1 )AB.

Hence, C2 (A + BK2C2 )AB = C2 (A + BKC)A B, and thus

C2 AAB = C2 (A + BKC)A B,

C1 (A + BK1C1 )AB = C1 (A + BKC)A B, C1 AA B = C1 (A + BKC)A B,

which implies CAA B = C(A + BKC)A B. Remark: If m = p ≥ 1, then ]A [ ][ ]A [ A + BKC B I 0 A B , = −KC I C 0 C 0 and thus

( [ A + BKC det C

B 0

])n+p−1

( [ A = det C

B 0

])n+p−1 .

Remark: If m ≥ 2 and p ≥ 2, then the result does not hold. Related: Proposition 16.10.10 and

Fact 16.24.15.

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CHAPTER 3

Fact 3.19.12. Let A1 ∈ Fn×n, A2 ∈ Fm×m, B1 ∈ Fn×1, B2 ∈ Fm×1, C1 ∈ F1×n, C2 ∈ F1×m, and define

[

] B1C2 . A2

A1 A= B2C1 △

Then,

det A = (det A1 ) det A2 − C1 AA1 B1C2 AA2 B2 .

If, in addition, A1 and A2 are nonsingular, then −1 det A = (det A1 )(det A2 )(1 − C1 A−1 1 B1 C 2 A2 B2 ).

Credit: K. Aljanaideh.

3.20 Facts on the Inverse n×n Fact 3.20.1. Let A, B,[C, D ] ∈ F and ABCD = I. Then, ABCD = DABC = CDAB = BCDA. Fact 3.20.2. Let A = ac db ∈ F2×2, where ad − bc , 0. Then,

A−1 = (ad − bc)−1 Furthermore, if A =

[a b c ] d e f g h i

A−1

[

] d −b . −c a

∈ F3×3 and β = a(ei − f h) − b(di − f g) + c(dh − eg) , 0, then   ei − f h  = β −1  −(di − f g)  dh − eg

 −(bi − ch) b f − ce   ai − cg −(a f − cd)  .  −(ah − bg) ae − bd

Fact 3.20.3. Let A ∈ Fn×n, and assume that I + A is nonsingular. Then,

(I + A)−1 = I − A(I + A)−1 = I − (I + A)−1A,

A(I + A)−1 = (I + A)−1A.

Fact 3.20.4. Let A ∈ Fn×n, and assume that A and I + A are nonsingular. Then,

(I + A)−1 + (I + A−1 )−1 = (I + A)−1 + (I + A)−1A = I. Fact 3.20.5. Let A, B ∈ Fn×n, and assume that B is nonsingular. Then, A = B[I + B−1(A − B)]. Fact 3.20.6. Let A ∈ Fn×m and B ∈ Fm×n. Then, I + AB is nonsingular if and only if I + BA is

nonsingular. Now, assume that these conditions hold. Then, (In + AB)−1A = A(Im + BA)−1, (In + AB)−1 = In − A(Im + BA)−1B,

(Im + BA)−1B = B(In + AB)−1, (Im + BA)−1 = Im − B(In + AB)−1A,

(In + AB)−1 = In − (In + AB)−1AB = In − A(Im + BA)−1B, (Im + BA)−1 = Im − (Im + BA)−1BA = Im − B(In + AB)−1A. Remark: The first equality is the push-through identity. Remark: Fact 10.11.15. Fact 3.20.7. Let A ∈ Fn×m. Then,

(In + AA∗ )−1 = In − A(Im + A∗A)−1A∗,

(Im + A∗A)−1 = Im − A∗ (In + AA∗ )−1A,

(Im + A∗A)−1A∗A = A∗A(Im + A∗A)−1 = Im − (Im + A∗A)−1 . Fact 3.20.8. Let A, B ∈ Fn×n, and assume that A + B is nonsingular. Then,

A(A + B)−1B = B(A + B)−1A = A − A(A + B)−1A = B − B(A + B)−1B. Now, assume that A is nonsingular. Then, (A + B)−1 = A−1 − (I + A−1B)−1A−1BA−1 = A−1 − A−1 (I + BA−1 )−1BA−1

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BASIC MATRIX PROPERTIES

= A−1 − A−1B(I + A−1B)−1A−1 = A−1 − A−1BA−1 (I + BA−1 )−1. Fact 3.20.9. Let A, B ∈ Fn×n, and assume that A and B are nonsingular. Then,

A + B = A(A−1 + B−1 )B,

A−1 + B−1 = A−1(A + B)B−1,

rank(A + B) = rank(A−1 + B−1 ).

In particular, A−1 + B−1 is nonsingular if and only if A + B is nonsingular. In this case, (A + B)−1 = B−1 − B−1 (A−1 + B−1 )−1B−1 = A−1 − A−1 (A−1 + B−1 )−1A−1, (A−1 + B−1 )−1 = A(A + B)−1B = B(A + B)−1A = A − A(A + B)−1A = B − B(A + B)−1B. Fact 3.20.10. Let A, B ∈ Fn×n, and assume that A and B are nonsingular. Then,

A − B = A(B−1 − A−1 )B,

A−1 − B−1 = A−1(B − A)B−1,

rank(A − B) = rank(A−1 − B−1 ).

In particular, A−1 − B−1 is nonsingular if and only if A − B is nonsingular. In this case, (A − B)−1 = B−1 (B−1 − A−1 )−1B−1 − B−1 = A−1 (A−1 − B−1 )−1A−1 − A−1, (A−1 − B−1 )−1 = A(B − A)−1B = B(B − A)−1A = A + A(B − A)−1A = B(B − A)−1B − B. [ ] Source: Fact 8.9.7 implies that rank AI BI−1 = n + rank(A − B) = n + rank(B−1 − A−1 ). See [2142]. Fact 3.20.11. Let A, B ∈ Fn×n, and assume A and A + B are nonsingular. Then, for all k ≥ 0, k ∑ (A + B)−1 = A−1 (−BA−1 )i + (−A−1B)k+1 (A + B)−1 i=0

=

k ∑

A−1 (−BA−1 )i + A−1 (−BA−1 )k+1 (I + BA−1 )−1.

i=0

Fact 3.20.12. Let A, B ∈ Fn×n , let α, β ∈ F, and assume that αI − A and βI − A are nonsingular.

Then,

(αI − A)−1 − (βI − A)−1 = (β − α)(αI − A)−1 (βI − A)−1 .

Fact 3.20.13. Let A, B ∈ Fn×n and α ∈ F, and assume that A, B, αA−1 + (1 − α)B−1, and

αB + (1 − α)A are nonsingular. Then,

αA + (1 − α)B − [αA−1 + (1 − α)B−1 ]−1 = α(1 − α)(A − B)[αB + (1 − α)A]−1(A − B). Related: iv) of Proposition 10.6.17. Fact 3.20.14. Let A ∈ Fn×n, assume that A is nonsingular, let B ∈ Fn×m, let C ∈ Fm×n, and

assume that A + BC and I + CA−1B are nonsingular. Then,

(A + BC)−1B = A−1B(I + CA−1B)−1. In particular, if A + BB∗ and I + B∗A−1B are nonsingular, then (A + BB∗ )−1B = A−1B(I + B∗A−1B)−1. Fact 3.20.15. Let A ∈ Fn×n, B ∈ Fn×m, C ∈ Fl×n, and D ∈ Fm×l, and assume that A and A + BDC

are nonsingular. Then, (A + BDC)−1 = A−1 − (In + A−1BDC)−1A−1BDCA−1 = A−1 − A−1 (In + BDCA−1 )−1BDCA−1 = A−1 − A−1B(Im + DCA−1B)−1DCA−1 = A−1 − A−1BD(Il + CA−1BD)−1CA−1 = A−1 − A−1BDC(In + A−1BDC)−1A−1 = A−1 − A−1BDCA−1 (In + BDCA−1 )−1.

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CHAPTER 3

Source: [1366]. Remark: Since D is not necessarily either square or nonsingular, the third equality

generalizes the matrix inversion lemma given by Corollary 3.9.8 in the form (A + BDC)−1 = A−1 − A−1B(D−1 + CA−1B)−1CA−1 . Fact 3.20.16. Let A, B ∈ Fn×m, let C, D ∈ Fn×m, and assume that I + DB is nonsingular. Then,

I + AC = (A + B)(I + DB)−1 (C + D) + (I − AD)(I + BD)−1 (I − BC). Source: Compare blocks after inverting both sides of

[

I + BD I − AD

] [ I − BC I = I + AC I

−B A

][

] I I . −D C

See [1851, 2184, 2933] and [2991, p. 233]. Remark: This result generalizes Hua’s matrix equality. See Fact 10.12.52. Related: Fact 3.17.10, Fact 3.20.17, and Fact 3.20.19. Fact 3.20.17. Let A, B, C, D ∈ Fn×m, and assume that I + B∗ D is nonsingular. Then, I + A∗C = (A − B)∗ (I + DB∗ )−1 (C − D) + (I + A∗D)(I + B∗D)−1 (I + B∗C). Remark: This is equivalent to Fact 3.20.16 with A and B replaced by A∗ and B∗ and with B and D

replaced by −B and −D. Fact 3.20.18. Let A, B, C, D ∈ Fn×m, let R, S ∈ Fn×n , and assume I + B∗D is nonsingular. Then, R∗S + A∗C = (A − BR)∗ (I + DB∗ )−1 (C − DS ) + (R∗ + A∗D)(I + B∗D)−1 (S + B∗C). Source: [1851]. Remark: This result generalizes Fact 3.20.17. Fact 3.20.19. Let A, B, C ∈ Fn×m. Then,

I + AC ∗ = (A + B)(I + B∗B)−1 (B + C)∗ + (I − AB∗ )(I + BB∗ )−1 (I − BC ∗ ). Source: Set D = B∗ and replace C with C ∗ in Fact 3.20.16. Fact 3.20.20. Let A ∈ Fn×n, assume that A is either upper triangular or lower triangular, let

D = I ⊙ A denote the diagonal part of A, and assume that D is nonsingular. Then, A−1 =

n ∑ (I − D−1A)iD−1. i=0

Fact 3.20.21. Let A ∈ F

△

, assume that A is nonsingular, and define A0 = In . Furthermore, for and, for all k ∈ {1, . . . , n − 1}, let Ak = AAk−1 − αk I. Then,

n×n

all k ∈ {1, . . . , n}, let αk =

1 k tr AAk−1 ,

A−1 =

1 αn An−1 .

Source: [2979, p. 198]. Credit: J. S. Frame. See [353, p. 99]. Fact 3.20.22. Let A ∈ Fn×n, assume that A is nonsingular, and define (Bi )∞ i=1 by △

Bi+1 = 2Bi − BiABi , where B0 ∈ Fn×n satisfies ρmax (I − B0 A) < 1. Then, Bi → A−1 as i → ∞. Source: [300, p. 167]. Remark: This sequence is given by a Newton-Raphson algorithm. Related: Fact 8.3.38 for the case where A is either singular or nonsquare. Fact 3.20.23. Let A ∈ Fn×n, and assume that A is nonsingular. Then, A + A−∗ is nonsingular. Source: Note that AA∗ + I is positive definite.

351

BASIC MATRIX PROPERTIES

3.21 Facts on Bordered Matrices Fact 3.21.1. Let x, y ∈ Fn. Then,

(I + xyT )A = (1 + yTx)I − xyT ,

det(I + xyT ) = det(I + yxT ) = 1 + xTy = 1 + yTx.

If, in addition, xTy , −1, then (I + xyT )−1 = I − (1 + xTy)−1xyT. Fact 3.21.2. Let A ∈ Fn×n and x, y ∈ Fn. Then,

yT (A + xyT )A = yTAA ,

A(A + xyT )A = (A − xyT )AA + yTAA xI.

If, in addition, A is singular, then det(A + xyT ) = yTAA x = yT (A + xyT )A x,

yTAA xAA = AA xyTAA .

Source: Use Fact 3.21.3 and the last equality in Fact 3.21.5. Fact 3.21.3. Let A ∈ Fn×n, assume that A is nonsingular, and let x, y ∈ Fn. Then,

det(A + xyT ) = (1 + yTA−1x) det A,

(A + xyT )A = AA +

1 (yTAA xAA − AA xyTAA ). det A

Furthermore, the following statements are equivalent: i) det(A + xyT ) , 0. ii) yTA−1x , −1. [ ] A x iii) T is nonsingular. y −1 If these statements hold, then (A + xyT )−1 = A−1 −

1 A−1xyTA−1. 1 + yTA−1x

Remark: The last equality is the Sherman-Morrison-Woodbury formula, which is a special case of the matrix inversion lemma given by Corollary 3.9.8. Related: Fact 3.17.2 and Fact 3.21.4. Problem: Obtain expressions for (A + xyT )A in the cases i) rank(A + xyT ) = n − 1, where n − 2 ≤ rank A ≤ n − 1, and ii) rank(A + xyT ) = n, where A is singular. Fact 3.21.4. Let A ∈ Fn×n, x, y ∈ Fn, and a ∈ F. Then,        I A−1x    A  I 0 0    det A , 0,   ,      T −1    1  yA 1   0 a − yTA−1x   0            [ ]     0   I A−1x  A x  I 0   A =  T    , det A , 0,     yT a  y 1   yT − yTA a − yTA−1x   0 1                 I a−1x   A − a−1xyT 0   I   0    a , 0.     −1 T  ,   0 1 0 a a y 1 Remark: The second factorization follows from Fact 8.9.34 in the case where A is nonsingular. Fact 8.9.34 provides a factorization in the case where A is singular and a = 0, but with the additional assumption that x ∈ R(A).

352

CHAPTER 3

Fact 3.21.5. Let A ∈ Fn×n, let x, y ∈ Fn, and let a ∈ F. Then,

[

Now, assume that

[

A yT

x a

A yT

x a

]A

[ A x]

 −AA x    . det A

  (a + 1)AA − (A + xyT )A =  −yTAA

is nonsingular. Then,      (a − yTA−1x)A−1 + A−1xyTA−1 −A−1x     1        , det A , 0,    T −1   a−yA x  T −1  −y A 1                (a + 1)AA − (A + xyT )A −AA x     1    =  , a , 0,    −1xyT )    adet(A − a T A   −y A det A                (A + xyT )A − AA AA x     1     a = 0.    ,   T A  y A x yTAA − det A

yT a

]−1

Source: Fact 3.17.2 and [991, 1394].

3.22 Facts on the Inverse of Partitioned Matrices Fact 3.22.1. Let A ∈ Fn×n, B ∈ Fn×m, C ∈ Fm×n, and D ∈ Fm×m, and assume that A and D are

nonsingular. Then, [ ]−1 [ −1 A A B = 0 D 0

] −A−1BD−1 , D−1

[

A C

0 D

]−1

[ =

A−1 −D−1CA−1

] 0 . D−1

Fact 3.22.2. Let A ∈ Fn×n , B ∈ Fm×m, and C ∈ Fm×n. Then,

[

] [ 0 A C det = det B C A

] B = (−1)nm (det A)(det B). 0

If, in addition, A and B are nonsingular, then [ ]−1 [ −1 −1 ] −B CA B−1 0 A = , B C A−1 0

[

C A

B 0

]−1

[ =

0 B−1

] A−1 . −B−1CA−1

Fact 3.22.3. Let A ∈ Fn×n, B ∈ Fn×m, and C ∈ Fm×m, and assume that C is nonsingular. Then,

][ ] ] [ A − BC −1BT B I 0 B = . C 0 C C −1BT I [ ] A B If, in addition, A − BC −1BT is nonsingular, then BT C is nonsingular and [

[

A BT

B C

]−1

A BT

[

(A − BC −1BT )−1 = −1 T −C B (A − BC −1BT )−1

Fact 3.22.4. Let A, B ∈ Fn×n. Then,

[

I det B

] −(A − BC −1BT )−1BC −1 . C −1BT (A − BC −1BT )−1BC −1 + C −1

] A = det(I − AB) = det(I − BA). I

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BASIC MATRIX PROPERTIES

If det(I − BA) , 0, then [ ]−1 [ ] [ I + A(I − BA)−1B −A(I − BA)−1 (I − AB)−1 I A = = −1 −1 B I −(I − BA) B (I − BA) −B(I − AB)−1 Fact 3.22.5. Let A, B ∈ Fn×m. Then,

[

] [ 1 In B = A 2 In

A B

[

Therefore, rank

A B

In −In

][

A+B 0

0 A−B

][

Im Im

] −(I − AB)−1A . I + B(I − AB)−1A

] Im . −Im

] B = rank(A + B) + rank(A − B). A

Now, assume that n = m. Then, [ ] A B det = det[(A + B)(A − B)] = det(A2 − B2 − [A, B]). B A [ ] Hence, AB AB is nonsingular if and only if A + B and A − B are nonsingular. If these conditions hold, then [ ] [ ]−1 1 (A + B)−1 + (A − B)−1 (A + B)−1 − (A − B)−1 A B , = B A 2 (A + B)−1 − (A − B)−1 (A + B)−1 + (A − B)−1 [ ]−1 [ ] [ ]−1[ ] In In 1 1 A B A B (A + B)−1 = [In In ] , (A − B)−1 = [In − In ] . B A B A 2 2 In −In Related: Fact 8.9.32. Fact 3.22.6. Let A, B ∈ Fn×m. Then,

  A   B B

B A B

   △  S n =  

where

Therefore,

  B   A + 2B  B  = S n  0   0 A

  A  rank  B  B

B A B

√ 3 In √3 3 In √3 3 3 In

√ 2 2 In

0

−

√ 2 2 In

0 A−B 0 √

 0   0  S mT ,  A−B 

6  In   √6  . 6 I n  3√ 6  − 6 In

−

 B   B  = rank(A + 2B) + 2 rank(A − B).  A

Now, assume that n = m. Then, S nT = S n−1 , and    A B B    det  B A B  = det(A + 2B)[det(A − B)]2 .   B B A [ ] A B B Hence, B A B is nonsingular if and only if A + 2B and A − B are nonsingular. If these conditions B B A hold, then  −1    A B B   (A + 2B)−1  0 0     T 0 (A − B)−1 0  B A B  = S n   S n . B B A 0 0 (A − B)−1

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CHAPTER 3

Source: [2667]. Related: Fact 3.17.30. Fact 3.22.7. Let A1 , . . . , Ak ∈ Fn×n, and assume that the kn × kn partitioned matrix below is

nonsingular. Then, A1 + · · · + Ak is nonsingular, and   A1   Ak 1 (A1 + · · · + Ak )−1 = [In · · · In ]  .  .. k  A2

A2

···

A1 .. .

··· .. .

A3

···

−1 Ak      In    Ak−1   .    .  . ..   .  .     In A1

Source: [2629]. Remark:[This matrix is block circulant. See Fact 8.9.33 and Fact 8.12.5. ] △ A B Fact 3.22.8. Let A = 0m×m C , where A ∈ Fn×m, B ∈ Fn×n, and C ∈ Fm×n, and assume that CA △

△

is nonsingular. Furthermore, define P = A(CA)−1C and P⊥ = I − P. Then, A is nonsingular if and only if P + P⊥BP⊥ is nonsingular. If these conditions hold, then    (CA)−1(C − CBD) −(CA)−1CB(A − DBA)(CA)−1  −1  , A =  D (A − DBA)(CA)−1 △

where D = (P + P⊥BP⊥ )−1P⊥ . Source: [1325]. Fact 3.22.9. Let A ∈ Fn×m and B ∈ Fn×(n−m), and assume that [A B] is nonsingular and A∗B = 0. Then,  ∗ −1 ∗   (A A) A  −1  . [A B] =   (B∗B)−1B∗ Related: Fact 8.9.21. Problem: Find an expression for [A B]−1 without assuming A∗B = 0. Fact 3.22.10. Let A ∈ Fn×m, B ∈ Fn×l, and C ∈ Fm×l. Then,

  In  0  0

−1   In B    C  =  0   0 Il

A Im 0

−A Im 0

 AC − B   −C  .  Il

Fact 3.22.11. Let A ∈ Fn×n, and assume that A is nonsingular. Then, X = A−1 is the unique

matrix satisfying

[

A rank I

] I = rank A. X

Source: [1043]. Related: Fact 8.3.34 and Fact 8.10.11.

3.23 Facts on Commutators Fact 3.23.1. Let A, B ∈ Fn×n. Then,

tr [A, B]2 = 2[tr (AB)2 − tr A2B2 ],

tr [A, B]3 = 3 tr(A2B2AB − B2A2BA) = −3 tr AB2A[A, B].

If, in addition, n = 3, then tr [A, B]3 = 3 det [A, B]. Fact 3.23.2. Let A, B ∈ Fn×n, assume that [A, B] = 0, and let k, l ∈ N. Then, [Ak, Bl ] = 0. Fact 3.23.3. Let A, B, C ∈ Fn×n. Then, the following statements hold: i) [A, A] = 0. ii) [A, B] = [−A, −B] = −[B, A]. iii) [A, B + C] = [A, B] + [A, C]. iv) [αA, B] = [A, αB] = α[A, B] for all α ∈ F. v) [A, [B, C]] + [B, [C, A]] + [C, [A, B]] = 0.

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BASIC MATRIX PROPERTIES

[A, B]T = [BT, AT ] = −[AT, BT ]. tr [A, B] = 0. tr Ak [A, B] = tr Bk [A, B] = 0 for all k ≥ 1. [[A, B], B − A] = [[B, A], A − B]. [A, [A, B]] = −[A, [B, A]]. Remark: v) is the Jacobi identity. Related: Fact 3.23.4. Fact 3.23.4. Let A, B, C ∈ Fn×n. Then, the following statements hold: i) adA (A) = 0. ii) For all α ∈ F, adA (αB) = α adA (αB). iii) adA (B + C) = adA (B) + adA (C). iv) adA (BC) = [adA (B)]C + B adA (C). v) ad[A,B] = [adA , adB ]. vi) ad[A,B](C) = [adA , adB ](C) = adA[adB (C)] − adB[adA (C)]. Remark: vi) is equivalent to the Jacobi identity given by Fact 3.23.3. Fact 3.23.5. Let A ∈ Fn×n and, for all X ∈ Fn×n, define    k = 1, adA(X), △  adAk (X) =   k−1  adA [adA(X)], k ≥ 2. vi) vii) viii) ix) x)

Then, for all X ∈ Fn×n and k ≥ 1, adA2 (X)

= [A, [A, X]] − [[A, X], A],

adAk (X)

( ) k ∑ k−i k i = (−1) A XAk−i. i i=0

Source: For the last equality, see [2263, pp. 176, 207]. Remark: The proof of Proposition 15.4.12 △ is based on g(et adA et adB ), where g(z) = (log z)/(z − 1). See [2379, p. 35]. Related: Fact 15.15.5. Fact 3.23.6. Let A, B ∈ Fn×n, and assume that [A, B] = A. Then, A is singular. Source: If A is

nonsingular, then tr B = tr ABA−1 = tr B + n, which is false. Fact 3.23.7. Let A, B ∈ Cn×n, and assume that there exist nonzero a, b ∈ C such that AB = aA + bB. Then, AB = BA. Source: [1158, p. 72]. Fact 3.23.8. Let A, B ∈ Rn×n , and assume that AB = BA. Then, there exists C ∈ Rn×n such that A2 + B2 = C 2. Source: [913]. Remark: This result does not hold for complex matrices. Fact 3.23.9. Let A ∈ Fn×n. Then, n ≤ dim {X ∈ Fn×n : AX = XA},

dim {[A, X]: X ∈ Fn×n } ≤ n2 − n.

Source: [860, pp. 125, 142, 493, 537]. Remark: The first set is the centralizer (also called the commutant) of A. See Fact 9.5.3. Remark: These quantities are the defect and rank, respectively, △ of the operator f : Fn×n 7→ Fn×n defined by f (X) = AX − XA. See Fact 9.5.3. Related: Fact 7.16.8 and Fact 7.16.9. Fact 3.23.10. Let A ∈ Fn×n. Then, the following statements are equivalent: i) There exists α ∈ F such that A = αI. ii) For all X ∈ Fn×n, AX = XA. Source: To prove sufficiency, note that AT ⊕ −A = 0. Hence, {0} = spec(AT ⊕ −A) = {λ − µ : λ, µ ∈ spec(A)}. Therefore, spec(A) = {α}, and thus A = αI + N, where N is nilpotent. Consequently, for all X ∈ Fn×n, NX = XN. Setting X = N ∗, it follows that N is normal. Hence, N = 0. Remark: This result determines the center subgroup of GL(n).

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CHAPTER 3 △

Fact 3.23.11. Define S ⊆ Fn×n by S = {[X, Y]: X, Y ∈ Fn×n }. Then, S is a subspace. Furthermore,

S = {Z ∈ Fn×n : tr Z = 0},

dim S = n2 − 1.

Consequently, if Z ∈ Fn×n and tr Z = 0, then there exist X, Y ∈ Fn×n such that Z = [X, Y]. Source: 2 [860, pp. 125, 493]. Alternatively, note tr: Fn 7→ F is onto, and use Corollary 3.6.5. Fact 3.23.12. Let A, B, C, D ∈ Fn×n. Then, there exist E, F ∈ Fn×n such that [E, F] = [A, B] + [C, D]. Source: Fact 3.23.11. Problem: Construct E and F.

3.24 Facts on Complex Matrices Fact 3.24.1. Let a, b ∈ R. Then,

[

]

is a representation of the complex number a + ȷb that preserves addition, multiplication, and inversion of complex numbers. In particular, if a2 + b2 , 0, then  a −b    [ ]−1  2 2 a2 + b2  a b a b  a + b  , (a + b ȷ)−1 = 2 =  −ȷ 2 . 2 −b a   b a + b a + b2 a   a2 + b2 a2 + b2 [ ] a b is a rotation-dilation. See Fact 4.32.6. Remark: −b a Fact 3.24.2. Let ν, ω ∈ R. Then, [ ][ ] [ ]∗ [ ] 1 1 1 1 1 1 ν + ωȷ 0 ν ω = √ √ 0 ν − ωȷ 2 ȷ −ȷ −ω ν 2 ȷ −ȷ [ ][ ] [ ]∗ 1 1 ȷ ν + ωȷ 1 1 ȷ 0 = √ √ 0 ν − ωȷ 2 ȷ 1 2 ȷ 1 [ ][ ] [ ] 1 1 −ȷ ν + ωȷ 1 1 −ȷ 0 = √ , √ 0 ν − ω ȷ 2 ȷ −1 2 ȷ −1 [ ]−1 [ ] 1 ν ω ν −ω = 2 . −ω ν ν + ω2 ω ν a b −b a

Remark: All three transformations are unitary. The third transformation is also Hermitian. Related: Fact 3.24.1. [ A B] [ A −B ] Fact 3.24.3. Let A, B ∈ Rn×m. Then, the real matrices −B A and B A are representations [ ] of

A the complex matrices A + ȷB and A + ȷB, respectively. Furthermore, the real matrices −B T [ T T] A −B are representations of the complex matrices (A + ȷB)T and (A + ȷB)∗, respectively. BT AT T

[

n×m Fact and C, D ∈ Rm×l. Then, for all α, β ∈ R, the real matrices ] 3.24.4. Let A, B ∈ R

C D −D C

, and

[

] [ αA + βC αB + βD A =α −(αB + βD) αA + βC −B

] [ B C +β A −D

D C

BT AT

and

[A

B −B A

]

,

]

are representations of the complex matrices A + ȷB, C + ȷD, and α(A + ȷB) + β(C + ȷD), respectively. [ A B] [ C D] Fact 3.24.5. Let A, B ∈ Rn×m and C, D ∈ Rm×l. Then, the real matrices −B A , −D C , and [ ] [ ][ ] AC − BD AD + BC A B C D = −(AD + BC) AC − BD −B A −D C are representations of the complex matrices A + ȷB, C + ȷD, and (A + ȷB)(C + ȷD), respectively.

357

BASIC MATRIX PROPERTIES

Fact 3.24.6. Let A, B ∈ Cn×m. Then,

[

A −B

[ ][ ][ ] ] 1 I I A + ȷB 0 I − ȷI B = 0 A − ȷB I ȷI A 2 ȷI − ȷI [ ][ ][ ] 1 I ȷI A − ȷB 0 I ȷI = 0 A + ȷB − ȷI −I 2 − ȷI −I [ =

Consequently,

[

and thus A + ȷB = Furthermore,

A + ȷB 0

I ȷI

0 I

][

A + ȷB 0

] [ 1 I 0 = A − ȷB 2 I

[ 1 A [I − ȷI] −B 2

B A

][

] I , ȷI

B A − ȷB

− ȷI ȷI

][

][

A −B

A − ȷB =

] 0 . I

I − ȷI

B A

][

] I , − ȷI

I ȷI

[ 1 A [I ȷI] −B 2 [

A rank(A + ȷB) + rank(A − ȷB) = rank −B

Finally, if A and B are real, then

[ 1 A rank(A + ȷB) = rank(A − ȷB) = rank −B 2

B A

][

] I . −ȷI

] B . A ] B . A

Source: Fact 6.10.32 and [181, 2628]. Fact 3.24.7. Let A, B ∈ Cn×n. Then, the following statements hold:

[

] B = det(A + ȷB) det(A − ȷB) = det[A2 + B2 ± ȷ(AB − BA)]. A [ ] A B ii) If AB = BA, then det = det(A2 + B2 ). −B A [ A B] iii) −B A is nonsingular if and only if A + ȷB and A − ȷB are nonsingular. If these conditions hold, then [ ]−1 [ ] [ ]−1 [ ] 1 1 A B I A B I , (A − ȷB)−1 = [I ȷI] . (A + ȷB)−1 = [I − ȷI] −B A ȷI −B A −ȷI 2 2 A i) det −B

iv) Assume that A is nonsingular. Then, [ ] A B det = det(A2 + ABA−1B). −B A Furthermore, A + ȷB and A − ȷB are nonsingular if and only if A + BA−1B is nonsingular. If these conditions hold, then  [ ]−1   (A + BA−1B)−1 −A−1B(A + BA−1B)−1  A B   , =  −1 −B A A B(A + BA−1B)−1 (A + BA−1B)−1 (A + ȷB)−1 = (A + BA−1B)−1 − ȷA−1B(A + BA−1B)−1, (A − ȷB)−1 = (A + BA−1B)−1 + ȷA−1B(A + BA−1B)−1.

358

CHAPTER 3

v) Assume that B is nonsingular. Then, [ ] A B det = det(B2 + BAB−1A). −B A Furthermore, A + ȷB and A − ȷB are nonsingular if and only if B + AB−1A is nonsingular. If these conditions hold, then  [ ]−1  −1  B A(B + AB−1A)−1 −(B + AB−1A)−1  A B  ,  =  −B A (B + AB−1A)−1 B−1A(B + AB−1A)−1 (A + ȷB)−1 = B−1A(B + AB−1A)−1 − ȷ(B + AB−1A)−1, (A − ȷB)−1 = B−1A(B + AB−1A)−1 + ȷ(B + AB−1A)−1. vi) Assume that A and B are nonsingular. Then, [ ] A B det = det(A2 + ABA−1B) = det(B2 + BAB−1A). −B A Furthermore, the following statements are equivalent: a) A + ȷB and A − ȷB are nonsingular. b) A + BA−1B is nonsingular. c) B + AB−1A is nonsingular. Now, assume that a)–c) hold. Then,  [ ]−1   (A + BA−1B)−1 −(B + AB−1A)−1  A B  ,  =  −B A (B + AB−1A)−1 (A + BA−1B)−1 (A + ȷB)−1 = (A + BA−1B)−1 − ȷ(B + AB−1A)−1, (A − ȷB)−1 = (A + BA−1B)−1 + ȷ(B + AB−1A)−1. [ ] A B vii) If A and B are real, then det = | det(A + ȷB)|2 ≥ 0. −B A viii) Assume that A and B are real. Then, the following conditions are equivalent: [ A B] a) −B A is nonsingular. b) A + ȷB is nonsingular. c) A − ȷB is nonsingular. Source: If A is nonsingular, then [ ] [ ][ ] [ ] A B A 0 I A−1B I A−1B = , det = det[I + (A−1B)2 ]. −B A 0 A −A−1B I −A−1B I Remark: See [2629]. Related: Fact 4.10.30, Fact 4.13.12, Fact 8.9.32, and Fact 10.16.6. Fact 3.24.8. Let A, B ∈ Fn×n. Then,

[

det

A −B

] B ≥ 0. A

If, in addition, A is nonsingular, then [ ] A B = | det A|2 det(I + A−1BA−1B). det −B A

359

BASIC MATRIX PROPERTIES

Source: [2982] and [2991, p. 106]. Remark: Fact 10.15.15 implies that det(I + A−1BA−1B) ≥ 0. Related: Fact 6.10.33. Fact 3.24.9. Let A, B ∈ Cn×n . Then,

(A + ȷAB)(A − ȷBA) = (A − ȷAB)(A + ȷBA) = A2 + AB2A, (A + ȷBA)(A − ȷAB) = (A − ȷBA)(A + ȷAB) = A2 + BA2B − ȷ(BA2 − A2B). Source: [1845].

  C11

 △  C21 

Fact 3.24.10. Let A, B ∈ Rn×n, and define C ∈ R2n×2n by C = △

[

A(i, j) B(i, j) −B(i, j) A(i, j)

]

  . . .

C12 ···

 · · ·    , 

where, for all

∈ R2×2 . Then, det C = | det(A + ȷB)|2. Source: [568] and note that [ ] A B C = A ⊗ I2 + B ⊗ J2 = P2,n (I2 ⊗ A + J2 ⊗ B)P2,n = P2,n P . −B A 2,n

i, j ∈ {1, . . . , n}, Ci j =

3.25 Facts on Majorization Fact 3.25.1. Let x, y ∈ Rn . Then, the following statements are equivalent: w

i) x ≺ y. s

ii) There exists z ∈ Rn such that x ≤≤ z and z ≺ y. s

iii) There exists z ∈ Rn such that x ≺ z and z ≤≤ y. Source: [2991, p. 328]. Fact 3.25.2. Let x, y ∈ Rn . Then, the following statements hold: w

i) x ≺ |x|. w

ii) |x + y| ≺ |x|↓ + |y|↓ . s

s

iii) x ≺ y if and only if −x ≺ −y. s

w

w

iv) x ≺ y if and only if x ≺ y and −x ≺ −y. s

v) x − y ≺ x↓ − y↑ . s

s

vi) x↓ + y↑ ≺ x + y ≺ x↓ + y↓ . vii) x↓T y↑ ≤ xT y ≤ x↓T y↓ . w

viii) If x ≺ y, then (x↓ )(1) ≤ (y↓ )(1) . s

ix) If x ≺ y, then (y↓ )(n) ≤ (x↓ )(n) ≤ (x↓ )(1) ≤ (y↓ )(1) . x) (x↓ + y↓ )↓ = x↓ + y↓ . xi) If x and y are nonnegative, then (x↓ ⊙ y↓ )↓ = x↓ ⊙ y↓ . w

xii) x ≺ y if and only if, for all z ∈ [0, ∞)n , x↓T z↓ ≤ y↓T z↓ . s

xiii) x ≺ y if and only if, for all z ∈ Rn , x↓T z↓ ≤ y↓T z↓ . s

xiv) If x ≺ y and z ∈ Rn , then z↓T y↑ ≤ z↓T x ≤ z↓T x↓ ≤ z↓T y↓ . w w w [ ] xv) If u, v ∈ Rm , x ≺ y, and u ≺ v, then [ ux ] ≺ yv . s s s [ ] xvi) If u, v ∈ Rm , x ≺ y, and u ≺ v, then [ ux ] ≺ yv . s [ ] s [ ] xvii) If z ∈ Rm and x ≺ y, then xz ≺ yz . w [ ] s [ ] xviii) If z ∈ Rm and xz ≺ yz , then x ≺ y.

360

CHAPTER 3 w

w

w

s

s

s

xix) If u, v ∈ Rn , x ≺ y, and u ≺ v, then x + u ≺ y↓ + v↓ . xx) If u, v ∈ Rn , x ≺ y, and u ≺ v, then x + u ≺ y↓ + v↓ . w w [ ] xxi) If z ∈ Rn and x ≺ 21 (y + z), then [ xx ] ≺ yz . s [ ] s xxii) If z ∈ Rn and x ≺ 21 (y + z), then [ xx ] ≺ yz .

w w [ ] xxiii) If z ∈ Rn , x, y, z ≥≥ 0, and x ≺ (y ⊙ z)⊙1/2 , then [ xx ] ≺ yz . ] w w [ xxiv) If z ∈ Rn , x, y, z ≥≥ 0, and x ≺ y ⊙ z, then [ xx ] ≺ y⊙y z⊙z . w

w

xxv) If x ≺ y, z ∈ Rn , and z ≥≥ 0, then x ⊙ z ≺ y↓ ⊙ z↓ . w

w

w

xxvi) If u, v ∈ Rn , u, v ≥≥ 0, x ≺ u, and y ≺ v, then x ⊙ y ≺ u↓ ⊙ v↓ . s

s

xxvii) If z ∈ Rn and x ≺ y↓ − z↑ , then x↓ + z↓ ≺ y↓ + z↓ − z↑ . Source: [1969, p. 95], [1971, p. 136], [2750], and [2991, pp. 327–333]. Fact 3.25.3. Let x, y ∈ Rn . Then, the following statement holds: s

s

i) x↓ + y↑ ≺ x + y ≺ x↓ + y↓ . Now, assume that x and y are nonnegative. Then, the following statements hold: slog

s

ii) If x ≺ y and x ≺ y, then x↓ = y↓ . wlog

w

iii) If x ≺ y, then x ≺ y. w

w

iv) x↓ ⊙ y↑ ≺ x ⊙ y ≺ x↓ ⊙ y↓ . Source: [1969, p. 117], [1971, p. 168, 224], [2750] and [2991, pp. 345, 348]. Related: Fact 2.12.8. s s ∑ Fact 3.25.4. Let x ∈ Rn be nonnegative, and assume that ni=1 x(i) = 1. Then, n1 1n×1 ≺ x ≺ e1,n . Source: [1969, p. 7] and [1971, p. 9]. Fact 3.25.5. Let a < b, let f : (a, b)n 7→ R, and assume that f is C1. Then, f is Schur-convex if and only if f is symmetric and, for all x ∈ (a, b)n, ( ) ∂ f (x) ∂ f (x) (x(1) − x(2) ) − ≥ 0. ∂x(1) ∂x(2) Furthermore, the following functions are Schur-convex: i) f : Rn 7→ R, where f (x) = maxi∈{1,...,n} |x(i) |. ∑ ii) f : Rn 7→ R, where f (x) = ni=1 |x(i) | p and p ≥ 1. )1/p (∑ and p ≥ 1. iii) f : Rn 7→ R, where f (x) = ni=1 |x(i) | p ∑k n ↓ p iv) f : R 7→ R, where f (x) = i=1 [(|x| )(i) ] , k ≤ n, and p ≥ 1. (∑ )1/p v) f : Rn 7→ R, where f (x) = ki=1 [(|x|↓ )(i) ] p , k ≤ n, and p ≥ 1. ∑n 1 n vi) f : [0, ∞) 7→ R, where f (x) = i=1 x(i) . ∏ ∏ vii) f : [0, ∞)n 7→ R, where f (x) = ni=1 xi − ni=1 (xi + 1). Source: [1835], [1969, p. 57], [1971, p. 84], and [2991, pp. 347, 376]. Remark: f is symmetric means that f (Ax) = f (x) for all x ∈ (a, b)n and every permutation matrix A ∈ Rn×n. Remark: See [1557]. s

Fact 3.25.6. Let x, y ∈ Rn be nonnegative, assume that x = x↓ , y = y↓ , and x ≺ y, and let

361

BASIC MATRIX PROPERTIES

p1 , . . . , pn be nonnegative numbers. Then, n ∑∏

1 ∑ ∏ y( j) ≤ pi j , n! j=1 n

x pi j( j)

j=1

where both sums are taken over all n! permutations {i1, . . . , in } of {1, . . . , n}. Source: [1146, p. 99], [1569], [1969, p. 87], [1971, p. 125], [1938, pp. 44–47], and [2264, pp. 81–83]. Remark: This is Muirhead’s theorem, which is based on a function that is Schur-convex. Remark: Let x = [5 0] and y = [3 2]. Then, x strongly majorizes y. Therefore, for all nonnegative a, b, it follows that a3 b2 + b3 a2 ≤ a5 + b5 . As another example, let x = [2 2 0] and y = [2 1 1]. Then, x strongly majorizes y. Therefore, for all nonnegative a, b, c, it follows that a2 bc + b2 ca + c2 ab ≤ s a2 b2 + b2 c2 + c2 a2 . See [1938, p. 44]. Remark: Let x = neT1,n and y = 11×n . Then, y ≺ x. Therefore, ∑ ∏ for all nonnegative z1 , . . . , zn , it follows that ni=1 zi ≤ n1 ni=1 zni , which is the arithmetic-mean– geometric-mean inequality. See Fact 2.11.114. s ∑ Fact 3.25.7. Let x, y ∈ Rn be positive, and assume that x ≺ y and ni=1 x(i) = 1. Then, n ∑ i=1

yi log y1(i) ≤

n ∑

xi log

1 x(i)

≤ log n.

i=1

Source: [1146, p. 102], [1969, pp. 71, 405], and [1971, pp. 101, 556]. Remark: For x(1) , x(2) > 0,

note that (x(1) − x(2) ) log(x(1) /x(2) ) ≥ 0. Hence, Fact 3.25.5 implies that the entropy function is Schurconcave. Related: Entropy bounds are given in Fact 2.15.27. s

Fact 3.25.8. Let x, y ∈ Rn, assume that x ≺ y, let f : R 7→ R, and assume that f is convex.

Then,

w

[ f (x(1) ) · · · f (x(n) )] ≺ [ f (y(1) ) · · · f (y(n) )]. Source: [449, p. 42], [1450, p. 173], [1969, p. 116], [1971, p. 167], and [2991, p. 342]. slog

Fact 3.25.9. Let x, y ∈ Rn be nonnegative, assume that x ≺ y, let f : [0, ∞) 7→ R, and assume △

that g: R 7→ R defined by g(z) = f (ez ) is convex. Then, w

[ f (x(1) ) · · · f (x(n) )] ≺ [ f (y(1) ) · · · f (y(n) )]. Source: Fact 3.25.8. w Fact 3.25.10. Let x, y ∈ Rn, assume that x ≺ y, let f : R 7→ R, and assume that f is convex and

nondecreasing. Then,

w

[ f (x(1) ) · · · f (x(n) )] ≺ [ f (y(1) ) · · · f (y(n) )]. Source: [449, p. 42], [1450, p. 173], [1969, p. 116], [1971, p. 167], and [2991, p. 342]. Related:

Fact 3.25.12. w

slog

△

Fact 3.25.11. Let x, y ∈ Rn, assume that x ≺ y, and define e x = [e x(1) · · · e x(n) ]T . Then, e x ≺ ey . Source: [2750]. s

Fact 3.25.12. Let x, y ∈ Rn be nonnegative, assume that x ≺ y, and let r ≥ 1. Then, w

r r [x(1) · · · x(n) ] ≺ [yr(1) · · · yr(n) ]. w

Source: Fact 3.25.10. Remark: The majorization can be written as x⊙r ≺ y⊙r . wlog

Fact 3.25.13. Let x, y ∈ Rn be positive, assume that x ≺ y, let f : [0, ∞) 7→ R, and assume △

that g: R 7→ R defined by g(z) = f (ez ) is convex and nondecreasing. Then, w

[ f (x(1) ) · · · f (x(n) )] ≺ [ f (y(1) ) · · · f (y(n) )].

362

CHAPTER 3

Source: Fact 3.25.10 and [449, p. 42]. s

Fact 3.25.14. Let x, y ∈ Rn be nonnegative, and assume that x ≺ y. Then, Source: [2991, p. 347]. Related: Fact 10.16.1.

∏n

i=1 y(i)

wlog

≤

∏n i=1

x(i) .

w

Fact 3.25.15. Let x, y ∈ Rn be nonnegative, and assume that x ≺ y. Then, x ≺ y. Source:

Use Fact 3.25.13 with f (t) = t and the fact that g(z) = ez is convex and increasing. See [2977, p. 19] and [2991, p. 345]. Example: Let x = [4 3] and y = [7 2]. Then, 4 ≤ 7 and 4 · 3 ≤ 7 · 2. Hence, y weakly log majorizes x. Furthermore, 4 ≤ 7 and 4 + 3 ≤ 7 + 2. Thus, y weakly majorizes x. Remark: The converse is false. Let x = [2 1] and y = [3 12 ]. Then, 2 ≤ 3 and 2 + 1 ≤ 3 + 21 . Thus, y weakly majorizes x. However, 2 ≤ 3 and 2 · 1 > 3 · 12 . Therefore, y does not weakly log majorize x. w

Fact 3.25.16. Let x, y ∈ Rn, be nonnegative, assume that x = x↓ and y = y↓ , assume that x ≺ y,

and let p ∈ [1, ∞). Then, for all k ∈ {1, . . . , n}, 1/p 1/p  k  k ∑  ∑  p p  x(i)  ≤  y(i)  .  i=1

i=1

△

Source: Fact 3.25.10, [1969, p. 96], and [1971, p. 138]. Remark: ϕ(x) = (

∑k i=1

p 1/p x(i) ) is a

symmetric gauge function. See Fact 11.9.59. Fact 3.25.17. Let x, y ∈ Rn. Then, the following statements are equivalent: s

i) x ≺ y. ∑ ∑ ii) For all convex functions f : R 7→ R, it follows that ni=1 f (xi ) ≤ ni=1 f (yi ). Furthermore, the following statements are equivalent: w

iii) x ≺ y. ∑ ∑ iv) For all nondecreasing functions f : R 7→ R, it follows that ni=1 f (xi ) ≤ ni=1 f (yi ). Source: [1569], [1969, p. 108], and [1971, p. 156]. Remark: This is Karamata’s inequality. See [1569].

3.26 Notes The theory of determinants is discussed in [2097, 2098, 2752]. A graph-theoretic interpretation is given in [588, Chapter 4]. Applications to physics are described in [2788, 2789]. Contributors to the development of this subject are highlighted in [1222]. The empty matrix is discussed in [847, 2118], [2314, pp. 462–464], and [2544, p. 3]. Recent versions of Matlab follow the properties of the empty matrix given in this chapter [1384, pp. 305, 306]. Convexity is the subject of [421, 523, 558, 980, 1769, 2213, 2319, 2544, 2767, 2846]. Convex optimization theory is developed in [362, 558]. In [523] the dual cone is called the polar cone. The development of rank properties is based on [1966]. Theorem 3.7.5 is based on [2139]. The term “subdeterminant” is used in [2221] and is equivalent to minor. The notation AA for adjugate is used in [2537]. Numerous papers on basic topics in matrix theory and linear algebra are collected in [641, 642]. A geometric interpretation of N(A), R(A), N(A∗ ), and R(AT ) is given in [2554]. Some reflections on matrix theory are given in [2593, 2618]. Applications of the matrix inversion lemma are discussed in [1291]. Some historical notes on the determinant and inverse of partitioned matrices as well as the matrix inversion lemma are given in [1366]. Combinatorial proofs of several matrix theorems are given in [2969]. A detailed treatment of 2 × 2 matrices is given in [2253].

Chapter Four Matrix Classes and Transformations This chapter presents definitions of various types of matrices as well as transformations for analyzing matrices.

4.1 Types of Matrices In this section we categorize various types of matrices based on their algebraic and patterned properties. The following definition introduces various types of square matrices. Note that, if F = R, then A∗ = AT , whereas, if F = C, then A∗ and AT may be different. Definition 4.1.1. For A ∈ Fn×n define the following types of matrices: i) A is group invertible if R(A) = R(A2 ). ii) A is involutory if A2 = I. iii) A is skew involutory if A2 = −I. iv) A is idempotent if A2 = A. v) A is skew idempotent if A2 = −A. vi) A is tripotent if A3 = A. vii) A is nilpotent if there exists k ∈ P such that Ak = 0. viii) A is unipotent if A − I is nilpotent. ix) A is range Hermitian if R(A) = R(A∗ ). x) A is range symmetric if R(A) = R(AT ). xi) A is range disjoint if R(A) ∩ R(A∗ ) = {0}. xii) A is range spanning if R(A) + R(A∗ ) = Fn . xiii) A is Hermitian if A = A∗. xiv) A is symmetric if A = AT. xv) A is skew Hermitian if A = −A∗. xvi) A is skew symmetric if A = −AT. xvii) A is normal if AA∗ = A∗A. xviii) A is positive semidefinite (A ≥ 0) if A is Hermitian and x∗Ax ≥ 0 for all x ∈ Fn. xix) A is negative semidefinite (A ≤ 0) if −A is positive semidefinite. xx) A is positive definite (A > 0) if A is Hermitian and x∗Ax > 0 for all x ∈ Fn such that x , 0. xxi) A is negative definite (A < 0) if −A is positive definite. xxii) A is semidissipative if A + A∗ is negative semidefinite. xxiii) A is dissipative if A + A∗ is negative definite. xxiv) A is unitary if A∗A = I. xxv) A is shifted unitary if A + A∗ = 2A∗A; equivalently, 2A − I is unitary. xxvi) A is orthogonal if ATA = I. xxvii) A is shifted orthogonal if A + AT = 2ATA; equivalently, 2A − I is orthogonal.

364

xxviii) xxix) xxx) xxxi) xxxii) xxxiii) xxxiv) xxxv) xxxvi) xxxvii) xxxviii)

CHAPTER 4

A is a projector if A is Hermitian and idempotent. A is a generalized projector if A2 = A∗ . A is a partial isometry if AA∗ is a projector. A is a reflector if A is Hermitian and unitary. A is a skew reflector if A is skew Hermitian and unitary. A is an elementary projector if there exists a nonzero vector x ∈ Fn such that A = I − (x∗x)−1xx∗. A is an elementary reflector if there exists a nonzero vector x ∈ Fn such that A = I − 2(x∗x)−1xx∗. A is an elementary matrix if there exist x, y ∈ Fn such that A = I − xyT and xTy , 1. A is reverse Hermitian if A = A∗ˆ. ˆ A is reverse symmetric if A = AT. A is a permutation matrix if each row of A and each column of A possesses one 1 and zeros otherwise. A is an (even, odd) permutation matrix if A is a permutation matrix and (det A = 1, det A = −1). A is a transposition matrix if it is a permutation matrix and A has exactly two off-diagonal entries that are nonzero. The cyclic permutation matrix Pn is defined by   0 · · · 0 0   0 1   ..   0 0 . 0 0  1   [ ]  ..   . 0 I 0 0 0 0 0   △ n−1 (4.1.1) Pn = (n−1)×1 =   , 1 01×(n−1)  .. . . . . . .. ..  . . . . . ..    ..  0 0 . 0 1  0   1 0 0 ··· 0 0 △

where P1 = 1. xxxix) A is reducible if either A = 01×1 or both [ B n C≥] 2 and there exist k ≥ 1 and a permutation (n−k)×(n−k) matrix S ∈ Rn×n such that SAS T = 0k×(n−k) , C ∈ F(n−k)×k, and D , where B ∈ F k×k D∈F . xl) A is irreducible if A is not reducible. xli) A is (totally nonnegative, totally positive) if every subdeterminant of A is (nonnegative, positive). Let A ∈ Fn×n be Hermitian. Then, the function f : Fn 7→ R defined by △

f (x) = x∗Ax is a quadratic form. The following definition concerns matrices that are not necessarily square. Definition 4.1.2. For A ∈ Fn×m define the following types of matrices: i) A is semicontractive if In − AA∗ is positive semidefinite. ii) A is contractive if In − AA∗ is positive definite. iii) A is left inner if A∗A = Im . iv) A is right inner if AA∗ = In . v) A is centrohermitian if A = Iˆn AIˆm .

(4.1.2)

MATRIX CLASSES AND TRANSFORMATIONS

365

vi) A is centrosymmetric if A = Iˆn AIˆm . vii) A is an outer-product matrix if there exist x ∈ Fn and y ∈ Fm such that A = xyT. The following definition introduces several types of structured and patterned matrices. Definition 4.1.3. For A ∈ Fn×m define the following types of matrices: i) A is diagonal if A(i, j) = 0 for all i , j. If n = m, then A is diagonal if and only if   0 ··· 0   A(1,1)   ..   . 0 A 0 △  (2,2)  . (4.1.3) A = diag(A(1,1) , . . . , A(n,n) ) =  .  .. ..  . . . 0   .  0 0 · · · A(n,n) ii) A is tridiagonal if A(i, j) = 0 for all |i − j| > 1. iii) A is reverse diagonal if A(i, j) = 0 for all i + j , min{n, m} + 1. If n = m, then A is reverse diagonal if and only if   ··· 0 A(1,n)   0    .. A  0 . 0  △  (2,n−1)   (4.1.4) A = revdiag(A(1,n) , . . . , A(n,1) ) =  . . . . .  . ..  .. ..  .  A(n,1) 0 ··· 0 iv) A is (upper triangular, strictly upper triangular) if A(i, j) = 0 for all (i > j, i ≥ j). v) A is (lower triangular, strictly lower triangular) if A(i, j) = 0 for all (i < j, i ≤ j). vi) A is (upper bidiagonal, lower bidiagonal) if A is tridiagonal and (upper triangular, lower triangular). vii) A is (upper Hessenberg, lower Hessenberg) if A(i, j) = 0 for all (i > j + 1, i < j + 1). viii) A is Toeplitz if A(i, j) = A(k,l) for all k − i = l − j; that is,   c · · ·   a b   . .    d a . b  A =  . .  . .  a  e d   . . .. .. ... ...  ix) A is Hankel if A(i, j) = A(k,l) for all i + j = k + l; that is,   c · · ·   a b   .   b c d . .   A =  .  . e . .   c d   . .. . . . . . . . . .  The following definition introduces several types of partitioned matrices whose blocks are structured or patterned. Definition 4.1.4. For A ∈ Fn×m define the following types of matrices:

366

i) A is block diagonal if

CHAPTER 4

  A1  △  A = diag(A1 , . . . , Ak ) =   0

..

 0    ,  Ak

.

where Ai ∈ Fni ×mi for all i ∈ {1, . . . , k}. ii) A is reverse block diagonal if

  0  A = revdiag(A1 , . . . , Ak ) =   Ak △

where Ai ∈ Fni ×mi for all i ∈ {1, . . . , k}. iii) A is upper block triangular if   A11  0  A =  ..  .  0

A12 A22 .. . 0

··· ··· .. . ···

where Ai j ∈ Fni ×n j for all i, j ∈ {1, . . . , k} such that i ≤ iv) A is lower block triangular if  0 ···  A11 ..  A . A 22  21 A =  . . . .. ..  ..  Ak1 Ak2 · · ·

..

.

 A1    ,  0

 A1k   A2k  ..  , .  Akk j. 0 0 .. . Akk

     ,  

where Ai j ∈ Fni ×n j for all i, j ∈ {1, . . . , k} such that j ≤ i. v) A is block Toeplitz if    A1 A2 A3 · · ·     A A A . . .  1 2   4 A =  . .  ,  A5 A4 A1 .     .. . . . . . .  . . . . where Ai ∈ Fni ×mi for all i ∈ {1, . . . , k}. vi) A is block Hankel if   A1   A  2 A =   A3   .. .

A2

A3

A3

A4

A4 . ..

A5 . ..

 · · ·   .  . .   .  , . .   .  ..

where Ai ∈ Fni ×mi for all i ∈ {1, . . . , k}. For x ∈ Fn , define △

diag(x) = diag(x(1) , . . . , x(n) ), △

revdiag(x) = revdiag(x(1) , . . . , x(n) ).

(4.1.5) (4.1.6)

367

MATRIX CLASSES AND TRANSFORMATIONS

Definition 4.1.5. For A ∈ Rn×m define the following types of matrices:

i) A is nonnegative (A ≥≥ 0) if A(i, j) ≥ 0 for all i ∈ {1, . . . , n} and j ∈ {1, . . . , m}. ii) A is row stochastic if A is nonnegative and A1m×1 = 1n×1 . iii) A is column stochastic if A is nonnegative and 11×n A = 11×m . iv) A is doubly stochastic if A is both row stochastic and column stochastic. v) A is positive (A >> 0) if A(i, j) > 0 for all i ∈ {1, . . . , n} and j ∈ {1, . . . , m}. Now, assume that n = m. Then, define the following types of matrices: vi) A is almost nonnegative if A(i, j) ≥ 0 for all i, j ∈ {1, . . . , n} such that i , j. vii) A is a Z-matrix if −A is almost nonnegative. Define the unit imaginary matrix J2n ∈ R2n×2n (or just J) by [ ] 0 In △ J2n = . −In 0 In particular,

] 0 1 J2 = . −1 0

(4.1.7)

[

(4.1.8)

Note that J2n is skew symmetric and orthogonal; that is, −1 T . = −J2n = J2n J2n

(4.1.9)

Hence, J2n is skew involutory, and J2n is a skew reflector. The following definition introduces two types of matrices of even size that are defined in terms of J. Note that F can represent either R or C. Definition 4.1.6. For A ∈ F2n×2n define the following types of matrices: i) A is Hamiltonian if J −1ATJ = −A. ii) A is symplectic if A is nonsingular and J −1ATJ = A−1. Proposition 4.1.7. Let A ∈ Fn×n. Then, the following statements hold: i) If A is either Hermitian, skew Hermitian, or unitary, then A is normal. ii) If A is either nonsingular or normal, then A is range Hermitian. iii) If A is either range Hermitian, idempotent, or tripotent, then A is group invertible. iv) If A is a reflector, then A is tripotent. v) If A is a permutation matrix, then A is orthogonal. Proof. i) is immediate. To prove ii), note that, if A is nonsingular, then R(A) = R(A∗ ) = Fn, and thus A is range Hermitian. If A is normal, then Theorem 3.5.3 implies that R(A) = R(AA∗ ) = R(A∗A) = R(A∗ ), which proves that A is range Hermitian. To prove iii), note that, if A is range Hermitian, then R(A) = R(AA∗ ) = AR(A∗ ) = AR(A) = R(A2 ), while, if A is idempotent, then R(A) = R(A2 ). If A is tripotent, then R(A) = R(A3 ) = A2 R(A) ⊆ R(A2 ) = AR(A) ⊆ R(A). Hence, R(A) = R(A2 ).  Proposition 4.1.8. Let A ∈ F2n×2n. Then, A is Hamiltonian if and only if there exist A, B, C ∈ Fn×n such that B and C are symmetric and [ ] A B . (4.1.10) A= C −AT

4.2 Matrices Related to Graphs Definition 4.2.1. Let G = (X, R) be a directed graph, where X = {x1 , . . . , xn }. Then, the following terminology is defined:

368

CHAPTER 4

i) Let {a1 , . . . , am } denote the set of directed edges in R that are not self-directed. Then, the incidence matrix B ∈ Rn×m of G is given by B(i, j) = 1 for all i ∈ {1, . . . , n} and j ∈ {1, . . . , m} such that xi is the tail of a j , B(i, j) = −1 for all i ∈ {1, . . . , n} and j ∈ {1, . . . , m} such that xi is the head of a j , and B(i, j) = 0 otherwise. ii) The adjacency matrix A ∈ Rn×n of G is given by A(i, j) = 1 for all i, j ∈ {1, . . . , n} such that (x j , xi ) ∈ R and A(i, j) = 0 for all i, j ∈ {1, . . . , n} such that (x j , xi ) < R. ∑ iii) The inbound Laplacian matrix Lin ∈ Rn×n of G is given by Lin(i,i) = nj=1, j,i A(i, j) for all i ∈ {1, . . . , n}, and Lin(i, j) = −A(i, j) for all distinct i, j ∈ {1, . . . , n}. ∑ iv) The outbound Laplacian matrix Lout ∈ Rn×n of G is given by Lout(i,i) = nj=1, j,i A( j,i) for all i ∈ {1, . . . , n}, and Lout(i, j) = −A(i, j) for all distinct i, j ∈ {1, . . . , n}. v) The indegree matrix Din ∈ Rn×n is the diagonal matrix such that Din(i,i) = indeg(xi ) for all i ∈ {1, . . . , n}. vi) The outdegree matrix Dout ∈ Rn×n is the diagonal matrix such that Dout(i,i) = outdeg(xi ) for all i ∈ {1, . . . , n}. △ vii) If G is symmetric, then the Laplacian matrix of G is given by L = Lin = Lout . △ viii) If G is symmetric, then the degree matrix D ∈ Rn×n of G is given by D = Din = Dout . ix) If G = (X, R, w) is a weighted directed graph, then the adjacency matrix A ∈ Rn×n of G is given by A(i, j) = w(x j , xi ) for all i, j ∈ {1, . . . , n} such that (x j , xi ) ∈ R, and A(i, j) = 0 for all i, j ∈ {1, . . . , n} such that (x j , xi ) < R. Note that the adjacency matrix is nonnegative, while the inbound Laplacian, outbound Laplacian, and Laplacian matrices are Z-matrices. Furthermore, note that the inbound Laplacian, outbound Laplacian, and Laplacian matrices are unaffected by the presence of self-directed edges. However, the indegree and outdegree matrices account for self-directed edges. For the directed edge (xk , xl ), the ith column of the incidence matrix B is given by coli (B) = el − ek . Finally, if G is a symmetric graph, then A and L are symmetric. Theorem 4.2.2. Let G = (X, R) be a directed graph, where X = {x1 , . . . , xn }, and let Lin , Lout , Din , Dout , and A denote the inbound Laplacian, outbound Laplacian, indegree, outdegree, and adjacency matrices of G, respectively. Then, Lin = Din − A, Lout = Dout − A.

(4.2.1) (4.2.2)

Theorem 4.2.3. Let G = (X, R) be a symmetric graph, where X = {x1 , . . . , xn }, and let A, L, D, and B denote the adjacency, Laplacian, degree, and incidence matrices of G, respectively. Then,

L = D − A.

(4.2.3)

Now, assume that G has no self-edges. Then, L = 12 BBT.

(4.2.4) △

Definition 4.2.4. Let M ∈ Fn×n and X = {x1 , . . . , xn }. Then, the directed graph of M is G(M) =

(X, R), where, for all i, j ∈ {1, . . . , n}, (x j , xi ) ∈ R if and only if M(i, j) , 0. Proposition 4.2.5. Let M ∈ Fn×n. Then, the adjacency matrix A of G(M) satisfies A = sign |M|.

(4.2.5)

4.3 Lie Algebras In the following definition, note that α and β are assumed to be real. Definition 4.3.1. Let S ⊆ Fn×n. Then, S is a Lie algebra if the following statements hold:

MATRIX CLASSES AND TRANSFORMATIONS

369

i) If A, B ∈ S and α, β ∈ R, then αA + βB ∈ S. ii) If A, B ∈ S, then [A, B] ∈ S. If F = R, then i) is equivalent to the statement that S is a subspace. However, if F = C and S contains matrices that are not real, then S is not a subspace. Proposition 4.3.2. The following sets are Lie algebras: △ i) glF (n) = Fn×n. △

ii) plC (n) = {A ∈ Cn×n : tr A ∈ R}. iii) iv) v) vi) vii) viii) ix)

△

slF (n) = {A ∈ Fn×n : tr A = 0}. △ u(n) = {A ∈ Cn×n : A is skew Hermitian}. △ su(n) = {A ∈ Cn×n : A is skew Hermitian and tr A = 0}. △ so(n) = {A ∈ Rn×n : A is skew symmetric}. △ su(n, m) = {A ∈ C(n+m)×(n+m): diag(In ,−Im )A∗ diag(In ,−Im ) = −A and tr A = 0}. △ so(n, m) = {A ∈ R(n+m)×(n+m): diag(In ,−Im )AT diag(In ,−Im ) = −A}. △ sympF (2n) = {A ∈ F2n×2n : A is Hamiltonian}. △

x) osympC (2n) = su(2n) ∩ sympC (2n). △

xi) osympR (2n) = so(2n) ∩ sympR (2n). {[ ] } A b △ n xii) affF (n) = : A ∈ glF (n), b ∈ F . 0 0 {[ ] } A b △ xiii) seC (n) = : A ∈ su(n), b ∈ Cn . 0 0 {[ ] } A b △ xiv) seR (n) = : A ∈ so(n), b ∈ Rn . 0 0 {[ ] } 0 b △ xv) transF (n) = : b ∈ Fn . 0 0

4.4 Abstract Groups Definition 4.4.1. Let S be a nonempty set, and let ϕ : S × S 7→ S. Then, (S, ϕ) is a group if the following statements hold: i) For all x, y, z ∈ S, ϕ[x, ϕ(y, z)] = ϕ[ϕ(x, y), z]. ii) There exists ı ∈ S such that, for all x ∈ S, ϕ(ı, x) = ϕ(x, ı) = x. iii) For all x ∈ S, there exists y ∈ S such that ϕ(x, y) = ϕ(y, x) = ı. Now, assume that (S, ϕ) is a group. Then, the following terminology is defined: iv) (S, ϕ) is a finite group if card(S) is finite. Otherwise, (S, ϕ) is an infinite group. v) (S, ϕ) is an Abelian group if, for all x, y ∈ S, ϕ(x, y) = ϕ(y, x). △ vi) (S, ϕ) is a cyclic group if there exists x ∈ S such that S = {ϕ(i) (x) : i ∈ Z}, where ϕ(0) (x) = ι, △ △ (1) (i+1) (i) (−i) (−i) (i) ϕ (x) = x, and, for all i ≥ 1, ϕ (x) = ϕ[ϕ (x), x] and ϕ (x) satisfies ϕ[ϕ (x), ϕ (x)] = ι. In this case, x is a generator of (S, ϕ). Now, let X ⊆ S be nonempty, let ϕX denote the restriction of ϕ to X × X, and assume that ϕX : X × X 7→ X. Then, the following terminology is defined: vii) (X, ϕX ) is a subgroup of (S, ϕ) if (X, ϕX ) is a group. viii) (X, ϕX ) is a nontrivial subgroup of (S, ϕ) if (X, ϕX ) is a subgroup of (S, ϕ) and X , {ι}. ix) (X, ϕX ) is a proper subgroup of (S, ϕ) if (X, ϕX ) is a subgroup of (S, ϕ) and X ⊂ S. x) (X, ϕX ) is a minimal subgroup of (S, ϕ) if (X, ϕX ) is a nontrivial subgroup of (S, ϕ) and there

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does not exist a nontrivial subgroup of (S, ϕ) that is also a proper subgroup of (X, ϕX ). xi) (X, ϕX ) is a maximal subgroup of (S, ϕ) if (X, ϕX ) is a proper subgroup of (S, ϕ) and there does not exist a proper subgroup of (S, ϕ) that contains (X, ϕX ) as a proper subgroup. xii) (X, ϕX ) is a normal subgroup of (S, ϕ) if (X, ϕX ) is a subgroup of (S, ϕ) and, for all x ∈ S, ϕ(x, X) = ϕ(X, x). xiii) (S, ϕ) is a simple group if card(S) ≥ 2 and the only normal subgroups of (S, ϕ) are ({ı}, ϕ{ı} ) and (S, ϕ). Proposition 4.4.2. Let (S, ϕ) be a group, and let X ⊆ S be nonempty. Then, the following statements hold: i) The following statements are equivalent: a) (X, ϕX ) is a subgroup of (S, ϕ). b) ϕX (X × X) ⊆ X, and ϕ(−1) (X) ⊆ X. c) For all x, y ∈ X, ϕ[ϕ(−1) (x), y] ∈ X. ii) If X is finite and ϕX (X × X) ⊆ X, then (X, ϕX ) is a subgroup of (S, ϕ). Proof. See [1713, pp. 4, 5] and [2336, p. 24].  The identity element ı of the group (S, ϕ) is unique, and it is contained in every subgroup (X, ϕX ) of (S, ϕ). Furthermore, for all x ∈ S, there exists a unique y ∈ S such that ϕ(x, y) = ϕ(y, x) = ı. In particular, y = ϕ(−1) (x). Note that ϕ(−1) (S) = {ϕ(−1) (x) : x ∈ S} = S. Every subgroup of an Abelian group is Abelian and normal. Proposition 4.4.3. Let (S, ϕ) be a group, and let x ∈ S. Then, exactly one of the following statements holds: i) For all distinct positive integers k and l, ϕ(k) (x) , ϕ(l) (x). ii) There exists a positive integer ℓ such that ι, ϕ(x), ϕ(2) (x), . . . , ϕ(ℓ−1) (x) of S are distinct and ϕ(ℓ) (x) = ι. Let (S, ϕ) be a group, and let x ∈ S. If i) holds, then S is infinite. By Definition 4.4.1, (S, ϕ) is cyclic if and only if there exists x ∈ S such that S = {ϕ(i) (x) : i ∈ Z}. The positive integer ℓ given by ii) of Proposition 4.4.3 is the order of x. In other words, the order ℓ of x is the smallest positive integer such that ϕ(ℓ) (x) = ι. If i) holds, then the order of x is infinite. If S is finite, then ii) holds and ℓ ≤ card(S). If there exists x ∈ S and a positive integer ℓ such that S = {ι, ϕ(x), ϕ(2) (x), . . . , ϕ(ℓ−1) (x)}, then S is a finite cyclic group. Consequently, the cyclic group (S, ϕ) with generator x is finite if and only if there exists n ≥ 1 such that ϕ(n) (x) = ι. △ △ △ △ △ △ As an example, let S = Z2 = {0, 1}, and define ϕ(0, 0) = ϕ(1, 1) = 0 and ϕ(0, 1) = ϕ(1, 0) = 1. △ △ The identity element ı of (Z2 , ϕ) is 0, and (Z2 , ϕ) is Abelian. As another example, let S = Z6 = △ {0, 1, 2, 3, 4, 5} and, for all k, l ∈ S, define ϕ(k, l) = rem6 (k + l). Then, ϕ(4, 0) = ϕ(0, 4) = 4, and ϕ(4, 5) = ϕ(5, 4) = 3. The identity element ı of (Z6 , ϕ) is 0, and (Z6 , ϕ) is Abelian. The following result is Lagrange’s theorem. Theorem 4.4.4. Let (S, ϕ) be a finite group, and let (X, ϕX ) be a subgroup of (S, ϕ). Then, card(S) = card(X) card({ϕ(x, X) : x ∈ S}).

(4.4.1)

Proof: See [79, Chapter 24], [966, pp. 89, 90], and [1560, pp. 77, 95].  Theorem 4.4.5. Let (S, ϕ) be a group, and let (X, ϕX ) be a subgroup of (S, ϕ). Then, {ϕ(x, X) :

△ △ ˜ define ϕ(A, ˜ x ∈ S} is a partition of S. Next, define S˜ = {ϕ(x, X) : x ∈ S}, and, for all A, B ∈ S, B) = {ϕ(x, y) : x ∈ A and y ∈ B}. Then, the following statements are equivalent: ˜ ϕ(A, ˜ ˜ B) ∈ S. i) For all (A, B) ∈ S˜ × S, ii) (X, ϕ) is a normal subgroup of (S, ϕ).

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˜ card(X). If these statements hold and S is finite, then card(S) = card(S) Proof. See [966, pp. 76–82].  ˜ ϕ) ˜ is a quotient group of (S, ϕ). This group If i) and ii) of Theorem 4.4.5 hold, then the group (S, ˜ is denoted by (S/X, ϕ). Definition 4.4.6. Let (S1 , ϕ1 ) and (S2 , ϕ2 ) be groups, and let Φ : S1 7→ S2 . Then, Φ is a homomorphism if, for all x, y ∈ S1 , Φ(ϕ1 (x, y)) = ϕ2 (Φ(x), Φ(y)). In addition, Φ is an isomorphism and (S1 , ϕ1 ) and (S2 , ϕ2 ) are isomorphic if Φ is one-to-one and onto. In this case, S1 ≃ S2 .

4.5 Addition Groups △

Note that (Fn×m , ϕ), where ϕ(A, B) = A+ B, is an Abelian group. An addition group is a subgroup of (Fn×m , ϕ). Definition 4.5.1. Let S ⊆ Fn×m. Then, S is an addition group if the following statements hold: i) If A ∈ S, then −A ∈ S. ii) If A, B ∈ S, then A + B ∈ S. Now, assume that S is an addition group, and let X ⊆ S. Then, X is an addition subgroup of S if X is an addition group. If S ⊆ Fn×m is an addition group, then S = −S and 0n×m ∈ S. Definition 4.5.2. Let S ⊆ Fn×m, assume that S is an addition group, and let X be an addition △ subgroup of S. Then, S˜ = {x + X : x ∈ S} is a quotient addition group of S. Definition 4.5.3. Let S1 ⊆ Fn×m and S2 ⊆ Fl×k be addition groups, and let Φ : S1 7→ S2 . Then, Φ is a homomorphism if, for all A, B ∈ S1 , Φ(A + B) = Φ(A) + Φ(B). In addition, Φ is an isomorphism and S1 and S2 are isomorphic if Φ is one-to-one and onto. In this case, S1 ≃ S2 . △ △ As an example, let S = Z ⊂ R1×1 , and define the even integers E = 2Z and the odd integers △ O = Z\E. Then, E is an addition subgroup of Z. Furthermore, −E = E, −O = O, E + E = E, O + O = E, and E + O = O. Therefore, S˜ = {E, O} is a quotient addition group of Z. Finally, Z2 ≃ Z/E, where Z2 is defined in the previous section. △ △ As another example, let S = Z, and define the addition subgroup X0 = {. . . , −18, −12, −6, 0, 6, ˜ 12, 18, . . .} = 6Z of Z. Consequently, S = {X0 , . . . , X5 } = Z/6Z is a quotient addition group of Z, △ where, for all i ∈ {0, . . . , 5}, Xi = i + 6Z. For example, X2 = {. . . , −16, −10, −4, 2, 8, 14, 20, . . .}. Hence, for all i ∈ {0, . . . , 5}, k ∈ Xi if and only if rem6 (k) = i, and k, l ∈ Z are in the same element of 6 S˜ if and only if k ≡ l. Furthermore, for all i, j ∈ {0, . . . , 6}, Xi + X j = Xrem (i+ j) . Finally, Z6 ≃ Z/6Z. 6

As a final example, let n ≥ 2, and let S ⊂ Fn be a subspace. Then, Fn is an addition group, and S △ is an addition subgroup of Fn . Furthermore, S˜ = {x + S : x ∈ Fn } is a quotient addition subgroup of n F .

4.6 Multiplication Groups Let GLF (n) denote the set of n × n nonsingular matrices with entries in F. Then, (GLF (n), ϕ), △ where ϕ(A, B) = AB, is a group. A multiplication group is a subgroup of (GLF (n), ϕ). Definition 4.6.1. Let S ⊂ Fn×n. Then, S is a multiplication group if the following statements hold: i) If A ∈ S, then A is nonsingular. ii) If A ∈ S, then A−1 ∈ S. iii) If A, B ∈ S, then AB ∈ S. Now, assume that S is a multiplication group. Then, S is an Abelian multiplication group if, for all A, B ∈ S, [A, B] = 0. Now, let X ⊆ S. Then, X is a multiplication subgroup of S if X is a multiplication group. Furthermore, X is a normal multiplication subgroup of S if X is a multiplication

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subgroup of S and, for all A ∈ S, AXA−1 = X. If S ⊂ Fn×n is a multiplication group, then S = S−1 , and In ∈ S. Definition 4.6.2. Let S ⊂ Fn×n, assume that S is a multiplication group, and let X be a normal △ multiplication subgroup of S. Then, S˜ = {xX : x ∈ S} is a quotient multiplication group of S. As an example, let S denote the nonzero real numbers. Then, (0, ∞) is a normal multiplication subgroup of S, and the corresponding quotient multiplication group of S is S˜ = {(−∞, 0), (0, ∞)}. Proposition 4.6.3. Let S ⊂ Fn×n be a multiplication group, and let A ∈ S. Then, exactly one of the following statements holds: i) For all distinct positive integers k and l, Ak , Al. ii) There exists a positive integer ℓ such that In , A, A2 , . . . , Aℓ−1 are distinct and Aℓ = In . Let S ⊂ Fn×n be a multiplication group, and let A ∈ S. If there exists A ∈ S such that S = {Ak : k ∈ Z}, then S is a cyclic multiplication group. If i) holds, then S is infinite. The positive integer ℓ given by ii) of Proposition 4.6.3 is the order of A. In other words, the order ℓ of A is the smallest positive integer such that Aℓ = In . If i) holds, then the order of A is infinite. If S is finite, then ii) holds and ℓ ≤ card(S). If there exists A ∈ Fn×n and a positive integer ℓ such that S = {In , A, A2 , . . . , Aℓ−1 }, then S is a finite cyclic multiplication group. Definition 4.6.4. Let S1 ⊂ Fn×n and S2 ⊂ Fm×m be multiplication groups, and let Φ : S1 7→ S2 . Then, Φ is a homomorphism if, for all A, B ∈ S1 , Φ(AB) = Φ(A)Φ(B). Furthermore, S1 and S2 are isomorphic and Φ is an isomorphism if Φ is one-to-one and onto. In this case, S1 ≃ S2 . Proposition 4.6.5. Let S1 ⊂ Fn×n and S2 ⊂ Fm×m be multiplication groups, and assume that S1 and S2 are isomorphic with isomorphism Φ : S1 7→ S2 . Then, Φ(In ) = Im , and, for all A ∈ S1 , Φ(A−1 ) = [Φ(A)]−1. The following result lists multiplication groups that arise in physics and engineering. For example, O(1, 3) is the Lorentz group, see [2379, p. 16] and [2423, p. 126]. The special orthogonal group SO(n) consists of the real n × n orthogonal matrices whose determinant is 1. In particular, each matrix in SO(2) and SO(3) is a rotation matrix. Furthermore, P(n), A(n), D(n), and C(n) are the n × n permutation group, alternating group, dihedral group, and cyclic group, respectively. Proposition 4.6.6. The following sets are multiplication groups: △ i) GLF (n) = {A ∈ Fn×n : det A , 0}. △ ii) PLF (n) = {A ∈ Fn×n : det A > 0}. △ iii) SLF (n) = {A ∈ Fn×n : det A = 1}. △ iv) U(n) = {A ∈ Cn×n : A is unitary}. △ v) O(n) = {A ∈ Rn×n : A is orthogonal}. △ vi) SU(n) = {A ∈ U(n): det A = 1}. △ vii) SO(n) = {A ∈ O(n): det A = 1}. △ viii) P(n) = {A ∈ Rn×n : A is a permutation matrix}. △ ix) A(n) = {A ∈ P(n): A is an even permutation matrix}. △ x) D(2) = {I2 , −I2 , Iˆ2 , −Iˆ2 }. △ ˆ ˆ ˆ 2 ˆ n−1 xi) D(n) = {In , Pn , P2n , . . . , Pn−1 n , In , In Pn , In Pn , . . . , In Pn }, where n ≥ 3. △

xii) C(n) = {In , Pn , P2n , . . . , Pn−1 n }. xiii) xiv) xv) xvi)

△

U(n, m) = {A ∈ C(n+m)×(n+m) : A∗ diag(In , −Im )A = diag(In , −Im )}. △ O(n, m) = {A ∈ R(n+m)×(n+m) : AT diag(In , −Im )A = diag(In , −Im )}. △ SU(n, m) = {A ∈ U(n, m): det A = 1}. △ SO(n, m) = {A ∈ O(n, m): det A = 1}.

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△

xvii) SympF (2n) = {A ∈ F2n×2n : A is symplectic}. △

xviii) OSympC (2n) = U(2n) ∩ SympC (2n). △

xix) OSympR (2n) = O(2n) ∩ SympR (2n). {[ ] } △ xx) Aff F (n) = A0 b1 : A ∈ GLF (n), b ∈ Fn . {[ ] } △ xxi) SEC (n) = A0 b1 : A ∈ SU(n), b ∈ Cn . {[ ] } △ xxii) SER (n) = A0 b1 : A ∈ SO(n), b ∈ Rn . {[ ] } △ xxiii) TransF (n) = 0I b1 : b ∈ Fn .

4.7 Matrix Transformations The following results use groups to define equivalence relations. Proposition 4.7.1. Let S1 ⊂ Fn×n and S2 ⊂ Fm×m be multiplication groups, and let M ⊆ Fn×m. Then, the subset of M × M defined by △

R = {(A, B) ∈ M × M: there exist S 1 ∈ S1 and S 2 ∈ S2 such that A = S 1BS 2 } is an equivalence relation on M. Proposition 4.7.2. Let S ⊂ Fn×n be a multiplication group, and let M ⊆ Fn×n. Then, the following subsets of M × M are equivalence relations: △ i) R = {(A, B) ∈ M × M: there exists S ∈ S such that A = SBS −1 }. △ ii) R = {(A, B) ∈ M × M: there exists S ∈ S such that A = SBS ∗ }. △ iii) R = {(A, B) ∈ M × M: there exists S ∈ S such that A = SBS T }. If, in addition, S is an Abelian multiplication group, then the following subset of M × M is an equivalence relation: △ iv) R = {(A, B) ∈ M × M: there exists S ∈ S such that A = SBS }. Various transformations can be employed for analyzing matrices. Propositions 4.7.1 and 4.7.2 imply that these transformations define equivalence relations. Definition 4.7.3. Let A, B ∈ Fn×m. Then, the following terminology is defined: i) A and B are left equivalent if there exists a nonsingular matrix S 1 ∈ Fn×n such that A = S 1B. ii) A and B are right equivalent if there exists a nonsingular matrix S 2 ∈ Fm×m such that A = BS 2 . iii) A and B are biequivalent if there exist nonsingular matrices S 1 ∈ Fn×n and S 2 ∈ Fm×m such that A = S 1 BS 2 . iv) A and B are unitarily left equivalent if there exists a unitary matrix S 1 ∈ Fn×n such that A = S 1B. v) A and B are unitarily right equivalent if there exists a unitary matrix S 2 ∈ Fm×m such that A = BS 2 . vi) A and B are unitarily biequivalent if there exist unitary matrices S 1 ∈ Fn×n and S 2 ∈ Fm×m such that A = S 1BS 2 . Definition 4.7.4. Let A, B ∈ Fn×n. Then, the following terminology is defined: i) A and B are similar if there exists a nonsingular matrix S ∈ Fn×n such that A = SBS −1. ii) A and B are congruent if there exists a nonsingular matrix S ∈ Fn×n such that A = SBS ∗. iii) A and B are T-congruent if there exists a nonsingular matrix S ∈ Fn×n such that A = SBS T. iv) A and B are unitarily similar if there exists a unitary matrix S ∈ Fn×n such that A = SBS ∗ = SBS −1.

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The transformations that appear in Definition 4.7.3 and Definition 4.7.4 are left equivalence, right equivalence, biequivalence, unitary left equivalence, unitary right equivalence, unitary biequivalence, similarity, congruence, T-congruence, and unitary similarity transformations, respectively. The following results summarize some matrix properties that are preserved under left equivalence, right equivalence, biequivalence, similarity, congruence, and unitary similarity. Proposition 4.7.5. Let A, B ∈ Fn×n. If the matrices A and B are similar, then the following statements hold: i) A and B are biequivalent. ii) tr A = tr B. iii) det A = det B. iv) Ak and Bk are similar for all k ≥ 1. v) Ak∗ and Bk∗ are similar for all k ≥ 1. vi) A is nonsingular if and only if B is; in this case, A−k and B−k are similar for all k ≥ 1. vii) A is (group invertible, involutory, skew involutory, idempotent, tripotent, nilpotent) if and only if B is. If A and B are congruent, then the following statements hold: viii) A and B are biequivalent. ix) A∗ and B∗ are congruent. x) A is nonsingular if and only if B is; in this case, A−1 and B−1 are congruent. xi) A is (range Hermitian, Hermitian, skew Hermitian, positive semidefinite, positive definite) if and only if B is. If A and B are unitarily similar, then the following statements hold: xii) A and B are similar. xiii) A and B are congruent. xiv) A is (range Hermitian, group invertible, normal, Hermitian, skew Hermitian, positive semidefinite, positive definite, unitary, involutory, skew involutory, idempotent, tripotent, nilpotent) if and only if B is.

4.8 Projectors, Idempotent Matrices, and Subspaces The following result shows that each subspace is associated with a unique projector. Proposition 4.8.1. Let S ⊆ Fn be a subspace. Then, there exists a unique projector A ∈ Fn×n such that S = R(A). Furthermore, x ∈ S if and only if x = Ax. Proof. See [2036, p. 386] and Fact 4.18.2.  n n×n For a subspace S ⊆ F , the projector A ∈ F given by Proposition 4.8.1 is the projector onto S. If, in addition, S′ ⊆ Fn, then AS′ is the projection of S′ into S. Let A ∈ Fn×n be a projector. Then, the complementary projector A⊥ is the projector defined by △

A⊥ = I − A. Proposition 4.8.2. Let S ⊆ F be a subspace, and let A ∈ F A⊥ is the projector onto S⊥. Furthermore, n

R(A)⊥ = N(A) = R(A⊥ ) = S⊥.

(4.8.1) n×n

be the projector onto S. Then, (4.8.2)

The following result shows that each pair of complementary subspaces is associated with a unique idempotent matrix. Proposition 4.8.3. Let S1 , S2 ⊆ Fn be complementary subspaces; that is, S1 + S2 = Fn and

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S1 ∩ S2 = {0}. Then, there exists a unique idempotent matrix A ∈ Fn×n such that R(A) = S1 and N(A) = S2 . Proof. See [423, p. 118] and [2036, p. 386].  For complementary subspaces S1 , S2 ⊆ Fn, the unique idempotent matrix A ∈ Fn×n given by Proposition 4.8.3 is the idempotent matrix onto S1 = R(A) along S2 = N(A). For an idempotent matrix A ∈ Fn×n, the complementary idempotent matrix A⊥ defined by (4.8.1) is also idempotent. Proposition 4.8.4. Let S1 , S2 ⊆ Fn be complementary subspaces, and let A ∈ Fn×n be the idempotent matrix onto S1 = R(A) along S2 = N(A). Then, R(A⊥ ) = S2 and N(A⊥ ) = S1 ; that is, the complementary idempotent matrix A⊥ is the idempotent matrix onto S2 along S1. Since, by Proposition 4.8.1 each subspace is associated with a unique projector, it follows that each pair of complementary subspaces is associated with a unique pair of projectors. Definition 4.8.5. Let A, B ∈ Fn×n , and assume that A and B are projectors. Then, A and B are complementary projectors if R(A) and R(B) are complementary subspaces. If A is a projector, then R(A) and R(A⊥ ) are orthogonally complementary subspaces. Consequently, R(A) and R(A⊥ ) are complementary subspace, and thus A and A⊥ are complementary projectors. The following result characterizes complementary pairs of projectors. Proposition 4.8.6. Let A, B ∈ Fn×n , and assume that A and B are projectors. Then, the following statements are equivalent: i) A and B are complementary projectors. ii) rank [A B] = rank A + rank B = n. Proposition 4.8.3 implies that every pair of complementary projectors can be associated with a unique idempotent matrix. In particular, for complementary projectors A, B ∈ Fn×n , Fact 8.8.14 provides an expression for the unique idempotent matrix onto R(A) along R(B). Conversely, for an idempotent matrix A ∈ Fn×n , the unique complementary projectors A, B ∈ Fn×n such that R(A) = R(A) and R(B) = N(A) are given by Fact 8.8.11. Definition 4.8.7. The index of A, denoted by ind A, is the smallest nonnegative integer k such that R(Ak ) = R(Ak+1 ). (4.8.3) Proposition 4.8.8. Let A ∈ Fn×n. Then, A is nonsingular if and only if ind A = 0. Furthermore,

A is group invertible if and only if ind A ≤ 1. Note that ind 0n×n = 1. Proposition 4.8.9. Let A ∈ Fn×n, and let k ≥ 1. Then, ind A ≤ k if and only if R(Ak ) and N(Ak ) are complementary subspaces. The following corollary of Proposition 4.8.9 shows that the null space and range of a groupinvertible matrix are complementary subspaces. Note that every idempotent matrix is group invertible. Corollary 4.8.10. Let A ∈ Fn×n. Then, A is group invertible if and only if R(A) and N(A) are complementary subspaces. Fact 4.9.6 states that the range and null space of a range-Hermitian matrix are orthogonally complementary subspaces. Furthermore, Proposition 4.1.7 states that every range-Hermitian matrix is group invertible. For a group-invertible matrix A ∈ Fn×n, the following result shows how to construct the idempotent matrix onto R(A) along N(A). This construction is based on the full-rank factorization given by Proposition 7.6.6. △ Proposition 4.8.11. Let A ∈ Fn×n, and let r = rank A. Then, A is group invertible if and only

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if there exist B ∈ Fn×r and C ∈ Fr×n such that A = BC and rank B = rank C = rank CB = r. If these △ conditions hold, then P = B(CB)−1C is the idempotent matrix onto R(A) along N(A). If, in addition, A is range Hermitian, then P is the projector onto R(A). Proof. See [2036, p. 634].  An alternative expression for the idempotent matrix onto R(A) along N(A) is given by Proposition 8.2.3.

4.9 Facts on Elementary, Group-Invertible, Range-Hermitian, Range-Disjoint, and Range-Spanning Matrices Fact 4.9.1. Let A ∈ Fn×m . Then, the following statements are equivalent:

i) A is an elementary matrix. ii) There exist x, y ∈ Fn such that A = I − xy∗ and x∗y , 1. iii) There exist x, y ∈ Fn such that A = I − xyT and xTy , 1. Fact 4.9.2. Let A ∈ Fn×n. Then, the following statements are equivalent: i) A is group invertible. ii) A∗ is group invertible. iii) AT is group invertible. iv) A is group invertible. v) R(A) = R(A2 ). vi) N(A) = N(A2 ). vii) N(A) ∩ R(A) = {0}. viii) N(A) + R(A) = Fn. ix) R(A) and N(A) are complementary subspaces. x) A and A2 are left equivalent. xi) A and A2 are right equivalent. xii) ind A ≤ 1. xiii) rank A = rank A2. xiv) rank(A − A4 ) = rank(A2 − A5 ). xv) def A = def A2. xvi) def A = amultA(0). Related: Fact 3.13.29, Corollary 4.8.10, Proposition 4.8.11, and Corollary 7.7.9. Fact 4.9.3. Let A ∈ Fn×n. Then, ind A ≤ k if and only if Ak is group invertible. Fact 4.9.4. Let A ∈ Fn×n. Then, the following statements hold: i) A is range disjoint if and only if N(A) + N(A∗ ) = Fn . ii) A is range spanning if and only if N(A) ∩ N(A∗ ) = {0}. iii) A is range Hermitian and range disjoint if and only if A = 0. iv) A is range Hermitian and range spanning if and only if A is nonsingular. v) A is range disjoint and range spanning if and only if R(A) and R(A∗ ) are complementary subspaces. vi) If A is range disjoint, then so is A2 . vii) If A2 is range spanning, then so is A. viii) A is (range disjoint, range spanning) if and only if A∗ is. Source: [250].

MATRIX CLASSES AND TRANSFORMATIONS

377

Fact 4.9.5. Let A ∈ Fn×n, and assume that AA∗ + A∗A = A + A∗ . Then, A is range Hermitian. Source: [2719]. Fact 4.9.6. Let A ∈ Fn×n. Then, the following statements are equivalent:

i) ii) iii) iv) v) vi) vii) viii) ix) x) xi) xii) xiii)

A is range Hermitian. A∗ is range Hermitian. R(A) = R(A∗ ). R(A) ⊆ R(A∗ ). R(A∗ ) ⊆ R(A). N(A) = N(A∗ ). A and A∗ are right equivalent. R(A)⊥ = N(A). N(A)⊥ = R(A). R(A) and N(A) are orthogonally complementary subspaces. rank A = rank [A A∗ ]. R(A2 ) = R(A∗ ). A is group invertible, and A2 is range Hermitian. Source: [257, 719, 2658]. Remark: Using Fact 4.18.2, Proposition 4.8.2, and Proposition 8.1.7, vi) is equivalent to A+A = I − (I − A+A) = AA+. See Fact 8.5.2, Fact 8.5.3, and Fact 8.5.8. Related: Fact 8.10.24. Fact 4.9.7. Let A, B ∈ Fn×n, and assume that A and B are range Hermitian. Then, R(A) ⊆ R(B) if and only if N(B) ⊆ N(A). Fact 4.9.8. Let A, B ∈ Fn×n, and assume that A and B are range Hermitian. Then, rank AB = rank BA. Source: [268]. Fact 4.9.9. Let A, B ∈ Fn×n, and assume that A and B are range Hermitian. Then, the following statements are equivalent: i) AB is range Hermitian. ii) R(AB) = R(A) ∩ R(B) and N(AB) = N(A) + N(B). Source: [916]. Fact 4.9.10. Let A, B ∈ Fn×n, and assume that A, B, AB, and BA are range Hermitian. Then, R(AB) = R(BA). Source: [845].

4.10 Facts on Normal, Hermitian, and Skew-Hermitian Matrices Fact 4.10.1. Let A ∈ Fn×m. Then, AAT and ATA are symmetric. Fact 4.10.2. Let α ∈ R and A ∈ Rn×n. Then, the matrix equation αA + AT = 0 has a nonzero

solution A if and only if either α = 1 or α = −1. Fact 4.10.3. Let A ∈ Fn×n, assume that A is Hermitian, and let k ≥ 1. Then, R(A) = R(Ak ) and N(A) = N(Ak ). Fact 4.10.4. Let A ∈ Rn×n. Then, the following statements hold: i) The following statements are equivalent: a) For all x ∈ Rn , xTAx = 0. b) For all x ∈ Cn , x∗Ax is imaginary. c) A is skew symmetric. ii) A is symmetric and, for all x ∈ Rn , xTAx = 0 if and only if A = 0. iii) x∗Ax = 0 for all x ∈ Cn if and only if A = 0.

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iv) x∗Ax is real for all x ∈ Cn if and only if A is symmetric. [ 1− ȷ ] [ ] 0 1 . Then, x∗Ax = 4 ȷ. Hence, “is imaginary” cannot be replaced Remark: Let x = 1+ ȷ and A = −1 0 by “= 0” in b) of i). Related: Fact 4.10.5. Fact 4.10.5. Let A ∈ Cn×n. Then, the following statements hold: i) x∗Ax = 0 for all x ∈ Cn if and only if A = 0. ii) x∗Ax is imaginary for all x ∈ Cn if and only if A is skew Hermitian. iii) x∗Ax is real for all x ∈ Cn if and only if A is Hermitian. iv) x∗Ax ≥ 0 for all x ∈ Cn if and only if A is positive semidefinite. v) Re(x∗Ax) ≥ 0 for all x ∈ Cn if and only if A + A∗ is positive semidefinite. vi) x∗Ax > 0 for all nonzero x ∈ Cn if and only if A is positive definite. [ ] 1 1 satisfies xTAx ≥ 0 but is not symmetric and thus is not positive Remark: For all x ∈ R2 , A = −1 0 [ ] 1 1 satisfies xTAx > 0 but is not symmetric semidefinite. Likewise, for all nonzero x ∈ R2 , A = −1 1 and thus is not positive definite. Hence, C cannot be replaced by R in ii) and iv). Fact 4.10.6. Let A ∈ Rn×n. Then, the following statements are equivalent: i) A is positive definite. ii) A is symmetric and, for all nonzero x ∈ Rn, xTAx > 0. iii) A is symmetric and, for all nonzero x ∈ Cn, x∗Ax > 0. Fact 4.10.7. Let A ∈ Fn×n, and assume that A is block diagonal. Then, A is (normal, Hermitian, skew Hermitian) if and only if every diagonally located block has the same property. Fact 4.10.8. Let A ∈ Cn×n. Then, the following statements hold: i) A is Hermitian if and only if ȷA is skew Hermitian. ii) A is skew Hermitian if and only if ȷA is Hermitian. iii) A is Hermitian if and only if Re A is symmetric and Im A is skew symmetric. iv) A is skew Hermitian if and only if Re A is skew symmetric and Im A is symmetric. v) If A is positive semidefinite, then Re A is positive semidefinite and Im A is skew symmetric. vi) If A is positive definite, then Re A is positive definite and Im A is skew symmetric. [ ] vii) A is symmetric if and only if A0 A0 is symmetric. [ ] viii) A is Hermitian if and only if A0 A0 is Hermitian. ] [ 0 A is skew symmetric. ix) A is symmetric if and only if −A 0 [ ] 0 A is skew Hermitian. x) A is Hermitian if and only if −A 0 [ ] 0 A = J ⊗ A, and J is a real representation of ȷ. Remark: x) is a real analogue of i) since −A 2 2 0 Fact 4.10.9. Let A ∈ Fn×n. Then, the following statements hold: i) If A is (nonsingular, range Hermitian, normal, Hermitian, skew Hermitian, unitary, positive semidefinite, positive definite, diagonal, diagonalizable over F, nilpotent), then so are A, AT , A∗ , and AA . ii) Assume that A is nonsingular. If A is (range Hermitian, normal, Hermitian, skew Hermitian, unitary, positive definite, diagonal, diagonalizable over F), then so is A−1 . iii) If A is skew Hermitian and n is odd, then AA is Hermitian. iv) If A is skew Hermitian and n is even, then AA is skew Hermitian. ∏ v) If A is diagonal, then, for all i ∈ {1, . . . , n}, (AA )(i,i) = nj=1, j,i A( j, j) . Source: Fact 3.19.9. Related: Fact 6.10.13.

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MATRIX CLASSES AND TRANSFORMATIONS △

Fact 4.10.10. Let A ∈ Fn×n, assume that A is Hermitian, define r = rank A, let λ1 , . . . , λr denote

the nonzero eigenvalues of A, let A1 ∈ F Then,

r×r

be a principal submatrix of A, and assume that det A1 , 0.

sign det A1 = sign

r ∏

λi .

i=1

Source: [2991, p. 259]. Fact 4.10.11. Let A ∈ Fn×n, assume that n is even and A is skew Hermitian, and let x ∈ Fn and α ∈ F. Then, det(A + αxx∗ ) = det A. Source: Fact 3.17.2 and Fact 4.10.9 imply that det(A + αxx∗ ) =

det A + αx∗AA x = det A. Fact 4.10.12. Let A ∈ Fn×n. Then, the following statements are equivalent: i) A is normal. ii) [A, AA∗ ] = 0. iii) [A, A∗A] = 0. iv) [A, A + A∗ ] = 0. v) [A, A − A∗ ] = 0. vi) [A + A∗ , A − A∗ ] = 0. vii) [A, [A, A∗ ]] = 0. viii) tr (AA∗ )2 = tr A2A2∗. ix) (AA∗ )2 = A2A2∗. x) There exists k ≥ 1 such that tr (AA∗ )k = tr AkAk∗. xi) There exist k, l ∈ P such that tr (AA∗ )kl = tr (AkAk∗ )l . xii) A is range Hermitian, and AA∗A2 = A2A∗A. xiii) AA∗ − A∗A is positive semidefinite. xiv) [ 21 (A + A∗ )]2 + [ 21ȷ (A − A∗ )]2 = AA∗ . xv) [ 12 (A + A∗ )]2 + [ 21ȷ (A − A∗ )]2 = A∗A.

There exists a unitary matrix S ∈ Fn×n such that A∗ = AS. There exists a unitary matrix S ∈ Fn×n such that A∗ = SA. For all p ∈ F[s], p(A) is normal. There exist µ1 , . . . , µr ∈ C and projectors A1 , . . . , Ar ∈ Fn×n such that, for all distinct i, j ∈ ∑ ∑ {1, . . . , r}, Ai A j = 0 and such that ri=1 Ai = I and A = ni=1 µi Ai . xx) If S ⊆ Fn is a subspace and AS ⊆ S, then AS⊥ ⊆ S⊥ . Source: [240, 719, 987, 990, 1242, 2476], [2238, pp. 345, 346], and [2991, pp. 294, 295]. Related: Fact 4.13.6, Fact 7.15.16, Fact 7.17.5, Fact 8.6.1, Fact 8.10.17, Fact 10.10.31, Fact 10.13.17, Fact 10.21.10, Fact 15.16.4, and Fact 15.16.12. Fact 4.10.13. Let A ∈ Fn×n. Then, the following statements are equivalent: i) A is Hermitian. ii) A2 = A∗A. iii) A2 = AA∗. iv) A∗2 = A∗A. v) A∗2 = AA∗. vi) There exists α ∈ F such that A2 = αA∗A + (1 − α)AA∗. vii) There exists α ∈ F such that A∗2 = αA∗A + (1 − α)AA∗. xvi) xvii) xviii) xix)

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viii) tr A2 = tr A∗A. ix) tr A2 = tr AA∗. x) tr A∗2 = tr A∗A. xi) tr A∗2 = tr AA∗. If, in addition, F = R, then the following statement is equivalent to i)–xi): xii) There exist α, β ∈ R such that αA2 + (1 − α)AT2 = βATA + (1 − β)AAT . Source: To prove viii) =⇒ i), use the Schur decomposition Theorem 7.5.1 to replace A with D + S, where D is diagonal and S is strictly upper triangular. Then, tr D∗D+tr S ∗S = tr D2 ≤ tr D∗D. Hence, S = 0, and thus tr D∗D = tr D2, which implies that D is real. See [240, 1718] and [2991, pp. 254, 255]. Remark: Fact 11.13.2 states that, for all A ∈ Fn×n, | tr A2 | ≤ tr A∗A. Related: Fact 4.17.4. Fact 4.10.14. Let A ∈ Fn×n, let α, β ∈ F, and assume that α , 0. Then, the following statements are equivalent: i) A is normal. ii) αA + βI is normal. Now, assume, in addition, that α, β ∈ R. Then, the following statements are equivalent: iii) A is Hermitian. iv) αA + βI is Hermitian. Remark: The function f (A) = αA + βI is an affine mapping. Fact 4.10.15. Let A ∈ Rn×n, and assume that A is skew symmetric. Then, the following statements hold: i) det A ≥ 0, and −A2 is positive semidefinite. ii) If n is odd, then det A = 0. iii) If α is a real number, then det(I + αA2 ) ≥ 0. iv) If α > 0, then det(αI + A) > 0. Source: iv) is given in [1158, p. 69]. Related: Fact 3.16.5 and Fact 4.13.20. Fact 4.10.16. Let A ∈ Fn×n, and assume that A is skew Hermitian. If n is even, then det A ≥ 0. If n is odd, then det A is imaginary. Source: The first statement follows from Proposition 7.7.21. △ Fact 4.10.17. Let x, y ∈ Fn, and define A = [x y]. Then, xy∗ − yx∗ = AJ2 A∗. Furthermore, ∗ ∗ ∗ ∗ xy − yx is skew Hermitian, and rank(xy − yx ) ∈ {0, 2}. Fact 4.10.18. Let x, y ∈ Fn. Then, the following statements hold: i) xyT is idempotent if and only if either xyT = 0 or xTy = 1. ii) xyT is Hermitian if and only if there exists α ∈ R such that either y = αx or x = αy. △ Fact 4.10.19. Let x, y ∈ Fn, and define A = I − xyT. Then, the following statements hold: i) det A = 1 − xTy. ii) A is nonsingular if and only if xTy , 1. iii) A is nonsingular if and only if A is elementary. iv) rank A = n − 1 if and only if xTy = 1. v) A is Hermitian if and only if there exists α ∈ R such that either y = αx or x = αy. vi) A is positive semidefinite if and only if A is Hermitian and xTy ≤ 1. vii) A is positive definite if and only if A is Hermitian and xTy < 1. viii) A is idempotent if and only if either xyT = 0 or xTy = 1. ix) A is orthogonal if and only if either x = 0 or y = 21 yTyx. x) A is involutory if and only if xTy = 2.

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A is a projector if and only if either y = 0 or x = x∗xy. A is a reflector if and only if either y = 0 or 2x = x∗xy. A is an elementary projector if and only if x , 0 and y = (x∗x)−1 x. A is an elementary reflector if and only if x , 0 and y = 2(x∗x)−1 x. Related: Fact 4.17.11. Fact 4.10.20. Let x, y ∈ Fn satisfy xTy , 1. Then, I − xyT is nonsingular, and xi) xii) xiii) xiv)

(I − xyT )−1 = I −

1 xyT. xTy − 1

Remark: The inverse of an elementary matrix is an elementary matrix. Fact 4.10.21. Let A ∈ Fn×n, and assume that A is Hermitian. Then, det A is real. Fact 4.10.22. Let A ∈ Fn×n, and assume that A is Hermitian. Then,

(tr A)2 ≤ (rank A) tr A2. Furthermore, equality holds if and only if there exists α ∈ R such that A2 = αA. Related: Fact 7.12.13 and Fact 11.15.14. Fact 4.10.23. Let A ∈ Rn×n, and assume that A is skew symmetric. Then, tr A = 0. If, in addition, B ∈ Rn×n is symmetric, then tr AB = 0. Fact 4.10.24. Let A ∈ Fn×n, and assume that A is skew Hermitian. Then, Re tr A = 0. If, in addition, B ∈ Fn×n is Hermitian, then Re tr AB = 0. Fact 4.10.25. Let A ∈ Fn×m. Then, A∗A is positive semidefinite. Furthermore, A∗A is positive definite if and only if A is left invertible. If these conditions hold, then AL ∈ Fm×n defined by △ AL = (A∗A)−1A∗ is a left inverse of A. Related: Fact 3.18.5, Fact 4.10.26, and Fact 4.17.8. Fact 4.10.26. Let A ∈ Fn×m. Then, AA∗ is positive semidefinite. Furthermore, AA∗ is positive definite if and only if A is right invertible. If these conditions hold, then AR ∈ Fm×n defined by △ AR = A∗(AA∗ )−1 is a right inverse of A. Related: Fact 3.18.5, Fact 4.10.25, and Fact 4.17.8. [ ∗] [ ] 0 A∗ is skew Fact 4.10.27. Let A ∈ Fn×m. Then, A∗A, AA∗, and A0 A0 are Hermitian, and −A 0 Hermitian. Fact 4.10.28. Let A ∈ Fn×n. Then, A + A∗ , ȷ(A − A∗ ), and 21ȷ (A − A∗ ) are Hermitian, and A − A∗ is skew Hermitian. Furthermore, A = 12 (A + A∗ ) + 12 (A − A∗ ) = 12 (A + A∗ ) + ȷ[ 21ȷ (A − A∗ )], [ 12 (A + A∗ )]2 + [ 21ȷ (A − A∗ )]2 = 12 (AA∗ + A∗A),

2[A, A∗ ] = [A − A∗ , A + A∗ ].

Related: Fact 7.20.2 and Fact 7.20.3. Fact 4.10.29. Let A, B ∈ Fn×n , assume that A and B are Hermitian, and assume that A + B is nonsingular. Then, A(A + B)−1B is Hermitian. Source: If A and B are nonsingular, then A(A +

B)−1B = B(A + B)−1A = A−1 + B−1 . In the case where either A or B is singular, use a continuity △ argument. Alternatively, define C = A + B. Then, A(A + B)−1B = AC −1 (C − A) = A − AC −1A = −1 −1 A − (C − B)C A = BC A = [A(A + B)−1B]∗ . Fact 4.10.30. Let A, B ∈ Fn×n , assume that A and B are Hermitian, and assume that either A or B is either positive definite or negative definite. Then, A + ȷB, A − ȷB, A + BA−1B, B + AB−1A, and [ A B] −B A are nonsingular. Source: Consider the case where B is either positive definite or negative definite. Let x ∈ Fn be nonzero, and assume that (A + ȷB)x = 0. Hence, x∗ (A + ȷB)x = 0, and thus − ȷ = x∗Ax/x∗Bx ∈ R, which is a contradiction. The remaining results follow from Fact 3.24.7. Fact 4.10.31. Let A, B ∈ Fn×n , and assume that A and B are Hermitian. Then, rank AB = rank BA,

R(A) + R(B) = R([A B]) = R(A2 + B2 ) = span[R(A) ∪ R(B)],

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dim[R(A) + R(B)] = rank [A B] = rank(A2 + B2 ) = rank

[ ] A . B

Furthermore, the following statements are equivalent: i) rank [A B] = n. [ ] A ii) def = 0. B iii) N(A) ∩ N(B) = {0}. Source: Fact 3.14.8 and Fact 3.14.9. Fact 4.10.32. Let A, B ∈ Cn×n, assume that A is either Hermitian or skew Hermitian, and assume that B is either Hermitian or skew Hermitian. Then, rank AB = rank BA. Source: AB and (AB)∗ = BA have the same singular values. See Fact 7.12.22. Related: Fact 3.13.33. Fact 4.10.33. Let A, B ∈ R3×3, and assume that A and B are skew symmetric. Then, tr AB3 = 12 (tr AB)(tr B2 ),

tr A3B3 = 14 (tr A2 )(tr AB)(tr B2 ) + 13 (tr A3 )(tr B3 ).

Source: [181]. Fact 4.10.34. Let A ∈ Fn×n and k ≥ 1. Then, the following statements hold:

i) If A is (normal, Hermitian, unitary, involutory, positive semidefinite, positive definite, idempotent, nilpotent), then so is Ak. ii) If A is (skew Hermitian, skew involutory), then so is A2k+1. iii) If A is Hermitian, then A2k is positive semidefinite. iv) If A is tripotent, then so is A3k. Fact 4.10.35. Let a, b, c, d, e, f ∈ R, and define the skew-symmetric matrix A ∈ R4×4 given by   a b c   0   −a 0 d e  △  . A =  0 f   −b −d −c −e − f 0 Then,

det A = (a f − be + cd)2.

Source: [2418, p. 63]. Related: Fact 6.8.16 and Fact 6.10.8. Fact 4.10.36. Let A ∈ Rn×n , and assume that A is skew symmetric, where every entry of A

above the diagonal is 1. If n is odd, then rank A = n − 1. If n is even, then det A = 1. Fact 4.10.37. Let A ∈ R2n×2n, and assume that A is skew symmetric. Then, there exists a nonsingular matrix S ∈ R2n×2n such that S TAS = J2n . Source: [218, p. 231]. Fact 4.10.38. Let A ∈ Fn×n . Then, the following statements are equivalent: i) A is reverse Hermitian. ˆ is Hermitian. ii) IA iii) AIˆ is Hermitian. Furthermore, the following statements are equivalent: iv) A is reverse symmetric. ˆ is symmetric. v) IA vi) AIˆ is symmetric. △ Fact 4.10.39. Let A ∈ Fn×m , assume that A is nonzero, define r = rank A, and let x ∈ Fm . Then, Ax = 0 if and only if there exist y1 , . . . , yr ∈ Fn and skew-Hermitian matrices S 1 , . . . , S r ∈ Fm×m ∑ such that A = ri=1 yi x∗ S i . Source: [2261]. Related: Fact 4.11.4.

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MATRIX CLASSES AND TRANSFORMATIONS

Fact 4.10.40. Let A ∈ Rn×n be skew symmetric, let α1 , . . . , αk be nonnegative numbers, and △

define β =

∏k

i=1

αi . Then,

[det(A + β1/k I)]k ≤

k ∏

det(A + αi I).

i=1

Now, assume that either α1 , . . . , αk ∈ [0, 1] or α1 , . . . , αk ∈ [1, ∞). Then, k ∏

det(A + αi I) ≤ [det(A + I)]k−1 det(A + βI).

i=1

Source: [1137].

4.11 Facts on Linear Interpolation Fact 4.11.1. Let y ∈ Fn and x ∈ Fm. Then, there exists A ∈ Fn×m such that y = Ax if and only if

either y = 0 or x , 0. If y = 0, then one such matrix is A = 0. If x , 0, then one such matrix is A = (x∗x)−1yx∗. Finally, if x , 0, then A ∈ Fn×m satisfies y = Ax if and only if there exists B ∈ Fn×m such that A = (x∗x)−1yx∗ + B(x∗xIm − xx∗ ).

Source: [2704]. Remark: This is a linear interpolation problem. See [1549, 2233]. Fact 4.11.2. Let x, y ∈ Fn, and assume that x , 0. Then, there exists a Hermitian matrix

A ∈ Fn×n such that y = Ax if and only if x∗y is real. One such matrix is A = (x∗x)−1[yx∗ + xy∗ − x∗yI].

Now, assume that x and y are real. Then, A = (xTx)−1[yxT + xyT − xTyI] satisfies σmax (A) =

∥y∥2 = min{σmax (B): B ∈ Rn×n is symmetric and y = Bx}. ∥x∥2

Source: The last statement is given in [2473]. Fact 4.11.3. Let x, y ∈ Fn, and assume that x , 0. Then, there exists a positive-definite matrix

A ∈ Fn×n such that y = Ax if and only if x∗y is real and positive. One such matrix is A = I + (x∗y)−1yy∗ − (x∗x)−1xx∗.

Source: To show that A is positive definite, note that the elementary projector I − (x∗x)−1xx∗ is

positive semidefinite and rank[I − (x∗x)−1 xx∗ ] = n − 1. Since (x∗y)−1yy∗ is positive semidefinite, it follows that N(A) ⊆ N[I − (x∗x)−1xx∗ ]. Next, since x∗y > 0, it follows that y∗x , 0 and y , 0, and thus x < N(A). Consequently, N(A) ⊂ N[I − (x∗x)−1xx∗ ] (note proper inclusion), and thus def A < 1. Hence, A is nonsingular. Fact 4.11.4. Let x, y ∈ Fn, where x , 0. Then, the following statements are equivalent: i) x∗ y = 0. ii) There exists a skew-Hermitian matrix A ∈ Fn×n such that y = Ax. iii) Re x∗ y = 0. If i) holds, then one such matrix satisfying ii) is A = (x∗x)−1(yx∗ − xy∗ ). Now, assume that F = R. Then, the following statements are equivalent: iv) xTy = 0. v) There exists a skew-symmetric matrix A ∈ Rn×n such that y = Ax. If these statements hold, then one such matrix satisfying v) is A = (xTx)−1(yxT − xyT ). Source: [1872]. Related: Fact 4.10.39.

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Fact 4.11.5. Let x, y ∈ Fn. Then, there exists a unitary matrix A ∈ Fn×n such that Ax = y if and

only if x∗x = y∗y. Now, let F = R. Then, one such matrix is given by a product of n plane rotations given by Fact 7.17.18. Another matrix is given by the product of elementary reflectors given by Fact 7.17.17. For n = 3, one such matrix is given by Fact 4.14.9, while another is given by the exponential of a skew-symmetric matrix given by Fact 15.12.7. Related: Fact 11.17.17. Problem: Construct A in the case where F = C. Fact 4.11.6. Let x, y ∈ Rn, where x(1) ≥ · · · ≥ x(n) and y(1) ≥ · · · ≥ y(n) . Then, the following statements are equivalent: s

i) x ≺ y. ii) x ∈ conv {Ay : A ∈ P(n)}. iii) There exists a doubly stochastic matrix A ∈ Rn×n such that y = Ax. Source: [449, p. 33], [1448, p. 197], [1969, p. 22], and [1971, p. 33]. Remark: The equivalence of i) and ii) is due to R. Rado. See [1969, p. 113] and [1971, p. 162]. The equivalence of i) and iii) is the Hardy-Littlewood-Polya theorem. Related: Fact 4.13.1 and Fact 10.21.11.

4.12 Facts on the Cross Product Fact 4.12.1. Let x, y, z, v, w ∈ R3, and define the cross-product matrix K(x) ∈ R3×3 by

  0  △  K(x) =  x(3)  −x(2)

−x(3) 0 x(1)

 x(2)   −x(1)  .  0

Then, the following statements hold: i) x × x = K(x)x = 0, xTK(x) = 0, K T(x) = −K(x), K 2(x) = xxT − (xT x)I. ii) tr K T(x)K(x) = − tr K 2 (x) = 2xT x, K 3(x) = −(xT x)K(x). iii) [I − K(x)]−1 = I + 1+x1 Tx [K(x) + K 2 (x)]. iv) [I + 12 K(x)][I − 12 K(x)]−1 = I + v) [I − K(x)][I + K(x)]−1 =

4 [K(x) 4+xTx

1 [(1 1+xTx

+ 12 K 2 (x)].

− xTx)I + 2xxT − 2K(x)].

△

Define H(x) = 21 [ 21 (1 − xTx)I + xxT + K(x)]. Then, H(x)H T(x) = For all α, β ∈ R, K(αx + βy) = αK(x) + βK(y). x × y = −(y × x) = K(x)y = −K(y)x = K T(y)x. If x × y , 0, then N[(x × y)T ] = {x × y}⊥ = R([x y]). K(x × y) = K[K(x)y] = [K(x), K(y)]. [ T] xi) K(x × y) = yxT − xyT = [x y] −y = −[x y]J2 [x y]T. xT

vi) vii) viii) ix) x)

xii) xiii) xiv) xv) xvi) xvii) xviii) xix) xx) xxi)

(x × y) × x = [(xTx)I − xxT ]y. K[(x × y) × x] = (xTx)K(y) − (xTy)K(x). (x × y)T(x × y) = det [x y x × y]. (x × y)Tz = xT(y × z) = det [x y z]. x × (y × z) = K(x)K(y)z = −K(x)K(z)y = (xTz)y − (xTy)z. (x × y) × z = −K(z)K(x)y = K(z)K(y)x = (xTz)y − (yTz)x. x × (y × z) + y × (z × x) + z × (x × y) = 0. K[(x × y) × z] = (xTz)K(y) − (yTz)K(x). K[x × (y × z)] = (xTz)K(y) − (xTy)K(z). (x × y)T (x × y) = xTxyTy − (xTy)2 .

1 16 (1

+ xTx)2 I.

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MATRIX CLASSES AND TRANSFORMATIONS

K(x)K(y) = [K(y)K(x)]T = yxT − (xTy)I. tr K(x)K(y) = −2xTy. K(x)K(y)K(x) = −(xTy)K(x). K 2 (x)K(y) + K(y)K 2 (x) = −(xTx)K(y) − (xTy)K(x). K 2 (x)K 2 (y) − K 2 (y)K 2 (x) = −(xTy)K(x × y). K(z)K(y)K(x) − K(y)K(z)K(x) = zxTK(y) − yxTK(z). K(x)K(y)K(z) + K(z)K(y)K(x) + (xTy)K(z) = K(x)K(z)K(y) + K(y)K(z)K(x) + (xTz)K(y). ∥x × y∥2 = ∥x∥2 ∥x∥2 sin θ, where θ is the angle between x and y. 2xxTK(y) + (xTy)K(x) = (x × y)xT + x(x × y)T + (xTx)K(y). If ∥x∥2 = ∥y∥2 = ∥z∥2 = 1, then 1 + 2(xTy)(yTz)(zTx) = [xT(y × z)]2 + (xTy)2 + (yTz)2 + (zTx)2. K(x)K(z)[(xTw)y − (xTy)w] = K(x)K(w)(xTz)y. xzTK(y) + yxTK(z) + zyTK(x) = −(det [x y z])I. [ T ] x z xTw xxxiv) (x × y)T(z × w) = xTzyTw − xTwyTz = det T . y z yTw

xxii) xxiii) xxiv) xxv) xxvi) xxvii) xxviii) xxix) xxx) xxxi) xxxii) xxxiii)

xxxv) xxxvi) xxxvii) xxxviii) xxxix)

(x × y) × (z × w) = [xT (y × w)]z − [xT(y × z)]w = [xT(z × w)]y − [yT(z × w)]x. (x × y) × (x × z) = [(x × y)Tz]x = [xT(y × z)]x. x × [y × (z × w)] = (yTw)(x × z) − (yTz)(x × w). x × [y × (y × x)] = y × [x × (y × x)] = (yTx)(x × y). Let A ∈ R3×3. Then, (tr A)K(x) = K(Ax) + ATK(x) + K(x)A, ATK(x)A = K(AA x), ATK(Ax)A = (det A)K(x),

K(Ax)A = AATK(x),

Ax × Ay = AAT (x × y), AT(Ax × Ay) = (det A)(x × y),

K(Ax × Ay) = AK(x × y)AT = K[AAT (x × y)]. xl) Let A ∈ R3×3, and assume that A is orthogonal. Then, K(Ax) = (det A)AK(x)AT ,

Ax × Ay = (det A)A(x × y).

xli) Let A ∈ R3×3, and assume that A is orthogonal and det A = 1. Then, K(Ax) = AK(x)AT ,

Ax × Ay = A(x × y).

xlii) K(x)A = xxT , [x y z]A = [y × z z × x x × y]T .    K(x) y  xliii) det  T  = (xTy)2 . −y 0  A    K(x) y   K(y) x   = −(xTy)   . xliv)  T −y 0 −xT 0 xlv) If xTy , 0, then

   −1  K(y) x   K(x) y  −1  .  =   T xTy  −xT 0  −y 0

xlvi) Let α and β be real numbers, and assume that either α , 0 or β∥x∥22 , 1. Then, [I + αK(x) + βK 2 (x)]−1 = I −

α2 + β2 ∥x∥22 − β α K(x) + K 2 (x). α2 ∥x∥22 + (β∥x∥22 − 1)2 α2 ∥x∥22 + (β∥x∥22 − 1)2

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CHAPTER 4

xlvii) [(x × y)T v]z × w + [(y × z)T v]x × w + [(z × x)T v]y × w = [(x × y)T z]v × w. △ xlviii) Let S be a triangle with vertices x, y, z ∈ R3 and sides given by the vectors a = x − y, △ △ △ △ b = y − z, and c = z − x. Then, the vectors a′ = a × (b × c), b′ = b × (c × a), and △ c′ = c × (a × b) are the sides of a triangle S′ that is similar to S. △ xlix) Let x, y, z ∈ R3 be linearly independent, and define w = x × y + y × z + z × x. Then, w is perpendicular to the plane passing through the points x, y, z. (√ )2 ( √ )2 ȷ ȷ ∥x∥2 ∥x∥2 1 1 T T √ √ √ 3 xx √ 3 xx . l) K(x) = = ȷ 2 I + 2∥x∥ K(x) − 2 I − 2∥x∥ K(x) − 2∥x∥2

2

li) lii) liii) liv) lv) lvi) lvii)

2∥x∥2

2

∥x × y∥2 = ∥K(x)y∥2 = ∥K(y)x∥2 = [∥x∥22 ∥y∥22 − (xTy)2 ]1/2 . ∥x×(y−z)∥22 +∥y×(z− x)∥22 +∥z×(x−y)∥22 = ∥x×y∥22 +∥y×z∥22 +∥z× x∥22 +∥x×y+y×z+z× x∥22 . ∥x × y∥22 I = ∥x∥22 yyT + ∥y∥22 xxT − (xT y)(xyT + yxT ) + (x × y)(x × y)T . 2n n T n−1 T T Let n ≥ 1. Then, K (x) = (−1) (x x) [(x x)I − xx ]. Let n ≥ 0. Then, K 2n+1 (x) = (−1)n (xTx)nK(x). Let n ≥ 0. Then, σmax [K n (x)] = ∥x∥n2 . Let A ∈ R3×3 , and assume that A is positive definite. Then,

∥K(x)Ax∥2 ≤ [λ2max (A) − λ2min (A)]1/2 ∥x∥22 . lviii) mspec[K(x)] = {0, ȷ∥x∥2 , − ȷ∥x∥2 }ms , and x is an eigenvector of K(x) associated with 0. √ lix) Assume that x , 0, xTy = 0, and ∥y∥2 = 2/2. Then,    0 0 0    0  S ∗ , K(x) = S  0 ȷ∥x∥2   0 0 − ȷ∥x∥2 where

△

S =

[

1 ∥x∥2

x

y−

ȷ ∥x∥2 K(x)y

Furthermore, S is a unitary matrix. lx) σ[K(x)] = [∥x∥2 ∥x∥2 0]T . lxi) Assume that x , 0, ∥y∥2 = 1, and xTy = 0. Then,   ∥x∥2 0  ∥x∥2 K(x) = U  0  0 0 where

△

U=

[

1 ∥x∥2 K(x)y

−y

1 ∥x∥2

] x ,

y+

ȷ ∥x∥2 K(x)y

]

.

 0   0  V T ,  0

[ △ V= y

1 ∥x∥2 K(x)y

1 ∥x∥2

] x .

Furthermore, U and V are orthogonal matrices. lxii) Let A ∈ R3×3 , and assume that A is symmetric. Then, tr AK(x) = 0. lxiii) Let A ∈ R3×3 , and assume that A − AT = K(y). Then, tr AK(x) = −xTy. ∑ ∑ lxiv) ∥x × y∥22 = 21 nj=1 ni=1 (x(i) y( j) − x( j) y(i) )2 . Source: iii), iv), and xxiv)–xxvi) are given in [1506, p. 363]; v) is given in [1961]; vi) is given in [2740]; xxi) is equivalent to sin2 θ + cos2 θ = 1; xxvii) and xxviii) are given in [1158, p. 203]; xxix) arises from quaternion multiplication |q3 | = |q2 ||q1 |, see Fact 4.14.8; xxxi) is due to N. Crasta; xxxii) follows from [1200, 1.10-7, p. 58]; xxxiii) is given in [2724]; xxxv) implies xxxvi); xxxix) is given in [1961]; xlii) is given in [1961, 2701]; xliii)–xlv) are given in [2723], see [908, 1028, 1506, 2181, 2435, 2596, 2711]; xlvii) is given in [1509]; xlviii) and xlix) are given in [1158, p. 203]; l) is given in [2698]; lii) is given in [1860, pp. 33, 216, 217]; lxiv) is given in [968, p. 113]. Remark: Cross

387

MATRIX CLASSES AND TRANSFORMATIONS

products of complex vectors are considered in [1253]. Remark: xviii) is the Jacobi identity, see Fact 3.23.3. Remark: For θ ∈ (0, π), xxx) gives twice the area of the triangle with vertices 0, x, and y. See Fact 5.2.6. Credit: The Schur decomposition in lix) and the singular value decomposition in lxi) are due to A. H. J. de Ruiter. Related: Fact 6.9.18, Fact 8.9.18, Fact 15.12.6, and Fact 15.12.11. △ Fact 4.12.2. Let x1 , . . . , xn−1 , y ∈ Rn , define M = [x1 · · · xn−1 ] ∈ Rn×(n−1) , for all i ∈ {1, . . . , n}, △ define αi = det M[i,·] , and define   α1     −α 2   △  α3  ∈ Rn . x1 × · · · × xn−1 =    ..   .   n+1 (−1) αn Then,

(x1 × · · · × xn−1 )T y = det [y x1 x2 · · · xn−1 ].

In addition, the following statements hold: i) For all i ∈ {1, . . . , n − 1}, (x1 × · · · × xn−1 )T xi = 0. ii) x1 × · · · × xn−1 = 0 if and only if x1 , . . . , xn−1 are linearly independent. iii) det [x1 × · · · × xn−1 x1 x2 · · · xn−1 ] = ∥x1 × · · · × xn−1 ∥22 . iv) ∥x1 × · · · × xn−1 ∥2 is the (n − 1)-dimensional volume of the parallellotope generated by x1 , . . . , xn−1 . Source: [774, 908]. Remark: An extension of the cross product to higher dimensions is given by the exterior product in Clifford algebras. See Fact 11.8.13 and [784, 908, 925, 1178, 1268, 1375, 1376, 1759, 1892, 1977, 2265]. A cross product on R7 is defined in [1035, pp. 297–299]. Remark: For n = 3, this definition coincides with the usual cross product. Fact 4.12.3. Let A ∈ R3×3, assume that A is orthogonal, let B ∈ C3×3, and assume that B is symmetric. Then, 3 ∑ (Aei ) × (BAei ) = 0. i=1

Source: For i = 1, 2, 3, multiply by ei A . Fact 4.12.4. Let α1 , α2 , α3 be distinct positive numbers, let A ∈ R3×3, assume that A is orthogT T

onal, and assume that

3 ∑

αi ei × Aei = 0.

i=1

Then, A ∈ {I, diag(1, −1, −1), diag(−1, 1, −1), diag(−1, −1, 1)}. Remark: This result characterizes equilibria for a dynamical system on SO(3). See [681].

4.13 Facts on Inner, Unitary, and Shifted-Unitary Matrices Fact 4.13.1. Let A ∈ Rn×n . Then, the following statements are equivalent:

i) A is a doubly stochastic matrix. ii) A ∈ conv P(n). If these statements hold, then A is a convex combination of not more than n2 − 2n + 2 permutation matrices. Source: [1448, p. 527]. Related: Fact 4.11.6 and Fact 6.11.11. Credit: G. Birkhoff. Fact 4.13.2. Let S ⊆ Fn, assume that S is a subspace, let A ∈ Fn×n , and assume that A is unitary. Then, (AS)⊥ = AS⊥ . Source: Fact 3.18.19.

388

CHAPTER 4

Fact 4.13.3. Let S1 , S2 ⊆ Fn, assume that S1 and S2 are subspaces, and assume that dim S1 ≤

dim S2 . Then, there exists a unitary matrix A ∈ Fn×n such that AS1 ⊆ S2 . Fact 4.13.4. Let S1 , S2 ⊆ Fn, assume that S1 and S2 are subspaces, and assume that dim S1 + dim S2 ≤ n. Then, there exists a unitary matrix A ∈ Fn×n such that AS1 ⊆ S⊥2 . Source: Fact 4.13.3. Fact 4.13.5. Let A ∈ Fn×n, and assume that A is unitary. Then, the following statements hold: i) A = A−∗. −1

∗

ii) AT = A = A . −∗ iii) A = A−T = A . iv) A∗ = A−1. Fact 4.13.6. Let A ∈ Fn×n, and assume that A is nonsingular. Then, the following statements are equivalent: i) A is normal. ii) A = A∗AA−∗ . iii) A−1A∗ is unitary. iv) [A, A∗ ] = 0. v) [A, A−∗ ] = 0. vi) [A−1, A−∗ ] = 0. vii) [A, A−1A∗ ] = 0. Source: [1242]. Related: Fact 4.10.12, Fact 7.17.5, Fact 8.6.1, and Fact 8.10.17. Fact 4.13.7. Let A ∈ Fn×m. If A is (left inner, right inner), then A is (left invertible, right invertible) and A∗ is a (left inverse, right inverse) of A. Fact 4.13.8. Let A ∈ Fn×m. If A is (left inner, right inner), then (AA∗ , A∗A) is idempotent. Fact 4.13.9. Let x, y ∈ Fn, let A ∈ Fn×n, and assume that A is unitary. Then, x∗y = 0 if and only if (Ax)∗Ay = 0. Fact 4.13.10. Let A ∈ Fn×n, and assume that A is block diagonal. Then, A is (unitary, shifted unitary) if and only if every diagonally located block has the same property. [ ] Fact 4.13.11. Let A ∈ Fn×n, and assume that A is unitary. Then, √1 AA −A A is unitary. 2

Fact 4.13.12. Let A, B ∈ Rn×n. Then, A + ȷB is (Hermitian, skew Hermitian, unitary) if and only ] if −B AB is (symmetric, skew symmetric, orthogonal). Related: Fact 3.24.7. Fact 4.13.13. Let A ∈ Fn×n, and assume that A is unitary. Then,

[A

|Re tr A| ≤ n,

|Im tr A| ≤ n,

| tr A| ≤ n.

Remark: The third inequality does not follow from the first two inequalities. Fact 4.13.14. Let A ∈ Rn×n, and assume that A is orthogonal. Then, −1n×n ≤≤ A ≤≤ 1n×n ; that

is, for all i, j ∈ {1, . . . , n}, |A(i, j) | ≤ 1. Hence, | tr A| ≤ n. Furthermore, the following statements are equivalent: i) A = I. ii) I ⊙ A = I. iii) tr A = n. Finally, if n is odd and det A = 1, then 2 − n ≤ tr A ≤ n. Related: Fact 4.13.15. Fact 4.13.15. Let A ∈ Rn×n, assume that A is orthogonal, let B ∈ Rn×n, and assume that B is diagonal and positive definite. Then, −B1n×n ≤≤ BA ≤≤ B1n×n ,

− tr B ≤ tr BA ≤ tr B.

389

MATRIX CLASSES AND TRANSFORMATIONS

Furthermore, the following statements are equivalent: i) BA = B. ii) I ⊙ (BA) = B. iii) tr BA = tr B. Related: Fact 4.13.14. Fact 4.13.16. Let x ∈ Cn, where n ≥ 2. Then, the following statements are equivalent:  A(1,1)    n×n i) There exists a unitary matrix A ∈ C such that x =  ...  . A(n,n) ∑ ii) For all j ∈ {1, . . . , n}, |x( j) | ≤ 1 and 2(1 − |x( j) |) + ni=1 |x(i) | ≤ n. Source: [2731]. Remark: This result is equivalent to the Schur-Horn theorem given by Fact 10.21.14. Remark: The inequalities in ii) define a polytope. Fact 4.13.17. Let A ∈ Cn×n, and assume that A is unitary. Then, |det A| = 1. Fact 4.13.18. Let A ∈ Rn×n, and assume that A is orthogonal. Then, either det A = 1 or det A = −1. Now, assume that A is a permutation matrix. Then, the following statements hold: i) A is either an even permutation matrix or an odd permutation matrix. ii) If A is a transposition matrix, then A is odd. iii) Let σ be a permutation of (1, . . . , n), and assume that [σ(1) · · · σ(n)]T = A[1 · · · n]T . Then, ∏ σ( j) − σ(i) det A = . j−i 1≤i< j≤n Related: Fact 3.16.1 and Fact 4.31.14. Fact 4.13.19. Let A, B ∈ SO(3). Then, det(A + B) ≥ 0. Source: [2064]. Fact 4.13.20. Let A ∈ Fn×n, and assume that A is unitary. Then, | det(I + A)| ≤ 2n. If, in addition, A is real, then 0 ≤ det(I + A) ≤ 2n. Related: Fact 3.16.5 and Fact 4.10.15. ] △ [ Fact 4.13.21. Let M = CA DB ∈ F(n+m)×(n+m), and assume that M is unitary. Then,

|det A| = |det D|. [ ] [ ] Source: Let = M , and take the determinant of M 0I DBˆˆ = CA 0I . See [24, 2426]. Related: Fact 3.17.7, Fact 3.17.32, Fact 4.13.22, and Fact 11.16.13. Fact 4.13.22. Let A ∈ Fn×n , assume that A is unitary, and let i ∈ {1, . . . , n}. Then, [ ˆ ˆ] A B Cˆ Dˆ

det A = (det M)det D,

△

−1

det A[i,i] = A(i,i) det A,

|det A[i,i] | = |A(i,i) |.

Source: Use (AA )(i,i) = (det A)(A∗ )(i,i) . See Fact 4.13.21 and [2991, pp. 43, 172]. Fact 4.13.23. Let A ∈ Fn×n, assume that A is unitary, and let x ∈ Fn satisfy x∗x = 1 and Ax = −x.

Then, the following statements hold: i) det(A + I) = 0. ii) A + 2xx∗ is unitary. iii) A = (A + 2xx∗ )(I − 2xx∗ ) = (I − 2xx∗ )(A + 2xx∗ ). iv) det(A + 2xx∗ ) = − det A. Fact 4.13.24. The following statements hold: △ i) If A ∈ Fn×n is Hermitian, then A + ȷI is nonsingular, B = ( ȷI − A)( ȷI + A)−1 is unitary, and I + B = 2 ȷ( ȷI + A)−1. ii) If B ∈ Fn×n is unitary and λ ∈ C is such that |λ| = 1 and I + λB is nonsingular, then

390

CHAPTER 4 △

A = ȷ(I − λB)(I + λB)−1 is Hermitian and ȷI + A = 2 ȷ(I + λB)−1. iii) If A ∈ Fn×n is Hermitian, then there exists a unique unitary matrix B ∈ Fn×n such that I + B is nonsingular and A = ȷ(I − B)(I + B)−1. In fact, B = ( ȷI − A)( ȷI + A)−1. iv) If B ∈ Fn×n is unitary and λ ∈ C is such that |λ| = 1 and I + λB is nonsingular, then there exists a unique Hermitian matrix A ∈ Fn×n such that λB = ( ȷI − A)( ȷI + A)−1. In fact, A = ȷ(I − λB)(I + λB)−1. v) If A is nonsingular and skew Hermitian, then A3 + I is nonsingular and (A3 − I)(A3 + I)−1 is unitary. Source: [1084, pp. 168, 169] and [2991, p. 258]. Remark: The linear fractional transformation f (s) = ( ȷ − s)/( ȷ + s) maps the closed upper half plane in C onto the closed inside unit disk in C, △ and the real line in C onto the unit circle in C. Remark: C(A) = (A − I)(A + I)−1 = I − 2(A + I)−1 is the Cayley transform of A. Related: Fact 4.14.9, Fact 4.13.25, Fact 4.13.26, Fact 4.28.12, Fact 10.10.35, and Fact 15.22.10. Fact 4.13.25. The following statements hold: △ i) If A ∈ Fn×n is skew Hermitian, then I + A is nonsingular, B = (I − A)(I + A)−1 is unitary, −1 and I + B = 2(I + A) . If, in addition, mspec(A) = mspec(A), then det B = 1. ii) If B ∈ Fn×n is unitary and λ ∈ C is such that |λ| = 1 and I + λB is nonsingular, then △ A = (I + λB)−1 (I − λB) is skew Hermitian and I + A = 2(I + λB)−1. iii) If A ∈ Fn×n is skew Hermitian, then there exists a unique unitary matrix B ∈ Fn×n such that I + B is nonsingular and A = (I + B)−1 (I − B). In fact, B = (I − A)(I + A)−1. iv) If B is unitary and λ ∈ C is such that |λ| = 1 and I + λB is nonsingular, then there exists a unique skew-Hermitian matrix A ∈ Fn×n such that B = λ(I − A)(I + A)−1. In fact, A = (I + λB)−1 (I − λB). Source: [1084, p. 184] and [1450, p. 440]. Fact 4.13.26. The following statements hold: △ i) If A ∈ Rn×n is skew symmetric, then I + A is nonsingular, B = (I −A)(I + A)−1 is orthogonal, I + B = 2(I + A)−1, and det B = 1. ii) If B ∈ Rn×n is orthogonal, C ∈ Rn×n is diagonal with diagonally located entries ±1, and △ I + CB is nonsingular, then A = (I + CB)−1 (I − CB) is skew symmetric, I + A = 2(I + CB)−1, and det CB = 1. iii) If A ∈ Rn×n is skew symmetric, then there exists a unique orthogonal matrix B ∈ Rn×n such that I + B is nonsingular and A = (I + B)−1 (I − B). In fact, B = (I − A)(I + A)−1. iv) If B ∈ Rn×n is orthogonal and C ∈ Rn×n is diagonal with diagonally located entries ±1, then there exists a unique skew-symmetric matrix A ∈ Rn×n such that CB = (I − A)(I + A)−1. In fact, A = (I + CB)−1 (I − CB). Remark: The Cayley transform is a one-to-one and onto map from the set of skew-symmetric matrices to the set of orthogonal matrices whose spectrum does not include −1. Credit: The last statement is due to P. L. Hsu. See [2263, p. 101]. Fact 4.13.27. Let A, B ∈ Fn×n , and assume that A and B are unitary. Then, √ 2 √ 2 √ 2 1 − 1n tr AB ≤ 1 − 1n tr A + 1 − 1n tr B . Source: [2816] and [2991, p. 197]. Related: Fact 3.15.1 and Fact 4.13.28. Fact 4.13.28. Let A, B ∈ Fn×n , and assume that A and B are unitary. Then,

2 2 2 2 n1 tr A + n1 tr B + 1n tr AB ≤ 1 + 2 n1 tr A 1n tr B n1 tr AB .

391

MATRIX CLASSES AND TRANSFORMATIONS

Source: [1897]. Related: Fact 4.13.27. △ Fact 4.13.29. If A ∈ Fn×n is shifted unitary, then B = 2A − I is unitary. Conversely, If B ∈ Fn×n △ △ 1 is unitary, then A = 2 (B + I) is shifted unitary. Remark: The affine mapping f (A) = 2A − I from

the shifted-unitary matrices to the unitary matrices is one-to-one and onto. See Fact 4.19.1 and Fact 4.20.3. Related: Fact 4.10.14 and Fact 4.17.14. Fact 4.13.30. If A ∈ Fn×n is shifted unitary, then A is normal. Furthermore, the following statements are equivalent: i) A is shifted unitary. ii) A + A∗ = 2A∗A. iii) A + A∗ = 2AA∗. Source: By Fact 4.13.29 there exists a unitary matrix B such that A = 21 (B + I). Since B is normal, it follows from Fact 4.10.14 that A is normal. Fact 4.13.31. The matrices   [ ] [ ] [ ] 0 −Iˆn  I  1 In In 1 In −Iˆn 1  n √ 1 In In , √ ˆ , √ ˆ , √  0 2 0  √ ˆ −I I I − I   I I n n n n 2 2 n 2 n 2 Iˆ 0 In n [ ] I I are orthogonal, and the matrix √12 ȷI − ȷI is unitary. Related: Fact 7.10.23, Fact 7.10.24, and Fact 7.10.25. Fact 4.13.32. Define A ∈ R5×5 by   1 0 4 −2   2   2 −2 4 0 1    △ 1  0 −2 1 4  . A =  2  5  −3 2 2 2 2   2 4 1 −2 0 Then, ATA = A5 = I5 ,

2π π π spec(A) = {1, cos 2π 5 ± sin 5 ȷ, − cos 5 ± sin 5 ȷ},

A15×1 = 15×1 .

Source: [1404]. Remark: A is used to prove the partition congruence 5|p5n+4 . See Fact 1.20.1. Fact 4.13.33. Let A, B ∈ O(n). Then, the following statements are equivalent:

i) AB is involutory. ii) tr (AB)2 = n. iii) AB is symmetric. Source: [2703]. Related: Fact 3.15.37.

4.14 Facts on Rotation Matrices Fact 4.14.1. Let θ ∈ R, and define the orthogonal matrix

[

cos θ A(θ) = − sin θ △

] sin θ . cos θ

Now, let θ1 , θ2 ∈ R. Then, A(θ1 )A(θ2 ) = A(θ1 + θ2 ), cos(θ1 + θ2 ) = (cos θ1 ) cos θ2 − (sin θ1 ) sin θ2 , sin(θ1 + θ2 ) = (cos θ1 ) sin θ2 + (sin θ1 ) cos θ2 .

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Furthermore, SO(2) = {A(θ): θ ∈ R} and

{[

O(2)\SO(2) =

cos θ sin θ

] } sin θ : θ∈R . − cos θ

Remark: See Proposition 4.6.6 and Fact 15.12.3. Fact 4.14.2. Let A ∈ R3×3 . Then, A ∈ O(3)\SO(3) if and only if −A ∈ SO(3). Fact 4.14.3. Let A ∈ F3×3 . Then, the following statements are equivalent:

A ∈ SU(3). ∥col1 (A)∥ = ∥col2 (A)∥ = 1, [col1 (A)]∗ col2 (A) = 0, and col3 (A) = col1 (A) × col2 (A). ∥col2 (A)∥ = ∥col3 (A)∥ = 1, [col2 (A)]∗ col3 (A) = 0, and col1 (A) = col2 (A) × col3 (A). ∥col3 (A)∥ = ∥col1 (A)∥ = 1, [col3 (A)]∗ col1 (A) = 0, and col2 (A) = col3 (A) × col1 (A). Credit: A. H. J. de Ruiter. Fact 4.14.4. Let A ∈ SO(3). Then, i) ii) iii) iv)

(tr A)2 = tr A2 + 2 tr A, (tr A)3 + 3 = tr A3 + 3 tr A2 + 6 tr A, (tr A)3 + 2 tr A3 = 3(tr A)(tr A2 ) + 6, (tr A)3 = (tr A)(tr A2 ) + 2 tr A2 + 4 tr A, (tr A)4 + 12 = tr A4 + 4 tr A3 + 10 tr A2 + 16 tr A, (tr A)5 + 45 = tr A5 + 5 tr A4 + 15 tr A3 + 30 tr A2 + 45 tr A, (tr A)6 + 153 = tr A6 + 6 tr A5 + 21 tr A4 + 50 tr A3 + 90 tr A2 + 126 tr A. Remark: The first equality can be written as

tr A + A(1,2) A(2,1) + A(1,3) A(3,1) + A(2,3) A(3,2) = A(1,1) A(2,2) + A(2,2) A(3,3) + A(3,3) A(1,1) . Remark: These equalities hold for all n × n matrices A that have exactly three nonzero eigenvalues of the form 1, λ, and 1/λ. [ ] △

Fact 4.14.5. Let A ∈ R3×3, and let z =

b c d

, where b2 + c2 + d2 = 1. Then, A ∈ SO(3), and A

rotates every vector in R3 by the angle π about z if and only if   2 2bc 2bd   2b − 1   2c2 − 1 2cd  = 2zzT − I3 = I3 + 2K 2 (z). A =  2bc   2bd 2cd 2d2 − 1 Source: This formula follows from the last expression for A in Fact 4.14.6 with θ = π. See [796, p. 30]. Remark: A is a reflector. Remark: z is uniquely determined up to a sign. Fact 4.14.6. Let A ∈ R3×3. Then, A ∈ SO(3) if and only if there exist real numbers a, b, c, d

such that a2 + b2 + c2 + d2 = 1, a ∈ (−1, 1], and  2 2(bc − ad)  a + b2 − c2 − d2  a2 − b2 + c2 − d2 A =  2(ad + bc)  2(bd − ac) 2(ab + cd) Assume that these conditions hold. Then,

√ a = ± 21 1 + tr A.

2(ac + bd) 2(cd − ab) a − b2 − c2 + d2 2

    . 

393

MATRIX CLASSES AND TRANSFORMATIONS

If, in addition, a , 0, then b, c, and d are given by A(3,2) − A(2,3) A(1,3) − A(3,1) A(2,1) − A(1,2) , c= , d= . 4a 4a 4a Furthermore, the following statements are equivalent: i) a = 1; ii) A = I3 ; iii) b = c = d = 0. △ Now, in the case where a , 1, define v = [b c d]T. If a , 1, then A represents a rotation about the △ θ unit-length vector z = (csc 2 )v through an angle θ ∈ (0, 2π), which satisfies a = cos 2θ , and where △ the direction of rotation about z is determined by the right-hand rule. If a = 1, then define θ = 0. Therefore, for all a ∈ (−1, 1], θ = 2 acos a. If a ∈ [0, 1], then √ θ = 2 acos( 12 1 + tr A) = acos( 12 [(tr A) − 1]), b=

whereas, if a ∈ (−1, 0], then

√ θ = 2 acos(− 12 1 + tr A) = π + acos( 21 [1 − tr A]).

In particular, a = 1 if and only if θ = 0, and a = 0 if and only if θ = π. Furthermore, A = (2a2 − 1)I3 + 2aK(v) + 2vvT = (cos θ)I3 + (sin θ)K(z) + (1 − cos θ)zzT = I3 + (sin θ)K(z) + (1 − cos θ)K 2 (z), A − AT = 4aK(v) = 2(sin θ)K(z). If θ , π, then A = I3 +

2 [K(αz) + K 2 (αz)], α2 + 1

△

sin θ θ 1 2 where α = 1+cos θ = tan 2 . If θ = 0, then v = z = 0, whereas, if θ = π, then K (z) = 2 (A − I). 3 T Conversely, let θ ∈ R be nonzero, let z ∈ R , assume that z z = 1, and define   z(1) z(2) (1 − cos θ) − z(3) sin θ z(1) z(3) (1 − cos θ) + z(2) sin θ   z2(1) + (z2(2) + z2(3) ) cos θ    2 2 2  z(2) + (z(1) + z(3) ) cos θ z(2) z(3) (1 − cos θ) − z(1) sin θ  . B =  z(1) z(2) (1 − cos θ) + z(3) sin θ     2 2 2 z(1) z(3) (1 − cos θ) − z(2) sin θ z(2) z(3) (1 − cos θ) + z(1) sin θ z(3) + (z(1) + z(2) ) cos θ

Then, B represents a rotation about the unit-length vector z through the angle θ, where the direction of rotation is determined by the right-hand rule. Finally, define    a     b  △  cos θ  2    =    c   (sin θ )z    2 d and A as above in terms of a, b, c, d. Then, A = B. Source: [1035, p. 162], [1178, p. 22], [2421, p. 19], [2716, 2722], and use Fact 4.14.9. Remark: The quadruples (a, b, c, d) are Euler parameters, and [a b c d]T is an element of the sphere S3 in R4. The Euler parameter (−1, 0, 0, 0) and point [−1 0 0 0]T ∈ S3 can be viewed as representing a rotation through the angle θ = 2π, which is equivalent to the case θ = 0. Each element of S3 can be represented by a unit quaternion in Sp(1), thus giving S3 a group structure. See Fact 4.31.8. Remark: A is unchanged if a, b, c, d are replaced by −a, −b, −c, −d. Replacing a by −a in A but keeping b, c, d unchanged yields AT . Remark: The entries of A are direction cosines. See [308, pp. 384–387] and Fact 4.32.1. Remark: For each rotation matrix A, there exist exactly two distinct Euler parameters (a, b, c, d) that parameterize A. Therefore, the Euler parameters, which parameterize the unit sphere S3 in R4, provide a double cover of SO(3). See [1967, p. 304] and Fact 4.32.1. Remark: Sp(1) ≃ SU(2) is a double cover of

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SO(3) (see Fact 4.32.4), Sp(1)×Sp(1) ≃ SU(2)×SU(2) is a double cover of SO(4), Sp(2) is a double cover of SO(5), and SU(4) is a double cover of SO(6). For each n, SO(n) is double covered by the spin group Spin(n). See [803, p. 141], [2588, p. 130], and [2889, pp. 42–47]. Sp(2) is defined in Fact 4.32.4. Remark: Rotation matrices in R2×2 are discussed in [2449]. Related: Fact 10.10.30 and Fact 15.16.11. Credit: O. Rodrigues. See [58]. Fact 4.14.7. Let n ≥ 4, let A ∈ SO(n), and let {λ ∈ mspec(A) : λ , 1}ms = {eθ1 ȷ , e−θ1 ȷ , . . . , eθm ȷ , −θm ȷ e }ms , where θ1 , . . . , θm ∈ (0, π]. Then, there exist skew-symmetric matrices A1 , . . . , Am ∈ Rn×n such that, for all i, j ∈ {1, . . . , m}, Ai A j = A j Ai , for all i ∈ {1, . . . , m}, A3i = −Ai , and A=I+

m ∑ [(sin θi )Ai + (1 − cos θi )A2i ]. i=1

Source: [1135, 2326]. Fact 4.14.8. Let θ1 , θ2 ∈ R, let z1, z2 ∈ R3, assume that zT1 z1 = zT2 z2 = 1, and, for i = 1, 2, let

Ai ∈ R3×3 be the rotation matrix that represents the rotation about the unit-length vector zi through △ the angle θi , where the direction of rotation is determined by the right-hand rule. Then, A3 = A2 A1 represents the rotation about the unit-length vector z3 through the angle θ3 , where the direction of rotation is determined by the right-hand rule, and where θ3 and z3 are given by cos θ23 = (cos θ22 ) cos θ21 − (sin θ22 ) sin θ21 zT2 z1 , z3 = (csc θ23 )[(sin θ22 )(cos θ21 )z2 + (cos θ22 )(sin θ21 )z1 + (sin θ22 )(sin θ21 )(z2 × z1 )] =

cot θ23 1 − zT2 z1 (tan θ22 ) tan θ21

[(tan θ22 )z2 + (tan θ21 )z1 + (tan θ22 )(tan θ21 )(z2 × z1 )].

Source: [58], [1178, pp. 22–24], and [2722]. Remark: These expressions are Rodrigues’s formulas, which follow from the quaternion multiplication formula given in Fact 4.32.1. In particular, for △ △ △ i = 1, 2, define qi = ai + bi ıˆ + ci ȷˆ + di kˆ and vi = [bi ci di ]T . Then, q3 = q2 q1 = a3 + b3 ıˆ + c3 ȷˆ + d3 kˆ is given by     a3      b   cos θ3    a1 a2 − vT2 v1 2 3  =    =   ,       c3  (sin θ23 )z3 a1v2 + a2 v1 + v2 × v1 d3

where

   a2       b   cos θ2   a2  2   2  =   =   ,  c2   (sin θ2 )z2   v2  2 d2

   a1       b   cos θ1   a1  2   1  =   =   .  c1   (sin θ1 )z1   v1  2 d1

Fact 4.14.9. Let z, w ∈ R3, assume that ∥z∥2 = ∥w∥2 = 1, let θ, ϕ ∈ R, and define Az (θ) ∈ R3×3

by

△

Az (θ) = (cos θ)I + (sin θ)K(z) + (1 − cos θ)zzT.

Then, the following statements hold: i) Az (θ) = I if and only if θ/π is an even integer. ii) Az (θ) = I + (sin θ)K(z) + (1 − cos θ)K 2 (z). iii) Az (θ) is a rotation matrix. iv) Both Az (θ) , I and A2z (θ) = I if and only if θ/π is an odd integer. If these conditions hold, then Az (θ) = −I + 2zzT.

MATRIX CLASSES AND TRANSFORMATIONS

395

T v) A−z (2π − θ) = Az (θ), A−1 z (θ) = Az (θ) = Az (−θ) = A−z (θ). vi) Let x, y ∈ R3 , assume that ∥x∥2 = ∥y∥2 , 0, and let θ ∈ [0, π] denote the angle between x 1 x × y, whereas, if and y. Furthermore, if θ ∈ (0, π), then let z ∈ R3 be given by z = ∥x×y∥ 2 ⊥ θ ∈ {0, π}, then let z ∈ {x} satisfy ∥z∥2 = 1. Then, y = Az (θ)x. vii) Let x ∈ R3 . Then, Az (θ)x = x if and only if either θ/π is an even integer or z × x = 0. viii) If cos 2θ , 0, then Az (θ) = [I + (tan 2θ )K(z)][I − (tan 2θ )K(z)]−1. ix) If cos 2θ , 0, then [I + Az (θ)]−1 [I − Az (θ)] = −(tan 2θ )K(z). x) [Az (θ), Aw (ϕ)] = (sin θ)(sin ϕ)K(z × w) + (sin θ)(1 − cos ϕ)[K(z), wwT ] + (sin ϕ)(1 − cos θ)[zzT, K(w)] + (1 − cos θ)(1 − cos ϕ)wTz(zwT − wzT ). △ △ θ xi) Assume that cos 2 , 0 and cos ϕ2 , 0, define A = Aw (ϕ)Az (θ) and ψ = acos 12 [(tr A) − 1], ψ 1 (A − AT ). Then, assume that cos 2 , 0, and let v ∈ R3 satisfy ∥v∥2 = 1 and K(v) = 2 √1+tr A

A = [I + (tan ψ2 )K(v)][I − (tan ψ2 )K(v)]−1 , (tan ψ2 )K(v) = [I + (tan 2θ )K(z)][I + (tan ϕ2 )K(w)(tan 2θ )K(z)]−1 · [(tan ϕ2 )K(w) + (tan 2θ )K(z)][I + (tan 2θ )K(z)]−1 . xii) [Az (θ), Aw (ϕ)] = 0 if and only if at least one of the following statements holds: a) z × w = 0. b) Either Az (θ) = I or Aw (ϕ) = I. c) A2z (θ) = I, A2w (ϕ) = I, and wTz = 0. xiii) mspec[Az (θ)] = {1, eθ ȷ , e−θ ȷ }ms , and z is an eigenvector of Az (θ) associated with 1. Source: viii), ix), and xi) are given in [2053, pp. 244, 245]; xii) is due to S. Bhat. Remark: If x T y . Furthermore, xxix) of Fact 4.12.1 implies and y in vi) are linearly independent, then θ = acos ∥x∥x2 ∥y∥ 2 that sin θ =

∥x×y∥2 ∥x∥2 ∥y∥2 .

Remark: In the notation of vi), Az (θ) can be written as

Az (θ) = (cos θ)I +

1 − cos θ 1 (yxT − xyT ) + (x × y)(x × y)T ∥x∥22 ∥x × y∥22

xTy 1 1 − cos θ I + T (yxT − xyT ) + T (x × y)(x × y)T T xx xx (x x sin θ)2 tan θ 1 xTy = T I + T (yxT − xyT ) + T 2 2 (x × y)(x × y)T xx xx (x x) sin θ T x y 1 1 = T I + T (yxT − xyT ) + T 2 (x × y)(x × y)T xx xx (x x) (1 + cos θ) 1 1 xTy (x × y)(x × y)T . = T I + T (yxT − xyT ) + T T xx xx x x(x x + xTy) =

Consequently, Az (θ)x = (cos θ)x + =

1 1 − cos θ (xTxy − yTxx) + (x × y)(x × y)Tx ∥x∥22 ∥x × y∥22

xTy 1 x+ (xTxy − yTxx) = y. 2 ∥x∥2 ∥x∥22

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Furthermore, Bz (θ) = −(tan 12 θ)K(z) can be written as Bz (θ) =

∥x∥2 ∥y∥2 − xT y T (xy − yxT ). ∥x × y∥2

These expressions satisfy Az (θ) + Bz (θ) + Az (θ)Bz (θ) = I. Remark: In vi), Az (θ) represents a righthand rule rotation of the nonzero vector x through the angle θ around z to yield y, which has the same length as x. The cases x = y and x = −y correspond, respectively, to θ = 0 and θ = π; in these cases, the axis of rotation z is not unique. Remark: Extensions of the Cayley transform are discussed in [2741]. Remark: vi) is a linear interpolation result. See Fact 4.11.5, Fact 15.12.7, and [285, 1549]. Related: Fact 15.12.6. Fact 4.14.10. Let x, y, z ∈ R2. If x is rotated according to the right-hand rule through an angle θ ∈ R about y, then the final point xˆ ∈ R2 has the coordinates [ ] [ ] cos θ − sin θ y(1) (1 − cos θ) + y(2) sin θ xˆ = x+ . sin θ cos θ y(2) (1 − cos θ) − y(1) sin θ If x is reflected across the line passing through 0 and z in the direction of the line passing through 0 and y, then the final point xˆ ∈ R2 has the coordinates     2  −z(1) (y2(1) − y2(2) − 1) − 2z(2) y(1) y(2)   y(1) − y2(2) 2y(1) y(2)   .     x +  xˆ =  −z(2) (y2(1) − y2(2) − 1) − 2z(1) y(1) y(2)  2y(1) y(2) y2(2) − y2(1)  Remark: These affine planar transformations are used in computer graphics. See [122, 1062, 2258]. Related: Fact 4.14.9 and Fact 4.14.11. Fact 4.14.11. Let x, y ∈ R3, and assume that yTy = 1. If x is rotated according to the right-hand rule through an angle θ ∈ R about the line passing through 0 and y, then the final point xˆ ∈ R3 has the coordinates xˆ = x + (sin θ)(y × x) + (1 − cos θ)[y × (y × x)]. Source: [44]. Related: Fact 4.14.9 and Fact 4.14.10.

4.15 Facts on One Idempotent Matrix Fact 4.15.1. Let A ∈ Fn×n, assume that A is idempotent, and let x ∈ Fn. Then, x ∈ R(A) if and only if Ax = x. Related: Fact 4.17.11. Fact 4.15.2. Let A ∈ Fn×n. Then, A is idempotent if and only if A is semisimple and there exists

a positive integer k such that Ak+1 = Ak. Fact 4.15.3. Let A ∈ Fn×m and B ∈ Fl×n . Then, the following statements are equivalent: i) R(A) and N(B) are complementary subspaces. ii) rank A = rank B = rank BA. Source: [659]. Related: Fact 8.7.9. Fact 4.15.4. Let S1 , S2 ⊆ Fn be complementary subspaces, and let A ∈ Fn×n . Then, the following statements are equivalent: i) A is the idempotent matrix onto S1 along S2 . ii) A⊥ is the idempotent matrix onto S2 along S1 . iii) A∗ is the idempotent matrix onto S⊥2 along S⊥1 . iv) A⊥∗ is the idempotent matrix onto S⊥1 along S⊥2 . Related: Fact 4.15.5. Fact 4.15.5. Let A ∈ Fn×n. Then, the following statements are equivalent: i) A is idempotent. ii) R(A) ⊆ N(A⊥ ).

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iii) R(A⊥ ) ⊆ N(A). iv) R(A) = N(A⊥ ). v) R(A⊥ ) = N(A). vi) R(A) and R(A⊥ ) are complementary subspaces. vii) For all x ∈ R(A), Ax = x. If these statements hold, then the following statements hold: viii) A is the idempotent matrix onto R(A) along N(A). ix) A⊥ is the idempotent matrix onto N(A) along R(A). x) A∗ is the idempotent matrix onto N(A)⊥ along R(A)⊥ . xi) A⊥∗ is the idempotent matrix onto R(A)⊥ along N(A)⊥ . Source: [257] and [1343, p. 146]. Related: Fact 3.13.2 and Fact 7.13.28. Fact 4.15.6. Let A ∈ Fn×n, and assume that A is idempotent. Then, R(I −AA∗ ) = R(2I − A − A∗ ). Source: [2635]. Fact 4.15.7. Let A ∈ Fn×n. Then, the following statements are equivalent: i) A is skew idempotent. ii) −A is idempotent. iii) rank A = − tr A, and rank(A + I) = n + tr A. Source: [2715]. Fact 4.15.8. Let A ∈ Fn×n. Then, A is idempotent and rank A = 1 if and only if there exist x, y ∈ Fn such that yTx = 1 and A = xyT. Fact 4.15.9. Let A ∈ Fn×n, and assume that A is idempotent. Then, AT, A, and A∗ are idempotent. Fact 4.15.10. Let A ∈ Fn×n, and assume that A is idempotent and skew Hermitian. Then, A = 0. Fact 4.15.11. Let A ∈ Fn×n. Then, the following statements are equivalent: i) A is idempotent. ii) rank A + rank(I − A) = n. iii) rank A = tr A, and rank(I − A) = tr(A − I). iv) I + A is nonsingular, and rank A + rank(I − A2 ) = n. v) A is tripotent, and I − A is tripotent. vi) A is tripotent, and I + A is nonsingular. vii) A is tripotent, and rank A = tr A. If these statements hold, then (I + A)−1 = I − 21 A and det(I + A) = 2tr A . Source: [257] and [2991, p. 130]. Fact 4.15.12. Let A ∈ Fn×m. If AL ∈ Fm×n is a left inverse of A, then AAL is idempotent and rank AL = rank A. Furthermore, if AR ∈ Fm×n is a right inverse of A, then ARA is idempotent and rank AR = rank A. Fact 4.15.13. Let A ∈ Fn×n, and assume that A is nonsingular and idempotent. Then, A = In . △ Fact 4.15.14. Let A ∈ Fn×n, and assume that A is idempotent. Then, so is A⊥ = I − A, and, furthermore, AA⊥ = A⊥A = 0. Fact 4.15.15. Let A ∈ Fn×n, assume that A is idempotent, and let k ≥ 1. Then, (A+I)k = 2kA+A⊥ . Source: [2238]. Fact 4.15.16. Let A ∈ Fn×n, and assume that A is idempotent. Then, the following statements hold: i) rank(A − A∗ ) = 2 rank [A A∗ ] − 2 rank A.

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rank(I − A − A∗ ) = rank(I + A − A∗ ) = n. rank(A + A∗ ) = rank(AA∗ + A∗A) = rank [A A∗ ]. R(A) ⊆ R(A + A∗ ) and R(A∗ ) ⊆ R(A + A∗ ). rank [A, A∗ ] = rank(A − A∗ ). rank(I − AA∗ ) = rank(2I − A − A∗ ). Source: [2672]. Fact 4.15.17. Let A ∈ Fn×n, B ∈ Fn×m, and C ∈ Fl×n, and assume that A is idempotent, rank [C ∗ B] = n, and CB = 0. Then, rank CAB = rank CA + rank AB − rank A. Source: [2674]. Related: Fact 4.15.22. Fact 4.15.18. Let A ∈ Fn×m and B ∈ Fm×n, and assume that AB is nonsingular. Then, B(AB)−1A is idempotent. △ Fact 4.15.19. Let A ∈ Fn×n, let r = rank A, and let B ∈ Fn×r and C ∈ Fr×n satisfy A = BC. Then, A is idempotent if and only if CB = I. Source: [2821, p. 16]. Remark: A = BC is a full-rank factorization. See Proposition 7.6.6. Fact 4.15.20. Let A, B ∈ Rn×n. Then, the following statements hold: i) Assume that A3 = −A and B = I + A+ A2. Then, B4 = I, B−1 = I −A+ A2, B3 − B2 + B−I = 0, A = 21 (B − B3 ), and I + A2 is idempotent. ii) iii) iv) v) vi)

ii) Assume that B3 − B2 + B − I = 0 and A = 12 (B − B3 ). Then, A3 = −A and B = I + A + A2. iii) Assume that B4 = I and A = 12 (B − B−1 ). Then, A3 = −A, and 14 (I + B + B2 + B3 ) is idempotent. Remark: The geometric meaning of these results is discussed in [1028, pp. 153, 212–214, 242]. Fact 4.15.21. Let A ∈ Fn×n and α ∈ F, where α , 0. Then, the matrices [ ] [ ] [ ] A A∗ A α−1A A α−1A , , A∗ A α(I − A) I − A −αA −A are, respectively, normal, idempotent, and nilpotent. [ A ] △ 12 Fact 4.15.22. Let A = AA11 ∈ F(n+m)×(n+m), and assume that A is idempotent. Then, 21 A22 [ ] [ ] A12 A11 rank A = rank + rank [A11 A12 ] − rank A12 = rank + rank [A21 A22 ] − rank A21 . A22 A21 Source: [2674] and Fact 4.15.17. Related: Fact 4.17.13 and Fact 8.9.14.

4.16 Facts on Two or More Idempotent Matrices Fact 4.16.1. Let A, B ∈ Fn×n, and assume that AB = A and BA = B. Then, A and B are idempotent. Source: [2418, p. 169]. Fact 4.16.2. Let A, B ∈ Fn×n, and assume that A2 = A = AB. Then, B2 = B = BA if and only if rank A = rank B. Source: [312]. Fact 4.16.3. Let A, B ∈ Fn×n, and assume that A and B are idempotent. Then, the following

statements hold: i) AB = B if and only if R(B) ⊆ R(A). ii) BA = A if and only if N(B) ⊆ N(A). Furthermore, the following statements are equivalent: iii) R(A) ⊆ R(B) and N(A) ⊆ N(B). iv) R(A) = R(B) and N(A) = N(B).

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v) A = B. Source: [778]. Fact 4.16.4. If A ∈ Fn×m and B ∈ Fn×(n−m) , assume that [A B] is nonsingular, and define △

P = [A 0][A B]−1 ,

△

Q = [0 B][A B]−1.

Then, the following statements hold: i) P and Q are idempotent. ii) P + Q = In . iii) PQ = 0. iv) P[A 0] = [A 0]. v) Q[0 B] = [0 B]. vi) R(P) = R(A) and N(P) = R(B). vii) R(Q) = R(B) and N(Q) = R(A). viii) A∗B = 0 if and only if P and Q are projectors. If these conditions hold, then P = A(A∗A)−1A∗ and Q = B(B∗B)−1B∗. ix) R(A) and R(B) are complementary subspaces. x) P is the idempotent matrix onto R(A) along R(B). xi) Q is the idempotent matrix onto R(B) along R(A). Source: [2996] and [2997, pp. 74, 75]. Related: Fact 4.18.14, Fact 4.18.19, Fact 8.8.14, and Fact 8.8.15. Fact 4.16.5. Let A, B ∈ Fn×n, and assume that A and B are idempotent. Then, rank(A + B) = rank A + rank(A⊥BA⊥ ) = n − dim[N(A⊥B) ∩ N(A)]   [ ]  0 A B  A B   − rank B = rank  A 0 0  − rank A − rank B = rank B 0   B 0 2B [ ] B A = rank − rank A = rank(B⊥AB⊥ ) + rank B = rank(A⊥BA⊥ ) + rank A A 0 = rank(A − AB − BA + BAB) + rank B = rank(B − AB − BA + ABA) + rank A = rank(A + A⊥B) = rank(A + BA⊥ ) = rank(B + B⊥A) = rank(B + AB⊥ ) = rank(I − A⊥B⊥ ) = rank(I − B⊥A⊥ ) = rank [AB⊥ B] = rank [BA⊥ A] [ ] [ ] [ ] B A A B A⊥ A⊥B⊥ = rank ⊥ = rank ⊥ = rank A + rank B − n + rank . B A B⊥A⊥ B⊥ Furthermore, the following statements hold: i) If α, β ∈ F are nonzero, γ ∈ F, and γ , α + β, then rank(A + B) = rank(αA + βB − γAB). ii) If α, β ∈ F are nonzero and α + β , 0, then rank(A + B) = rank(αA + βB). iii) If AB = 0, then rank(A + B) = rank BA⊥ + rank A = rank B⊥A + rank B. iv) If BA = 0, then rank(A + B) = rank AB⊥ + rank B = rank A⊥B + rank A. v) If AB = BA, then rank(A + B) = rank(A − AB) + rank B = rank(B − AB) + rank A. vi) A + B is idempotent if and only if AB = BA = 0. If these conditions hold, then A + B is the idempotent matrix onto R(A) + R(B) along N(A) ∩ N(B). vii) A + B = I if and only if AB = BA = 0 and rank(A + B) = n. viii) If either AB = 0 or BA = 0, then rank(A + B) = rank A + rank B.

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Finally, the following statements are equivalent: ix) A + B is nonsingular. x) R(A) ∩ R(BA⊥ ) = N(A) ∩ N(B) = {0}. xi) R(AB⊥ ) ∩ R(BA⊥ ) = N(A) ∩ N(B) = {0}. xii) R(A) ∩ R(BA⊥ ) = N(A) ∩ N(A⊥ B) = {0}. xiii) R(A) + R(BA⊥ ) = N(A) + N(A⊥ B) = Fn . xiv) R(A) and R(BA⊥ ) are complementary subspaces, and N(A) and N(A⊥ B) are complementary subspaces. ([ ]) ([ ]) A B xv) rank [A B] = n and R ∩R = {0}. B 0 xvi) There exist α, β, γ ∈ F such that α , 0, β , 0, γ , α + β, and αA + βB − γAB is nonsingular. xvii) For all α, β, γ ∈ F such that α , 0, β , 0, and γ , α + β, αA + βB − γAB is nonsingular. xviii) There exist nonzero α, β ∈ F such that α + β , 0 and αA + βB is nonsingular. xix) For all nonzero α, β ∈ F such that α + β , 0, αA + βB is nonsingular. Source: [221, 1251], [1275, p. 18], and [1659, 1661, 1662, 2672, 2673, 2676, 3023]. To prove necessity in vi), note that (A + B)2 = A + B implies AB + BA = 0, which implies AB + ABA = ABA + BA = 0. Hence, AB − BA = 0, and thus AB = 0. See [1305, p. 250] and [1343, p. 435]. Related: Fact 8.8.4. Fact 4.16.6. Let A, B ∈ Fn×n, assume that A and B are idempotent, and let α ∈ F be nonzero. Then,   [ ]  0 A B  A   + rank [A B] − rank A − rank B rank(A − B) = rank  A 0 0  − rank A − rank B = rank B   B 0 0 = rank(A − AB) + rank(AB − B) = rank(A − BA) + rank(BA − B) = rank[A − AB + α(AB − B)] = rank(A + B − 2AB) = n − dim[N(A) ∩ N(B)] − dim[R(A) ∩ R(B)] = rank(AB⊥ ) + rank(A⊥B) ≤ rank(A + B) ≤ rank A + rank B. Furthermore, the following statements hold: i) If either AB = 0 or BA = 0, then rank(A − B) = rank(A + B) = rank A + rank B. ii) If AB = 0, then rank(A − BA) + rank(BA − B) = rank A + rank B. iii) If BA = 0, then rank(A − AB) + rank(AB − B) = rank A + rank B. iv) If α, β ∈ F are nonzero, then rank(A − B) = rank[αA + βB − (α + β)AB]. Finally, the following statements are equivalent: v) rank(A − B) = rank A − rank B. vi) ABA = B. vii) R(B) ⊆ R(A) and R(B∗ ) ⊆ R(A∗ ). Source: [1251, 1662, 2657, 2672, 2673, 2676, 3023]. rank(A − B) ≤ rank(A + B) follows from Fact 3.14.19 and the block 3 × 3 expressions in this result and in Fact 4.16.5. To prove i) in the case AB = 0, note that rank A + rank B = rank(A − B), which yields rank(A − B) ≤ rank(A + B) ≤ rank A + rank B = rank(A − B). Related: Fact 8.8.4. Fact 4.16.7. Let A ∈ Fn×n, B ∈ Fm×m, and C ∈ Fn×m, and assume that A and B are idempotent.

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Then,

[

] AC rank(AC − CB) = rank + rank [A CB] − rank A − rank B B = rank(AC − ACB) + rank(ACB − CB).

Furthermore, the following statements hold: i) If ACB = 0, then rank(AC − CB) = rank AC + rank CB. ii) AC = CB if and only if R(CB) ⊆ R(A) and R[(AC)∗ ] ⊆ R(B∗ ). iii) Assume that n = m. Then, AC = CA if and only if R(CA) ⊆ R(A) and R[(AC)∗ ] ⊆ R(A∗ ). Source: [2628, 2657, 2672]. Fact 4.16.8. Let A, B ∈ Fn×n, and assume that A and B are idempotent. Then, the following statements are equivalent: i) A − B is idempotent. ii) rank(A⊥ + B) + rank(A − B) = n. iii) rank(A − B) = rank A − rank B. iv) R(B) ⊆ R(A) and R(B∗ ) ⊆ R(A∗ ). v) ABA = B. vi) AB = BA = B. If these statements hold, then A − B is the idempotent matrix onto R(A) ∩ N(B) along N(A) + R(B). Source: [1275, p. 19] and [2672, 2675]. Credit: R. E. Hartwig and G. P. H. Styan. Fact 4.16.9. Let A, B ∈ Fn×n, and assume that A and B are idempotent. Then, the following statements are equivalent: i) R(A) ∩ R(B) = {0}. ii) I − AB is nonsingular. iii) I − BA is nonsingular. Furthermore, the following statements are equivalent: iv) A − B is nonsingular. [ ] v) rank AB = rank [A B] = rank A + rank B = n. vi) I − AB is nonsingular, and there exist nonzero α, β ∈ F such that αA + βB is nonsingular. vii) I − AB is nonsingular, and, for all nonzero α, β ∈ F, αA + βB is nonsingular. viii) There exist nonzero α, β ∈ F such that αA + βB − (α + β)AB is nonsingular. ix) For all nonzero α, β ∈ F, αA + βB − (α + β)AB is nonsingular. x) I − AB and A + A⊥B are nonsingular. xi) I − BA and A + BA⊥ are nonsingular. xii) I − AB and A + B are nonsingular. xiii) R(A) and R(B) are complementary subspaces, and R(A∗ ) and R(B∗ ) are complementary subspaces. xiv) R(A) and R(B) are complementary subspaces, and N(A) and N(B) are complementary subspaces. xv) R(A) ∩ R(B) = N(A) ∩ N(B) = {0}. xvi) R(AB⊥ ) and R(A⊥ B) are complementary subspaces. xvii) N(AB⊥ ) and N(A⊥ B) are complementary subspaces. If iv)–xvii) hold, then the following statements hold:

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xviii) A(A − B)−1A = A, A(A − B)−1B = 0, and B(B − A)−1B = B. xix) A(A − B)−1 = (I − BA)−1 (I − B) = A(A + BA⊥ )−1 is the idempotent matrix onto R(A) along R(B). xx) A∗ (A∗ − B∗ )−1 is the idempotent matrix onto R(A∗ ) along R(B∗ ). Source: [221, 248, 1251, 1660, 1662, 2673]. Related: Fact 4.18.19. Fact 4.16.10. Let A, B ∈ Fn×n, assume that A and B are idempotent, and assume that A − B is nonsingular. Then, A + B is nonsingular. Now, define F, G ∈ Fn×n by △

F = A(A − B)−1 = (A − B)−1 B⊥ ,

△

G = (A − B)−1A = A⊥ (A − B)−1.

Then, F and G are idempotent. In particular, F is the idempotent matrix onto R(A) along R(B), G is the idempotent matrix onto N(B) along N(A), and G∗ is the idempotent matrix onto R(A∗ ) along R(B∗ ). Furthermore, F⊥ A = A⊥ F = FB = B⊥ F⊥ = GA⊥ = AG⊥ = BG = G⊥ B⊥ = 0, −1

(A − B)

= F − G⊥ = G − F⊥ = (A + B)−1 (A − B)(A + B)−1,

(I − AB)−1 = FG + F⊥ ,

(A + B)−1 = I − G⊥F − GF⊥ = (2G − I)(F − G⊥ ) = (A − B)−1 (A + B)(A − B)−1 . Now, let α, β ∈ F be nonzero. Then, αA + βB − (α + β)AB is nonsingular, and [αA + βB − (α + β)AB]−1 =

1 1 F⊥ + G. β α

If, in addition, γ ∈ F and γ , α + β, then αA + βB − γAB is nonsingular, and (αA + βB − γAB)−1 =

1 γ−α α+β−γ F⊥ + G+ GF. β αβ αβ

Source: [873, 1662, 3023]. Remark: See [1662] for an explicit expression for (A + B)−1 in the case where A − B is singular. Related: Proposition 4.8.3 and Fact 8.11.14. Fact 4.16.11. Let A, B ∈ Fn×n, assume that A and B are idempotent, and assume that AB = BA.

Then, the following statements are equivalent: i) A − B is nonsingular. ii) (A − B)2 = I. iii) A + B = I. Source: [1251]. Fact 4.16.12. Let A, B ∈ Fn×n, and assume that A and B are idempotent. Then, rank(I − A − B) = rank AB + rank BA − rank A − rank B + n = rank(I − A − B + AB) + rank AB = rank(I − A − B + BA) + rank BA. Furthermore, the following statements hold: i) A + B = I if and only if AB = BA = 0 and rank(A + B) = rank A + rank B = n. ii) I − A − B is nonsingular if and only if rank AB = rank BA = rank A = rank B. iii) rank(I + A − B) = rank BAB − rank B + n. iv) rank(2I − A − B) = rank(B − BAB) − rank B + n = rank(A − ABA) − rank A + n. v) rank(2I − A − B) = rank(I − AB). vi) I + A − B is nonsingular if and only if rank BAB = rank B. vii) 2I − A − B is nonsingular if and only if rank(A − ABA) = rank A.

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viii) If α ∈ C and α , 1, then rank(I − A − B + αAB) = rank(I − A − B). Source: [1251, 2672]. Fact 4.16.13. Let A, B ∈ Fn×n, and assume that A and B are idempotent. Then, the following statements hold: i) R(A) = R(B) if and only if AB = B and BA = A. ii) R(A) ∩ R(B) ⊆ R(AB) = R(A) ∩ [N(A) + R(B)]. iii) R(A) ⊆ R(B) if and only if R(B) = R(A) + R(B − AB) and R(A) ∩ R(B − AB) = {0}. iv) N(B) + [N(A) ∩ R(B)] ⊆ N(AB) = N(B) + [N(A) ∩ R(B)] ⊆ R(I − AB) ⊆ N(A) + N(B). v) If AB = BA, then AB is the idempotent matrix onto R(A) ∩ R(B) along N(A) + N(B). vi) If B⊥ A⊥ = 0, then AB is idempotent, R(AB) = R(A) ∩ R(B), N(AB) = N(A) + N(B), and N(A) ∩ N(B) = {0}. vii) R(AB) = R(BA) if and only if ABA = BA and BAB = AB. viii) N(AB) = N(BA) if and only if ABA = AB and BAB = BA. ix) AB = 0 if and only if R(B) ⊆ R(B − AB). x) AB = B if and only if R(B) ⊆ R(AB). xi) R([A, B]) = [R(A) + R(B)] ∩ [R(A) + N(B)] ∩ [N(A) + R(B)] ∩ [N(A) + N(B)]. xii) R([A, B]) = R(A − B) ∩ R(A⊥ − B). xiii) N([A, B]) = [R(A) ∩ R(B)] + [R(A) ∩ N(B)] + [N(A) ∩ R(B)] + [N(A) ∩ N(B)]. xiv) N([A, B]) = N(A − B) + N(I − A − B). xv) N(A − B) ∩ N(I − A − B) = {0}. The following statements are equivalent: xvi) R(A) = R(AB) + [R(A) ∩ N(B)] and R(AB) ∩ [R(A) ∩ N(B)] = {0}. xvii) rank AB = rank BA and R(AB) ∩ N(B) = {0}. The following statements are equivalent: xviii) N(A) = N(BA) ∩ [N(A) + R(B)] and R(AB) ∩ [R(A) ∩ R(B)] = {0}. xix) rank AB = rank BA and N(BA) + R(B) = Fn . The following statements are equivalent: xx) AB = BA. xxi) rank AB = rank BA, and AB is the idempotent matrix onto R(A) ∩ R(B) along N(A) + N(B). xxii) rank AB = rank BA, and A + B − AB is the idempotent matrix onto R(A) + R(B) along N(A) ∩ N(B). The following statements are equivalent: xxiii) AB is idempotent. xxiv) R(AB) ⊆ N(A − AB). xxv) R(AB) ∩ R(AB⊥ A) = {0}. xxvi) R(AB) ⊆ R(B) + [N(A) ∩ N(B)]. xxvii) R(AB) = R(A) ∩ (R(B) + [N(A) ∩ N(B)]). xxviii) N(B) + [N(A) ∩ R(B)] = R(I − AB). xxix) rank AB + rank AB⊥ A = rank A. xxx) tr A = rank B and tr AB + rank AB⊥ A = rank A. The following statements are equivalent: xxxi) R(AB) = R(A) ∩ R(B).

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xxxii) rank AB + rank(A − BA) = rank A. The following statements are equivalent: xxxiii) N(AB) = N(A) + N(B). xxxiv) rank AB + rank(A − AB) = rank A. The following statements are equivalent: xxxv) AB is the idempotent matrix onto R(A) ∩ R(B) along R(A) + N(B). xxxvi) R(AB) = R(A) ∩ R(B) and N(AB) = N(A) + N(B). The following statements are equivalent: xxxvii) AB is the idempotent matrix onto R(A) along R(B). xxxviii) R(A) ⊆ N(A − AB) and N(A) ∩ R(B) = {0}. xxxix) R(B − AB) ⊆ N(B) and N(A) + R(B) = Fn . The following statements are equivalent: xl) rank AB = rank A + rank B − n. xli) N(A) ⊆ R(B). xlii) A + B − AB = I. xliii) R(BA) + N(A) = R(B). Finally, assume that AB is idempotent. Then, the following statements are equivalent: xliv) R(AB) = N(A − AB). xlv) N(A − AB) ⊆ R(A). xlvi) N(A − AB) = N(I − AB). xlvii) rank(A − AB) = rank(I − AB). Source: [256], [1133, p. 53], [1250], [1275, p. 19], and [2640, 2870]. Related: Fact 7.13.29. Fact 4.16.14. Let A, B ∈ Fn×n, and assume that A and B are idempotent. Then, the following statements hold: i) R(A − B) = [R(A) + R(B)] ∩ [N(A) + N(B)]. ii) N(A − B) = [R(A) ∩ R(B)] + [N(A) ∩ N(B)] = N(A − BA) ∩ N(B − BA). iii) R(I − A − B) = [R(A) + N(B)] ∩ [N(A) + R(B)]. iv) N(I − A − B) = [R(A) ∩ N(B)] + [N(A) ∩ R(B)]. v) N(I − A − B) = N(AB) ∩ N(A⊥ B⊥ ). The following statements are equivalent: vi) A + B is idempotent. vii) A is the idempotent matrix onto R(A) ∩ N(B) along N(A) + R(B). viii) R(B) ⊆ R(B − AB) and N(B − BA) ⊆ N(B). The following statements are equivalent: ix) A − B is idempotent. x) B is the idempotent matrix onto R(A) ∩ R(B) along N(A) + N(B). xi) R(B) ⊆ (AB) and N(BA) ⊆ N(B). The following statements are equivalent: xii) R(A + B) = R(A) + R(B). xiii) R(A) + R(A⊥ BA⊥ ) = R(A) + R(B). The following statements are equivalent: xiv) N(A + B) = N(A) ∩ N(B).

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xv) N(AB + BA) = [R(A) ∩ N(B)] + [N(A) ∩ R(B)] + [N(A) ∩ N(B)]. xvi) rank A⊥ BA⊥ = rank(B − BA). xvii) N(A⊥ BA⊥ ) = N(B − BA). xviii) R(A) ∩ R(B − BA) = {0}. The following statements are equivalent: xix) N(I − AB) = R(A) ∩ R(B). xx) rank AB⊥ A = rank(A − BA). xxi) N(AB⊥ A) = N(A − BA). xxii) N(A) ∩ R(A − BA) = {0}. The following statements are equivalent: xxiii) R(A) + R(B − BA) = R(A) + R(B). xxiv) R(A⊥ BA⊥ ) = R(B − AB). xxv) rank A⊥ BA⊥ = rank(B − AB). The following statements are equivalent: xxvi) R(A) ∩ N(A − AB) = R(A) ∩ R(B). xxvii) N(AB⊥ A) = N(A − BA). xxviii) rank AB⊥ A = rank(A − BA). Source: [256]. Fact 4.16.15. Let A, B ∈ Fn×n, and assume that A and B are idempotent. Then, the following statements hold: i) (A − B)2 + (A⊥ − B)2 = I. ii) [A, B] = [B, A⊥ ] = [B⊥ , A] = [A⊥ , B⊥ ]. iii) A − B = AB⊥ − A⊥B. iv) AB⊥ + BA⊥ = AB⊥A + A⊥BA⊥ . v) A[A, B] = [A, B]A⊥ . vi) B[A, B] = [A, B]B⊥ . Source: [2138]. Fact 4.16.16. Let A, B ∈ Fn×n, and assume that A and B are idempotent. Then, rank [A, B] = rank(A − B) + rank(I − A − B) − n = rank(A − B) + rank AB + rank BA − rank A − rank B [ ] A = rank + rank [A B] + rank AB + rank BA − 2 rank A − 2 rank B B = rank(A − AB) + rank(AB − B) + rank AB + rank BA − rank A − rank B = rank(AB − ABA) + rank(BA − ABA) [ ] AB = rank + rank [A BA] − 2 rank A A [ ] AB = rank + rank [AB BA] − rank AB − rank BA. BA Source: [1662, 2657, 2672]. Fact 4.16.17. Let A, B ∈ Fn×n, and assume that A and B are idempotent. Then, following

statements are equivalent:

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i) AB = BA. ii) [A, B] = 0. iii) R(AB) = R(BA) and R[(AB)∗ ] = R[(BA)∗ ]. iv) rank(A − B) + rank(I − A − B) = n. v) rank(A − B) = rank A + rank B − rank AB − rank BA. vi) rank(A − AB) = rank A − rank AB and rank(B − AB) = rank B − rank AB. The following statements are equivalent: vii) rank [A, B] = rank(A − B). viii) rank AB = rank BA = rank A = rank B. ix) I − A − B is nonsingular. The following statements are equivalent: x) [A, B] is nonsingular. xi) A − B and I − A − B are nonsingular. xii) rank AB = rank BA = rank A = rank B, R(A) and R(B) are complementary subspaces, and R(A∗ ) and R(B∗ ) are complementary subspaces. Source: [2672]. Fact 4.16.18. Let A, B ∈ Fn×n, and assume that A and B are idempotent. Then, rank(AB + BA) = rank(A + B) + rank(I − A − B) − n = rank(A + B) + rank AB + rank BA − rank A − rank B = rank(A − AB − BA + BAB) + rank AB + rank BA − rank A = rank [A, B] + rank(A + B) − rank(A − B). Furthermore, the following statements hold: i) AB + BA = 0 if and only if AB = BA = 0. ii) If α, β ∈ F are nonzero and α + β , 0, then rank(AB + BA) = rank(αAB + βBA). iii) max {rank AB, rank BA} ≤ rank(AB + BA). iv) rank [A, B] ≤ rank(AB + BA). The following statements are equivalent: v) rank(AB + BA) = rank(A + B). vi) rank [A, B] = rank(A − B). vii) rank(I − A − B) = n. viii) rank A = rank B = rank AB = rank BA. Finally, the following statements are equivalent: ix) AB + BA is nonsingular. x) A + B and I − A − B are nonsingular. xi) R(B− BA)∩R(A) = N(B−AB)∩N(A) = {0}, R(A) and N(B) are complementary subspaces, and N(A) and R(B) are complementary subspaces. Source: [221, 1251, 1662, 2642, 2657, 2672]. Fact 4.16.19. Let A, B ∈ Fn×n, and assume that A and B are idempotent. Then, rank[AB − (AB)2 ] = rank(2I − A − B) + rank AB − n, rank[(A − B)2 − (A − B)] = rank(I − A + B) + rank(A − B) − n

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= rank ABA + rank(A − B) − rank A. Source: [1251, 2672]. Fact 4.16.20. Let A, B ∈ Fn×n, assume that A and B are idempotent, and assume that A − B is group invertible. Then, AB⊥ and B⊥ A are group invertible. Source: [873]. Fact 4.16.21. Let A1 , . . . , Ar ∈ Fn×n, assume that A1 , . . . , Ar are idempotent, and assume that ∑r i=1 Ai = I. Then, for all distinct i, j ∈ {1, . . . , r}, Ai A j = 0. Source: [2991, p. 132]. Related: Fact

4.18.24.

4.17 Facts on One Projector Fact 4.17.1. Let A ∈ Fn×n, and assume that A is Hermitian. Then, the following statements are

equivalent: i) A is a projector. ii) rank A = tr A = tr A2. Source: [2418, p. 55]. Related: Fact 4.17.3 and Fact 4.17.4. Fact 4.17.2. Let A ∈ Fn×n, and assume that A is a projector. Then, R(A⊥ ) = R(A)⊥ = N(A). Fact 4.17.3. Let A ∈ Fn×n. Then, the following statements are equivalent: i) A is a projector. ii) A is idempotent and Hermitian. iii) A is idempotent and normal. iv) A is idempotent and range Hermitian. v) A is idempotent, and R(A) and N(A) are mutually orthogonal. vi) A is idempotent, and R(A)⊥ = N(A). vii) A is idempotent, and, for all x ∈ Fn, x∗Ax ≥ 0. viii) A is idempotent, and, for all x ∈ Fn, x∗Ax ≤ x∗x. ix) A is idempotent, and, for all x ∈ Fn , ∥Ax∥2 ≤ ∥x∥2 . x) A is idempotent, and rank A + rank(I − A∗A) = n. xi) A is idempotent and AA∗A = A. xii) A is idempotent, and AA∗ + A∗A = A + A∗. xiii) A is tripotent, range Hermitian, and I − A is tripotent. xiv) A is tripotent, range Hermitian, and I + A is nonsingular. xv) A is tripotent, range Hermitian, and rank A = tr A. xvi) A is tripotent and positive semidefinite. xvii) R(A∗ ) ⊆ N(I − A). Source: [257], [2238, p. 308], [2263, p. 105], [2705, 2725], and [2991, pp. 131, 322]. Related: Fact 4.17.1 and Fact 4.17.4. Fact 4.17.4. Let A ∈ Fn×n. Then, the following statements are equivalent: i) A is a projector. ii) A = AA∗ . iii) A = A∗A. iv) A is normal, and A3 = A2 . v) A and A∗A are idempotent. vi) A and 21 (A + A∗ ) are idempotent.

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A is tripotent, and A2 = A∗. AA∗ = A∗AA∗. 3 tr A∗A + tr A2A2∗ = 2 Re tr(A2 + A2 A∗ ). A is range Hermitian, and rank A + tr A∗A = tr(A + A∗ ). Source: ix)] is given in [2238, p. 307] and [2700]; x) is given in [234]. Remark: The matrix [ 1/2 1/2 A = 0 0 satisfies tr A = tr A∗A but is not a projector. See Fact 4.10.13. Related: Fact 4.17.1, Fact 4.17.3, and Fact 8.7.6. Fact 4.17.5. Let A ∈ Fn×n, and assume that A is a projector. Then, A is positive semidefinite. Fact 4.17.6. Let n ≥ 2, let A ∈ Fn×n, and assume that A is a nonzero projector. Then, √ ˆ ≤ min {rank A, n + 1 [1 − (−1)n ] − rank A} < n rank A. | tr IA| 2 vii) viii) ix) x)

Source: [254, p. 55]. Fact 4.17.7. Let A ∈ Fn×n, assume that A is a projector, and let x ∈ Fn. Then, x ∈ R(A) if and

only if x = Ax.

△

Fact 4.17.8. Let A ∈ Fn×m. If rank A = m, then B = A(A∗A)−1A∗ is a projector and rank B = m. If △ rank A = n, then B = A∗ (AA∗ )−1A is a projector and rank B = n. Related: Fact 3.18.5, Fact 4.10.25,

and Fact 4.10.26. △

Fact 4.17.9. Let x ∈ Fn, assume that x is nonzero, and define the elementary projector A = ∗

−1

∗

I − (x x) xx . Then, the following statements hold: i) R(A) = {x}⊥. ii) rank A = n − 1. iii) N(A) = span {x}. iv) def A = 1. v) 2A − I is the elementary reflector I − 2(x∗x)−1xx∗. Remark: If y ∈ Fn, then Ay is the projection of y into {x}⊥. Related: Fact 3.13.1. Fact 4.17.10. Let n ≥ 2, let S ⊂ Fn, and assume that S is a hyperplane. Then, there exists a unique elementary projector A ∈ Fn×n such that R(A) = S and N(A) = S⊥. Furthermore, if x ∈ Fn is △ nonzero and S = {x}⊥, then A = I − (x∗x)−1xx∗. Fact 4.17.11. Let A ∈ Fn×n. Then, A is a projector and rank A = n − 1 if and only if there exists a nonzero vector x ∈ N(A) such that A = I − (x∗x)−1xx∗. Now, assume that these conditions hold. Then, for all y ∈ Fn, |y∗x|2 y∗y − y∗Ay = ∗ . xx n Furthermore, for all y ∈ F , the following statements are equivalent: i) y∗x = 0. ii) y∗Ay = y∗y. iii) Ay = y. iv) y ∈ R(A). Related: Fact 4.10.19, Fact 4.15.1, and Fact 4.17.12. Fact 4.17.12. Let A ∈ Fn×n, assume that A is a projector, and let x ∈ Fn. Then, x∗Ax ≤ x∗x. Furthermore, the following statements are equivalent: i) x∗Ax = x∗x. ii) ∥Ax∥2 = ∥x∥2 . iii) Ax = x.

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iv) x ∈ R(A). Related: Fact 4.15.1 and Fact ] [ 4.17.11. △ A11 A12 (n+m)×(n+m) , and assume that A is a projector. Then, Fact 4.17.13. Let A = A∗12 A22 ∈ F rank A = rank A11 + rank A22 − rank A12 . Source: [2675] and Fact 4.15.22. Related: Fact 8.9.14. Fact 4.17.14. Let A ∈ Fn×n, and assume that A satisfies two out of the three properties (Hermi-

tian, shifted unitary, idempotent). Then, A satisfies the remaining property. Furthermore, A satisfies all three properties if and only if A is a projector. Source: If A is idempotent and shifted unitary, then (2A − I)−1 = 2A − I = (2A∗ − I)−1. Hence, A is Hermitian. Related: Fact 4.13.30, Fact 4.19.2, and Fact 4.19.6.

4.18 Facts on Two or More Projectors △

Fact 4.18.1. Let A, B ∈ Fn×n, assume that A and B are projectors, and define S1 = R(A) and

△

S2 = R(B). Then, the following statements are equivalent: i) AB is a projector. ii) S1 ∩ S2 = S1 ∩ (S⊥1 + S2 ). iii) S1 ∩ (S⊥1 + S2 ) = S2 ∩ (S⊥2 + S1 ). The following statements are equivalent: iv) A + B is a projector. v) S1 ∩ (S⊥1 + S2 ) = {0}. vi) S2 ⊆ S⊥1 . The following statements are equivalent: vii) A − B is a projector. viii) S2 ∩ (S⊥1 + S⊥2 ) = {0}. ix) S⊥1 ∩ (S1 + S2 ) = {0}. x) S2 ⊆ S1 . The following statement holds: xi) rank(A − B) = dim(S1 + S2 ) − dim(S1 ∩ S2 ) = dim[(S1 + S2 ) ∩ (S⊥1 + S⊥2 )]. Source: [255]. Related: Fact 3.12.19 and Fact 7.13.27. Fact 4.18.2. Let A, B ∈ Fn×n, and assume that A and B are projectors. Then, the following statements are equivalent: i) R(A) = R(B). ii) A = B. Related: Proposition 4.8.1. Fact 4.18.3. Let A ∈ Fn×m, let B ∈ Fn×n, and assume that B is a projector. Then, the following statements are equivalent: i) R(A) = R(BA). ii) A = BA. Source: To prove i) =⇒ ii), note that 0 = R(B⊥ BA) = B⊥R(BA) = B⊥R(A) = R(B⊥A). Hence, B⊥A = 0. Consequently, BA = (B + B⊥ )A = A. Related: Fact 8.8.3. Fact 4.18.4. Let A, B ∈ Fn×n, and assume that A and B are projectors. Then, the following statements are equivalent: i) A < B.

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ii) A = 0 and B = I. Fact 4.18.5. Let A, B ∈ Fn×n, and assume that A and B are projectors. Then, the following statements are equivalent: i) A ≤ B. ii) R(A) ⊆ R(B). iii) R(A) = R(BA). iv) AB = BA and R(A) = R(AB). v) A = AB. vi) A = BA. vii) A = ABA. viii) B − A is idempotent. ix) B − A is a projector. x) A + B⊥A is idempotent. xi) A + B⊥A is a projector. xii) B⊥ ≤ A⊥ . xiii) R(B⊥ ) ⊆ R(A⊥ ). xiv) B⊥ = B⊥ A⊥ . xv) B⊥ = A⊥B⊥ . xvi) B⊥ = B⊥ A⊥B⊥ . xvii) A⊥ − B⊥ is a projector. xviii) B⊥ + AB⊥ is idempotent. xix) For all x ∈ Fn , ∥Ax∥2 ≤ ∥Bx∥2 . If these statements hold, then AB = BA. Source: [2418, pp. 24, 169]. Related: Fact 4.18.7 and Fact 10.11.9. Fact 4.18.6. Let A, B ∈ Fn×n, and assume that A and B are projectors. Then, tr (AB)2 ≤ tr AB ≤ min {tr A, tr B, rank AB}. In addition, the following statements are equivalent: i) Either A − B is a projector or B − A is a projector. ii) tr AB = min {tr A, tr B}. Source: [234, 252] and [2942, p. 48]. Related: Fact 4.18.7. Fact 4.18.7. Let A, B ∈ Fn×n, and assume that A and B are projectors. Then, the following statements are equivalent: i) AB = BA. ii) AB is a projector. iii) AB is idempotent. iv) AB is Hermitian. v) AB is normal. vi) AB is range Hermitian. vii) [AB(AB)∗ , (AB)∗AB] = 0. viii) AB = ABA. ix) AB = BAB.

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x) AB = (AB)2 . xi) There exist distinct tuples with alternating components A and B whose products are equal. xii) A + A⊥B is a projector. xiii) A + A⊥B is idempotent. xiv) A + A⊥B is Hermitian. xv) A + B⊥A is Hermitian. xvi) AB is the projector onto R(A) ∩ R(B). xvii) A + A⊥B is the projector onto R(A) + R(B). xviii) tr (AB)2 = tr AB. xix) R(AB) = R(A) ∩ R(B). xx) R(A) ∩ R(B) and R(A) ∩ N(B) are complementary subspaces. xxi) R(AB) ⊆ R(B). xxii) R(BA) ⊆ R(A). xxiii) R(AB) = R(BA). xxiv) N(AB) = N(BA). xxv) R(AB) and N(AB) are complementary subspaces. xxvi) R(A − B) ∩ R(AB) = {0}. xxvii) rank [A B] + rank AB = rank A + rank B. xxviii) rank(A + B) + rank AB = rank A + rank B. xxix) rank(I − AB) + rank AB = n. xxx) rank(A − B) + rank AB = rank(A + B). xxxi) rank AB = rank(AB + BA). xxxii) tr (AB)2 = rank AB. xxxiii) tr AB ≤ rank AB. xxxiv) tr AB = rank AB. xxxv) tr AB = rank ABA. xxxvi) 21 tr(AB + BA) = rank(AB + BA). tr(AB + BA)2 = tr(AB + BA). ABA is a projector. rank ABA = tr ABA. rank ABA = tr (ABA)2 . tr ABA = tr (ABA)2 . AB − BA is a projector. rank(AB − BA) = tr(AB − BA). rank(AB − BA) + tr (AB − BA)2 = 0. A + B − AB is a projector. rank(A + B − AB) = tr(A + B − AB). Source: [234, 242, 252, 261], [1124, pp. 42–44], [2238, p. 308], and [2657, 2706, 2869]. Remark: To illustrate xi), consider (A, B, A, B, A) and (B, A, B). Then, ABABA = BAB implies that AB = BA. See [235]. Related: Fact 4.18.5, Fact 7.13.5, Fact 8.4.32, and Fact 8.8.5.

xxxvii) xxxviii) xxxix) xl) xli) xlii) xliii) xliv) xlv) xlvi)

1 2

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Fact 4.18.8. Let A, B ∈ Fn×n, and assume that A and B are projectors. Then, the following

statements are equivalent: i) AB = 0. ii) For all α, β ∈ F, R(I − αA − βB) = R[(I − αA)(I − βB)]. iii) For all α, β ∈ F, rank(I − αA − βB) = rank (I − αA)(I − βB). iv) For all α, β ∈ F, tr(I − αA − βB) = tr (I − αA)(I − βB). Source: [259]. Related: Fact 3.16.14. Fact 4.18.9. Let A, B ∈ Fn×n, and assume that A and B are projectors. Then, the following statements are equivalent: i) A + B is a projector. ii) AB = 0. iii) BA = 0. iv) AB = BA = 0. v) AB + BA = 0. vi) (A + B)2 = A + B. vii) tr (A + B)2 = tr(A + B). viii) R(A) ⊆ R(B)⊥. ix) R(B) ⊆ R(A)⊥. x) R(A) and R(B) are mutually orthogonal. xi) AB is a projector, and rank(A + B) = tr(A + B). xii) AB is a projector, and rank(A + B) = tr (A + B)2 . xiii) AB is a projector, and rank(I − A − B) = tr(I − A − B). Source: [261], [1124, pp. 42–44], and [2657]. Remark: See [244, 1134]. Fact 4.18.10. Let A, B ∈ Fn×n, and assume that A and B are projectors. Then, the following statements are equivalent: i) A − B is a projector. ii) tr(A − B) = tr (A − B)2 . iii) AB is a projector, and rank(A − B) = tr(A − B). Source: [261]. Fact 4.18.11. Let A, B ∈ Fn×n, and assume that A and B are projectors. Then, the following statements hold: i) R(A + B) = R(A) + R(B) = R([A B]) = span[R(A) ∪ R(B)]. ii) R(A + B) = R(A + A⊥B) = R(A) + R(A⊥ B). iii) R(A + B) = R(A − B) + R(AB + BA) = R(I − A⊥ B⊥ ). iv) R(A + B) = R(A − B) + R(AB). v) R(A + B) = R(B) + R(B⊥ AB⊥ ). vi) R(A + B) = R(A) + R(B − AB − BA + ABA). vii) N(A + B) = N(A) ∩ N(B). viii) R(A − B) + R(A⊥ − B) = Fn . ix) R(A − B) ∩ R(A⊥ − B) = R([A, B]) ⊆ R(A + B). x) R(A − B) = [R(A) + R(B)] ∩ [N(A) + N(B)]. xi) R(AB) = R(A) ∩ [R(A) ∩ N(B)]⊥ .

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413

xii) N(AB) = N(B) + [N(A) ∩ N(B)]. xiii) R(ABA⊥ ) = R(A) ∩ [R(A) ∩ R(B)]⊥ ∩ [R(A) ∩ N(B)]⊥ . xiv) N(ABA⊥ ) = R(A) + [N(A) ∩ R(B)] + [N(A) ∩ N(B)]. xv) R(I − AB) = N(A) + N(B). xvi) N(I − AB) = R(A) ∩ R(B). xvii) R([A, B]) = R(ABA⊥ ) + R(A⊥ BA). xviii) R(AB + BA) = [R(A) + R(B)] ∩ [R(A) + N(B)] ∩ [N(A) + R(B)]. xix) N(AB + BA) = [R(A) ∩ N(B)] + [N(A) ∩ R(B)] + [N(A) ∩ N(B)]. Furthermore, the following statements are equivalent: xx) R(A + B) = Fn . xxi) R(B⊥ ) = R(B⊥ AB⊥ ). Source: [242, 256, 1247, 2239, 2712]. Fact 4.18.12. Let A, B ∈ Fn×n, and assume that A and B are projectors. Then, ([ ]) A⊥ R(A) ∩ R(B) = N . B⊥ Furthermore, the following statements are equivalent: i) R(A) ∩ R(B) = {0}. ii) R(A − B) = R(A) + R(B). iii) rank(A − B) = rank A + rank B. iv) R(AB⊥ ) + R(A⊥ B) = R(A) + R(B). v) rank AB⊥ + rank A⊥ B = rank A + rank B. Source: Fact 8.9.2 and [242]. Related: Fact 8.8.17. Fact 4.18.13. Let A, B ∈ Fn×n, and assume that A and B are projectors. Then, A + A⊥ B is range Hermitian. Source: [258]. Fact 4.18.14. Let A, B ∈ Fn×n, assume that A and B are projectors, assume that rank A+rank B = △ rank(A + B) = n, and define P = A(A + B)−1 . Then, the following statements hold: i) P is the idempotent matrix onto R(A) along R(B). ii) A(A + B)−1B = 0. iii) A(A + B)−1A = A. Related: Fact 4.16.4. Fact 4.18.15. Let A, B ∈ Fn×n, where A and B are projectors. Then, AB is group invertible. Source: N(BA) ⊆ N(BABA) ⊆ N(ABABA) = N(ABAABA) = N(ABA) = N(ABBA) = N(BA). Remark: See [2869]. Remark: Fact 10.11.23 shows that AB is semisimple. Fact 4.18.16. Let A, B ∈ Fn×n, where A and B are projectors. Then, [A, B] is skew Hermitian, and rank [A, B] is even. Source: [406]. Fact 4.18.17. Let A, B ∈ Fn×n, assume that A and B are projectors, and assume that rank A = rank B. Then, there exists a reflector S ∈ Fn×n such that A = SBS . If, in addition, A + B − I is nonsingular, then one such reflector is given by S = ⟨A + B − I⟩(A + B − I)−1. Source: [726]. Remark: ⟨·⟩ is defined in Chapter 10. Fact 4.18.18. Let A, B ∈ Fn×n, assume that A and B are projectors, and let α, β ∈ R, where αβ , 0 and α + β , 0. Then, rank(A + B) = rank [A B] = rank A + rank B − n + rank(A⊥ + B⊥ )

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= rank A + rank(B − AB) = rank B + rank(A − BA) = rank(αA + βB), R(αA + βB) = R([A B]),

sig(A − B) = rank A − rank B,

rank(A − B) = 2 rank [A B] − rank A − rank B = rank(A − AB) + rank(AB − B), rank(I − A − B) = 2 rank AB + n − rank A − rank B = rank AB + rank A⊥ B⊥ , rank [A B] = rank A + rank B − n + rank [A⊥ B⊥ ], rank [A, B] = 2(rank [AB BA] − rank AB) = 2 rank [AB − (AB)2 ] = 2 rank(AB − ABA) = 2 rank(AB − BAB) = rank(A − B) + rank(A + B − I) − n = 2(rank [A B] + rank AB − rank A − rank B), [A, B] = (A − B)(A + B − I) = (A + B − I)(B − A), AB + BA = (A + B)(A + B − I) = (A + B − I)(A + B), rank(I − AB) = rank(I − BA) = rank [A B] − rank A − rank B + n, N(I − AB) = N(I − BA) = R(A) ∩ R(B), rank [AB BA] = rank(AB + BA) = rank[AB − (AB)2 ] + rank AB = rank(A + B) + rank(A + B − I) − n = rank [A B] + 2 rank AB − rank A − rank B, rank(AB − ABA) = rank AB + rank [A B] − rank A − rank B = rank AB + rank(B − AB) − rank B = rank BA + rank(A − AB) − rank A, rank [AB − (AB)2 ] = rank(I − AB) + rank AB − n = rank [A⊥ B⊥ − (A⊥ B⊥ )2 ] = rank [A BA] − rank A = rank [B AB] − rank B = rank[(AB)+ − BA] = rank [A B] + rank AB − rank A − rank B = rank[2A(A + B)+B − AB], rank [A BA] = rank [A B] + rank AB − rank B. Hence, 2A(A + B)+B = AB if and only if (AB)2 = AB. Furthermore, AB = BA if and only if rank(A + B) + rank AB = rank A + rank B. Finally, if AB = BA, then rank(A − B) + 2 rank AB = rank A + rank B. Source: [406, 2657, 2659, 2660, 2672, 2673, 2676]. Fact 4.18.19. Let A, B ∈ Fn×n, and assume that A and B are projectors. Then, the following

statements are equivalent: i) R(A) + R(B) = Fn . ii) A + B − AB is nonsingular. iii) B + A − BA is nonsingular. Furthermore, the following statements are equivalent: iv) A − B is nonsingular. v) rank [A B] = rank A + rank B = n. vi) R(A) and R(B) are complementary subspaces.

MATRIX CLASSES AND TRANSFORMATIONS

415

If these statements hold, then the following statements hold: vii) I − AB and I − BA are nonsingular. viii) A + B − AB and B + A − BA are nonsingular. ix) The idempotent matrix M ∈ Fn×n onto R(A) along R(B) is given by M = (I − AB)−1A(I − AB) = A(I − BA)−1 (I − AB) = (I − BA)−1 (I − B) = A(A + B − BA)−1. x) M satisfies M + M ∗ = (A − B)−1 + I, and thus (A − B)−1 = M + M ∗ − I = M − M⊥∗ . Source: Fact 7.13.27 and [15, 248, 601, 1134, 1232, 1251, 1504, 2296]. ix) follows from Fact 7.13.28. Remark: Fact 8.8.14 provides an alternative expression for M involving the MoorePenrose generalized inverse. Related: Fact 4.16.4, Fact 4.16.9, and Fact 8.8.15. Fact 4.18.20. Let A, B ∈ Fn×n, assume that A and B are projectors, assume that A , 0, B , 0, and A , B, and let α, β ∈ F, where α and β are nonzero. Then, αA + βB is a projector if and only if exactly one of the following statements holds: i) α = β = 1 and AB = BA = 0. ii) α = −β = 1 and AB = BA = B. iii) −α = β = 1 and AB = BA = A. iv) α + β = 1, AB , BA, and (A − B)2 = 0. Source: [220, 251]. Related: Fact 8.5.13 and Fact 8.8.8. Fact 4.18.21. Let A, B ∈ Fn×n, and assume that A and B are projectors. Then, tr AB ≤ rank AB ≤ min {tr A, tr B}. Furthermore, the first and third terms are equal if and only if either A − B is a projector or B − A is a projector. In addition, for all k ≥ 1, dim[R(A) ∩ R(B)] ≤ tr (AB)k ≤ rank (AB)k . Source: [252]. Fact 4.18.22. Let A, B ∈ Fn×n, and assume that A and B are projectors. Then,

[

rank

A B

] B = 3 rank [A B] − rank A − rank B. A

Furthermore, the following statements are equivalent: [ ] i) AB AB is nonsingular. ii) A + B and A − B are nonsingular. iii) 3 rank [A B] = rank A + rank B + 2n. Source: [2667]. Related: Fact 7.9.25. Fact 4.18.23. Let A, B ∈ Fn×n, and assume that A and B are projectors. Then,   AB  A + B      .. . A+B  AB      .. .. ..   . . . rank   = l rank(A + B),      ..   . A+B AB      AB A+B

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where the size of the matrix is ln × ln. Source: [2676]. Fact 4.18.24. Let A1 , . . . , Ar ∈ Fn×n, assume that A1 , . . . , Ar are Hermitian, and consider the following statements: i) A1 , . . . , Ar are projectors. ∑ ii) ri=1 Ai is a projector. iii) For all distinct i, j ∈ {1, . . . , r}, it follows that Ai Aj = 0. ∑ ∑ iv) rank ri=1 Ai = ri=1 rank Ai . Then, if at least one of the pairs of statements [i),ii)], [i),iii)], [ii),iii)], [ii),iv)] holds, then i)–iv) ∑ hold. In particular, if A1 , . . . , Ar are projectors and ri=1 Ai = I, then, for all distinct i, j ∈ {1, . . . , r}, Ai Aj = 0. Source: [2403, pp. 400–402]. Remark: The last result is Cochran’s theorem. A stronger version is given by Fact 4.16.21. Problem: Extend this result to the case where A1 , . . . , Ar are idempotent but not necessarily projectors.

4.19 Facts on Reflectors △

Fact 4.19.1. If A ∈ Fn×n is a projector, then B = 2A − I is a reflector. Conversely, if B ∈ Fn×n is △ △ a reflector, then A = 21 (B + I) is a projector. Remark: The affine mapping f (A) = 2A − I from the projectors to the reflectors is one-to-one and onto. Related: Fact 4.13.29 and Fact 4.20.3. Fact 4.19.2. Let A ∈ Fn×n, and assume that A satisfies two out of the three properties (Hermitian,

unitary, involutory). Then, A also satisfies the remaining property. Furthermore, A satisfies all three properties if and only if A is a reflector. Related: Fact 4.17.14 and Fact 4.19.6. Remark: Properties of reflectors are discussed in [2084]. △ Fact 4.19.3. Let x ∈ Fn be nonzero, and define the elementary reflector A = I − 2(x∗x)−1xx∗. Then, the following statements hold: i) det A = −1. ii) If y ∈ Fn, then Ay is the reflection of y across {x}⊥. iii) Ax = −x. iv) 21 (A + I) is the elementary projector I − (x∗x)−1xx∗. Fact 4.19.4. Let x, y ∈ Fn. Then, there exists a unique elementary reflector A ∈ Fn×n such that Ax = y if and only if x∗y is real and x∗x = y∗y. If, in addition, x , y, then A is given by A = I − 2[(x − y)∗ (x − y)]−1(x − y)(x − y)∗. Remark: This is the reflection theorem. See [1184, pp. 16–18] and [2314, p. 357]. See Fact 4.11.5. Fact 4.19.5. Let n > 1, let S ⊂ Fn, and assume that S is a hyperplane. Then, there exists a

unique elementary reflector A ∈ Fn×n such that, for all y = y1 + y2 ∈ Fn, where y1 ∈ S and y2 = S⊥, it follows that Ay = y1 − y2 . Furthermore, if S = {x}⊥, then A = I − 2(x∗x)−1xx∗. Fact 4.19.6. Let A ∈ Fn×n, and assume that A satisfies two out of the three properties (skew Hermitian, unitary, skew involutory). Then, A also satisfies the remaining property. Furthermore, these matrices are the skew reflectors. Remark: Properties of skew reflectors are discussed in [2084]. Related: Fact 4.17.14, Fact 4.19.2, and Fact 4.19.7. Fact 4.19.7. Let A ∈ Cn×n. Then, A is a reflector if and only if ȷA is a skew reflector. Remark: △ The mapping f (A) = ȷA relates Fact 4.19.2 to Fact 4.19.6. Problem: Assuming A is real and n is even, determine a real transformation between the reflectors and the skew reflectors. Fact 4.19.8. Let A ∈ Fn×n. Then, the following statements are equivalent: i) A is a reflector. ii) A = AA∗ + A∗ − I. iii) A = 21 (A + I)(A∗ + I) − I.

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MATRIX CLASSES AND TRANSFORMATIONS

4.20 Facts on Involutory Matrices Fact 4.20.1. Let A ∈ Fn×n, and assume that A is involutory. Then, rank(A + I) + rank(A − I) = n. Fact 4.20.2. Let A ∈ Fn×n, and assume that A is involutory. Then, either det A = 1 or det A = −1. Fact 4.20.3. The following statements hold: △

△

i) If A ∈ Fn×n is idempotent, then B1 = 2A − I and B2 = I − 2A are involutory. △ △ ii) If B ∈ Fn×n is involutory, then A1 = 12 (I + B) and A2 = 12 (I − B) are idempotent. △

△

Remark: The affine mappings f1 (A) = 2A−I and f2 (A) = I−2A from the idempotent matrices to the

involutory matrices are one-to-one and onto with inverses f1inv (B) = 21 (I + B) and f2inv (B) = 21 (I − B) from the involutory matrices to the idempotent matrices. See Fact 4.13.29 and Fact 4.19.1. Fact 4.20.4. Let A ∈ Fn×n. Then, A is involutory if and only if (A + I)(A − I) = 0. Fact 4.20.5. Let n ≥ 1. Then, Iˆn is involutory. Fact 4.20.6. Let A ∈ Fn×n, and assume that A is involutory. Then, A + A∗ is nonsingular, and rank(A − A∗ ) = rank [A, A∗ ] = 2 rank [I + A I + A∗ ] − 2 rank(I + A) = 2 rank [I − A I − A∗ ] − 2 rank(I − A).

Source: [2672]. Fact 4.20.7. Let n ≥ 1. Then, Pn Iˆn Pn = Iˆn . Consequently, Pn Iˆn is involutory. Remark: This

equality helps to define the generators for the dihedral group D(n). See [1560, p. 169]. Fact 4.20.8. Let A, B ∈ Fn×n, and assume that A and B are involutory. Then, R([A, B]) = R(A − B) ∩ R(A + B),

R(A − B) + R(A + B) = Fn ,

N([A, B]) = N(A − B) + N(A + B),

N(A − B) ∩ N(A + B) = {0}

Source: [2640]. Fact 4.20.9. Let A, B ∈ Fn×n, and assume that A and B are involutory. Then,

[

rank(A + B) = rank

] I+A + rank [I + A I − B] − rank(I + A) − rank(I − B) I−B

= rank[(I + A)(I + B)] + rank[(I − A)(I − B)] = rank[(I + A)(I + B)] + rank[(I + B)(I + A)] − rank(I + A) − rank(I + B) + n, rank(A − B) = rank[(I + A)(I − B)] + rank[(I − B)(I + A)] − rank(I + A) − rank(I − B) + n, rank [A, B] = rank(A + B) + rank(A − B) − n. Consequently, AB = BA if and only if rank(A + B) + rank(A − B) = n. Source: [2672]. Fact 4.20.10. Let A ∈ Fn×m and B ∈ Fm×n, and define [ ] I − BA B △ C= . 2A − ABA AB − I Then, C is involutory. Source: [2036, p. 113]. Fact 4.20.11. Let A ∈ Rn×n, and assume that A is skew involutory. Then, n is even.

4.21 Facts on Tripotent Matrices 2 Fact 4.21.1. Let A ∈ Fn×n[, and ] assume that A is tripotent. Then, A is idempotent. Remark:

The converse is false. Let A = 00 10 . Fact 4.21.2. Let A ∈ Fn×n. Then, A is nonsingular and tripotent if and only if A is involutory. Fact 4.21.3. Let A ∈ Fn×n. Then, the following statements are equivalent: i) A is tripotent.

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R(A) and R(I − A2 ) are complementary subspaces. R(A) ⊆ N(I − A2 ). N(A) ∩ N(I − A2 ) = {0}. R(I − A2 ) ⊆ N(A). For all x ∈ R(A), A2 x = x. rank A = rank(A + A2 ) + rank(A − A2 ). rank A + rank(I − A2 ) = n. rank A + rank(I − A) + rank(I + A) = 2n. A is group invertible, and A2 is idempotent. Source: [257, 585, 2675]. Fact 4.21.4. Let A ∈ Fn×n, and assume that A is tripotent. Then, the following statements are equivalent: i) A is Hermitian. ii) A is normal. iii) A is range Hermitian and a partial isometry. Source: [257]. Fact 4.21.5. Let A ∈ Rn×n be tripotent. Then, rank A = rank A2 = tr A2 . Fact 4.21.6. Let A, B ∈ Fn×n , and assume that A and B are tripotent. Then, the following statements hold: i) If AB = BA, then AB is tripotent. ii) If A and B are Hermitian, then the following statements are equivalent: a) AB is tripotent and Hermitian. b) AB = BA. iii) If A and B are Hermitian and tripotent, then the following statements are equivalent: a) (AB)+ = B+A+ . b) A2B2 is Hermitian. c) (A2B2 )2 = B2A2 . iv) If A and B are Hermitian and tripotent, then the following statements are equivalent: ii) iii) iv) v) vi) vii) viii) ix) x)

∗

a) A ≤ B. b) ABA = A and BAB = A. ∗

c) ABA = A and A2 ≤ B2 . Source: [257]. Fact 4.21.7. If A, B ∈ Fn×n are idempotent and AB = 0, then A + BA⊥ is idempotent and △ △ △ C = A − B is tripotent. Conversely, if C ∈ Fn×n is tripotent, then A = 21 (C 2 + C) and B = 12 (C 2 − C) are idempotent and satisfy C = A − B and AB = BA = 0. Source: [1998, p. 215].

4.22 Facts on Nilpotent Matrices Fact 4.22.1. Let A ∈ Fn×n. Then, the following statements are equivalent:

i) R(A) = N(A). ii) A is similar to a block-diagonal matrix each of whose diagonal blocks is N2 . Source: To prove i) =⇒ ii), let S ∈ Fn×n transform A into its Jordan form. Then, it follows from Fact 3.13.3 that R(SAS −1 ) = S R(AS −1 ) = S R(A) = S N(A) = S N(AS −1S ) = N(AS −1 ) = N(SAS −1 ). The only Jordan block J that satisfies R(J) = N(J) is J = N2 . Using R(N2 ) = N(N2 ) and reversing these

419

MATRIX CLASSES AND TRANSFORMATIONS

steps yields the converse result. Remark: The fact that n is even follows from rank A + def A = n and rank A = def A. Related: Fact 4.22.2 and Fact 4.22.3. Fact 4.22.2. Let A ∈ Fn×n. Then, the following statements are equivalent: i) N(A) ⊆ R(A). ii) A is similar to a block-diagonal matrix each of whose diagonal blocks is either nonsingular or N2 . Related: Fact 4.22.1 and Fact 4.22.3. Fact 4.22.3. Let A ∈ Fn×n. Then, the following statements are equivalent: i) R(A) ⊆ N(A). ii) A is similar to a block-diagonal matrix each of whose diagonal blocks is either zero or N2 . Related: Fact 4.22.1 and Fact 4.22.2. Fact 4.22.4. Let n ∈ P and k ∈ {0, . . . , n}. Then, rank Nnk = n − k. Fact 4.22.5. Let A ∈ Rn×n. Then, the following statements hold: i) rank Ak is a nonincreasing function of k ≥ 1. ii) If there exists k ∈ {1, . . . , n} such that rank Ak+1 = rank Ak, then rank Al = rank Ak for all l ≥ k. iii) If A is nilpotent and Al , 0, then rank Ak+1 < rank Ak for all k ∈ {1, . . . , l}. Fact 4.22.6. Let A ∈ Fn×n. Then, A is nilpotent if and only if, for all k ∈ {1, . . . , n}, tr Ak = 0. Source: [2263, p. 103] or use Fact 6.8.4 with p = χA and µ1 = · · · = µn = 0. Fact 4.22.7. Let λ ∈ F and n, k ∈ P. Then,  () ()   λk In + 1k λk−1Nn + · · · + kk Nnk , k < n − 1,    (λIn + Nn )k =  ( ) ( )    λk I + k λk−1N + · · · + k λk−n+1N n−1 , k ≥ n − 1. n

Equivalently, for all k ≥ n − 1,   λ   0   ..  .   0  0

1 λ .. . 0 0

··· .. . .. . .. . ···

k 0    0   ..  .   1  λ

0 0 .. . λ 0

1

 k  λ     0    =  ...     0    0

n

(k ) 1

n

n−1

(

)

λk−1

···

λk

..

.

..

.

..

.

..

0

..

.

λk

0

···

0

(

k n−2

k n−3

)

λk−n+1 λk−n+2 .

(

)

 λk−n+1     ( k ) k−n+2   λ n−2    ..  . .    (k ) k−1   1 λ    k λ k n−1

Fact 4.22.8. Let A ∈ Rn×n, assume that A is nilpotent, and let k ≥ 1 satisfy Ak = 0. Then,

det(I − A) = 1,

(I − A)−1 =

k−1 ∑

Ai.

i=0

Fact 4.22.9. Let A ∈ R

, and assume that A = 0. Then, 21 A2 + A + I is nonsingular, and

n×n

3

( 12 A2 + A + I)−1 = 12 A2 − A + I. Fact 4.22.10. Let A ∈ Fn×n, and assume that A is nilpotent. Then, there exist idempotent matrices B, C ∈ Fn×n such that A = [B, C]. Source: [951]. Remark: A necessary and sufficient

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CHAPTER 4

condition for a matrix to be a commutator of a pair of idempotents is given in [951]. Related: Fact 11.11.2 in the case of projectors. Fact 4.22.11. Let A, B ∈ Fn×n, assume that B is nilpotent, and assume that AB = BA. Then, det(A + B) = det A. Source: Assuming A is nonsingular, apply Fact 4.22.8 with A replaced by −A−1B. Then, use continuity to remove the assumption that A is nonsingular. Alternatively, use Fact 7.19.5. Fact 4.22.12. Let A, B ∈ Rn×n, assume that A and B are nilpotent, and assume that AB = BA. Then, A + B is nilpotent. Source: If k, l ≥ 1 and Ak = Bl = 0, then (A + B)k+l = 0. Fact 4.22.13. Let A, B ∈ Fn×n, and assume that A and B are either both upper triangular or both lower triangular. Then, [A, B] is nilpotent. Source: [1065, 1066]. Related: Fact 7.19.7. Fact 4.22.14. Let A, B ∈ Fn×n, and assume that there exist k ∈ P and nonzero α ∈ R such that [Ak, B] = αA. Then, A is nilpotent. Source: For all l ∈ N, Ak+lB − AlBAk = αAl+1, and thus tr Al+1 = 0. The result now follows from Fact 4.22.6. See [2340]. Remark: If [A, B] = A, then A is nilpotent. Fact 4.22.15. Let A, B ∈ Fn×n. Then, the following statements hold: i) If [AA , B] = A, then A is nilpotent. ii) [AA , B] = AA , then (AA )2 = 0. iii) If either [A, [AA , BA ]] = 0 or [AA , [AA , BA ]] = 0, then ([A, B]A )2 = 0. Source: [1899].

4.23 Facts on Hankel and Toeplitz Matrices Fact 4.23.1. Let A ∈ Fn×m. Then, the following statements hold:

ˆ and AIˆ are Hankel. If A is Toeplitz, then IA ˆ and AIˆ are Toeplitz. If A is Hankel, then IA ˆ Iˆ is Toeplitz. A is Toeplitz if and only if IA ˆ Iˆ is Hankel. A is Hankel if and only if IA n×n Fact 4.23.2. Let A ∈ C . Then, the following statements hold: i) If A is Hankel, then A is symmetric. ii) If A is Hermitian and symmetric, then A is real. iii) If A is Hankel and Hermitian, then A is symmetric and real. Fact 4.23.3. Let A ∈ Fn×n, and assume that A is a partitioned matrix all of whose blocks are k×k (circulant, Hankel, Toeplitz) matrices. Then, A is similar to a block-(circulant, Hankel, Toeplitz) matrix. Source: [296]. △ Fact 4.23.4. For all i, j ∈ {1, . . . , n}, define A ∈ Rn×n by A(i, j) = 1/(i + j − 1). Then, A is Hankel, positive definite, and [1!2! · · · (n − 1)!]4 . det A = 1!2! · · · (2n − 1)! i) ii) iii) iv)

Furthermore, for all i, j ∈ {1, . . . , n}, A−1 has integer entries given by ( )( )( )2 n + i −1 n + j −1 i + j − 2 −1 i+ j (A )(i, j) = (−1) (i + j − 1) . n− j n−i i −1 Finally, as n → ∞, det A ∼ 4−n . Remark: A is the Hilbert matrix, which is a Cauchy matrix. See [724] and [1389, p. 513]. Related: Fact 2.2.63, Fact 4.27.5, Fact 4.27.6, Fact 10.9.5, Fact 10.9.8, and Fact 16.22.18. Fact 4.23.5. Let A ∈ Fn×n, and assume that A is Toeplitz. Then, A is reverse symmetric. 2

421

MATRIX CLASSES AND TRANSFORMATIONS

Fact 4.23.6. Let A ∈ Fn×n. Then, A is Toeplitz if and only if there exist a0 , . . . , an ∈ F and

b1 , . . . , bn ∈ F such that

A=

n ∑

bi NniT +

i=1

n ∑

ai Nni .

i=0

Fact 4.23.7. Let A ∈ F , let k ≥ 1, and assume that A is (lower triangular, strictly lower triangular, upper triangular, strictly upper triangular). Then, so is Ak. If, in addition, A is Toeplitz, then so is Ak. Remark: If A is Toeplitz, then A2 is not necessarily Toeplitz. Related: Fact 15.14.1. Fact 4.23.8. Let n ≥ 2 and m ≥ 2, and define A ∈ Fn×m by n×n

  sin θ   sin 2θ   △  A =  sin 3θ  .  ..   sin nθ

sin 2θ sin 3θ . ..

sin 3θ . .. ..

.

..

.

..

.

..

.

..

.

··· . .. . .. . .. . ..

sin mθ . .. . .. . .. sin (m + n − 1)θ

       .    

Then, rank A = 2. Source: Proposition 16.9.13. Fact 4.23.9. Let A ∈ Fn×n be the tridiagonal, Toeplitz matrix    b c 0 · · · 0 0     a b c · · · 0 0     0 a b . . . 0 0   △  , A =  . . . . . . . . . . . ...   .. ..     0 0 0 . . . b c      0 0 0 ··· a b √ √ △ △ and define α = 12 (b + b2 − 4ac) and β = 12 (b − b2 − 4ac). Then,  n  b, ac = 0,       det A =  (n + 1)(b/2)n , b2 = 4ac,      (αn+1 − βn+1 )/(α − β), b2 , 4ac. Source: [2983, pp. 101, 102]. Remark: The square root is the principal square root. Related:

Fact 4.24.3 and Fact 7.12.46. Fact 4.23.10. Let A ∈ Rn×n be the Toeplitz matrix  a a2 · · ·  1   b 1 a ···  ..  2 . b 1  b △  A =  . . . .  .. .. .. ..  .  n−2 .. bn−3 bn−4  b  bn−1 bn−2 bn−3 · · ·

an−2 an−1 an−2 .. . 1 b

 an−1   an−2    an−3   ..  . .   a   1

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Then, A is nonsingular if and only if ab , 1. If these conditions hold, then  −ac 0 ··· 0 0  c   −ac ··· 0 0  −bc (ab + 1)c   ..  0 . −bc (ab + 1)c 0 0   −1 A =  . . .. .. .. ..  .. .. . . . .   ..  . (ab + 1)c −ac 0 0  0   0 0 0 ··· −bc c

          ,       

△

where c = (1 − ab)−1. Now, assume that a = b. Then, A is nonsingular if and only if |a| < 1. If these conditions hold, then A is positive definite. Source: [2403, pp. 348, 349]. Fact 4.23.11. Let A ∈ Fn×m , B ∈ Fl×m , C ∈ Fn×p , D ∈ Fl×p , and E ∈ Fl×q , and assume that A has full column rank and R(A) ∩ R(C) = {0}. Then, ([ ]) ([ ]) A C 0 R ∩ = {0}. B D E Fact 4.23.12. Let r ≥ 3, for all i ∈ {1, . . . , r}, let Ai ∈ Fn×m , and define the block-Toeplitz matrix

  A1   A 2   .  ..    Ar−1  Ar

0 A1 .. . Ar−2 Ar−1

··· .. . .. . .. . ···

0 0 .. . A1 A2

 0    0   [ ..  = C r .   0   A1

Cr−1

···

C2

] C1 ,

where, for all i ∈ {1, . . . , r}, Ci ∈ Frn×m . Furthermore, let l ∈ {2, . . . , r−1}, and assume that Cl has full column rank and R(Cl ) ∩ R([Cl−1 · · · C1 ]) = {0}. Then, [Cr · · · Cl ] has full column rank, and R([Cr · · · Cl ]) ∩ R([Cl−1 · · · C1 ]) = {0}. Credit: A. Ansari.

4.24 Facts on Tridiagonal Matrices Fact 4.24.1. Let A ∈ Fn×n. Then, the following statements hold:

i) If A is upper triangular, then A is upper Hessenberg and upper bidiagonal. ii) If A is lower triangular, then A is lower Hessenberg and lower bidiagonal. iii) The following statements are equivalent: a) A is diagonal. b) A is upper bidiagonal and lower bidiagonal. c) A is upper triangular and lower triangular. iv) The following statements are equivalent: a) A is upper bidiagonal. b) A is tridiagonal and upper triangular. c) A is lower Hessenberg and upper triangular. v) The following statements are equivalent: a) A is lower bidiagonal. b) A is tridiagonal and lower triangular.

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c) A is upper Hessenberg and lower triangular. vi) A is tridiagonal if and only if A is upper Hessenberg and lower Hessenberg. Fact 4.24.2. Let A ∈ Fn×n, assume that A is nonsingular, and let l ∈ {0, . . . , n} and k ∈ {1, . . . , n}. Then, the following statements are equivalent: i) Every submatrix B of A whose entries are located above the lth superdiagonal of A satisfies rank B ≤ k − 1. ii) Every submatrix C of A−1 whose entries are located above the lth subdiagonal of A−1 satisfies rank C ≤ l + k − 1. Specifically, the following statements hold: iii) A is lower triangular if and only if A−1 is lower triangular. iv) A is upper triangular if and only if A−1 is upper triangular. v) A is diagonal if and only if A−1 is diagonal. vi) A is lower Hessenberg if and only if every submatrix C of A−1 whose entries are located either on or above above the diagonal of A−1 satisfies rank C ≤ 1. vii) A is upper Hessenberg if and only if every submatrix C of A−1 whose entries are located either on or below the superdiagonal of A−1 satisfies rank C ≤ 1. viii) A is tridiagonal if and only if every submatrix C of A−1 whose entries are located either on or above the diagonal of A−1 satisfies rank C ≤ 1 and every submatrix C of A−1 whose entries are located either on or below the diagonal of A−1 satisfies rank C ≤ 1. ix) Every submatrix B of A whose entries are located above the diagonal of A satisfies rank B ≤ 1 if and only if every submatrix C of A−1 above the diagonal of A−1 satisfies rank C ≤ 1. Source: [2557]. Remark: The 0th subdiagonal and the 0th superdiagonal are the diagonal. Remark: iii) corresponds to l = 0 and k = 1, iv) corresponds to l = 0 and k = 1 applied to A and AT, v) corresponds to l = 1 and k = 1, and vi) corresponds to l = 1 and k = 1 applied to A and AT. Related: Fact 3.14.27. Remark: Extensions to generalized inverses are considered in [280, 2317]. Fact 4.24.3. Let A ∈ Fn×n be the tridiagonal, Toeplitz matrix   ab 0 ··· 0 0   a + b     1 a + b ab · · · 0 0    ..  . 1 a+b 0 0   0  △ A =  . .. ..  . .. .. ..  .. . . . . .     ..   0 . a+b 0 0 ab     0 0 0 ··· 1 a+b Then,

   (n + 1)an, a = b,      det A =    an+1 − bn+1    , a , b.  a−b

Source: [1674, pp. 401, 621]. Fact 4.24.4. Let A ∈ Rn×n, assume that A is tridiagonal with positive diagonal entries, and

assume that, for all i ∈ {2, . . . , n},

A(i,i−1) A(i−1,i) < 14 A(i,i) A(i−1,i−1) sec2

π n+1 .

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Then, det A > 0. If, in addition, A is symmetric, then A is positive definite. Source: [1542]. Related: Fact 4.24.5 and Fact 10.9.22. Fact 4.24.5. Let A ∈ Rn×n, assume that A is tridiagonal and symmetric with positive diagonal △

△

entries, define α = min {A(1,1) , A(3,3) , . . .} and β = min {A(2,2) , A(4,4) , . . .}, and assume that ∑ ∑ 8αβ 0 for all i ∈ {1, . . . , n}}. △

iii) UT±1 (n) = {A ∈ UT(n): A(i,i) = ±1 for all i ∈ {1, . . . , n}}. △

iv) SUT(n) = {A ∈ UT(n): A(i,i) = 1 for all i ∈ {1, . . . , n}}. v) {In }. Remark: The matrices in SUT(n) are unipotent. See Fact 7.17.10. Remark: SUT(3) with F = R is the Heisenberg group. Related: Fact 4.25.2. Fact 4.31.12. Let n ≥ 1. Then, (SL(n), SO(n), SU(n), SUT(n)) is a normal subgroup of (GL(n), O(n), U(n), UT(n). Source: [2336, p. 53]. △ Fact 4.31.13. Let P ∈ Fn×n, and define S = {A ∈ GLF (n): ATPA = P}. Then, S is a multiplication group. If, in addition, P is nonsingular and skew symmetric, then every matrix A ∈ S satisfies det A = 1. Source: [765]. Remark: If F = R, n is even, and P = Jn , then S = SympR (n). Remark: Necessary and sufficient conditions are given in [765] under which every matrix A ∈ S satisfies det A = 1. Fact 4.31.14. Let A ∈ P(n). Then, there exist transposition matrices T 1, . . . , T k ∈ Rn×n such that A = T 1 · · · T k . Furthermore, the following statements hold: i) det A = (−1)k. ii) A is an even permutation matrix if and only if k is even. iii) A is an odd permutation matrix if and only if k is odd. Remark: Every permutation of n objects can be realized as[a finite ] sequence [ ] [ of transpositions. ] [ ] [ See ] [966, pp. 106, 107] and [1061, p. 82]. Example: P3 =

010 001 100

=

001 010 100

100 001 010

=

100 001 010

010 100 001

,

which represents a 3-cycle. Remark: As the above example shows, factorization in terms of transpositions is not unique. However, Fact 7.18.14 shows that every permutation matrix corresponds to a unique collection of disjoint cycles. Related: Fact 3.16.1 and Fact 4.13.18. Fact 4.31.15. For all n ≥ 2, the following statements hold: i) card[P(n)] = n!. ii) card[A(n)] = 21 n!. iii) card[D(n)] = 2n. iv) card[C(n)] = n.

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MATRIX CLASSES AND TRANSFORMATIONS

In addition, the following statements hold: v) A(2) ⊂ C(2) = P(2) ⊂ D(2). vi) A(3) = C(3) ⊂ P(3) = D(3). vii) If n ≥ 4, then } C(n) ⊂ D(n) ⊂ P(n) ⊂ O(n), A(n) viii) If n ≥ 5 and n is odd, then

{ C(n) ⊂

A(n) ⊂ SO(n) ⊂ O(n).

} D(n) ⊂ P(n) ⊂ O(n). A(n)

ix) If n ≥ 4 and n(n − 1)/2 is even, then C(n) ⊂ D(n) ⊂ A(n) ⊂ P(n) ⊂ O(n). Source: Fact 3.16.1 and Fact 3.16.2. Fact 4.31.16. Let k be a positive integer, define Rk ∈ R2×2 by    cos 2π  sin 2π △  k k    , Rk =  2π 2π  − sin k cos k and note that R1 = I2 , R2 = −I2 , and Rkk = I2 . Furthermore, define △

ˆ ˆ ˆ k−1 Ok (2) = {I, Rk , . . . , Rk−1 k , I2 , I2 Rk , . . . , I2 Rk },

△

SOk (2) = {I, Rk , . . . , Rk−1 k },

△

SUk (1) = {1, e2πȷ/k , e4πȷ/k , . . . , e2(k−1)πȷ/k }. Then, the following statements hold: i) C(1) = SU1 (1) = {1} ≃ SO1 (2) = {I2 }. ii) O1 (2) = P(2) = C(2) = {I2 , Iˆ2 } ≃ SO2 (2) = {I2 , −Iˆ2 } ≃ {I2 , −I2 } ≃ SU2 (1) = {1, −1}. iii) O2 (2) = D(2) = {I2 , −I2 , Iˆ2 , −Iˆ2 }. √

√

iv) C(3) = {I3 , P3 , P23 } ≃ SO3 (2) = {I2 , R3 , R23 } ≃ SU3 (1) = {1, − 12 + 23 ȷ, − 12 − 23 ȷ}. v) P(3) = {I3 , Iˆ3 , P3 , P3 Iˆ3 , Iˆ3 P3 , Iˆ3 P3 Iˆ3 }. ] [ ]} ] [ ] [ ] [ {[ ] [ 0 1 , −1 −1 , 0 1 , −1 −1 , 1 0 . vi) P(3) ≃ 10 01 , −1 0 1 1 0 10 −1 −1 −1 √ √ √ √           [ ]  − 1 3 0   − 1 − 3 0   − 1 3 0   − 1 − 3 0     1 0 0 2    √2 2   √2 2   √2 2   √2  0 −1 0 ,  I , . vii) D(3) ≃  , , ,         3 3 3 3 3 1 1 1 1             − − − 0 − 0 0 0 0 0 −1           2 2 2 2 2 2 2 2 0

0

1

0

0

1

0

0

−1

0

0

−1

viii) C(4) = {I4 , P4 , P24 , P34 } ≃ SO4 (2) = {I2 , R4 , R24 , R34 } ≃ SU4 (1) = {1, −1, ȷ, − ȷ}. ix) A(4) ≃ {I3 , D1, D2 , D3 , P3 , D1P3 , D2 P3 , D3 P3 , P23 , D1P23 , D2 P23 , D3 P23 }, △ △ △ where D1 = diag(1, −1, −1), D2 = diag(−1, 1, −1)}, and D3 = diag(−1, −1, 1). x) P(4) ≃ {A ∈ SO(3) : |A| ∈ P(3)}. { [ ] [ ] [I 0 ] [ ] [ P 0 ]} xi) C(6) ≃ I5 , P02 I03 , I02 P03 , 02 P2 , P02 P03 , 02 P2 . 3

3

xii) For all k ≥ 2, Ok (2) ≃ D(k) and card[Ok (2)] = 2k. xiii) For all k ≥ 2, C(k) ≃ SOk (2) ≃ SUk (1) and card[SOk (2)] = k. Remark: The multiplication groups P(k), A(k), D(k), and C(k) are isomorphic to symmetry groups, which are abstract groups consisting of transformations that map a set onto itself. Specifically, P(k), A(k), D(k) ≃ Ok (2), and C(k) ≃ SOk (2) are isomorphic to the symmetric group Sk , the alternating group Ak , the dihedral group Dk , and the cyclic group Ck , respectively. The elements of Sn ≃ P(n) permute n-tuples arbitrarily, while the elements of An ≃ A(n) permute n-tuples evenly. See Fact 7.18.14 for the decomposition of a permutation matrix in terms of cyclic permutation matrices.

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The elements of SOk (2) perform counterclockwise rotations of planar figures by the angle 2π/k about a line perpendicular to the plane and passing through 0, while the elements of Ok (2) perform the rotations of SOk (2) and reflect planar figures across the line y = x. See [966, pp. 41, 845]. Matrix representations of groups are discussed in [707, 1098, 1257, 1306, 1436, 2424]. Remark: Every finite subgroup of O(2) is isomorphic to either Dk or Ck for some k. Furthermore, every finite subgroup of SO(3) is isomorphic to either Dk for some k, Ck for some k, A4 , S4 , or A5 . The symmetry group D3 is isomorphic to the group of bijective transformations of an equilateral triangle, namely, three planar rotations and three 180-degree out-of-plane rotations about the medians. The symmetry groups A4 , S4 , and A5 are isomorphic to the group of bijective transformations of regular solids. Specifically, A4 is represented by the tetrahedral group, which consists of 4!/2 = 12 rotation matrices that map a regular tetrahedron onto itself; S4 is represented by the octahedral group, which consists of 4! = 24 rotation matrices that map either an octahedron or a cube onto itself; and A5 is represented by the icosahedral group, which consists of 5!/2 = 60 rotation matrices that map either a regular icosahedron or a regular dodecahedron onto itself. See [170, p. 184], [773, p. 32], [1199, pp. 176–193], [1259, pp. 9–23], [2345, p. 69], [2424, pp. 35–43], [2588, pp. 45–47], and [2589, Chapter 7]. Remark: The dihedral group D2 is the Klein four group. Remark: viii) is given in [1199, p. 180]. Remark: For all k ≥ 3, the permutation group Sk is not Abelian. The alternating group A3 is Abelian, whereas, for all k ≥ 4, Ak is not Abelian. For all k ≥ 5, Ak is simple; see [1560, p. 145] and [2336, pp. 50, 51]. This result is related to the classical result of H. Abel and E. Galois that there exist polynomials of degree 5 and greater whose roots cannot be expressed in terms of radicals involving the coefficients. Two such polynomials are p(x) = x5 − x − 1 and p(x) = x5 − 16x + 2. See [170, p. 574], [173, Chapter 13], [966, pp. 32, 625–639], [1069, pp. 488–494], [1290, Theorem 34], [1618], and [2264, pp. 199–203]. Quintic polynomials that can be solved in terms of radicals are discussed in [965]. Remark: The 24 elements of the octahedral group representing either S4 or P(4) are given in ix) by the 3 × 3 signed permutation matrices with determinant 1, where a signed permutation matrix has exactly one nonzero entry, which is either 1 or −1, in each row and column. Remark: x) shows that C6 ≃ C2 × C3 . See [1560, p. 169]. Remark: The converse of Lagrange’s theorem is not true. Eleven proofs are given in [566] of the fact that A4 , which (has) 12 elements, has no subgroup with 6 elements. In particular, it can be shown that none of the 11 5 = 462 subsets of A4 consisting of the identity and 5 nonidentity elements is closed under composition. See also [565]. Fact 4.31.17. The following statements hold: i) There exists exactly one isomorphically distinct group consisting of one element. ii) Let n ≥ 1. Then, there exists exactly one isomorphically distinct group consisting of n elements if and only if n is a cyclic number. This group is Cn . iii) The cyclic group C2 is isomorphic to the permutation group S2 and the multiplication groups P(2), C(2), O1 (2), SO2 (2), and SU2 (1) = {1, −1}. iv) The cyclic group C3 is isomorphic to the alternating group A3 and the multiplication groups A(3), C(3), SO3 (2), and SU3 (1). v) There exist exactly two isomorphically distinct groups consisting of four elements, namely, the cyclic group C4 and the dihedral group D2 . C4 is isomorphic to the multiplication groups C(4), SO4 (2), and SU4 (1) = {1, −1, ȷ, − ȷ}. D2 is isomorphic to the multiplication group O2 (2). vi) The cyclic group C5 is isomorphic to the multiplication groups C(5), SO5 (2), and SU5 (1). vii) There exist exactly two isomorphically distinct groups consisting of six elements, namely, the cyclic group C6 and the dihedral group D3 . C6 is isomorphic to the multiplication groups C(6), SO6 (2), and SU6 (1). D3 is isomorphic to the multiplication groups P(3) and O3 (2). D3 is the smallest group that is not Abelian.

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MATRIX CLASSES AND TRANSFORMATIONS

viii) The cyclic group C7 is isomorphic to the multiplication groups C(7), SO7 (2), and SU7 (1). ix) There exist exactly five isomorphically distinct groups containing eight elements, namely, ˆ FurC8 , D2 × C2 , C4 × C2 , D4 , and the quaternion multiplication group {±1, ±ˆı, ± ȷˆ, ±k}. thermore, C8 is isomorphic to C(8), SO8 (2), and SU8 (1); D4 is isomorphic to O4 (2); and ˆ is isomorphic to the multiplication group given by v) of Fact 4.32.6. {±1, ±ˆı, ± ȷˆ, ±k} Source: [1178, pp. 4–7] and [1560, pp. 168–172]. ii) is given in [2337, p. 7]. Remark: Cyclic numbers are defined in Fact 1.20.4. Remark: SUk (1) is defined in Fact 4.31.16. Remark: The Euler totient function is defined in Fact 1.20.4. Remark: There are 267 isomorphically distinct groups consisting of 64 elements, 2328 isomorphically distinct groups consisting of 128 elements, 56092 isomorphically distinct groups consisting of 264 elements, and 10494213 isomorphically distinct groups consisting of 512 elements. See [1560, p. 168] and [2336, p. 294]. There are 73 nonprime positive integers n between 1 to 520 inclusive for which there is exactly one isomorphically distinct group consisting of n elements. These 73 values of n are cyclic numbers that are not prime. See Fact 1.20.4. Remark: There are 173 isomorphically distinct simple groups whose cardinality is less than 1000. Five of these isomorphically distinct groups are not Abelian. Remark: The collection of finite simple groups, that is, groups that contain no normal subgroups other than the identity subgroup and the group itself, consists of 18 countably infinite sets of finite groups (one of which is the set of cyclic groups with a prime number of elements, while another is the set of alternating groups with either 5 or more elements) along with 26 finite groups called sporadic groups [2890, Chapter 5]. The largest sporadic group is the monster group M, which includes 19 sporadic groups as proper subgroups and has 246 · 320 · 59 · 76 · 112 · 133 · 17 · 19 · 23 · 29 · 31 · 41 · 47 · 59 · 71 = 808, 017, 424, 794, 512, 875, 886, 459, 904, 961, 710, 757, 005, 754, 368, 000, 000, 000 ≃ 8 × 1053 elements. None of the sporadic groups are Abelian. Remark: The Rubik’s cube group has 43, 252, 003, 274, 489, 856, 000 = 227 · 314 · 53 · 72 · 11 elements. This group is not simple and not Abelian, and the largest order of its elements is 1260. Rubik’s cube can be solved from an arbitrary orientation in either 20 or fewer moves.

4.32 Facts on Quaternions Fact 4.32.1. Let ıˆ, ȷˆ, kˆ satisfy

ıˆ2 = ȷˆ2 = kˆ 2 = −1, and define

ıˆȷˆ = kˆ = − ȷˆıˆ,

ȷˆkˆ = ıˆ = −kˆ ȷˆ,

ˆ ı = ȷˆ = −ˆık, ˆ kˆ

△ H = {a + bˆı + c ȷˆ + dkˆ : a, b, c, d ∈ R}. △

△

△

ˆ and |q| = ˆ q = a − bˆı − c ȷˆ − dk, Furthermore, for all a, b, c, d ∈ R, define q = a + bˆı + c ȷˆ + dk, √ 2 2 2 2 a + b + c + d = |q|. Then, qI4 = UQ(q)U, where

   a −b −c −d   b a −d c  △  , Q(q) =  d a −b   c d −c b a

  ıˆ ȷˆ k   1   ı 1 kˆ − ȷˆ △ 1  −ˆ U =  , ıˆ  2  − ȷˆ −kˆ 1  −kˆ ȷˆ −ˆı 1

and U satisfies U 2 = I4 . In addition, det Q(q) = (a2 + b2 + c2 + d2 )2.

√ qq =

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Furthermore, if |q| = 1, then △

[ a −b

−c −d b a −d c c d a −b d −c b a △

] is orthogonal. Next, for i = 1, 2, let ai , bi , ci , di ∈ R and △

ˆ Then, define qi = ai + bi ıˆ + ci ȷˆ + di kˆ and vi = [bi ci di ]T , and define q3 = q2 q1 = a3 + b3 ıˆ + c3 ȷˆ + d3 k. Q(q3 ) = Q(q2 )Q(q1 ),

q3 = q2 q1 , |q3 | = |q2 q1 | = |q1 q2 | = |q1 q2 | = |q1 q2 | = |q1 q2 | = |q1 | |q2 |,       a1    a3    b   a2 a1 − vT2 v1  b3  1  .     = Q(q2 )   =    c1   c3  a1v2 + a2 v1 + v2 × v1 d1 d3

Remark: q is a quaternion. See [1035, pp. 287–294]. Note the analogy between ıˆ, ȷˆ, kˆ and the unit vectors in R3 under cross-product multiplication. See [218, p. 119]. Remark: The group Sp(1) of unit-length quaternions is isomorphic to SU(2). See [803, p. 30], [2588, p. 40], Fact 4.28.11, and Fact 4.32.4. Remark: The unit-length quaternions, which are called Euler parameters, comprise the unit sphere S3 ⊂ R4 and provide a double cover of SO(3) as shown by Fact 4.14.6. See [57], [308, p. 380], and [773, 1709, 2448]. Remark: An equivalent formulation of quaternion multiplication is given by Rodrigues’s formulas. See Fact 4.14.8. Remark: Determinants and inverses of matrices with quaternion entries are discussed in [182, 2076], [2588, p. 31], and [2981]. Solutions of quaternion equations are given in [2323, 2753]. Calculus with quaternions is developed in [2076]. Remark: The Clifford algebras include the quaternion algebra H and the octonion algebra O, which involves the Cayley numbers. See [1035, pp. 295–300]. These ideas form the basis for geometric algebra. See [2505, p. 100] and [207, 773, 784, 808, 909, 925, 926, 1035, 1268, 1277, 1315, 1374, 1375, 1376, 1392, 1656, 1759, 1892, 2077, 2263, 2421, 2580, 2588, 2626]. △ Fact 4.32.2. Let a, b, c, d ∈ R, and let q = a + bˆı + c ȷˆ + dkˆ ∈ H. Then, q = a + bˆı + (c + dˆı) ȷˆ. Remark: H denotes the quaternion algebra. For all q ∈ H, there exist z, w ∈ C such that q = z + w ȷˆ, where we interpret C as {a + bˆı : a, b ∈ R}. This observation √ is analogous to the fact that, for all △ z ∈ C, there exist a, b ∈ R such that z = a + b ȷ, where ȷ = −1. See [2588, p. 10]. Fact 4.32.3. The following sets are groups: △ ˆ i) Q = {±1, ±ˆı, ± ȷˆ, ±k}. △ ˆ a, b, c, d ∈ R and a2 + b2 + c2 + d2 > 0}. ii) GLH (1) = H\{0} = {a + bˆı + c ȷˆ + dk: △ ˆ iii) Sp(1) = {a + bˆı + c ȷˆ + dk: a, b, c, d ∈ R and a2 + b2 + c2 + d2 = 1}. { [ 0 −1 0 0 ] [ 0 0 −1 0 ] [ 0 0 0 −1 ]} △ 0 , ± 0 0 0 1 , ± 0 0 −1 0 iv) QR = ±I4 , ± 10 00 00 −1 . 1 0 0 0 01 0 0 0 0 1 0 10 0 0 {[ a −b −c −d ] 0 −1 0 0 } △ 2 2 2 2 c v) GLH,R (1) = bc da −d a −b : a + b + c + d > 0 . d −c b a {[ a −b −c −d ] } △ ′ 2 2 2 2 b a −d c vi) GLH,R (1) = c d a −b : a + b + c + d = 1 . d −c b

a

Furthermore, Q ≃ QR , GLH (1) ≃ GLH,R (1), Sp(1) ≃ GL′H,R (1), and GL′H,R (1) ⊂ SO(4) ∩ SympR (4). Remark: J4 ∈ SympR (4) ∩ SO(4) but is not an element of GL′H,R (1). Related: Fact 4.32.1. △

△

T

Fact 4.32.4. Define Sp(n) = {A ∈ Hn×n : A∗A = I}, where H is the quaternion algebra, A∗ = A , △

ˆ Then, the groups Sp(n) and U(2n) ∩ SympC (2n) and, for q = a + bˆı + c ȷˆ+ dkˆ ∈ H, q = a − bˆı − c ȷˆ− dk. are isomorphic. In particular, Sp(1) and U(2) ∩ SympC (2) = SU(2) are isomorphic. Source: [206]. Remark: U(n) ≃ O(2n) ∩ SympR (2n). Related: Fact 4.32.3. Fact 4.32.5. Let n be a positive integer. Then, SO(2n) ∩ SympR (2n) is a multiplication group whose Lie algebra is so(2n) ∩ sympR (2n). Now, let A ∈ R2n×2n . Then, the following statements are equivalent: i) A ∈ SO(2n) ∩ SympR (2n).

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MATRIX CLASSES AND TRANSFORMATIONS

ii) A ∈ SympR (2n) and AJ2n = J2n A. Furthermore, the following statements are equivalent: iii) A ∈ so(2n) ∩ sympR (2n). iv) A ∈ sympR (2n) and AJ2n = J2n A. Source: [445]. Fact 4.32.6. Define Q0 , Q1, Q2 , Q3 ∈ C2×2 by [ ] [ ] [ ] −1 0 0 −ȷ △ △ 0 △ −ȷ △ Q0 = I2 , Q1 = , Q2 = , Q3 = . 1 0 0 ȷ −ȷ 0 Then, the following statements hold: i) Q∗0 = Q0 and Q∗i = −Qi for all i ∈ {1, 2, 3}. ii) Q20 = Q0 and Q2i = −Q0 for all i ∈ {1, 2, 3}. iii) Qi Qj = −Qj Qi for all 1 ≤ i < j ≤ 3. iv) Q1Q2 = Q3 , Q2 Q3 = Q1, and Q3 Q1 = Q2 . v) {±Q0 , ±Q1, ±Q2 , ±Q3 } is a group. △ For β = [β0 β1 β2 β3 ]T ∈ R4 define   3 ∑  β0 + β1 ȷ −(β2 + β3 ȷ)  △  .  Q(β) = βi Qi =  β − β ȷ β − β ȷ 2 3 0 1 i=0 Then,

Q(β)Q∗(β) = βTβI2 ,

det Q(β) = βTβ.

Hence, if βTβ = 1, then Q(β) is unitary. Furthermore, the complex matrices Q0 , Q1, Q2 , Q3 , and Q(β) have the real representations [ ] −J2 0 Q0 = I4 , Q1 = , 0 −J2       0 −1  0 −1 0   β0 −β1 −β2 −β3   0 0  0    β  0 0 −1  0 β0 −β3 β2  0  0 0 1  .  , Q(β) =  1  , Q3 =  Q2 =  β3 β0 −β1  0 0  0 0 0   β2  0 1  1 β3 −β2 β1 β0 1 0 0 0 0 −1 0 0 Hence,

Q(β)QT(β) = βTβI4 ,

det Q(β) = (βTβ)2.

ˆ See Fact 4.32.1. An alternative repreRemark: Q0 , Q1 , Q2 , Q3 represent the quaternions 1, ıˆ, ȷˆ, k.

sentation is given by the Pauli spin matrices σ0 = I2 , σ1 = ȷQ3 , σ2 = ȷQ1 , σ3 = ȷQ2 . See [1315, pp. 143–144], [1555]. Remark: For applications of quaternions, see [57, 1277, 1315, 1709]. Remark: [ A B] ˆ Q(β) has the form −B A , where A and IB are rotation-dilations. See Fact 3.24.1. △ n×m ˆ Fact 4.32.7. Let A, B, C, D ∈ R , define ıˆ, ȷˆ, kˆ as in Fact 4.32.1, and let Q = A + ıˆB + ȷˆC + kD. Then, [ ] [ ] In −ˆıIn 1 △ ∗ A + ıˆB −C − ıˆD diag(Q, Q) = Un Um , Un = √ . C − ıˆD A − ıˆB kIn 2 − ȷˆIn Furthermore, Un Un∗ = I2n . Source: [2670, 2671]. Remark: In the case n = m, this equality uses a similarity transformation to construct a complex representation of quaternions. Remark: The complex conjugate Un∗ is constructed as in Fact 4.32.7.

440

CHAPTER 4 △

ˆ Fact 4.32.8. Let A, B, C, D ∈ Rn×n, define ıˆ, ȷˆ, kˆ as in Fact 4.32.1, and let Q = A + ıˆB + ȷˆC + kD. Then,    A −B −C −D   B A −D C  U , diag(Q, Q, Q, Q) = Un  D A −B  n  C D −C B A

  In  −ˆıI 1 △ Un =  n 2  − ȷˆIn ˆn −kI

ıˆIn In ˆn −kI ȷˆIn

ȷˆIn ˆkIn In −ˆıIn

ˆ n  kI  − ȷˆIn  . ıˆIn   In

Furthermore, Un∗ = Un and Un2 = I4n . Source: [182, 568, 1017, 1254, 2670, 2671, 2981]. Remark: In the case n = m, this equality uses a similarity transformation to construct a real representation of quaternions. See Fact 3.17.38. Remark: The complex conjugate Un∗ is constructed by replacing ˆ ıˆ, ȷˆ, kˆ in UnT by −ˆı, − ȷˆ, −k. Fact 4.32.9. Let A ∈ C2×2. Then, A is unitary if and only if there exist θ ∈ R and β ∈ R4 such that A = eθ ȷQ(β), where Q(β) is defined in Fact 4.32.6. Source: [2314, p. 228].

4.33 Notes In the literature on generalized inverses, range-Hermitian matrices are traditionally called EP matrices. The name “EP” originated in [2408, p. 130], where “P” stands for “principal,” and “E” apparently stands for “equal.” However, since AA+ is the projector onto R(A) and A+A is the projector onto R(A∗ ), it is widely believed (see [405]) that “EP” stands for “equal projectors.” Elementary reflectors are also called either Householder matrices or Householder reflections. An alternative term for irreducible is indecomposable, see [1952, p. 147]. Diagonal, bidiagonal, and tridiagonal matrices are examples of sparse matrices, which have a high percentage of zero entries. Left equivalence, right equivalence, and biequivalence are treated in [2314]. Each of the groups defined in Proposition 4.6.6 is a Lie group; see Definition 15.6.1. Elementary treatments of Lie algebras and Lie groups are given in [170, 172, 218, 803, 998, 1023, 1176, 1177, 1471, 2210, 2343, 2421], while an advanced treatment appears in [2781]. Some additional groups of matrices are given in [1916]. Applications of group theory are discussed in [1560]. Almost nonnegative matrices are called ML-matrices in [2418, p. 208], essentially nonnegative matrices in [423, 434, 1288], and Metzler matrices in [2451, p. 402]. The terminology “idempotent” and “projector” is not standardized in the literature. Some writers use either “projector,” “oblique projector,” or “projection” [1133] for idempotent, and either “orthogonal projector” or “orthoprojector” for projector. Matrices with set-valued entries are discussed in [1170]. Matrices with entries having physical dimensions are discussed in [1327, 2187]. Graphs with specialized nodes are used to analyze the structure of molecules in [1518, 2730]. Connections between groups are illustrated in [646] as directed graphs with specialized edges in the form of Cayley diagrams.

Chapter Five Geometry 5.1 Facts on Angles, Lines, and Planes △

Fact 5.1.1. Let X = {x1 , . . . , xn } be a set of points in R2 , and assume that no three points in △

X lie in a single line. Furthermore, let L = {L1 , . . . , Ln(n−1)/2 } be the set of lines passing through all pairs of points in X, and assume that no pair of lines in L is parallel and no three of the lines in L intersect at a point that is not in X. Let P denote the set of polygons whose boundaries are subsets of ∪n(n−1)/2 Li and whose interiors are disjoint from ∪n(n−1)/2 Li . Then, the lines in L intersect i=1 i=1 at exactly 18 n(n − 1)(n − 2)(n − 3) points that are not in X, and the number of polygons in P is 1 3 2 8 (n − 1)(n − 5n + 18n − 8), of which n(n − 1) are not bounded. Source: [771, p. 72]. △

Fact 5.1.2. Let L = {L1 , . . . , Ln } be a set of lines in R2 , assume that no pair of lines in L is

parallel, and assume that the intersection of each triple of lines in L is empty. Let P denote the set of polygons whose boundaries are subsets of ∪ni=1 Li and whose interiors are disjoint from ∪ni=1 Li . Then, the number of polygons in P is 21 (n2 + n + 2), of which 21 (n − 1)(n − 2) are bounded. Source: [771, p. 72]. △ Fact 5.1.3. Let P = {P1 , . . . , Pn } be a set of planes in R3 , assume that no pair of planes in P is parallel, and assume that the intersection of each triple of planes in P is a single point. Let H denote the set of polyhedra in R3 whose boundaries are subsets of (∪)ni=1 Pi and whose interiors are ∑ disjoint from ∪ni=1 Pi . Then, the number of polyhedra in H is 3i=0 ni = 16 (n3 + 5n + 6), of which (n−1) = 31 (n − 1)(n − 2)(n − 3) are bounded. Source: [771, p. 72]. Remark: Extensions to 3 hyperplanes in Rn are discussed in [771, p. 72]. Fact 5.1.4. The points x, y, z ∈ R2 lie on one line if and only if [ ] x y z det = 0. 1 1 1 Fact 5.1.5. Let x, y, z ∈ Rn , assume that ∥x∥2 = ∥y∥2 = ∥z∥2 = 1, and let ϕ x,y ∈ (−π, π] denote

the angle between x and y. Then, ϕ x,z ≤ ϕ x,y + ϕy,z ,

sin ϕ x,z ≤ sin ϕ x,y + sin ϕy,z .

Furthermore, equality holds in each inequality if and only if x, y, z lie in a single plane. Source: [2991, pp. 31, 198]. Related: Fact 3.15.1 and Fact 11.8.9. Fact 5.1.6. The points w, x, y, z ∈ R3 lie in one plane if and only if [ ] w x y z det = 0. 1 1 1 1 Fact 5.1.7. Let x1 , . . . , xn ∈ Rn. Then,

[

rank

1 x1

··· ···

] [ 1 1 = rank xn x1

0 x2 − x1

··· ···

] 0 . xn − x1

442

CHAPTER 5

[

Hence,

1 rank x1

if and only if

··· ···

] 1 =n xn

rank [x2 − x1 · · · xn − x1 ] = n − 1.

If these conditions hold, then affin {x1 , . . . , xn } = x1 + span {x2 − x1 , . . . , xn − x1 }, and thus affin {x1 , . . . , xn } is an affine hyperplane. Finally, [ 1 1 n affin {x1 , . . . , xn } = {x ∈ R : det x x1

··· ···

] 1 = 0}. xn

Source: [2418, p. 31]. Related: Fact 5.1.8. Fact 5.1.8. Let x1 , . . . , xn+1 ∈ Rn. Then, the following statements are equivalent:

conv {x1 , . . . , xn+1 } is a simplex. conv {x1 , . . . , xn+1 } has nonempty interior. affin {x1 , . . . , xn+1 } = Rn. span {x2 − x1 , . . . , xn+1 − x1 } = Rn . [ ] 1 ··· 1 v) is nonsingular. x1 · · · xn+1 Source: The equivalence of i) and ii) follows from Fact 12.11.20. The equivalence of i) and iv) follows from Fact 3.11.14. Finally, the equivalence of iv) and v) follows from    1 1 1 · · · 1  [ ] [ ]  0 1 0 · · · 0    1 ··· 1 1 0 ··· 0  0 0 1 · · · 0  . =  x1 · · · xn+1 x1 x2 − x1 · · · xn+1 − x1  .. .. . . .  ..  . . . . ..    0 0 ··· ··· 1 i) ii) iii) iv)

Related: Fact 5.1.7 and Fact 12.11.13. Fact 5.1.9. Let z1, z2 , z be complex numbers, and assume that z1 , z2 . Then, the following

statements are equivalent: i) z lies on the line passing through z1 and z2 . z − z1 is real. ii) z2 − z1 [ ] z − z1 z − z1 iii) det = 0. z2 − z1 z2 − z1    z z 1    iv) det  z1 z1 1  = 0.   z2 z2 1 Furthermore, the following statements are equivalent: v) z lies on the line segment connecting z1 and z2 excluding the endpoints. z − z1 vi) is a positive number. z2 − z Source: [110, pp. 54–56].

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GEOMETRY

5.2 Facts on Triangles Fact 5.2.1. Let z1, z2 , z3 be distinct complex numbers. Then, the following statements are equivalent: i) z1, z2 , z3 are the vertices of an equilateral triangle. ii) |z1 − z2 | = |z2 − z3 | = |z3 − z1 |. iii) z21 + z22 + z23 = z1z2 + z2 z3 + z3 z1 . z3 − z1 z3 − z2 = . iv) z3 − z2 z1 − z2 Source: [110, pp. 70, 71] and [1757, p. 316]. Fact 5.2.2. Let n ≥ 2, let x1 , . . . , xn be positive numbers, assume that x1 = max {x1 , . . . , xn }, ∑ ∑ and assume that x1 ≤ ni=2 xi . Then, there exist complex numbers z1 , . . . , zn such that ni=1 zi = 0 and, for all i ∈ {1, . . . , n}, |zi | = xi . Source: [2979, pp. 189, 190]. [ ] [ ] [ ] Fact 5.2.3. Let S ⊂ R2 denote the triangle with vertices 00 , yx11 , yx22 ∈ R2. Then, [ ] 1 x1 x2 area(S) = det . y1 y2 2 [ ] [ ] [ ] Fact 5.2.4. Let S ⊂ R2 denote the triangle with vertices yx11 , yx22 , yx33 ∈ R2. Then,    1 1 1  1   area(S) = det  x1 x2 x3  . 2  y y y  1 2 3 Source: [2418, p. 32]. Fact 5.2.5. Let z1, z2 , z3 be complex numbers. Then, the area of the triangle S formed by z1, z2 , z3

 z 1  1 area(S) = det  z2 4  z

is given by

3

z1 z2 z3

 1   1  .  1

Source: [110, p. 79]. Fact 5.2.6. Let S ⊂ R3 denote the triangle with vertices x, y, z ∈ R3. Then,

area(S) = Furthermore,

√

1 2

[(y − x) × (z − x)]T [(y − x) × (z − x)] = 21 ∥(y − x) × (z − x)∥2 . area(S) = 21 ∥y − x∥2 ∥z − x∥2 sin θ,

where θ ∈ (0, π) is the angle between y − x and z − x. Now, assume that x = 0. Then, √ area(S) = 12 [y × z]T [y × z] = 12 ∥y × z∥2 . Furthermore,

area(S) = 12 ∥y∥2 ∥z∥2 sin θ,

where θ ∈ (0, π) is the angle between y and z. Remark: The connection between the norm of the cross product of two vectors and the angle between the vectors is given by xxviii) of Fact 4.12.1. Fact 5.2.7. Let S ⊂ R2 denote a triangle whose sides a, b, c have lengths a, b, c, let A, B, C denote the vertices opposite a, b, c with radian measure A, B, C, respectively, define the semiperimeter △ s = 12 (a + b + c), let r denote the inner radius of S, that is, the radius of the incircle, which is the largest inscribed circle, and let R denote the outer radius of S, that is, the radius of the circumcircle, which is the smallest circumscribed circle. The triangle S is acute if all of its angles are less than

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π 2;

right if one of its angles is π2 ; obtuse if one of its angles is greater than π2 ; equilateral if all of its sides are equal (equivalently, all of its angles are equal); and isosceles if two of its sides are equal (equivalently, two of its angles are equal). Then, the following statements hold: i) A + B + C = π. ii) a2 + b2 = c2 + 2ab cos C. iii) a2 = (b + c)2 sin2 A2 + (b − c)2 cos2 A2 . iv) a = b cos C + c cos B, ab cos C + bc cos A + ca cos B = 21 (a2 + b2 + c2 ). 1 sin A sin B sin C = = = . v) a b c 2R A+B cot C2 a + b tan 2 vi) = = . a − b tan A−B tan A−B 2 2 vii) viii) ix) x) xi) xii) xiii) xiv) xv) xvi) xvii) xviii) xix) xx) xxi) xxii)

A−B A−B A+B tan sin A − sin B 2 cos 2 sin 2 a − b sin 2 = = = = C C C c sin C cos 2 2 sin 2 cos 2 tan

A 2 A 2

− tan + tan

B 2 B 2

.

A−B a2 − b2 a + b cos 2 a2 − b2 sin(A − B) , = = = a cos B − b cos A, . C c c sin(A + B) c2 sin 2 cot A + cot B = bc csc A, a cos A + b cos B = b cos(A − B). a cos A + b cos B + c cos C = 2a(sin B) sin C. √ √ (s − b)(s − c) s(s − a) A A , cos 2 = . sin 2 = bc bc √ r (s − b)(s − c) = . tan A2 = s(s − a) s−a 2 √ b2 + c2 − a2 sin A = s(s − a)(s − b)(s − c), cos A = . bc√ 2bc 4 s(s − a)(s − b)(s − c) . tan A = b2 + c2 − a2 c2 (sin A) sin B 1 2 (sin A) sin B a2 − (b − c)2 area(S) = 21 ab sin C = . = 2 (a − b2 ) = 2 sin C sin(A − B) 4 tan A2 √3 abc 1 area(S) = rs = = 4 (a + b + c)2 (tan A2 )(tan B2 ) tan C2 = 12 (abc)2 (sin A)(sin B) sin C. 4R √ (s − a)(s − b)(s − c) . area(S) = s(s − a)(s − b)(s − c) = r √ area(S) = 14 (a + b + c)(a + b − c)(b + c − a)(c + a − b). √ area(S) = 41 (a2 + b2 + c2 )2 − 2(a4 + b4 + c4 ). √ area(S) = 41 2(a2 b2 + b2 c2 + c2 a2 ) − (a4 + b4 + c4 ). 2abc ab + bc + ca = (cos A2 )(cos B2 ) cos C2 . area(S) = 2(csc A + csc B + csc C) a + b + c △ Let S = 21 (sin A + sin B + sin C). Then, √ area(S) = 4R2 S (S − sin A)(S − sin B)(S − sin C).

xxiii) sin A, sin B, and sin C are the lengths of the sides of a triangle whose area is area(S)/(4R2 ). xxiv) If A , π2 , then area(S) = 41 (tan A)(b2 + c2 − a2 ).

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GEOMETRY

xxv) r = xxvi) xxvii) xxviii) xxix) xxx)

c(sin A2 ) sin cos

C 2

B 2

=

ab sin C = (s − a) tan 2s

A 2

= s(tan A2 )(tan B2 ) tan C2 .

√ area(S) (s − a)(s − b)(s − c) (a + b − c)(b + c − a)(c + a − b)R r= = = . s s 2abc abc a abc abc a = = = = √ R= . A A 2 sin A 4(sin 2 ) cos 2 4rs 4 area(S) 4 s(s − a)(s − b)(s − c) 2abc 1 R = = . r (a + b − c)(b + c − a)(c + a − b) cos A + cos B + cos C − 1 a = r cot B2 + r cot C2 = r cot B2 + r tan A+B 2 . a, b, c are the roots of the cubic equation x3 − 2sx2 + (s2 + r2 + 4Rr)x − 4Rrs = 0. That is,

xxxi) a, b, c satisfy

a + b + c = 2s,

ab + bc + ca = s2 + r2 + 4rR,

abc = 4Rrs.

ab + bc + ca = 4Rr + r2 + s2 , a2 + b2 + c2 = 2(s2 − r2 − 4Rr), a3 + b3 + c3 = 2s(s2 − 3r2 − 6Rr), (a + b)(b + c)(c + a) = 2s(s2 + r2 + 2Rr), a2 b2 c + b2 c2 a + c2 a2 b = 4Rrs(s2 + 4Rr + r2 ), a2 b2 + b2 c2 + c2 a2 = (s2 + 4Rr + r2 )2 − 16Rrs2 , a4 + b4 + c4 = 2s4 − 4(4Rr + 3r2 )s2 + 2(4Rr + r2 )2 , a(b − c)2 + b(c − a)2 + c(a − b)2 = 2s(s2 − 14Rr + r2 ), a2 b + b2 c + c2 a + ab2 + bc2 + ca2 = 2s(s2 − 2Rr + r2 ),

a4 b + b4 c + c4 a + ab4 + bc4 + ca4 = 2s(s4 − 3r4 − 14Rr3 − 8R2 r2 − 2r2 s2 − 6Rrs2 ), a3 b2 + b3 c2 + c3 a2 + a2 b3 + b2 c3 + c2 a3 = 2s(s4 + r4 + 6Rr3 + 8R2 r2 + 2r2 s2 − 10Rrs2 ), 1 1 1 1 1 1 1 s2 + r2 + 4Rr + + = , + + = , a b c 4Rrs ab bc ca 2Rr s2 + 10Rr + r2 (a + b)2 (b + c)2 (c + a)2 + + = , ab bc ca 2Rr )2 ( 2 1 a + b b + c c + a s2 + r2 − 2Rr 1 1 1 s + 4Rr + r2 − , + + = , + + = 2 2 2 4Rrs Rr c a b 2Rr a b c 1 1 4R + r a b c 4R − 2r 1 + + = , + + = , s−a s−b s−c rs s−a s−b s−c r a2 b2 c2 4s(R − r) + + = , s−a s−b s−c r a b c 2(4R + r) + + = , (s − b)(s − c) (s − c)(s − a) (s − a)(s − b) rs a2 b2 c2 4(R + r) + + = . (s − b)(s − c) (s − c)(s − a) (s − a)(s − b) r

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xxxii) S is equilateral if and only if a2 + b2 + c2 √= ab + bc + ca. If√ these conditions hold, then √ √ √ s = 32 a, r = 63 a, R = 33 a, and area(S) = 43 a2 = 3 3r2 = 3 4 3 R2 . Source: ii) is the law of cosines; v) is the law of sines; vi) is the law of tangents; vii) is Mollweide’s formula; viii) is Newton’s formula; xvii) is Heron’s formula, see [3024, pp. 318–320, 512–515]; iii) is given in [757]; ix) is given in [757, 759]; the last expression in xv) is given in [2527, p. 102]; the last expression in xvi) is given in [1559]; xxii) is given in [47, p. 124]; xxii) follows from v), see [2056]; xxviii) is given in [1938, p. 74]; xxix) is given in [215]; xxxi) is given in [1026, 1533] and [2062, pp. 52–54]. Remark: The role of triangle geometry in mathematics is discussed in [840]. Geometric constructions are described in [534, 1443]. Fact 5.2.8. Let S ⊂ R2 denote a triangle with the notation defined in Fact 5.2.7. Then, the following statements hold: 3R r √ 5 R r + , R(R − 2r) ≤ R − r, (4R + r)2 (2R − r) < 32R3 . i) 2r ≤ R, ≤ + , 2 ≤ 2 r R 4r R √   ii)   2 3s             9 r     √4 √   ( )   3/2   2  2 ( s )2 2 3 s  2 4Rr + r  R  ≤ . ≤ ≤ 2≤     3 r  r 27 r 3 r2 s       √     (     abc R) 3       − 2 4 2 +   r (a + b + 2c)(b + c + 2a)(c + a + 2b) √ 16R4 − (13/2)r3 R r 4R2 2R + , s≤ < , iii) 1 < s 2R 2R − r 2R − r √ ( ) √ √ 3 11 s ≤ 2R + (3 3 − 4)r ≤ R + r ≤ 3(2R − r), 4 2 √ √ √ (4R + r)2 R area(S) 1 abc 2rs 3 6Rr ≤ 3 3r ≤ ≤s= = (a + b + c) = ≤ R 2 r 2 4Rr 2(2R − r) √ √ √ √ √ √ 3 3 3 3R2 3 (4R + r) ≤ R ≤ 7R2 − r2 ≤ . ≤ 4R2 + 4Rr + 3r2 ≤ 2R + (3 3 − 4)r ≤ 3 2 4r ( ) iv) 2rs 2 (4R + r)2 r 27 (4R + r)2 r ≤ 27r2 ≤ ≤ 3r(4R + r) ≤ Rr ≤ 14Rr − r2 ≤ R 2R − r 2 R+r 2 2 √ 4r(12R − 11Rr + r ) 1 ≤ 16Rr − 5r2 ≤ ≤ 2 r[20R − r + 3(12R + r)(4R − 5r)] 3R − 2r √ √ ≤ 2R2 + 10Rr − r2 − 2(R − 2r) R(R − 2r) ≤ s2 ≤ 2R2 + 10Rr − r2 + 2(R − 2r) R(R − 2r) √ (4R + r)2 R ≤ ≤ 4R2 + 4Rr + 3r2 ≤ [2R + (3 3 − 4)r]2 ≤ 4R2 + 5Rr + r2 2(2R − r) 9 9 14 2 ≤ 4R2 + 6Rr − r2 ≤ R2 + 4Rr + r2 ≤ (R2 + Rr) ≤ (R + Rr) − r2 2 2 3 1 27 2 27R4 ≤ (4R + r)2 ≤ R ≤ 7R2 − r2 ≤ (4R + r)(2R − r) ≤ . 3 4 16r2 √ √ v) 8Rr + 119 3 area(S) ≤ s2 ≤ 4R2 + 119 3 area(S), √ √ 2 2 2 2 2 2 2 1 2 [(a−b) +(b−c) +(c−a) ]+3 3area(S) ≤ s ≤ 2[(a−b) +(b−c) +(c−a) ]+3 3area(S).

447

GEOMETRY

vi)

3r(4R + r) r 3r r(4R + r) r(16R + 3r) ≤ ≤ ≤ ≤ 2 2R − r 4R + r (2R − r)(2R + 5r) (4R − r)(4R + 7r) (7R − 5r) r(16R − 5r) 4r(12R2 − 11Rr + r2 ) ( s )2 r ≤ ≤ ≤ ≤ R+r 4R + r (4R + r)2 (3R − 2r)(4R + r)2 2 2 R 1 4R + r 4R + 4Rr + 3r R2 ≤ ≤ ≤ ≤ ≤ . 4R − 2r 3 27r 4r(R + r) (4R + r)2

vii)

27 Rr + r2 ≤ 14Rr ≤ 16Rr − 4r2 ≤ r2 + s2 ≤ 4(R2 + Rr + r2 ) ≤ 4R2 + 6Rr 2 27 2 27R4 14 2 (R + Rr) ≤ R + r2 ≤ 7R2 ≤ ≤ + r2 . 3 4 16r2 √   √  1 + 3 3  (1 + 3 3)r ≤ r + s ≤   R ≈ 3.098R. 2 √ △ 4s2 3 3 Define δ = 1 − 1 − 2r R . Then, δ(4 − δ) ≤ R2 ≤ (2 − δ)(2 + δ) . ( r ) p ( s ) p 1 + 33p/2 + ≤ . Let p ≥ 0. Then, R R 2p a b R 2≤ + ≤ . b( a r ) ( ) 2 a b c a b c 1 a2 b2 c2 R 2≤ + + ≤ + + −1≤ 1+ + + ≤ . 3 b c a b c a 2 bc ca ab r R 2abc = . 2≤ (a + b − c)(b + c − a)(c + a − b) r √ a(4r − R) R√ ≤ 2 (s − b)(s − c) ≤ a ≤ (s − b)(s − c). R r ( )5/2 1 2 r≤ (a + b) ≈ 0.1501(a + b). √ 2 1+ 5 1 1 1 If a ≤ b ≤ c, then − < . a c √2r √ √ √ 2 3 3 abc 1 abc 3 3 2 3 3 3 3r ≤ area(S) = rs ≤ Rr = ≤ Rs = ≤ R . 2 4 a+b+c 2 8r 4 )5/2 ( 1 2 area(S) ≤ (a + b)s. √ 2 1+ 5 √ bc s bcs bcs area(S) ≤ ≤ = . 4 s − a 2(b + c − a) 4(s − a) √ area(S) ≤ 21 (b + c) s(s − a). √ √ √ 3 3 2 3 2 2 area(S) ≤ (ab + bc + ca) − (a + b + c ) ≤ (ab + bc + ca). 6 12 12 √  3 1 2 2 2 2 2 2 2    4 min, {a + b , b + c , c + a } ≤ 24 (a + b + c) area(S) ≤    1 min, {a2 + b2 + 4c2 , b2 + c2 + 4a2 , c2 + a2 + 4b2 }.  √  12  3(a + b − c)(b + c − a)(c + a − b) abc 3 3  0<  a2  ≤ √ 2 2 b c 4 4 a 2 + b 2 + c2 b+c−a + c+a−b + a+b−c √4 3[(a + b − c)(b + c − a)(c + a − b)(a + b + c)]3/4 ≤ √ 4 a2 + b2 + c2

viii) ix) x) xi) xii) xiii) xiv) xv) xvi) xvii) xviii) xix) xx) xxi) xxii) xxiii)

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CHAPTER 5

√ abc √ = s(s − a)(s − b)(s − c) = a2 b2 + b2 c2 + c2 a2 − 12 (a4 + b4 + c4 ) ≤ area(S) = rs = 4R   3abc       √       2 + b2 + c2 4 a ≤      √ √     3 [2(ab + bc + ca) − (a2 + b2 + c2 )] = 3 (s2 − 1 [(a − b)2 + (b − c)2 + (c − a)2 ])   12 9 2   √3 √3 √    a2 b2 c2   4 √ √  √3 √  √ 3 3abc    √ ≤ 12 (a bc + b ca + c ab) ≤ 41 2abcs ≤ ≤  3abcs     8s     2(ab + bc + ca)     √ √     3 2 3 2     √ √     s s     9 6 3 2 2 2 (a + b + c ) ≤ ≤ . ≤ 123 (ab + bc + ca) ≤     √ √    12     1 a2 b2 + b2 c2 + c2 a2   1 a4 + b4 + c4    4 4 √ √ √ √ 2 √ √ xxiv) 3r 6Rr r(12R2 − 11Rr + r2 ) 2 2 0 < 3 3r ≤ 3r 4Rr + r ≤ ≤ r 16Rr − 5r ≤ 2r 2 3R − 2r √ √ abc 1 ≤ r 2R2 + 10Rr − r2 − 2(R − 2r) R(R − 2r) ≤ area(S) = rs = (a + b + c)r = 2 4R √ √ √ √ ≤ r 2R2 + 10Rr − r2 + 2(R − 2r) R(R − 2r) ≤ r 4R2 + 4Rr + 3r2 ≤ 2Rr + (3 3 − 4)r2 √ √ ≤ r 4R2 + 5Rr + r2 ≤ r 4R2 + 11 2 Rr √     √ √   14R2 + 14Rr − 3r2   2 2 2 2    ≤ r 5R + 4Rr − r  r 4R + 6Rr − r ≤ r       3   √ √ √   3 3 3 2 2 2 ≤  (4Rr + r ) ≤ r 6R + 3r ≤ Rr ≤   3 2           √     3 3   2 (R + 3r) 25   1 2  √ √    (R + 2 r)   25 2 √ ≤ 33 (R + r)2 ≤ 1225 3 (R + 12 r)2 ≤  ≤ R.    3 3 R2   16 4

xxv) (s − a)(s − b) ≤ ab, 8(s − a)(s − b)(s − c) ≤ abc, (a + b − c)(b + c − a)(c + a − b) ≤ abc. xxvi) (s − a)(s − b) + (s − b)(s − c) + (s − c)(s − a) ≤ 41 (ab + bc + ca). 9 1 1 1 4R + r xxvii) ≤ + + = . s s − a s −√b s − c rs √ √ √ √ xxviii) s ≤ s − a + s − b + s − c ≤ 3s. √ √ √ √ xxix) a(s − a) + b(s − b) + c(s − c) ≤ 2s. xxx) a(s − a) + b(s − b) + c(s − c) ≤ 9Rr. xxxi) abc < a2 (s − a) + b2 (s − b) + c2 (s − c) ≤ 23 abc. √ √ 3 3R3 s3 2 xxxii) a3 (s−a)+b3 (s−b)+c3 (s−c) ≤ abcs, 3 3r3 s3 ≤ s3 (s−a)(s−b)(s−c) ≤ 27 . 64 (abc) ≤ 8 n n n n+1 n n 1+n/2 1+n n xxxiii) Let n ≥ 0. Then, a + b + c ≤ 2 R + 2 (3 − 2 )r . √ √3 √ √ xxxiv) abc abc ≤ 6 3r ≤ 3 3 ≤ 3 abc ≤ a + b + c a+b+c R2   √     √ 3(a2 + b2 + c2 )   abc   ≤ ≤ 3 3R ≤ 2 . √  √    4r  4R + (6 3 − 8)r ≤ 2 3 (4R + r)   3

GEOMETRY

449

√ √ √     9(a2 bc + b2 ca + c2 ab) 2s       (a + b + c) ≤ √ √   √ √   2 2 3 ( a + b + c) 36r ≤ 2 3r(a + b + c) ≤ 18Rr ≤        √    24Rr − 12r2 ≤ 4R2 + 12Rr − 4r2 − 4(R − 2r) R2 − 2Rr     √   72R4  2 2 2 2 2 − 2Rr ≤     4R + 12Rr − 4r + 4(R − 2r) ≤ 8R + 4r R 2 − 4r 2 ≤ a2 + b2 + c2 ≤   9R   √     (s − 3r)(a + b + c) √ √ 4 3 rs ≤ 9R2 , a2 + b2 + c2 ≤ 64R4 + 48r2 s2 . ≤ 8R2 + 9   √3 √ √ √ 2  xxxvi) √     3 (abc) ≤ a bc + b ca + c ab ≤ ab + bc + ca  4 3rs ≤  √      2 2 2 4 3rs + (a − b) + (b − c) + (c − a) √ √ 4(R − 2r) 2 2 ≤ 4r 4R + 4Rr + 3r 3+ + (a − b)2 + (b − c)2 + (c − a)2 4R + r √ ≤ a2 + b2 + c2 ≤ 4 3rs + 3[(a − b)2 + (b − c)2 + (c − a)2 ]. xxxv)

If, in addition, S is acute, then

xxxvii) xxxviii) xxxix)

xl)

xli)

√ √ 2− √3 4(R + r)2 ≤ a2 + b2 + c2 ≤ 4 3rs + 3−2 [(a − b)2 + (b − c)2 + (c − a)2 ] 2 √ ≤ 4 3rs + 2[(a − b)2 + (b − c)2 + (c − a)2 ]. ( ) ( ) 16 2 2 1 1 1 1 1 16r2 s2 2 2 2 2 2 4 1 r s ≤ 36r ≤ + + ≤ a + b + c ≤ 9R ≤ 9R + + . 3 3R2 a2 b2 c2 a2 b2 c2 √ 27abc 9 3R 2 2 54Rr = ≤ (a + b + c) ≤ (a + b2 + c2 ). 2s √ 2s √ 36r2 ≤ 4 3rs ≤ 18Rr ≤ 20Rr − 4r2 ≤ 2R2 + 14Rr − 2(R − 2r) R2 − 2Rr √ ≤ ab + bc + ca ≤ 2R2 + 14Rr + 2(R − 2r) R2 − 2Rr ≤ 4(R + r)2 ≤ 9R2 . √ 8r(R − 2r) ≤ 4(R − 2r)(R + r − R(R − 2r)) ≤ (a − b)2 + (b − c)2 + (c − a)2 √ ≤ 4(R − 2r)(R + r + R(R − 2r)) ≤ 8R(R − 2r). √ √ √ 192 3r3 ≤ 64r2 s ≤ 96 3r2 R ≤ 12 3r2 (9R − 2r) ≤ 4rs(9R − 2r)

≤ (a + b)(b + c)(c + a) ≤ 4s(2R2 + 3Rr + 2r2 )   √     4(2R + (3 3 − 4)r)(2R2 + 3Rr + 2r2 )   ≤      8R(R + 2r)s √ √ ≤ 6 3R(2R2 + 3Rr + 2r2 ) ≤ 24 3R3 . { } √ √ √ √ √ 4Rr[2R + (3 3 − 4)r] ≤ 6 3R2 r xlii) 24 3r3 ≤ 12 3Rr2 ≤ abc ≤ ≤ 3 3R3 . 8 3 s  √ √27      xliii) 4 3rs(s − 3r)     √ 3 √     2 2 12Rrs ≤ 8rs(2R − r) ≤ 4rs(5R − 4r) 72 3r ≤ 24r s ≤ 36 3Rr ≤      √     12 3r2 (5R − 4r)     √ √    4R[2R + (3 3 − 4)r](2R − r) ≤ 6 3R2 (2R − r) 3 3 3 ≤ a + b + c ≤ 4R(2R − r)s ≤   18R2 s − 8r2 s − 20Rrs. 

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CHAPTER 5

 5R − r 5R − r      √ √   ≤ √     Rs  3 2 3(5R − r)    R[2R + (3 3 − 4)r] √ ≤ ≤     2   R 9R 3 3         2(R + r)   (R + r)2     √ √       3(R + r)2 3R 1 1 1    Rrs  ≤ ≤ , ≤ + + ≤ √   2 2   a b c  9Rr 4r   3       2r √ √ 2 R + 7Rr − (R − 2r) R(R − 2r) 1 1 1 R2 + 7Rr + (R − 2r) R(R − 2r) ≤ + + ≤ . 2Rrs a b c 2Rrs ( ) 2(5R − r) 1 1 1 2(R + r)2 9R xlv) ≤ (a + b + c) + + ≤ ≤ . R a b c Rr 2r xlvi) a4 + b4 + c4 ≤ 38 R(R − r)(4R + r)2 ≤ 54R3 (R − r).

xliv)

xlvii) xlviii) xlix) l) li)

lii)

liii) liv) lv)

lvi) lvii) lviii) lix) lx) lxi)

16r2 s2 2 (a + b2 + c2 ) ≤ 16r2 s2 ≤ a2 b2 + b2 c2 + c2 a2 ≤ 4R2 s2 ≤ 3R2 (a2 + b2 + c2 ). 9R2 ( )2 1 1 1 1 1 1 1 1 1 (R2 + r2 )2 + Rr3 1 ≤ + + ≤ 2+ 2+ 2 ≤ 2 3 ≤ ≤ 2. 2 2Rr 3 a b c R a b c R r (16R − 5r) 4r 7R − 2r a + b b + c c + a 2R2 + Rr + 2r2 R 2≤ ≤ + + ≤ ≤ . 3R 3c 3a 3b 3Rr r (5R − 2r)s ab bc ca ≤ + + . R s−c s−a s−b 6R s−a s−b s−c R2 + 3Rr + 2r2 + + ≤ . ≤ 2R2 + 3Rr + 2r2 b + c c + a a + b 9R − 2r  2  R    2r(R − 2r) ≤    4 min {(a − b)2 , (b − c)2 , (c − a)2 } ≤   2 2   s − 8Rr − 2r    . 5 R 16a2 b2 c2 ≤ . 2≤ 2 2 r [2a − (b − c) ][2b2 − (c − a)2 ][2c2 − (a − b)2 ] 2 2 a bc b ca c2 ab 18r2 ≤ + + . (b + c)(b + c − a) (c + a)(c + a − b) (a + b)(a + b − c) △ Let α = (log 9 − log 4)/(log 4 − log 3) ≈ 2.8, and let p ∈ [−1, α], where p , 0. Then, √ √ 2 3r ≤ [ 31 (a p + b p + c p )]1/p ≤ 3R. ( s )p 1 1 1 Let p ∈ [0, 1]. Then, p + p + p ≤ 31−2p . a b c Rr Let α, β, γ ≥ 0, and assume that α + β + γ = 1. Then,√2r ≤ min {a, b, c} ≤ aα bβ cγ ≤ √3 √3 √ √3 max {a, b, c} ≤ 2R. Furthermore, 2r ≤ 4Rr2 ≤ 2Rr ≤ 33 abc ≤ 2R2 r ≤ R. √ a < b + c ≤ a + 2b2 + 2c2 − a2 . 2bc . a 0. Then, 3 4 33rs ≤ ap + bp + cp.

lxii) 5184r6 ≤ a6 + b6 + c6 , lxiii) lxiv) lxv) lxvi) lxvii) lxviii) lxix)

576r6 ≤

a3

lxx) Let p ∈ R. Then, a p (s − a) + b p (s − b) + c p (s − c) ≤ 12 abc(a p−2 + b p−2 + c p−2 ). √ 9r 1 1 1 1 4− ≤ + + . lxxi) 2r 4R + r a + b − c b + c − a c + a − b √ √ c2 a2 b2 9r ≤ + + . lxxii) 3 3R ≤ 3R 4 − ( ) 4R + r a + b − c b + c − a c + a − b 1 9r 1 1 1 lxxiii) 5− ≤ + + . 8Rr 4R + r c(a + b − c) a(b + c − a) b(c + a − b) ) ( 1 1 9r 1 1 1 lxxiv) 2 − ≤ + + . 2 2 r 2 4(4R + r) (a + b − c) (b + c − a) (c + a − b)2 lxxv) Let x, y, z ∈ R, and assume that x + y, y + z, and z + x are positive. Then, √ x 2 y 2 z 2 2 3rs ≤ a + b + c . y+z z+x x+y √ √ lxxvi) 4 3rs + 12 (|a − b| + |b − c| + |c − a|)2 ≤ a2 + b2 + c2 ≤ 4 3rs + 23 (|a − b| + |b − c| + |c − a|)2 . 3 5s2 + r2 + 4Rr s2 + r2 − 8Rr 2(s2 − r2 − Rr) 3 s2 + r2 + 4Rr ≤ ≤ ≤ . ≤ < 2, 2 3(s2 + r2 + 2Rr) 2 12Rr 4Rr s2 + r2 + 2Rr a + 2b + 3c b + 2c + 3a c + 2a + 3b lxxviii) 18 ≤ + + . −a + b + c −b + c + a −c + a + b a2 b2 c2 1 + 2 + 2 . lxxix) ≤ 2 3 4a + 5bc 4b + 5ca 4c + 5bc 4 4 4 4 4 4 +b b +c c +a 14 2 2 2 lxxx) a2a+ab+b 2 + b2 +bc+c2 + c2 +ca+a2 < 2(a + b + c ) < 4(ab + bc + ca) < 3 (ab + bc + ca). lxxvii)

lxxxi) Let c = max {a, b, c}. Then, 3ab(a+b+2c) (a+c)(b+c) ≤ a + b + c. ( ) (s − a)(s − b) (s − b)(s − c) (s − c)(s − a) sr 5R − r ≤ + + . lxxxii) R 4R + r c a b 1 r 8Rr ≤ . lxxxiii) ≤ 2 2 R s + r + 2Rr 2 lxxxiv) (R + r)(s2 + r2 + 4Rr) ≤ 3R(s2 − r2 − 4Rr), 2r(R + r)(4R + r) ≤ 3R(s2 − 2r2 − 8Rr), 16s2 r(R + r) ≤ 3[(s2 + r2 + 4Rr)2 − 16s2 Rr], (R + r)(s2 + r2 − 8Rr) ≤ 3R(8R2 + r2 − s2 ),

2(R + r)s2 ≤ 3R[(4R + r)2 − 2s2 ],

(R + r)[s2 + (4R + r)2 ] ≤ 3R[(4R + r)2 − s2 ].

452

CHAPTER 5

3R a + b b + c c + a 3R b c + +1≤ , + + ≤ , 432r4 ≤ a2 b2 + b2 c2 + c2 a2 ≤ 27R4 . c b 2r c a b r√ √ √ √ )2 ( b c 9R 3 2R a a b c + + ≤ , + + ≤ . lxxxvi) b+c c+a a+b 8r s−a s−b s−c 2r 1 1 1 1 1 1 1 1 1 1 lxxxvii) 2 ≤ + + ≤ , ≤ + + ≤ . ab bc ca 4r2 R 3R4 a2 b2 b2 c2 c2 a2 48r4 (a + b)(b + c)(c + a) 4R ≤ . lxxxviii) 8 ≤ abc r Source: i) is given in [47, p. 112], [1730], and [2062, p. 165]; ii) is given in [47, p. 60], [367, 889], [1964], and [1938, p. 68]; iii) is given in [47, p. 60] and [1730]; iv)–viii) are given in [917] and [2062, pp. 166, 177, 189, 190]; ix) is given in [2927]; x) is given in [2062, p. 165]; xi) and xii) are given in [2791]; xiii) follows from Fact 5.2.7 and ix); xiv) is given in [2244] and [2341]; xv) and xvii) are given in [215]; xvi) is given in [316]; xx) is given in [1533]; xxii) is given in [535, p. 42] and [2062, p. 110]; xxiii) is given in [47, p. 91], [110, p. 145], [535, p. 42], and [1938, pp. 73, 74]; xxiv) is given in [535, pp. 49, 50], [1860, p. 122], [1938, p. 168], [2062, pp. 50, 150, 168, 189], and [2377, 2922, 2926]; xxv) and xxvi) are given in [917] and [1938, p. 58]; xxvii) and xxviii) are given in [47, p. 18]; xxix) is given in [2062, p. 679]; xxx) is given in [806, p. 207]; xxxi) is given in [2062, p. 683]; xxxii) is given in [47, p. 38] and [917]; xxxiii) is given in [2377]; xxxiv) and xxxv) are given in [497], [2062, pp. 170, 171], and [2292]; xxxvi) is given in [933, 1906, 1905, 1907, 2292]; xxxvii)–lii) are given in [395, 917] and [2062, pp. 172–174, 177, 178, 192]; liii) is given in [2262]; liv) is given in [1822]; lv)–lvii) are given in [2062, p. 178]; lviii) follows from an inequality involving the medians of a triangle in Fact 5.2.12; lix) is given in [1938, p. 54]; lx) and lxi) are given in [2062, p. 177]; lxii) and lxvi) are given in [2467, 2925]; lxiii)–lxv) are given in [917]; lxvii) is given in [2002]; lxviii) is given in [109, pp. 314, 315]; lxix) and lxx) are given in [917]; lxxi)–lxxiv) are given in [1907]. lxxv) is given in [1956, 2739]; lxxvi) is given in [1956]; lxxvii) and lxxviii) are given in [397]; lxxix) is given in [751]; lxxx) is given in [1732]; lxxxi) is given in [109, pp. 110, 111]; lxxxii) is given in [109, pp. 198, 199]; lxxxiii) is given in [373]; lxxxiv) is given in [392]; lxxxv) is given in [395, 2995]; lxxxvi)–lxxxviii) are given in [395]. Remark: In xiii), the inequality (a √+ b − c)(b + c − a)(c + a − b) ≤ abc is Padoa’s inequality. See Fact 5.2.25. Remark: area(S) ≤ 123 (a2 + b2 + c2 ) in xxiii) is Weitzenbock’s inequality. See [47, lxxxv)

√

pp. 84, 85]. Remark: area(S) ≤ 123 [2(ab + bc + ca) − (a2 + b2 + c2 )] in xxiii) can be rewritten as √ 4 3 area(S) + (a − b)2 + (b − c)2 + (c − a)2 ≤ a2 + b2 + c2 . This is the Hadwiger-Finsler inequality. This is interpolated in xxxvi). Extensions are given in [1690]. See [993, p. 174] and [1907]. Remark: The adjacent upper and lower bounds for s2 in iv) are Blundon’s inequalities. See [119]. According to lvi), these inequalities are necessary and sufficient for the existence of a triangle with semiperimeter s, inner radius r, and outer radius R. Remark: In xxiv), area(S) ≤ (R + 12 r)2 in iii) ´ is Mircea’s inequality. The adjacent √ and lower bounds for area(S) are due √ to E. Rouch´e. See √ 2 upper 2 and the upper bounds r 4R2 + 4Rr + 3r 2 and 3r and r 16Rr − 5r [2922]. The lower bounds 3 √ 2Rr +(3 3−4)r2 are due to W. J. Blundon. See [927,√2377]. Remark: 2r ≤ R is Euler’s inequality. See [47, pp. 56–60]. If S is a right triangle, then (1 + 2)r ≤ R. See [2366, p. 19]. The interpolation in xi) is Bandila’s inequality. Remark: a2 + b2 + c2 ≤ 9R2 in xxxv) is Leibniz’s inequality. See [1938, pp. 68–70]. Remark: The upper bounds for a + b + c and a2 + b2 + c2 in xxxiv) and xxxv) are due to W. J. Blundon. See [497, 535] and [2062, pp. 170, 171]. Versions of the adjacent upper and lower bounds in xxxv) given in [497] and quoted in [535, p. 53], [917], and [2062, pp. 170, 171] are incorrect. Credit: i) is due to L. Bankoff. See [47, p. 112] and [2062, p. 165]. Fact 5.2.9. Let S ⊂ R2 denote a triangle with the notation defined in Fact 5.2.7. Then, the following statements hold:

453

GEOMETRY

i) sin A + sin B + sin C = 4(cos A2 )(cos B2 ) cos C2 =

s . R

ii) sin A + sin B − sin C = 4(sin A2 )(sin B2 ) cos C2 . iii) sin2 A + sin2 B − sin2 C = 2(sin A)(sin B) cos C. iv) (sin2 A + sin2 B − sin2 C)(sin2 B + sin2 C − sin2 A)(sin2 C + sin2 A − sin2 B) r2 s2 (s2 − 4R2 − 4Rr − r2 ) . = 8(sin2 A)(sin2 B)(sin2 C)(cos A)(cos B) cos C = 2R6 2 2 2 2 2 a +b +c s − 4Rr − r v) a sin A + b sin B + c sin C = = . 2R R rs vi) (sin A)(sin B) sin C = . 2R2 vii) 4(sin A)(sin B) sin C = sin(A + B − C) + sin(B + C − A) + sin(C + A − B) − sin(A + B + C). viii) 4(sin A)(cos B) cos C = sin(A + B − C) + sin(A − B + C) + sin(A − B − C) + sin(A + B + C). ix) (sin A)(cos B) cos C + (sin B)(cos C) cos A + (sin C)(cos A) cos B = sin(A + B − C) + (sin A)(sin B) sin C. s2 + 4Rr + r2 . x) (sin A) sin B + (sin B) sin C + (sin C) sin A = 4R2 2 2 s − 4Rr − r xi) sin2 A + sin2 B + sin2 C = . 2R2 2 s(s − 6Rr − 3r2 ) . xii) sin3 A + sin3 B + sin3 C = 4R3 s4 − (8Rr + 6r2 )s2 + (4R + r)2 r2 xiii) sin4 A + sin4 B + sin4 C = . 8R4 s(s2 + 2Rr + r2 ) . xiv) (sin A + sin B)(sin B + sin C)(sin C + sin A) = 4R3 r xv) sin2 A2 + sin2 B2 + sin2 C2 = 1 − . 2R 2Rs2 + 4Rr2 + r3 + rs2 xvi) (a + b)2 sin2 C2 + (b + c)2 sin2 A2 + (c + a)2 sin2 B2 = . 2R 2 2 2 8R + r − s xvii) sin4 A2 + sin4 B2 + sin4 C2 = . 8R2 s2 + r2 − 8Rr xviii) (sin2 A2 ) sin2 B2 + (sin2 B2 ) sin2 C2 + (sin2 C2 ) sin2 A2 = . 16R2 sin A + sin B sin B + sin C sin C + sin A s2 + r2 − 2Rr + + = . xix) 2Rr ( sin C ) ( sin A ) sin B sin A + sin B sin B + sin C sin C + sin A s xx) = . cos A + cos B cos B + cos C cos C + cos A r r . xxi) (sin A2 )(sin B2 ) sin C2 = 14 (cos A + cos B + cos C − 1) = 4R r2 xxii) (1 − cos A)(1 − cos B)(1 − cos B) = 8(sin2 A2 )(sin2 B2 ) sin2 C2 = . 2R2 2rs xxiii) sin 2A + sin 2B + sin 2C = 4(sin A)(sin B) sin C = 2 . R rs[s2 − (2R + r)2 ] xxiv) (sin 2A)(sin 2B) sin 2C = . R4 xxv) sin A, sin B, sin C are the roots of 4R2 x3 − 4rsx2 + (s2 + 4Rr + r2 )x − 2rs = 0. xxvi) sin2 A2 , sin2 B2 , sin2 C2 are the roots of 16R2 x3 − 8R(2R − r)x2 + (s2 − 8Rr + r2 )x − r2 = 0. r xxvii) cos A + cos B + cos C = 4(sin A2 )(sin B2 ) sin C2 + 1 = 1 + . R xxviii) cos A + cos B − cos C = 4(cos A2 )(cos B2 ) sin C2 − 1.

454

CHAPTER 5

s2 − (2R + r)2 . 4R2 s2 − 4R2 + r2 xxx) (cos A) cos B + (cos B) cos C + (cos C) cos A = . 4R2 2 6R + 4Rr + r2 − s2 xxxi) cos2 A + cos2 B + cos2 C = 1 − 2(cos A)(cos B) cos C = . 2R2 3 2 (2R + r) − 3s r − 1. xxxii) cos3 A + cos3 B + cos3 C = 4R3 rs2 + 2Rr2 + r3 xxxiii) (cos A + cos B)(cos B + cos C)(cos C + cos A) = . 4R3 r 2 B 2 C 2 A . xxxiv) cos 2 + cos 2 + cos 2 = 2 + 2R s A B C 1 xxxv) (cos 2 )(cos 2 ) cos 2 = 4 (sin A + sin B + sin C) = . 4R s2 . xxxvi) (1 + cos A)(1 + cos B)(1 + cos B) = 8(cos2 A2 )(cos2 B2 ) cos2 C2 = 2R2 2 3R + 4Rr + r2 − s2 xxxvii) cos 2A + cos 2B + cos 2C = −4(cos A)(cos B) cos C − 1 = . R2 s2 + (4R + r)2 xxxviii) (cos2 A2 ) cos2 B2 + (cos2 B2 ) cos2 C2 + (cos2 C2 ) cos2 A2 = . 16R2 cos A + cos B cos B + cos C cos C + cos A (R + r)(s2 + r2 − 4R2 ) + + = xxxix) − 3. cos C cos A cos B R[s2 − (2R + r)2 ] xl) 2(cos A2 )(cos B2 ) cos C2 − 2(sin A2 )(sin B2 ) sin C2 = 1 + (cos A2 − sin A2 )(cos B2 − sin B2 )(cos C2 − sin C2 ). C−A A−B 3 B 3 C xli) (cos3 A2 ) sin B−C 2 + (cos 2 ) sin 2 + (cos 2 ) sin 2 = 0. xxix) (cos A)(cos B) cos C =

√

xlii) (cos A4 )(cos B4 ) cos C4 = 82 + 14 [cos( π4 − A2 ) + cos( π4 − B2 ) + cos( π4 − C2 )] = cos B2 + cos C2 + sin A2 + sin B2 + sin C2 ]. xliii) xliv) xlv) xlvi) xlvii) xlviii) xlix) l) li) lii) liii) liv) lv)

√ 2 8 [1

+ cos

A 2

+

(cos A)(sin B) sin C = 41 (cos 2A − cos 2B − cos 2C + 1). cos A, cos B, cos C are the roots of 4R2 x3 −4R(R+r)x2 +(s2 +4R2 +r2 )x +(2R+r)2 − s2 = 0. cos2 A2 , cos2 B2 , cos2 C2 are the roots of 16R2 x3 − 8R(4R + r)x2 + [s2 + (4R + r)2 ]x − s2 = 0. a cos A + b cos B + c cos C = 4R(sin A)(sin B) sin C. 2rs . tan A + tan B + tan C = (tan A)(tan B) tan C = 2 s − (2R + r)2 s2 − r2 − 4Rr (tan A) tan B + (tan B) tan C + (tan C) tan A = 2 . s − (2R + r)2 2rs (tan A)(tan B) tan C = 2 . s − (2R + r)2 4s2 r2 − 2(s2 − r2 − 4Rr)[s2 − (2R + r)2 ] . tan2 A + tan2 B + tan2 C = [s2 − (2R + r)2 ]2 8rs(s2 r2 − 3R2 [s2 − (2R + r)2 ]) tan3 A + tan3 B + tan3 C = . [s2 − (2R + r)2 ]3 8R2 rs (tan A + tan B)(tan B + tan C)(tan C + tan A) = 2 . [s − (2R + r)2 ]2 4R + r tan A2 + tan B2 + tan C2 = . s A B C a tan 2 + b tan 2 + c tan 2 = 4R − 2r. (a + b) tan C2 + (b + c) tan A2 + (c + a) tan B2 = 4(R + r).

455

GEOMETRY

lvi) tan2

A 2

+ tan2

B 2

+ tan2

=

C 2

r lvii) (tan A2 )(tan B2 ) tan C2 = . s lviii) (tan

A 2

+ tan B2 )(tan

lix) (tan A2 ) tan lx) tan3

A 2

B 2

B 2

(4R + r)2 − 2s2 . s2

+ (tan B2 ) tan C2 + (tan C2 ) tan

+ tan3

B 2

4R . s = 1.

+ tan C2 )(tan C2 + tan A2 ) =

+ tan3

C 2

=

A 2

(4R + r) − 12s2 R . s3 3

lxi) tan

A 2

+ tan

tan C2

B 2

+

tan

B 2

+ tan C2

tan

A 2

+

tan C2 + tan tan

B 2

A 2

=

cot

A 2

+ cot

B 2

+

cot

B 2

+ cot C2

+

cot C2 + cot

cot C2 cot A2 a b c 4R − 2r = + + . = r s−a s−b s−c

cot

lxii) tan A, tan B, tan C are the roots of [s2 − (2R + r)2 ]x3 − 2rsx2 + (s2 − 4Rr − r2 )x − 2rs = 0. lxiii) tan A2 , tan B2 , tan C2 are the roots of sx3 − (4R + r)x2 + sx − r = 0. s2 + r2 + 4Rr . 2rs 2 2 (s + r + 4Rr)2 − 16s2 Rr . csc2 A + csc2 B + csc2 C = 4s2 R2 2R . (csc A) csc B + (csc B) csc C + (csc C) csc A = r 2R2 (csc A)(csc B) csc C = . rs s2 + r2 − 8Rr csc2 A2 + csc2 B2 + csc2 C2 = . r2 8R(2R − r) (csc2 A2 ) csc2 B2 + (csc2 B2 ) csc2 C2 + (csc2 C2 ) csc2 A2 = . r2 3 2 2 2 csc A, csc B, csc C are the roots of 2rsx − (s + r + 4Rr)x + 4Rsx − 4R2 = 0. csc2 A2 , csc2 B2 , csc2 C2 are the roots of r2 x3 − (s2 + r2 − 8Rr)x2 + 8R(2R − r)x − 16R2 = 0.

lxiv) csc A + csc B + csc C = lxv) lxvi) lxvii) lxviii) lxix) lxx) lxxi)

s2 + r2 − 4R2 . s2 − (2R − r)2 (s2 + r2 − 4R2 )2 − 8R(R + r)[s2 − (2R + r)2 ] . sec2 A + sec2 B + sec2 C = [s2 − (2R + r)2 ]2 4R(R + r) (sec A) sec B + (sec B) sec C + (sec C) sec A = 2 . s − (2R + r)2 s2 + (4R + r)2 sec2 A2 + sec2 B2 + sec2 C2 = . s2 8R(4R + r) (sec2 A2 ) sec2 B2 + (sec2 B2 ) sec2 C2 + (sec2 C2 ) sec2 A2 = . s2 sec A, sec B, sec C are the roots of [s2 −(2R+r)2 ]x3 −(s2 +r2 −4R2 )x2 +4R(R+r)x−4R2 = 0. sec2 A2 , sec2 B2 , sec2 C2 are the roots of s3 x3 − [s2 + (4R + r)2 ]x2 + 8R(4R + r)x − 16R2 = 0.

lxxii) sec A + sec B + sec C = lxxiii) lxxiv) lxxv) lxxvi) lxxvii) lxxviii)

a2 + b2 + c2 s2 − r2 − 4Rr = . 4rs 2rs lxxx) (cot A) cot B + (cot B) cot C + (cot C) cot A = 1.

lxxix) cot A + cot B + cot C =

B 2

A 2

456

CHAPTER 5

s2 − (2R + r)2 . 2rs (s2 − r2 − 4Rr)2 − 2. lxxxii) cot2 A + cot2 B + cot2 C = 4s2 r2 2R2 lxxxiii) (cot A + cot B)(cot B + cot C)(cot C + cot A) = . rs (s2 − r2 − 4Rr)3 − 48s2 R2 r2 lxxxiv) cot3 A + cot3 B + cot3 C = . 8s3 r3 2 2 1 − (cot A)(cot B) cot C 1 − (cot B)(cot C) cot A 1 − (cot2 C)(cot A) cot B lxxxv) + + = 2. 1 + cot2 A 1 + cot2 B 1 + cot2 C s lxxxvi) cot A2 + cot B2 + cot C2 = (cot A2 )(cot B2 ) cot C2 = . r 4R + r . lxxxvii) (cot A2 ) cot B2 + (cot B2 ) cot C2 + (cot C2 ) cot A2 = r s2 − 2r(4R + r) lxxxviii) cot2 A2 + cot2 B2 + cot2 C2 = . r2 s(s2 − 12Rr) . lxxxix) cot3 A2 + cot3 B2 + cot3 C2 = r3 4Rs xc) (cot A2 + cot B2 )(cot B2 + cot C2 )(cot C2 + cot A2 ) = 2 . r s2 − r2 − 4Rr rs xci) cot 2A + cot 2B + cot 2C = . + 4rs (2R + r)2 − s2 [2s2 − (2R + r)2 − r2 − 4Rr]2 − 16s2 r2 xcii) (cot 2A)(cot 2B) cot 2C = . 16rs[s2 − (2R + r)2 ] xciii) cot A, cot B, cot C are the roots of 2rsx3 − (s2 − r2 − 4Rr)x2 + 2rsx + (2R + r)2 − s2 = 0. xciv) cot A2 , cot B2 , cot C2 are the roots of rx3 − sx2 + 2rsx + (4R + r)2 − s = 0. Source: [47, p. 124], [107, p. 117], [108, pp. 11, 99], [161, 292], [644, pp. 166, 182], [806, p. 79], [1026], [1860, p. 135], [2062, pp. 54–60, 89, 90], [2162], and [2413]. Remark: The first equality in xxviii) is the Cayley cosine cubic. See [1307]. Fact 5.2.10. Let S ⊂ R2 denote a triangle with the notation defined in Fact 5.2.7. Then, the following statements hold: √ √ i) a + (2 − 2 − 2 cos A) min {b, c} ≤ b + c ≤ a + (2 − 2 − 2 cos A) max {b, c}. ii) If a < 12 (b + c), then A < 21 (B + C). lxxxi) (cot A)(cot B) cot C =

iii) π3 (a + b + c) ≤ aA + bB + cC ≤ 21 (π − min {A, B, C})(a + b + c) < π2 (a + b + c). iv) If A ≤ B ≤ C, then   π−A         π aA + bB + cC   π 2 ≤ ≤ < .    [ ]  3 a+b+c    π 1 − (tan A ) tan B   2 2 2 2 ( ) ( π )3 1 1 1 3 R + + , 64ABC ≤ (A + π)(B + π)(C + π). v) 2 ≤ ≤ 2 r 9 A B C 3 9 8 1 1 1 vi) ≤ + 3 (A2 + B2 + C 2 ) ≤ + + . π π√ π A B C 3 3 A3 B3 C3 vii) π < + 2 + + . 4 A + π2 B2 + π2 C 2 + π2

457

GEOMETRY

) a b c π2 − A2 π2 − B2 π2 − C 2 ≤ + + . + + A B C π2 + A2 π2 + B2 π2 + C 2 ( ) ( ) ( ) 3 a b c 1 1 1 1 1 1 (a + b + c) ≤ + + ≤ 2a + + 2b + + 2c + . π A B C B C C A A B a+b b+c c+a 6 (a + b + c) ≤ + + . π C A B 3 a+b−c b+c−a c+a−b (a + b + c) ≤ + + . π C A B 9 a+b−c b+c−a c+a−b ≤ + + . π cC aA bB   ( ) ( )2/3       π 3 2r             ( π )3         29 10π 7     3 R  ( π )3 2r   ≈ 1.14838 < < <  ≤       π  ( )   3 3 25 27 6   π ≤ ABC ≤  ≤ .            abc 3 R 2       2s     πr       R √ 6s a abc b c 3 ≤3 ≤ √ + √ + √ . π ABC BC CA AB (

viii) 2R ix) x) xi) xii)

xiii)

xiv) xv)

27ABCR     2≤   π3 r   ( ) ≤ (1 −   π2 R     1−  12 r

1 24 [(A

− B)2 + (B − C)2 + (C − A)2 ])

≤e xvi) xvii) xviii) xix) xx) xxi) xxii) xxiii) xxiv) xxv) xxvi) xxvii)

(

2π3 ≤ r 27ABC

1 − 24 [(A−B)2 +(B−C)2 +(C−A)2 ] R

√ 16 3 ( π )3 s s 2 2 2 ≤ (π − A)(π − B)(π − C), < 4e−(A +B +C )/8 . 9 3 R R (A − B)2 + (B − C)2 + (C − A)2 < 2π2 ≈ 19.739. π2 (aB − bA)2 + (bC − cB)2 + (cA − aC)2 < (a + b + c)2 . 4 2bc cos A 2bc 0. Then, ( 2 ) p/3 ( r )p r 31+p/2 ≤ 31+p/2 ≤ 3[(sin A)(sin B) sin C] p/3 ≤ sin p A + sin p B + sin p C. R 2R2 In particular, √ √ ( r )3 3 3 ( r )2 1 ≤ ≤ (sin A)(sin B) sin C ≤ (sin3 A + sin3 B + sin3 C). 3 3 R 2 R 3 xl) 0 ≤ sin 2A + sin 2B + sin 2C ≤ sin A + sin B + sin C < sin( A2 + B) + sin( B2 + C) + sin( C2 + A). √ √ √ r 3 r 3s 1 3s A B C ≤ − + ≤ sin 2 + sin 2 + sin 2 ≤ 2 + ≤ . xli) 3R 2 4R 12R 2R 2 xlii) Let p ≥ 1. Then, sin πp ≤ sin Ap + sin Bp + sin Cp ≤ πp . ( r )2 3 2Rr − r2 r xliii) 2 A 2 B 2 C ≤ ≤ 1 − ≤ sin + sin + sin = 1 − 2 2 2 4 2R R R2 2r 1 7 2r 15 9r A−B B−C ≤ − ≤2− − (cos 2 )(cos 2 ) cos C−A ≤ − 2 4 R 8 4R R 4 2r 2r 17 11r ≤ − 0. Then, sin

A−B 2

+

B−C A 2 ≤ cos 2 . 1 3A sin A−C 2 + p sin 2

≤

p2 +2 2p .

cxix) sin2 B + sin2 C ≤ 1 + 2(sin B)(sin C) cos A. √ sin A sin B sin C 9 3 + + ≤ . cxx) 2 < A B C 2π sin A sin B sin C 1 cxxi) 0 < + + < . (A − B)(C − A) (B − C)(A − B) (C − A)(B − C) 2 7R − 2r sin A + sin B sin B + sin C sin C + sin A 2R2 + Rr + 2r2 3R ≤ + + ≤ ≤ . cxxii) 6 ≤ R sin C sin A sin B Rr r sin A sin B sin C 3 cxxiii) + + ≤ (csc A)(csc B) csc C. (sin B + sin C)2 (sin C + sin A)2 (sin A + sin B)2 4 (sin A) sin B (sin B) sin C (sin C) sin A cxxiv) 9 ≤ + + . sin2 C2 sin2 A2 sin2 B2 cxxv) cxxvi) cxxvii) cxxviii) cxxix) cxxx) cxxxi) cxxxii) cxxxiii)

sin

A 2

+

sin

B 2

+

sin C2

≤1≤

1 2 + 2 sin

+

1

+

A 1 + sin 1 + sin 1 + sin 2 + 2 sin B2 2 + 2 (cos A2 ) cos B2 (cos B2 ) cos C2 (cos C2 ) cos A2 9 r (R + r)2 ≤5− ≤ . + + ≤ 2 R Rr sin C2 sin A2 sin B2 √ sin A2 sin B2 sin C2 2 3≤ + + . (sin B2 ) cos C2 (sin C2 ) cos A2 (sin A2 ) cos B2 A+B cos B+C cos C+A 3 cos 2 2 2 ≤ + + < 2. B−C C−A 2 cos A−B cos cos 2 2 2 A 2

B 2

C 2

cos A cos B cos C 3 ≤ + + . 2 cos(B − C) cos(C − A) cos(A − B) √ 3 3 cos A − cos B cos B − cos C cos C − cos A 4 − ≤ + + 0, then

( 3

r ≤3

p+1 p

p+1

Rr2 2

) p/3 ≤ lap + lbp + lcp .

478

CHAPTER 5

If p ∈ (0, 1], then

( lap + lbp + lcp ≤ 31−p/2 s p ≤ 3

If p ∈ R, then

3R 2

)p .

a p la2 + b p lb2 + c p lc2 ≤ 21 abcs(a p−2 + b p−2 + c p−2 ).

iv) The medians ma , mb , mc , altitudes ha , hb , hc , and angle bisectors la , lb , lc satisfy 9R 9r ≤ ha + hb + hc ≤ la + lb + lc ≤ ma + mb + mc ≤ , 2 √ 2 { √ 3 2 2 2 2 3s ≤ 2 (a + b + c ) 6rs ≤ ala + blb + clc ≤ ama + bmb + cmc ≤ 3 3Rs, ha ≤ la ≤ ma ,

ma + mb + mc + min {a, b, c} ≤ la + lb + lc + max {a, b, c}, la − ha ≤ R − 2r, ( ) ma mb mb mc mc ma C A B ma + mb + mc ≤ 2 sin 2 + sin 2 + sin 2 , hc ha hb ma ha + mb hb + mc hc ≤ ma mb sin C2 + mb mc sin

A 2

+ mc ma sin B2 ,

ma ha sec A2 + mb hb sec B2 + mc hc sec C2 ≤ ma mb sec C2 + mb mc sec A2 + mc ma sec B2 , √ √ √ ha ha hb hb hc hc ha hb hc + + ≤ √ + √ + √ ≤ 2(sin A2 + sin B2 + sin C2 ), ma mb mb mc mc ma mb mc mc ma ma mb ma

√

√ √ √ (cos B2 ) cos C2 + mb (cos C2 ) cos A2 + mc (cos A2 ) cos B2 ( ) ma mb mb mc mc ma ≤2 sin C + sin A + sin B , hc ha hb

ma ha (cos B2 ) cos C2 +

√

mb hb (cos C2 ) cos

A 2

+

√

mc hc (cos A2 ) cos

B 2

≤ ma mb sin C + mb mc sin A + mc ma sin B, √ ( ) √ r ma mb mb mc mc ma A B C + + , ma sin 2 + mb sin 2 + mc sin 2 ≤ R hc ha hb √ √ √ √ r (ma mb + mb mc + mc ma ) , ma ha sin A2 + mb hb sin B2 + mc hc sin C2 ≤ R √ √ √ √ la + lb + mc ≤ s(s − a) + s(s − b) + mc ≤ 2(s2 − m2c ) + mc ≤ 3s, √ √ 1 ha + mb + lc 3 ha + mb + mc 1 la + lb + m c 3 3 ≤ ≤ , ≤ ≤ , ≤ ≤ 1, 2 a+b+c 2 4 a+b+c 2 8 a+b+c 2(m2a + m2b + m2c ) ha + hb + hc ≤ , ma ha + mb hb + mc hc ≤ s2 , 3R [area(S)]2 (b − c)2 1 s2 ≤ s(s − a) ≤ ma la ≤ s(s − a) + ≤ s(s − a) + (b − c)2 , 8 2s(s − a)(b + c)2 √

√

{ s ≤ ma la + mb lb + mc lc ≤ 2

s2 + 81 [(a − b)2 + (b − c)2 + (c − a)2 ] = 45 s2 − 3Rr − 34 r2 s2 + 2Rr − 4r2

≤ 5R2 + 2Rr + 3r2 ≤ 6R2 + 3r2 ≤

27R2 , 4

}

479

GEOMETRY

R ha hb hc ≤ ma mb mc , 2r (b + c)ma la + (c + a)mb lb + (a + b)mc lc ≤ 21 s(5s2 − 22Rr − 7r2 ),

ama la + bmb lb + cmc lc ≤ 14 s(s2 + 18Rr + 9r2 ), h2a + h2b + h2c ≤ la2 + lb2 + lc2 ≤ s2 ≤ m2a + m2b + m2c , (b + c)2 ma ≤ , 4bc la

b2 + c2 ma R ≤ ≤ , 2bc ha 2r

la lb lc ≤ ma mb mc ,

la6 + lb6 + lc6 ≤ s4 (s2 − 12Rr) ≤ m6a + m6b + m6c , √ √ √ √ ma mb mc R 3≤ + + ≤3 , ha hb hc 2r

ma mb mc 9R2 − 14Rr + 4r2 9R − 12r + + ≤ ≤ , hb hc ha 2Rr 2r

6r ha hb hc ≤ + + ≤ 3, 3R − 4r mb mc ma

 4r    1+   R    √ √     2 3s  6R2 + 5Rr + 2r2  ha hb hc + + ≤ ≤    3R  la lb lc R     (9R − 2r)r 9    √   4 (13R2 + 14Rr − 8r2 )R  √ 48R3 + 16R2 r − 7Rr2 − 2r3 1 ≤ 2R 2R − r √ √ 6[R + ( 6 − 2)r] 5R + 2r ≤ ≤ ≤ 3, R 2R   13 r       −      la lb lc 6r 2R   4  s2 + 10Rr + r2 ma mb ≤ + + ≤ 3, 3 ≤  ≤ + + ≤   2   R ma mb mc 8Rr la lb 8s  5        +  3 81Rr √ √ ( 9R la lb lc R(13R2 + 14Rr − 8r2 ) 3≤ √ ≤ + + ≤4 ≤ 3 1+ (9R − 2r)r 6R2 + 5Rr + 2r2 ha hb hc √ ma lb mb lc mc la 35/4 area(S) ≤ + + , 0 ≤ m2a − la2 ≤ 21 (b − c)2 , hc ha hb √ √ √ √ √ √ la + lb + lc ≤ ma + mb + mc ,

mc 3R ≤ , lc 2r R ) 3R ≤ , r 2r

480

CHAPTER 5

√ √  √ √ √ 3 6   ma mb mc la lb lc ≤ 33 3 ma mb mc la lb lc   3     √√    3  3 2/9    12 (abcma mb mc la lb lc )         8s2 la lb lc       27abc     √ √ √   1 3 1 3    3 √3     2 abcla lb lc ≤ 2 abcma mb mc   ha hb hc la lb lc       3  √  ≤ area(S) ≤ ( ) ma mb mc la lb lc ma mb mc      la lb lc 2 2 2  la lb lc 2la lb lc ≤      ≤ ≤ √3 + + s s      s 9 a b c √  3 abc     (abc)2 ma mb mc  3      (a + b)(b + c)(c + a)     √ ( )2/3    3 (a + b)(b + c)(c + a)la lb lc       12 abc       3abcma mb mc     a(b + c)m m + b(c + a)m m + c(a + b)m m . b c c a a b

If a ≤ b ≤ c, then mc ≤ mb ≤ ma , hc ≤ hb ≤ ha , and lc ≤ lb ≤ la . Source: [47, pp. 45–47, 51, 72, 109], [54, 56], [59, p. 70], [107, pp. 47, 99], [108, pp. 3, 26, 66, 67, 166, 167], [109, p. 343], [160], [289, pp. 7, 13, 69, 90], [317, 322, 395, 396, 401], [806, p. 207], [813], [993, p. 185], [1026, 1533, 1681], [1938, pp. 53, 54, 66, 70, 98, 108, 109, 141], [1994, 2011], [2062, pp. 110, 163, 200–206, 210–223, 229, 231, 680, 683, 684], [2376, 2444, 2602, 2603, 2924, 3013]. Fact 5.2.13. Let S ⊂ R2 denote a triangle with the notation defined in Fact 5.2.7. Then, the following statement holds: s

s

s

s

i) [A B C] ≺ [π 0 0], and [ π3 π3 π3 ] ≺ [A B C]. Furthermore, the following statements are equivalent: ii) S is acute. iii) 1 < (tan A) tan B. iv) [A B C] ≺ [ π2 π2 0], and [ π3 π3 π3 ] ≺ [A B C]. v) 2R + r < s. vi) 8R2 < a2 + b2 + c2 . a2 b2 c2 3 2 vii) + + ≤ (a + b2 + c2 ). A B C π viii) 0 < (a2 + b2 − c2 )(b2 + c2 − a2 )(c2 + a2 − b2 ). ix) 0 < (5m2a − m2b − m2c )(5m2b − m2c − m2a )(5m2c − m2a − m2b ). Furthermore, the following statements are equivalent: x) S is right. xi) 2R + r = s. xii) 8R2 = a2 + b2 + c2 . xiii) cos A2 cos B2 cos C2 = 21 + sin A2 sin B2 sin C2 . xiv) (5m2a − m2b − m2c )(5m2b − m2c − m2a )(5m2c − m2a − m2b ) = 0. Furthermore, the following statements are equivalent: xv) S is obtuse.

481

GEOMETRY s

s

xvi) [A B C] ≺ [π 0 0], and [ π2 π4 π4 ] ≺ [A B C]. xvii) s < 2R + r. xviii) a2 + b2 + c2 < 8R2 . xix) (5m2a − m2b − m2c )(5m2b − m2c − m2a )(5m2c − m2a − m2b ) < 0. The following statements hold: xx) If S is acute, n ≥ 1, and p is a nonnegative number, then π ABC ≤ 27 , ( √ )3 3 3 4 ABC ≤ (sin A)(sin B) sin C ≤ ABC ≤ 2 2π π

min {|A − B|, |B − C|, |C − A|} ≤ π6 ,

tan A > 0,

3

12R a b c < + + , π A B C

√ 3 3 8 ,

( )3 4 r 5 r 3 + ≤ + ≤ sin A2 +sin B2 +sin C2 , π12 ABC ≤ (sin A2 )(sin B2 ) sin C2 ≤ 2π ABC ≤ 18 , 4 2R 3 3R A B B C C A 3 1 2 < sin2 A + sin2 B + sin2 C ≤ 49 , 2 < (sin 2 ) sin 2 + (sin 2 ) sin 2 + (sin 2 ) sin 2 ≤ 4 , cos A + cos B + cos C ≤ 23 ,

3 4

0 ≤ (cos A)(cos B) cos C ≤ 18 ,

≤ cos2 A + cos2 B + cos2 C < 1, 1 2

< (cos A2 )(cos B2 ) cos C2 ≤

√ 3 3 8 ,

R(a cos3 A + b cos3 B + c cos3 C) ≤ area(S), 1 2

≤ cos3 A + cos3 B + cos3 C + (cos A)(cos B) cos C, 3+

3n 2

< tann A + tann B + tann C,

9 ≤ tan2 A + tan2 B + tan2 C,

31+p/2 < tan p A + tan p B + tan p C,

(tan A + tan B)(tan B + tan C)(tan C + tan A) > 0,    2(sin 2A + sin 2B + sin 2C)    √     3(cos A + cos B + cos C) ≤ 2(sin A + sin B + sin C) ≤ 3 3 0 0. iv) x > |y − z|, y > |z − x|, and z > |x − y|. v) |y − z| < x < y + z. vi) 2(x4 + y4 + z4 ) < (x2 + y2 + z2 )2. vii) (x2 + y2 + z2 )2 < 4(x2 y2 + y2 z2 + z2 x2 ). viii) There exist positive numbers a, b, c such that x = a + b, y = b + c, and z = c + a. ix) For all p ∈ R, f (p) > 0. x) f [(y2 + z2 − x2 )/(2z2 )] > 0. If these statements hold, then a, b, c in viii) are given by a = 21 (z + x − y),

b = 12 (x + y − z),

c = 21 (y + z − x).

Source: [993, p. 164]. To prove the equivalence of iii) and vi), note that (x2 + y2 + z2 )2 − 2(x4 +

y4 + z4 ) = (x + y + z)(x + y − z)(x − y + z)(−x + y + z). v) and vi) are given in [1371, p. 125]; vii) is given in [47, p. 71] and [1158, p. 38]. To prove iii) =⇒ ix), note that f (p) = z2 p2 + (x2 − y2 − z2 )p + y2 )2 ( (x + y + z)(x + y − z)(y + z − x)(z + x − y) x2 − y2 − z2 2 + . =z p+ 2z2 4z2 To prove x) =⇒ iii), note that 0 < f [(y2 + z2 − x2 )/(2c2 )] = (x + y + z)(x + y − z)(y + z − x)(z + x − y). Remark: The expressions x = a + b, y = b + c, and z = c + a can be used to recast inequalities

involving the lengths x, y, z of the sides of a triangle into inequalities involving arbitrary positive

485

GEOMETRY

numbers a, b, c. Conversely, each inequality that holds for arbitrary positive numbers a, b, c necessarily holds in the case where a, b, c are the lengths of the sides of a triangle, while the expressions a = 21 (z + x − y), b = 12 (x + y − z), and c = 21 (y + z − x) can be used to obtain possibly different inequalities that are valid only for x, y, z that represent the lengths of the sides of a triangle. Related: Fact 10.10.7. Fact 5.2.15. Let S ⊂ R2 denote a triangle with the notation defined in Fact 5.2.7, and define x = 12 (c + a − b), y = 21 (a + b − c), and z = 12 (b + c − a). Then, a = x + y,

b = y + z,

c = z + x,

(a − b) + (b − c) + (c − a) = (x − y)2 + (y − z)2 + (z − x)2 , √ s = x + y + z, area(S) = (x + y + z)xyz, √ xyz (x + y)(y + z)(z + x) r= , R= . √ x+y+z 4 (x + y + z)xyz 2

2

2

Source: [1938, pp. 56, 57, 74]. Remark: The numbers x, y, z determine the points along the sides of the triangle at which the incircle is tangent to the sides of the triangle. See [47, p. 57]. Fact 5.2.16. Let n ≥ 2 be an integer, let x, y, z be positive numbers, and assume that xn +yn = zn. Then, x, y, z represent the lengths of the sides of a triangle. Source: [1371, p. 112]. Remark: If x √ and y are positive numbers and n ≥ 1, then x, y, and n xn + yn represent the lengths of the sides of a triangle. Fact 5.2.17. Let a, b, c be positive numbers that represent the lengths of the sides of a triangle. Then, 1/(a + b), 1/(b + c), 1/(c + a) represent the lengths of the sides of a triangle. Source: [1757, p. 44]. Related: Fact 5.2.14 and Fact 5.2.18. Fact 5.2.18. Let a, b, c be positive that represent the lengths of the sides of a triangle √ √ √ numbers whose circumradius is R. √ Then, a, b, c represent the lengths of the sides of a triangle whose circumradius R2 satisfies 3R22 ≤ R. Source: [1371, p. 99] and Fact 5.2.19. Related: Fact 5.2.14 and Fact 5.2.17. Fact 5.2.19. Let a, b, c be positive numbers that represent the lengths of the sides of a triangle 1/p whose circumradius is R, and let p > 1. Then, √ a1/p , b1/p , and √ c , represent the lengths of the sides p of a triangle whose circumradius R p satisfies ( 3R p ) ≤ 3R. Source: [1860, p. 119]. Credit: A. Oppenheim. Related: Fact 5.2.14 and Fact 5.2.17. Fact 5.2.20. Let a, b, c be positive numbers that represent the lengths of the sides of triangle S1 . Then, a + b, b + c, and c + a represent the lengths of the sides of triangle S2 . Furthermore, 4 area(S1 ) ≤ area(S2 ). Source: [806, p. 207]. Fact 5.2.21. Let x, y, z be positive numbers. Then, the following statements hold: i) x, y, z represent the lengths of the sides of a triangle if and only if x4 + y4 + z4 < 2(x2 y2 + y2 z2 + z2 x2 . ii) If z2 ≥ x2 + y2 , then z = max {x, y, z}. iii) Let z = max {x, y, z}. Then, x, y, z represent the lengths of the sides of a triangle if and only if z < x + y. iv) Let z = max√{x, y, z}. Then, x, y, z represent the lengths of the sides of an obtuse triangle if and only if x2 + y2 < z < x + y. v) Let z = max {x, y, z}. Then, x, y, z represent the lengths of the sides of a right triangle if and only if z2 = x2 + y2 . vi) Let z = max {x,√ y, z}. Then, x, y, z represent the lengths of the sides of an acute triangle if and only if z < x2 + y2 .

486

CHAPTER 5

vii) x, y, z represent the lengths of the sides of an acute triangle if and only if x2 < y2 + z2 , y2 < z2 + x2 , and z2 < x2 + y2 . viii) If x, y, z represent the lengths of the sides of an acute triangle, then x2 , y2 , z2 represent the lengths of the sides of a triangle. ix) If, for every positive integer n, xn , yn , zn represent the lengths of the sides of a triangle S, then S is isosceles. √ √ √ √ x) x + y, y + z, z + x represent the sides of a triangle whose area is 21 xy + yz + zx. Source: [47, pp. 88, 89] and [1007]. To prove sufficiency in vii), note that x2 , y2 , z2 represent the lengths of the sides of a triangle. It thus follows from Fact 5.2.18 that x, y, z represent the lengths of the sides of a triangle. However, z is shorter than the hypotenuse of a right triangle whose remaining sides are x and y, and likewise for x and y. Therefore, the triangle is acute. ix) is given in [1158, p. 38]. x) is given in [1956]. See Fact 2.3.40. Fact 5.2.22. Let S1 and S2 be triangles with sides a1 , b1 , c1 and a2 , b2 , c2 , respectively. Then, 16 area(S1 ) area(S2 ) ≤ a21 (−a22 + b22 + c22 ) + b21 (a22 − b22 + c22 ) + c21 (a22 + b22 − c22 ). Furthermore, equality holds if and only if the triangles are similar. Source: [47, p. 108]. Remark: This is the Neuberg-Pedoe inequality. Fact 5.2.23. Let a, b, c be positive numbers that represent the lengths of the sides of a triangle, and let f : [0, ∞) 7→ [0, ∞) be nonincreasing and subadditive. Then, f (a), f (b), f (c) denote the lengths of the sides of a triangle. Source: [47, p. 120]. Remark: “Subadditive” means that, for all x, y ≥ 0, f (x + y) ≤ f (x) + f (y). Fact 5.2.24. Let x, y, z > 0 represent the lengths of the sides of a triangle. Then, ( ) x − y + y − z + z − x < 1 , x + y + z − y + z + x < 1. x + y y + z z + x 8 y z x x y z Source: [993, pp. 181, 183]. Fact 5.2.25. Let x, y, z be positive numbers that represent the lengths of the sides of a triangle

S. Then, 1≤

x 2 + y 2 + z2 < 2, xy + yz + zx

2
0, and define the hypersphere of radius r by Sn = {x ∈ Rn : ∥x∥2 = △

r} and the hyperball of radius r by Bn = {x ∈ Rn : ∥x∥2 ≤ r}. Then, area(Sn ) =

In particular,

2πn/2 rn−1 ( ) , Γ n2

area(S1 ) = 2,

volume(Bn ) =

πn/2 rn ( ). Γ n2 + 1

volume(B1 ) = 2r,

area(S2 ) = 2πr,

volume(B2 ) = πr2 ,

area(S3 ) = 4πr2 ,

volume(B3 ) = 34 πr3 ,

area(S4 ) = 2π2 r3 ,

volume(B4 ) = 12 π2 r4 ,

area(S5 ) = 38 π2 r4 ,

volume(B5 ) =

8 2 5 15 π r .

Furthermore, area(Sn+2 ) 2πr2 = , area(Sn ) n

volume(Bn+2 ) 2πr2 = , volume(Bn ) n+2

volume(Bn ) r = , area(Sn ) n

area(Sn+2 ) = 2πr. volume(Bn )

498

CHAPTER 5

Finally,

 n/2 n  π r    , n even,     (n/2)! volume(Bn ) =      2n π(n−1)/2[(n − 1)/2]!rn    , n odd. n! Fact 5.5.14. Let A ∈ Rn×n, assume that A is positive definite, and define the hyperellipsoidal solid △ E = {x ∈ Rn : xTAx ≤ 1}. Then, volume(E) =

volume(Bn ) , √ det A

where volume(Bn ) is the volume of the hyperball in Rn . In particular, the area of the ellipse {x ∈ R2 : xTAx ≤ 1} is π/det A. Source: [1600, p. 36]. Related: Fact 5.5.15 and Fact 14.13.13. △ ∑ Fact 5.5.15. Let α1 , . . . , αn > 0, define β = ni=1 α1i , let r > 0, and define   n   ∑    △  n αi . x ∈ R : |x | ≤ r S=  (i)     i=1

Then, volume(S) =

2n

∏n i=1

Γ(1 +

Γ(1 + β)

Source: [1141]. Related: Fact 5.5.14 and Fact 14.13.13.

1 αi ) β

r .

Chapter Six Polynomial Matrices and Rational Transfer Functions In this chapter we consider matrices whose entries are either polynomials or rational functions. The decomposition of a polynomial matrix in terms of a Smith matrix provides the foundation for developing canonical matrices in Chapter 7. In this chapter we also present some basic properties of eigenvalues and eigenvectors as well as the minimal and characteristic polynomials of a square matrix. Finally, we consider the extension of the Smith matrix to the Smith-McMillan matrix for rational transfer functions.

6.1 Polynomials A function p: C 7→ C of the form p(s) = βk sk + βk−1 sk−1 + · · · + β1 s + β0 ,

(6.1.1)

where k ∈ N and β0 , . . . , βk ∈ F, is a polynomial. The set of polynomials is denoted by F[s]. If the coefficient βk ∈ F is nonzero, then the degree of p, denoted by deg p, is k. If, in addition, βk = 1, then p is monic. If k = 0, then p is constant. The degree of a nonzero constant polynomial is zero, while the degree of the zero polynomial is defined to be −∞. Let p1 and p2 be polynomials. Then, deg p1 p2 = deg p1 + deg p2 .

(6.1.2)

If either p1 = 0 or p2 = 0, then deg p1 p2 = deg p1 + deg p2 = −∞. If p2 is a nonzero constant, then deg p2 = 0, and thus deg p1 p2 = deg p1. Furthermore, deg(p1 + p2 ) ≤ max {deg p1 , deg p2 }.

(6.1.3)

Therefore, deg(p1 + p2 ) = max {deg p1 , deg p2 } if and only if either i) deg p1 , deg p2 or ii) p1 = △ p2 = 0 or iii) r = deg p1 = deg p2 , −∞ and the sum of the coefficients of sr in p1 and p2 is not △ zero. Equivalently, deg(p1 + p2 ) < max {deg p1 , deg p2 } if and only if r = deg p1 = deg p2 , −∞ r and the sum of the coefficients of s in p1 and p2 is zero. Let p ∈ F[s] be a polynomial of degree k ≥ 1. Then, it follows from the fundamental theorem of algebra that p has k possibly repeated complex roots λ1 , . . . , λk and thus can be factored as p(s) = β

k ∏

(s − λi ),

(6.1.4)

i=1

where β ∈ F. The multiplicity of a root λ ∈ C of p is denoted by mult p (λ). If λ is not a root of p, then mult p (λ) = 0. The multiset consisting of the roots of p including multiplicity is mroots(p) = {λ1 , . . . , λk }ms , while the set of roots of p ignoring multiplicity is roots(p) = {λˆ 1 , . . . , λˆ l }, where ∑l ˆ i=1 mult p(λi ) = k. If F = R, then the multiplicity of a root λi whose imaginary part is nonzero is equal to the multiplicity of its complex conjugate λi . Hence, mroots(p) is self-conjugate; that is, mroots(p) = mroots(p).

500

CHAPTER 6

Let p ∈ F[s], and let mroots(A) = {λ1 , . . . , λn }ms . The spread of p is defined by △

δ(p) = max |λi − λ j |. i, j∈{1,...,n}

(6.1.5)

Then, the root moduli of p are the nonnegative numbers ρn (p) ≤ · · · ≤ ρ1 (p) such that {ρ1 (p), . . . , ρn (p)}ms = {|λ1 |, . . . , |λn |}ms . In particular, define the minimum root modulus of p by △

ρmin (p) = ρn (p)

(6.1.6)

and the root radius of p by △

ρmax (p) = ρ1 (p).

(6.1.7)

The root real parts of p are the real numbers αn (p) ≤ · · · ≤ α1 (p) such that {α1 (p), . . . , αn (p)}ms = {Re λ1 , . . . , Re λn }ms . In particular, define the minimum root real part of A by △

αmin (p) = αn (p)

(6.1.8)

and the root real abscissa of p by △

αmax (p) = α1 (p).

(6.1.9)

The root imaginary parts of p are the real numbers βn (p) ≤ · · · ≤ β1 (p) such that {β1 (p), . . . , βn (p)}ms = {Im λ1 , . . . , Im λn }ms . In particular, define the minimum root imaginary part of A by △

βmin (p) = βn (p)

(6.1.10)

and the root imaginary abscissa of p by △

βmax (p) = β1 (p).

(6.1.11)

Let p ∈ F[s]. If p(−s) = p(s) for all s ∈ C, then p is even, while, if p(−s) = −p(s) for all s ∈ C, then p is odd. If p is either odd or even, then mroots(p) = −mroots(p). If p ∈ R[s] and there exists a polynomial q ∈ R[s] such that p(s) = q(s)q(−s) for all s ∈ C, then p has a spectral factorization. If p has a spectral factorization, then p is even and deg p is an even integer. Proposition 6.1.1. Let p ∈ R[s]. Then, the following statements are equivalent: i) p has a spectral factorization. ii) p is even, and every imaginary root of p has even multiplicity. iii) For all ω ∈ R, p(ω ȷ) ∈ [0, ∞). Proof. i) ⇐⇒ ii) is immediate. To prove i) =⇒ iii), note that, for all ω ∈ R, p(ω ȷ) = q(ω ȷ)q(−ω ȷ) = |q(ω ȷ)|2 ≥ 0. Conversely, to prove iii) =⇒ i) write p = p1 p2 , where every root of p1 is imaginary and none of the roots of p2 are imaginary. Now, let z be a root of p2 . Then, −z, z, and −z are also roots of p2 with the same multiplicity as z. Hence, there exists a polynomial p20 ∈ R[s] such that p2 (s) = p20 (s)p20 (−s) for all s ∈ C. ∏ Next, assuming that p has at least one imaginary root, write p1 (s) = ki=1 (s2 + ω2i )mi , where △ 0 ≤ ω1 < · · · < ωk and mi = mult p (ωi ȷ). Let ωi0 denote the smallest element of the set {ω1 , . . . , ωk } ∏ such that mi is odd. Then, it follows that p1 (ω ȷ) = ki=1 (ω2i − ω2 )mi < 0 for all ω ∈ (ωi0 , ωi0 +1 ), △ where ωk+1 = ∞. However, note that p1 (ω ȷ) = p(ω ȷ)/p2 (ω ȷ) = p(ω ȷ)/|p20 (ω ȷ)|2 ≥ 0 for all ω ∈ R, which is a contradiction. Therefore, mi is even for all i ∈ {1, . . . , k}, and thus p1(s) = p10 (s)p10 (−s) △ ∏ for all s ∈ C, where p10 (s) = ki=1 (s2 + ω2i )mi/2. Consequently, p(s) = p10 (s)p20 (s)p10 (−s)p20 (−s) for all s ∈ C. Finally, if p has no imaginary roots, then p1 = 1, and p(s) = p20 (s)p20 (−s) for all s ∈ C. 

POLYNOMIAL MATRICES AND RATIONAL TRANSFER FUNCTIONS

501

The following division algorithm is essential to the study of polynomials. Lemma 6.1.2. Let p1 , p2 ∈ F[s], and assume that p2 is not the zero polynomial. Then, there exist unique polynomials q, r ∈ F[s] such that deg r < deg p2 and p1 = qp2 + r. △

(6.1.12)

△

Proof. Define n = deg p1 and m = deg p2 . If n < m, then q = 0 and r = p1. Hence, deg r =

deg p1 = n < m = deg p2 . Now, assume that n ≥ m ≥ 0, and write p1(s) = βn sn + · · · + β0 and p2 (s) = γm sm + · · · + γ0 . If n = 0, then m = 0, γ0 , 0, q = β0 /γ0 , and r = 0. Hence, −∞ = deg r < 0 = deg p2 . If n = 1, then either m = 0 or m = 1. If m = 0, then p2 (s) = γ0 , 0, and (6.1.12) holds with q(s) = p1 (s)/γ0 and r = 0, in which case −∞ = deg r < 0 = deg p2 . If m = 1, then (6.1.12) holds with q(s) = β1/γ1 and r(s) = β0 − β1 γ0 /γ1. Hence, deg r ≤ 0 < 1 = deg p2 . Now, suppose that n = 2. Then, pˆ 1(s) = p1(s) − (β2 /γm )s2−m p2 (s) has degree 1. Applying (6.1.12) with p1 replaced by pˆ 1 , it follows that there exist polynomials q1, r1 ∈ F[s] such that pˆ 1 = q1 p2 + r1 and such that deg r1 < deg p2 . It thus follows that p1(s) = q1(s)p2 (s) + r1(s) + (β2 /γm )s2−m p2 (s) = q(s)p2 (s) + r(s), where q(s) = q1(s) + (β2 /γm )sn−m and r = r1, which verifies (6.1.12). Similar arguments apply to successively larger values of n. To prove uniqueness, suppose there exist polynomials qˆ and rˆ such that deg rˆ < deg p2 and p1 = qp ˆ 2 + rˆ. Then, it follows that (qˆ − q)p2 = r − rˆ. Next, note that deg(r − rˆ) < deg p2 . If qˆ , q, then deg p2 ≤ deg[(qˆ − q)p2 ] so that deg(r − rˆ) < deg[(qˆ − q)p2 ], which is a contradiction. Thus, qˆ = q, and, hence, r = rˆ.  In Lemma 6.1.2, q is the quotient of p1 and p2 , while r is the remainder. If r = 0, then p2 divides p1 ,; equivalently, p1 is a multiple of p2 . Note that, if p2 (s) = s − α, where α ∈ F, then r is constant and is given by r(s) = p1(α). If a polynomial p3 ∈ F[s] divides two polynomials p1 , p2 ∈ F[s], then p3 is a common divisor of p1 and p2 . Given polynomials p1 , p2 ∈ F[s], there exists a unique monic polynomial p3 ∈ F[s], the greatest common divisor of p1 and p2 , such that p3 is a common divisor of p1 and p2 and such that every common divisor of p1 and p2 divides p3 . In addition, there exist polynomials q1, q2 ∈ F[s] such that the greatest common divisor p3 of p1 and p2 is given by p3 = q1 p1 + q2 p2 . See [2221, p. 113] for proofs of these results. Finally, p1 and p2 are coprime if their greatest common divisor is p3 = 1, while a polynomial p ∈ F[s] is irreducible if there do not exist nonconstant polynomials p1 , p2 ∈ F[s] such that p = p1 p2 . For example, if F = R, then p(s) = s2 + s + 1 is irreducible. If a polynomial p3 ∈ F[s] is a multiple of two polynomials p1 , p2 ∈ F[s], then p3 is a common multiple of p1 and p2 . Given nonzero polynomials p1 and p2 , there exists (see [2221, p. 113]) a unique monic polynomial p3 ∈ F[s] that is a common multiple of p1 and p2 and that divides every common multiple of p1 and p2 . The polynomial p3 is the least common multiple of p1 and p2 . The polynomial p ∈ F[s] given by (6.1.1) can be evaluated with a square matrix argument A ∈ Fn×n by defining △

p(A) = βk Ak + βk−1 Ak−1 + · · · + β1 A + β0 I.

(6.1.13)

6.2 Polynomial Matrices The set F[s]n×m of polynomial matrices consists of matrix functions P: C 7→ Cn×m whose entries are elements of F[s]. A polynomial matrix P ∈ F[s]n×m can thus be written as P(s) = skBk + sk−1Bk−1 + · · · + sB1 + B0 ,

(6.2.1)

where B0 , . . . , Bk ∈ Fn×m. If Bk is nonzero, then the degree of P, denoted by deg P, is k, whereas, if P = 0, then deg P = −∞. If n = m and Bk is nonsingular, then P is regular, while, if Bk = I, then P

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is monic. The following result, which generalizes Lemma 6.1.2, provides a division algorithm for polynomial matrices. Lemma 6.2.1. Let P1, P2 ∈ F[s]n×n , where P2 is regular. Then, there exist unique polynomial ˆ Rˆ ∈ F[s]n×n such that deg R < deg P2 , deg Rˆ < deg P2 , matrices Q, R, Q, P1 = QP2 + R, ˆ P1 = P2 Qˆ + R.

(6.2.2) (6.2.3)

 ˆ If R = 0, then P2 right divides P1, while, if R = 0, then P2 left divides P1. Let the polynomial matrix P ∈ F[s]n×m be given by (6.2.1). Then, P can be evaluated with a square matrix argument in two different ways, either from the right or from the left. For all A ∈ Cm×m , define Proof. See [1186, p. 90] and [2221, pp. 134–135].

△

PR (A) = Bk Ak + Bk−1 Ak−1 + · · · + B1 A + B0 , while, for all A ∈ C

(6.2.4)

, define

n×n

△

PL (A) = Ak Bk + Ak−1Bk−1 + · · · + AB1 + B0 .

(6.2.5)

PR(A) and PL (A) are matrix polynomials. If n = m, then PR(A) and PL (A) can be evaluated for all A ∈ Fn×n, although these matrices may be different. The following result is useful. △ Lemma 6.2.2. Let Q, Qˆ ∈ F[s]n×n and A ∈ Fn×n. Furthermore, define P, Pˆ ∈ F[s]n×n by P(s) = △ ˆ = (sI − A)Q(s). ˆ Q(s)(sI − A) and P(s) Then, PR (A) = 0 and Pˆ L (A) = 0. Let p ∈ F[s] be given by (6.1.1), and define △

P(s) = p(s)In = sk βk In + sk−1 βk−1In + · · · + sβ1In + β0 In ∈ F[s]n×n . For A ∈ Cn×n it follows that p(A) = P(A) = PR (A) = PL (A). The following result specializes Lemma 6.2.1 to polynomial matrix divisors of degree 1. Corollary 6.2.3. Let P ∈ F[s]n×n and A ∈ Fn×n. Then, there exist unique polynomial matrices ˆ Q, Q ∈ F[s]n×n and unique matrices R, Rˆ ∈ Fn×n such that P(s) = Q(s)(sI − A) + R, ˆ + R. ˆ P(s) = (sI − A)Q(s)

(6.2.6) (6.2.7)

Furthermore, R = PR (A) and Rˆ = PL (A). Proof. In Lemma 6.2.1 set P1 = P and P2 (s) = sI − A. Since deg P2 = 1, it follows that deg R = deg Rˆ = 0, and thus R and Rˆ are constant. The last statement follows from Lemma 6.2.2.  Definition 6.2.4. Let P ∈ F[s]n×m . Then, rank P is defined by △

rank P = max rank P(s). s∈C

(6.2.8)

Let P ∈ F[s]n×n . Then, P(s) ∈ Cn×n for all s ∈ C. Furthermore, det P is a polynomial in s; that is, det P ∈ F[s]. Definition 6.2.5. Let P ∈ F[s]n×n . Then, P is nonsingular if det P is not the zero polynomial; otherwise, P is singular. Proposition 6.2.6. Let P ∈ F[s]n×n , and assume that P is regular. Then, P is nonsingular.

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POLYNOMIAL MATRICES AND RATIONAL TRANSFER FUNCTIONS

Let P ∈ F[s]n×n . If P is nonsingular, then the inverse P−1 of P can be constructed according to −1 (3.8.22). In general, rational [ s+2 ] the entries of P [are ] functions of s (see Definition 6.7.1). For example, s+1 1 s−1 −s−1 −1 if P(s) = s−2 s−1 , then P (s) = 2s . In certain cases, P−1 is also a polynomial matrix. −s+2 s+2 [ s ] [ s+1 −1 ] 1 For example, if P(s) = s2 +s−1 s+1 , then P−1(s) = −s2 −s+1 s . The following result extends Proposition 3.8.7 from constant matrices to polynomial matrices. Proposition 6.2.7. Let P ∈ F[s]n×m . Then, rank P is the size of the largest nonsingular polynomial matrix that is a submatrix of P. Proof. For all s ∈ C it follows from Proposition 3.8.7 that rank P(s) is the size of the largest nonsingular submatrix of P(s). Now, let s0 ∈ C be such that rank P(s0 ) = rank P. Then, P(s0 ) has a nonsingular submatrix of maximal size rank P. Therefore, P has a nonsingular polynomial submatrix of maximal size rank P.  A polynomial matrix can be transformed by performing elementary row and column operations of the following types: i) Multiply either a row or a column by a nonzero constant. ii) Interchange either two rows or two columns. iii) Add a polynomial multiple of one (row, column) to another (row, column). These operations correspond, respectively, to left multiplication and right multiplication by the elementary matrices    Ii−1 0 0    (6.2.9) In + (α − 1)Ei,i =  0 α 0  ,   0 0 In−i where α ∈ F is nonzero,

In + Ei, j + E j,i − Ei,i − E j, j

    =   

Ii−1 0 0 0 0

0 0 0 1 0

where i , j, and the elementary polynomial matrix  0  Ii−1 0  0 1 0  In + pEi, j =  0 0 I j−i−1  0 0 0  0 0 0

0 0 I j−i−1 0 0

0 p 0 1 0

0 0 0 0 In− j

0 1 0 0 0      ,  

0 0 0 0 In− j

     ,  

(6.2.10)

(6.2.11)

where i , j and p ∈ F[s]. The matrices shown in (6.2.10) and (6.2.11) illustrate the case i < j. Applying these operations sequentially corresponds to forming products of elementary matrices and elementary polynomial matrices. Note that the elementary polynomial matrix I + pEi, j is nonsingular, and that (I + pEi, j )−1 = I − pEi, j . Therefore, the inverse of an elementary polynomial matrix is an elementary polynomial matrix.

6.3 The Smith Form and Similarity Invariants Definition 6.3.1. Let P ∈ F[s]n×n . Then, P is unimodular if P is the product of elementary

matrices and elementary polynomial matrices. The following result provides a canonical matrix, known as the Smith matrix, for polynomial matrices under unimodular transformation.

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Theorem 6.3.2. Let P ∈ F[s]n×m , and let r = rank P. Then, there exist unimodular matrices

S 1 ∈ F[s] and S 2 ∈ F[s] and unique monic polynomials p1 , . . . , pr ∈ F[s] such that pi divides pi+1 for all i ∈ {1, . . . , r − 1} and such that   0   p1   ..   . (6.3.1) P = S 1   S 2 .   pr   0 0(n−r)×(m−r) n×n

m×m

Furthermore, for all i ∈ {1, . . . , r}, the polynomial △

∆i =

i ∏

pj

(6.3.2)

j=1

is the monic greatest common divisor of all i × i subdeterminants of P. Proof. This result is obtained by applying elementary row and column operations to P. See [1573, pp. 390–392] and [2221, pp. 125–128].  The diagonal matrix in (6.3.1) is a Smith matrix and the Smith form of P. Definition 6.3.3. Let P ∈ F[s]n×m . Then, the monic polynomials p1 , . . . , pr ∈ F[s] in the Smith form of P are the Smith polynomials of P. The Smith zeros of P are the roots of p1 , . . . , pr ; that is, △

Szeros(P) = roots(pr ), r ∪ △ mSzeros(P) = mroots(pi ).

(6.3.3) (6.3.4)

i=1

Proposition 6.3.4. Let P ∈ R[s]n×m , and assume that there exist unimodular matrices S 1 ∈

F[s]n×n and S 2 ∈ F[s]m×m and monic polynomials p1 , . . . , pr ∈ F[s] satisfying (6.3.1). Then, rank P = r. △ Proposition 6.3.5. Let P ∈ F[s]n×m , and let r = rank P. Then, r is the largest size of all nonsingular submatrices of P. Proof. Let r0 denote the largest size of all nonsingular submatrices of P, and let P0 ∈ F[s]r0 ×r0 be a nonsingular submatrix of P. First, assume that r < r0 . Then, there exists s0 ∈ C such that rank P(s0 ) = rank P0 (s0 ) = r0 . Thus, r = rank P = max s∈C rank P(s) ≥ rank P(s0 ) = r0 , which is a contradiction. Next, assume that r > r0 . Then, it follows from (6.3.1) that there exists s0 ∈ C such that rank P(s0 ) = r. Consequently, P(s0 ) has a nonsingular r × r submatrix. Let Pˆ 0 ∈ F[s]r×r denote the corresponding submatrix of P. Thus, Pˆ 0 is nonsingular, which implies that P has a nonsingular submatrix whose size is greater than r0 , which is a contradiction. Consequently, r = r0 .  n×m Proposition 6.3.6. Let P ∈ F[s] . Then, rank P(s) < rank P if and only if s ∈ Szeros(P). Proposition 6.3.7. Let P ∈ F[s]n×m , and let S ⊂ C be a countable set. Then, rank P = max rank P(s). s∈C\S

Proposition 6.3.8. Let P ∈ F[s]n×n . Then, the following statements are equivalent:

i) ii) iii) iv) v)

P is unimodular. det P is a nonzero constant. The Smith form of P is the identity matrix. P is nonsingular, and P−1 is a polynomial matrix. P is nonsingular, and P−1 is unimodular.

(6.3.5)

POLYNOMIAL MATRICES AND RATIONAL TRANSFER FUNCTIONS

505

Proof. To prove i) =⇒ ii), note that every elementary matrix and every elementary polynomial matrix has a constant nonzero determinant. Since P is a product of elementary matrices and elementary polynomial matrices, its determinant is a constant. To prove ii) =⇒ iii), note that it follows from (6.3.1) that rank P = n and det P = (det S 1 )(det S 2 )p1 · · · pn , where S 1 , S 2 ∈ Fn×n are unimodular and p1 , . . . , pn are monic polynomials. From the result i) =⇒ ii), it follows that det S 1 and det S 2 are nonzero constants. Since det P is a nonzero constant, it follows that p1 · · · pn = det P/[(det S 1 )(det S 2 )] is a nonzero constant. Since p1 , . . . , pn are monic polynomials, it follows that p1 = · · · = pn = 1. Next, to prove iii) =⇒ iv), note that P is unimodular, and thus it follows that det P is a nonzero constant. Furthermore, since PA is a polynomial matrix, it follows that P−1 = (det P)−1PA is a polynomial matrix. To prove iv) =⇒ v), note that det P−1 is a polynomial. Since det P is a polynomial and det P−1 = 1/det P it follows that det P is a nonzero constant. Hence, P is unimodular, and thus P−1 = (det P)−1PA is unimodular. Finally, to prove v) =⇒ i), note that det P−1 is a nonzero constant, and thus P = [det P−1 ]−1 [P−1 ]A is a polynomial matrix. Furthermore, since det P = 1/det P−1 , it follows that det P is a nonzero constant. Hence, P is unimodular.  Proposition 6.3.9. Let A1 , B1 , A2 , B2 ∈ Fn×n, where A2 is nonsingular, and define the polyno△ △ mial matrices P1, P2 ∈ F[s]n×n by P1 (s) = sA1 + B1 and P2 (s) = sA2 + B2 . Then, P1 and P2 have the same Smith polynomials if and only if there exist nonsingular matrices S 1 , S 2 ∈ Fn×n such that P2 = S 1 P1 S 2 . Proof. The sufficiency result is immediate. To prove necessity, note that it follows from Theorem 6.3.2 that there exist unimodular matrices T1, T2 ∈ F[s]n×n such that P2 = T2 P1T1 . Now, since P2 is regular, it follows from Lemma 6.2.1 that there exist polynomial matrices Q, Qˆ ∈ F[s]n×n and ˆ Next, we have constant matrices R, Rˆ ∈ Fn×n such that T1 = QP2 + R and T2 = P2 Qˆ + R.

ˆ 1 (QP2 + R) + P2 QT ˆ 2−1P2 ˆ 1T1 = RP ˆ 1T1 + P2 QT ˆ 2−1P2 = RP P2 = T2 P1T1 = (P2 Qˆ + R)P ˆ 1 R + T2P1QP2 + P2 (−QP ˆ 1Q + QT ˆ 2−1 )P2 ˆ 1 R + (T2 − P2 Q)P ˆ 1QP2 + P2 QT ˆ 2−1P2 = RP = RP ˆ 1 R + P2 (T1−1Q − QP ˆ 1Q + QT ˆ 2−1 )P2 . = RP ˆ 1Q + QT ˆ −1 is not zero, then Since P2 is regular and has degree 1, it follows that, if T 1−1Q − QP 2 −1 −1 ˆ 1Q + QT ˆ ˆ deg P2 (T 1 Q − QP 2 )P2 ≥ 2. However, since P2 and RP1 R have degree less than 2, it ˆ 1Q + QT ˆ −1 = 0. Hence, P2 = RP ˆ 1 R. follows that T 1−1Q − QP 2 Next, to show that Rˆ and R are nonsingular, note that, for all s ∈ C, ˆ 1(s)R = sRA ˆ 1R + RB ˆ 1 R, P2 (s) = RP which implies that A2 = S 1 A1 S 2 , where S 1 = Rˆ and S 2 = R. Since A2 is nonsingular, it follows that S 1 and S 2 are nonsingular.  n×n Definition 6.3.10. Let A ∈ F . Then, the similarity invariants of A are the Smith polynomials of sI − A. The following result provides necessary and sufficient conditions for two matrices to be similar. Theorem 6.3.11. Let A, B ∈ Fn×n. Then, A and B are similar if and only if they have the same similarity invariants. Proof. To prove necessity, assume that A and B are similar. Then, the matrices sI − A and sI − B have the same Smith form and thus the same similarity invariants. To prove sufficiency, it follows from Proposition 6.3.9 that there exist nonsingular matrices S 1 , S 2 ∈ Fn×n such that sI − A = S 1 (sI − B)S 2 . Thus, S 1 = S 2−1, and, hence, A = S 1BS 1−1. 

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Corollary 6.3.12. Let A ∈ Fn×n. Then, A and AT are similar.

A stronger form of Corollary 6.3.12 is given by Corollary 7.4.9.

6.4 Eigenvalues Let A ∈ Fn×n. Then, the polynomial matrix sI − A ∈ F[s]n×n is monic and has degree 1. Definition 6.4.1. Let A ∈ Fn×n. Then, the characteristic polynomial of A is the polynomial χA ∈ F[s] given by △

χA (s) = det(sI − A).

(6.4.1)

Since sI − A is a polynomial matrix, its determinant is the product of its Smith polynomials, which are the similarity invariants of A. Proposition 6.4.2. Let A ∈ Fn×n, and let p1 , . . . , pn ∈ F[s] denote the similarity invariants of A. Then, n ∏ χA = pi . (6.4.2) i=1

Proposition 6.4.3. Let A ∈ F

. Then, χA is monic and deg χA = n. Let A ∈ Fn×n, and write the characteristic polynomial of A as n×n

χA (s) = sn + βn−1 sn−1 + · · · + β1 s + β0 ,

(6.4.3)

where β0 , . . . , βn−1 ∈ F. The eigenvalues of A are the n possibly repeated roots λ1 , . . . , λn ∈ C of χA , which are the solutions of the characteristic equation χA (s) = 0.

(6.4.4)

It is often convenient to denote the eigenvalues of A ∈ Fn×n by λ1 , . . . , λn . This notation, however, does not specify which eigenvalue of A is denoted by λi . If, however, every eigenvalue of A is real, then we denote the eigenvalues of A unambiguously by λ1 (A), . . . , λn (A), where λn (A) ≤ · · · ≤ λ1 (A).

(6.4.5)

Furthermore, we define △

△

λmin (A) = λn (A), and the eigenvalue vector of A by

λmax (A) = λ1 (A)

   λ1 (A)    △  λ(A) =  ...  .   λn (A)

(6.4.6)

(6.4.7)

Definition 6.4.4. Let A ∈ Fn×n. The algebraic multiplicity of an eigenvalue λ of A, denoted by

amultA(λ), is the algebraic multiplicity of λ as a root of χA ; that is, △

amultA(λ) = multχA(λ).

(6.4.8)

The multiset consisting of the eigenvalues of A including their algebraic multiplicity, denoted by mspec(A), is the multispectrum of A; that is, △

mspec(A) = mroots(χA ).

(6.4.9)

Ignoring algebraic multiplicity, spec(A) denotes the spectrum of A; that is, △

spec(A) = roots(χA ).

(6.4.10)

POLYNOMIAL MATRICES AND RATIONAL TRANSFER FUNCTIONS

507

Note that Szeros(sI − A) = spec(A), mSzeros(sI − A) = mspec(A).

(6.4.11) (6.4.12)

We can thus write mspec(A) = {λ1 , . . . , λn }ms to denote the multiset of repeated eigenvalues of A, and spec(A) = {λ1 , . . . , λr } to denote the set of distinct eigenvalues of A. However, as noted above, this notation is generic. If λ < spec(A), then λ < roots(χA ), and thus amultA (λ) = multχA(λ) = 0. Let A ∈ Fn×n and mroots(χA ) = {λ1 , . . . , λn }ms . Then, χA (s) =

n ∏

(s − λi ).

(6.4.13)

i=1

If F = R, then χA (s) has real coefficients, and thus the eigenvalues of A occur in complex conjugate pairs; that is, mroots(χA ) = mroots(χA ). Now, let spec(A) = {λ1 , . . . , λr }, and, for all i ∈ {1, . . . , r}, let ni denote the algebraic multiplicity of λi . Then, χA (s) =

r ∏

(s − λi )ni .

(6.4.14)

i=1

The following result gives some basic properties of the spectrum of a matrix. Proposition 6.4.5. Let A, B ∈ Fn×n. Then, the following statements hold: i) χAT = χA . ii) For all s ∈ C, χ−A (s) = (−1)n χA (−s). iii) mspec(AT ) = mspec(A). iv) mspec(A) = mspec(A). v) mspec(A∗ ) = mspec(A). vi) 0 ∈ spec(A) if and only if det A = 0. vii) If either k ∈ N or both A is nonsingular and k ∈ Z, then mspec(Ak ) = {λk : λ ∈ mspec(A)}ms .

(6.4.15)

If α ∈ F, then χαA+I (s) = χA (s − α). If α ∈ F, then mspec(αI + A) = α + mspec(A). If α ∈ F, then mspec(αA) = α mspec(A). If A is Hermitian, then spec(A) ⊂ R. If A and B are similar, then χA = χB and mspec(A) = mspec(B). Proof. To prove i), note that det(sI − AT ) = det (sI − A)T = det(sI − A). To prove ii), note that χ−A (s) = det(sI + A) = (−1)n det(−sI − A) = (−1)n χA (−s). Next, iii) follows from i), iv) follows from det(sI − A) = det(sI − A) = det(sI − A), v) follows from iii) and iv), and vi) follows from χA (0) = (−1)n det A. To prove “⊇” in vii), let λ ∈ spec(A) and let x ∈ Cn be an eigenvector of A associated with λ (see Section 6.5). Then, A2x = A(Ax) = A(λx) = λAx = λ2 x. Similarly, in the case where A is nonsingular, Ax = λx implies that A−1x = λ−1x, and thus A−2x = λ−2x. Similar arguments apply to arbitrary k ∈ Z. The reverse inclusion follows from the Jordan decomposition given by Theorem 7.4.2. To prove viii), note that χαI+A(s) = det[sI −(αI +A)] = det[(s−α)I −A] = χA (s−α). ix) follows immediately. x) is true for α = 0. For α , 0, χαA(s) = det(sI −αA) = αn det[(s/α)I − A] = αn χA (s/α). To prove xi), assume that A = A∗, let λ ∈ spec(A), and let x ∈ Cn be an eigenvector of A associated with λ. Then, λ = x∗Ax/x∗x, which is real. Finally, xii) is immediate.  viii) ix) x) xi) xii)

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The following result characterizes the coefficients of χA in terms of the eigenvalues of A. Proposition 6.4.6. Let A ∈ Fn×n, and let mspec(A) = {λ1 , . . . , λn }ms . Then, for all i ∈ {0, . . . , n− 1}, the coefficient βi of si in (6.4.3) is given by ∑ βi = (−1)n−i λ j1 · · · λ jn−i . (6.4.16) 1≤ j1 j. Then, there exists a polynomial p: Rn(n−1)/2 7→ R such that, for all α ∈ R and x ∈ Rn(n−1)/2 , p(αx) = αn/2 p(x),

det A = p2 (xA ).

In particular, [ det

0 −a

]

a = a2 , 0

 a  0  −a 0  det   −b −d −c −e

b d 0 −f

 c   e   = (af − be + cd)2. f  0

Source: [1768, p. 224] and [2263, pp. 125–127]. Remark: p is the Pfaffian, and this is Pfaff’s theorem. Remark: An extension to the product of a pair of skew-symmetric matrices is given in [948]. Related: Fact 4.10.35. Fact 6.8.17. Let G ∈ F(s)n×m , and let G(i, j) = ni j/di j , where ni j ∈ F[s] and di j ∈ F[s] are coprime for all i ∈ {1, . . . , n} and j ∈ {1, . . . , m}. Then, q1 given by the Smith-McMillan form is the least common multiple of d11 , d12 , . . . , dnm .

POLYNOMIAL MATRICES AND RATIONAL TRANSFER FUNCTIONS

523

Fact 6.8.18. Let G ∈ F(s)n×m , assume that rank G = m, and let λ ∈ C, where λ is not a pole of

G. Then, λ is a transmission zero of G if and only if there exists a nonzero vector u ∈ Cm such that G(λ)u = 0. Furthermore, if G is square, then λ is a transmission zero of G if and only if det G(λ) = 0. Fact 6.8.19. Let G ∈ Rn×m(s), let ω ∈ R, and assume that ω ȷ is not a pole of G. Then, Im G(−ω ȷ) = −Im G(ω ȷ). Fact 6.8.20. Let p ∈ R[s], and assume that all of the coefficients of p are nonnegative. Then, p is increasing on [0, ∞). ∑ Fact 6.8.21. Let n ≥ 2, let p ∈ F[s], where p(s) = ni=0 βi si , let mroots(p) = {λ1 , . . . , λn }m , and ∏ ∏ define △ dis(p) = β2n−2 (λi − λ j )2 = (−1)n(n−1)/2 β2n−2 (λi − λ j ), n n

where the first product is taken over all i, j ∈ {1, . . . , n} such that i < j, and the second product is taken over all distinct i, j ∈ {1, . . . , n}. Then, the following statements are equivalent: i) p has distinct roots. ii) dis(p) , 0. iii) mroots(p) ∩ mroots(p′ ) = ∅. iv) p and p′ are coprime. Furthermore, the following statements hold: v) If p(s) = as2 + bs + c, then dis(p) = b2 − 4ac. vi) If p(s) = as3 + bs2 + cs + d, then dis(p) = b2 c2 − 4ac2 − 4b3 d − 27a2 d2 + 18abcd. vii) If p(s) = s3 + cs + d, then dis(p) = −4c3 − 27d2 . viii) If p(s) = as4 + bs3 + cs2 + ds + e, then dis(p) = 256a3 e3 − 192a2 bde2 − 128a2 c2 e2 + 144a2 cd2 e − 27a2 d4 + 144ab2 ce2 − 6ab2 d2 e − 80abc2 de + 18abcd3 + 16ac4 e − 4ac3 d2 − 27b4 e2 + 18b3 cde − 4b3 d3 − 4b2 c3 e + b2 c2 d2 . Source: [128], [1083, pp. 70, 119]. Remark: dis(p) is the discriminant of p. Related: Fact 6.8.6 Fact 6.8.22. Let p1 , p2 ∈ C[s], assume that p1 and p2 are coprime, and assume that p1 , p2 , and

p1 + p2 are not constant. Then,

max {deg p1 , deg p2 } + 1 ≤ card(roots[p1 p2 (p1 + p2 )]), deg[p1 p2 (p1 + p2 )] + 3 ≤ 3 card(roots[p1 p2 (p1 + p2 )]). Source: [331]. Remark: This is Stothers’s theorem. Related: Fact 6.8.23. Fact 6.8.23. Let p1 , p2 , p3 ∈ C[s], and assume that p1 , p2 , p3 , p1 + p2 + p3 are coprime and not

constant. Then, max {deg p1 , deg p2 , det p3 , deg(p1 + p2 + p3 )} + 3 ≤ 2 card(roots[p1 p2 p3 (p1 + p2 + p3 )]). Furthermore, deg[p1 p2 p3 (p1 + p2 + p3 )] + 12 ≤ 8 card(roots[p1 p2 (p1 + p2 )]). If, in addition, p1 , p2 , p3 , p1 + p2 + p3 are linearly dependent, then max {deg p1 , deg p2 , det p3 , deg(p1 + p2 + p3 )} + 5 ≤ 2 card(roots[p1 p2 p3 (p1 + p2 + p3 )]). Source: [331]. Related: This result extends Fact 6.8.22.

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CHAPTER 6

6.9 Facts on the Characteristic and Minimal Polynomials Fact 6.9.1. Let A =

i) mspec(A) =

{ [ 1 2

[

]

∈ R2×2. Then, the following statements hold: ]} { [ ]} √ √ a + d ± (a − d)2 + 4bc = 12 tr A ± (tr A)2 − 4 det A . a b c d

ms

ms

χA (s) = s2 − (tr A)s + det A. A2 = (tr A)A − (det A)I. tr A2 = (tr A)2 − 2 det A. det A = 21 [(tr A)2 − tr A2 ]. (sI − A)A = sI + A − (tr A)I. AA = (tr A)I − A. mspec(AA ) = mspec(A). χAA (s) = χA (s). tr AA = tr A. det AA = det A. tr A If A is nonsingular, then A−1 = (det A)−1 [(tr A)I − A] and tr A−1 = det A. 2 2 2 If A is singular, then A = (tr A)A and tr A = (tr A) . If tr A = 0, then tr A2 + 2 det A = 0 and A2 = −(det A)I = 21 (tr A2 )I. If A is singular and tr A = 0, then A = 0. Fact 6.9.2. Let A ∈ R3×3. Then, the following statements hold: i) χA (s) = s3 − (tr A)s2 + (tr AA )s − det A. ∑ ii) tr AA = 21 [(tr A)2 − tr A2 ] = 3i=1 det A[i,i] .

ii) iii) iv) v) vi) vii) viii) ix) x) xi) xii) xiii) xiv) xv)

iii) A3 = (tr A)A2 − (tr AA )A + (det A)I = (tr A)A2 + 12 [tr A2 − (tr A)2 ]A + (det A)I. iv) tr A3 = (tr A) tr A2 − (tr AA ) tr A + 3(det A) = 23 (tr A) tr A2 − 12 (tr A)3 + 3(det A). v) det A = 13 [tr A3 − (tr A) tr A2 + (tr AA ) tr A] =

1 3

tr A3 − 12 (tr A)tr A2 + 61 (tr A)3.

vi) (sI − A)A = s2I + s[A − (tr A)I] + A2 − (tr A)A + 21 [(tr A)2 − tr A2 ]I. vii) AA = A2 − (tr A)A + 12 [(tr A)2 − tr A2 ]I. viii) det AA = (det A)2 . ix) If A is singular, then A3 = (tr A)A2 − (tr AA )A = (tr A)A2 + 21 [tr A2 − (tr A)2 ]A. x) If tr A = 0, then tr AA = − 12 tr A2 and tr A3 = 3(det A). xi) If A is nonsingular, then A−1 = (det A)−1 [A2 − (tr A)A + (tr AA )I] = (det A)−1 [A2 − (tr A)A + 21 [(tr A)2 − tr A2 ]I, tr A−1 =

tr A2 − (tr A)2 + 3 tr AA (tr A)2 − tr A2 tr AA = = . det A 2 det A det A

Related: Fact 3.19.1 and Fact 9.5.18. Fact 6.9.3. Let A ∈ R4×4. Then, the following statements hold:

i) χA (s) = s4 − (tr A)s3 + 12 [(tr A)2 − tr A2 ]s2 − (tr AA )s + det A. ∑ ii) 12 [(tr A)2 − tr A2 ] = 41≤i< j≤4 det A({i, j},{i, j}) . ∑ iii) tr AA = 16 [(tr A)3 − 3(tr A) tr A2 + 2 tr A3 ] = 4i=1 det A[i,i] . iv) A4 = (tr A)A3 − 12 [(tr A)2 − tr A2 ]A2 + 16 [(tr A)3 − 3(tr A) tr A2 + 2 tr A3 ]A − (det A)I. v) tr A4 = (tr A) tr A3 − 21 [(tr A)2 − tr A2 ]A2 + 61 [(tr A)3 − 3(tr A) tr A2 + 2 tr A3 ] tr A − 4(det A).

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POLYNOMIAL MATRICES AND RATIONAL TRANSFER FUNCTIONS

vi) If tr A = 0, then tr AA = 13 tr A3 and tr A4 = 12 (tr A2 )2 − 4(det A). Fact 6.9.4. Let A ∈ Fn×n, let χA (s) = sn + βn−1 sn−1 + · · · + β0 , and let mspec(A) = {λ1 , . . . , λn }ms . Then, AA = (−1)n−1 (An−1 + βn−1An−2 + · · · + β1I). Furthermore, ∑

tr AA = (−1)n−1χ′A(0) = (−1)n−1 β1 =

λ j1 · · · λ jn−1 =

1≤ j1 0 if and only if

√ λ1 = 13 tr A + 2 p cos ϕ, √ √ λ2 = 13 tr A + 3p sin ϕ − p cos ϕ, √ √ λ3 = 31 tr A − 3p sin ϕ − p cos ϕ,

where ϕ ∈ [0, π/3] is given by

ϕ = 31 acos

q . p3/2

iv) ϕ = 0 if and only if q = p3/2 > 0. If these conditions hold, then √ √ λ1 = 31 tr A + 2 p, λ2 = λ3 = 13 tr A − p. v) ϕ = π/6√if and only if p > 0 and q = 0. If these conditions hold, then sin ϕ = 1/2, cos ϕ = 3/2, and √ √ λ1 = 31 tr A + 3p, λ2 = 13 tr A, λ3 = 31 tr A − 3p.

532

CHAPTER 6

vi) ϕ = π/3 if and only if q = −p3/2 < 0. If these conditions hold, then sin ϕ = cos ϕ = 1/2, and √ √ λ1 = λ2 = 13 tr A + p, λ3 = 31 tr A − 2 p.

√

3/2,

Source: [2471]. Remark: This result is based on Cardano’s trigonometric solution for the roots of a cubic polynomial. See [502], [1083, Lecture 4], and [2471]. Remark: q2 ≤ p3 follows from Fact 2.2.27. Related: Fact 2.21.2. Fact 6.10.8. Let a, b, c, d, ω ∈ R, and define the skew-symmetric matrix A ∈ R4×4 given by

  0  −ω △  A =   −a −b

ω a 0 c −c 0 −d −ω

 b   d  . ω  0

Then, χA (s) = s4 + (2ω2 + a2 + b2 + c2 + d2 )s2 + [ω2 − (ad − bc)]2 , det A = [ω2 − (ad − bc)]2 . √ Hence, A is singular if and only if bc ≤ ad and ω = ad − bc. Furthermore, A has a repeated eigenvalue if and only if either i) A is singular or ii) a = −d and b = c. In√case i), A has the repeated eigenvalue 0, whereas, in case ii), A has the repeated eigenvalues ω2 + a2 + b2 ȷ and √ 2 − ω + a2 + b2 ȷ. Finally, cases i) and ii) cannot occur simultaneously. Related: Fact 4.10.35, Fact 6.9.19, Fact 15.12.16, and Fact 15.12.18. Fact 6.10.9. Define A, B ∈ Rn×n by     0  0   1 −2 0 · · · 0  1 −2 0 · · · 0        .. ..    0 1 −2 . 0 . 0 0  0   0 1 −2     .. ..  0 0  0 0 . 0 . 0 1 0  1 0  △  △     A =  . .. ..  , .. ..  , B =  .. .. .. .. .. .. ..  ..    . . . . . . . . . . .        . .  0 0 . . 1 −2  . . 1 −2  0 0  0 0     α 0 0 ··· 0 1 0 0 0 ··· 0 1 △

where α = −1/2n−1. Then, spec(A) = {1} and det B = 0. Fact 6.10.10. Let A ∈ Fn×n. Then, |αmax (A)| ≤ ρmax (A). Fact 6.10.11. Let A ∈ Fn×n, assume that A is nonsingular, and assume that ρmax (I − A) < 1. Then, ∞ ∑ A−1 = (I − A)k. k=0

Fact 6.10.12. Let A ∈ F

and B ∈ F . If tr Ak = tr Bk for all k ∈ {1, . . . , max {m, n}}, then A and B have the same nonzero eigenvalues with the same algebraic multiplicity. Now, assume, in addition, that n = m. Then, tr Ak = tr Bk for all k ∈ {1, . . . , n} if and only if mspec(A) = mspec(B). Source: Fact 6.8.4. Remark: Since, for all k ≥ 1, tr (AB)k = tr (BA)k , this result yields Proposition 6.4.10. Remark: Setting B = 0n×n yields necessity in Fact 3.15.15. Fact 6.10.13. Let A ∈ Fn×n, and let mspec(A) = {λ1 , . . . , λn }ms . Then,       ∏ ∏ ∏     A mspec(A ) =  λi , λi , . . . , λi  .     i∈{2,...,n} i∈{1,3,...,n}  i∈{1,...,n−1} n×n

m×m

ms

533

POLYNOMIAL MATRICES AND RATIONAL TRANSFER FUNCTIONS

Consequently,

} {  det A det A   , . . . , , rank A = n,    λ1 λn ms         n    ∑  mspec(AA ) =      det A , 0, . . . , 0 , rank A = n − 1,   [i,i]          i=1 ms      {0}, rank A ≤ n − 2.

In particular, the following statements hold: i) If n = 2, then mspec(AA ) = mspec(A). ii) If n = 3, then mspec(AA ) = {λ2 λ3 , λ1 λ3 , λ1 λ2 }ms . iii) If n = 4, then mspec(AA ) = {λ2 λ3 λ4 , λ1 λ3 λ4 , λ1 λ2 λ3 }ms . Furthermore, n n ∑ ∑ tr AA = det A[i,i] = (−1)n−1 χ′A (0) = i=1

∏

λ j.

i=1 j∈{1,...,n}\{i}

Finally, if A is singular and λn = 0, then tr AA =

n ∑

det A[i,i] = (−1)n−1 χ′A (0) =

i=1

n−1 ∏

λi .

i=1

Source: [2263, p. 68]. The expression for tr AA is given by (6.4.20). Remark: If rank A = n − 1, then mspec(A) = {0} is possible. For example, N2A = −N2 . See Fact 3.19.3. Remark: If rank A ≤ ∑ n − 2, then 2 ≤ n − rank A = def A ≤ amultA (0), and thus tr AA = ni=1 det A[i,i] = 0. Related: Fact

3.19.1, Fact 3.19.3, Fact 4.10.9, Fact 6.9.4, and Fact 7.12.39. Fact 6.10.14. Let A ∈ Fn×n, and assume that A is either upper triangular or lower triangular. Then, n ∏ χA (s) = (s − A(i,i) ), mspec(A) = {A(1,1) , . . . , A(n,n) }ms . i=1

Related: Fact 4.25.1. Fact 6.10.15. Let A ∈ Fn×n, B ∈ Fn×m, and C ∈ Fm×m, and let p ∈ F[s]. Then, there exists

([

Bˆ ∈ Fn×m such that

p

A B 0 C

])

[

=

p(A) 0

] Bˆ . p(C)

Fact 6.10.16. Let A1 ∈ Fn×n, A12 ∈ Fn×m, and A2 ∈ Fm×m, and define A ∈ F(n+m)×(n+m) by △

A=

[

A1 0

] A12 . A2

Then, χA = χA1 χA2 . Furthermore, there exist B1 , B2 ∈ Fn×m such that [ ] [ 0 B1 χ (A ) χA1 (A) = , χA2 (A) = A2 1 0 χA1 (A2 ) 0 Therefore,

Hence,

([ ]) I R[χA2 (A)] ⊆ R n ⊆ N[χA1 (A)], 0

] B2 . 0

χA2 (A1 )B1 + B2 χA1 (A2 ) = 0.

χA(A) = χA1 (A)χA2 (A) = χA2 (A)χA1 (A) = 0.

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CHAPTER 6

Fact 6.10.17. Let A1 ∈ Fn×n, A12 ∈ Fn×m, and A2 ∈ Fm×m, assume that spec(A1 ) and spec(A2 ) are

disjoint, and define A ∈ F(n+m)×(n+m) by

△

A=

[

] A12 . A2

A1 0

Furthermore, let µ1 , µ2 ∈ F[s] satisfy µA = µ1 µ2 ,

roots(µ1 ) = spec(A1 ),

Then, there exist B1 , B2 ∈ Fn×m such that [ ] 0 B1 µ1 (A) = , 0 µ1(A2 ) Therefore, R[µ2 (A)] ⊆ R Hence,

roots(µ2 ) = spec(A2 ). [

] B2 . 0

µ (A ) µ2(A) = 2 1 0

([ ]) In ⊆ N[µ1 (A)], 0

µ2 (A1 )B1 + B2 µ1(A2 ) = 0.

µA(A) = µ1 (A)µ2(A) = µ2(A)µ1 (A) = 0.

Fact 6.10.18. Let A1 , A2 , A3 , A4 , B1 , B2 ∈ Fn×n, and define A ∈ F4n×4n by

  A1  0 △  A =   0 0

B1 A2 0 0

Then, mspec(A) =

4 ∪

0 0 A3 B2

 0   0  . 0  A4

mspec(Ai ).

i=1

Fact 6.10.19. Let A ∈ Fn×n . Then,

([

mspec

0 A

A 0

])

([

= mspec(A) ∪ mspec(−A),

mspec

0 A∗

A 0

])

([ = − mspec

0 A∗

A 0

]) .

Related: Fact 6.9.20 and Fact 7.10.25. Fact 6.10.20. Let A ∈ Fn×m and B ∈ Fm×n, and assume that m < n. Then,

mspec(In + AB) = mspec(Im + BA) ∪ {1, . . . , 1}ms . Fact 6.10.21. Let a, b ∈ F, and define the symmetric, Toeplitz matrix A ∈ Fn×n by

  a  b   △  A =  b  .  ..  b

b a b .. .

b b a .. .

b

b

··· ··· ··· .. . ···

 b   b   b  . ..  .  a

Then, mspec(A) = {a + (n − 1)b, a − b, . . . , a − b}ms , A1n×1 = [a + (n − 1)b]1n×1 , △

△

mspec(aIn + b1n×n ) = {a + nb, a, . . . , a}ms , A2 + a1A + a0 I = 0,

where a1 = −2a + (2 − n)b and a0 = a2 + (n − 2)ab + (1 − n)b2. Remark: For the remaining eigenvectors of A, see [2418, pp. 149, 317]. Related: Fact 3.16.19, Fact 10.9.1, and Fact 10.10.39.

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POLYNOMIAL MATRICES AND RATIONAL TRANSFER FUNCTIONS

Fact 6.10.22. Let A ∈ Fn×n. Then,

     n n    ∑ ∪     . s ∈ C: |s − A | ≤ |A | spec(A) ⊂   (i,i) (i, j)      j=1   i=1  j,i

△

Now, let λ ∈ spec(A), and let r = gmultA (λ). Then, there exist 1 ≤ i1 < · · · < ir ≤ n such that      n r    ∑ ∩     . s ∈ C: |s − A | ≤ |A | λ∈   (i ,i ) (i , j) k k k       j=1   k=1

j,ik

Source: [590, 2786]. The last statement is given in [1972]. Remark: This is the Gershgorin circle theorem. Remark: This result yields Corollary 11.4.5 for ∥ · ∥col and ∥ · ∥row . ∑ Fact 6.10.23. Let A ∈ Fn×n, and assume that, for all i ∈ {1, . . . , n}, nj=1, j,i |A(i, j) | < |A(i,i) |. Then, A is nonsingular. Source: Fact 6.10.22 and [2979, p. 188]. Remark: This is the diagonal dominance theorem, and A is diagonally dominant. See [2397]. Related: Fact 6.10.25. Fact 6.10.24. Let A ∈ Fn×n, assume that, for all i ∈ {1, . . . , n}, A(i,i) , 0, and assume that ∑n j=1, j,i |A(i, j) | △ < 1. αi = |A(i,i) |

Then, |A(1,1) |

n ∏

(|A(i,i) | − li + Li ) ≤ | det A|,

i=2

where △

li =

i−1 ∑

α j |A(i, j) |,

j=1

n A(i,1) ∑ Li = |A(i, j) |. A(1,1) j=i+1 △

Source: [567]. Remark: Note that, for all i ∈ {1, . . . , n},

li =

i−1 ∑ j=1

α j |A(i, j) | ≤

n ∑

α j |A(i, j) | ≤

j=1, j,i

n ∑

|A(i, j) | = αi |A(i,i) | < |A(i,i) |.

j=1, j,i

Hence, the lower bound for | det A| is positive. △ △ ∑ Fact 6.10.25. Let A ∈ Fn×n, and, for all i ∈ {1, . . . , n}, define ri = nj=1, j,i |A(i, j) | and ci = ∑n j=1, j,i |A( j,i) |. Furthermore, assume that at least one of the following statements holds: i) For all distinct i, j ∈ {1, . . . , n}, ri c j < |A(i,i) A( j, j) |. ii) A is irreducible, for all i ∈ {1, . . . , n} it follows that ri ≤ |A(i,i) |, and there exists i ∈ {1, . . . , n} such that ri < |A(i,i) |. ∑ iii) There exist positive integers k1, . . . , kn such that ni=1 (1 + ki )−1 ≤ 1 and such that, for all i ∈ {1, . . . , n}, ki max j∈{1,...,n}, j,i |A(i, j) | < |A(i,i) |. iv) There exists α ∈ [0, 1] such that, for all i ∈ {1, . . . , n}, riα c1−α < |A(i,i) |. i Then, A is nonsingular. Source: [214]. Remark: Each statement is stronger than Fact 6.10.23. Fact 6.10.26. Let A ∈ Rn×n, assume that A is symmetric, and, for all i ∈ {1, . . . , n}, define △ ∑n αi = j=1, j,i |A(i, j) |. Then, n ∪ spec(A) ⊂ [A(i,i) − αi , A(i,i) + αi ]. i=1

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△

Now, for all i ∈ {1, . . . , n}, let βi = max {0, max j∈{1,...,n}, j,i A(i, j) } and γi = min {0, min j∈{1,...,n}, j,i A(i, j) }. Then,    n   n n ∑ ∪   ∑    A(i, j) − nγi  . A(i, j) − nβi ,  spec(A) ⊂ j=1

j=1

i=1

Source: The first statement is the specialization of the Gershgorin circle theorem to real, symmetric

matrices. See Fact 6.10.22. The second result is given in [291]. Fact 6.10.27. Let A ∈ Fn×n. Then,        n n n    ∑ ∑ ∪     ∈ A ||s − | ≤ | | spec(A)⊂ . s C: |s − A |A |A   (i,i) ( j, j) (i,k) ( j,k)        i, j=1 k=1 k=1   k,i k, j i, j

Remark: The inclusion region is the ovals of Cassini. See [1448, p. 380]. Credit: A. Brauer. Fact 6.10.28. Let A ∈ Fn×n. Then,

ρmin (A) ≤ max | tr Ai |1/i , i∈{1,...,n}

ρmax (A) ≤

max

i∈{1,...,2n−1}

| tr Ai |1/i ,

ρmax (A) ≤

5 max | tr Ai |1/i . n i∈{1,...,n}

Remark: These are Turan’s inequalities. See [2061, p. 657]. △ ∑ Fact 6.10.29. Let A ∈ Fn×n, and, for all j ∈ {1, . . . , n}, define b j = ni=1 |A(i, j) |. Then, n ∑ |A( j, j) | j=1

≤ rank A.

bj

Source: [2263, p. 67]. Remark: Interpret 0/0 as 0. Related: Fact 6.10.23. Fact 6.10.30. Let A1 , . . . , Ar ∈ Fn×n, assume that A1 , . . . , Ar are normal, and let A ∈

conv {A1 , . . . , Ar }. Then,

spec(A)⊆ conv

∪

spec(Ai ).

i∈{1,...,r}

Source: [2829]. Remark: The spectrum of a polytope of matrices is considered in [992, 2075, 2282]. Related: Fact 10.17.8. Fact 6.10.31. Let A, B ∈ Rn×n. Then,

([

mspec

A B

B A

])

= mspec(A + B) ∪ mspec(A − B).

Source: [2418, p. 93]. Related: Fact 3.17.30. Fact 6.10.32. Let A, B ∈ Rn×n. Then,

([

mspec

A −B

B A

])

= mspec(A + ȷB) ∪ mspec(A − ȷB).

Now, assume, in addition, that A is symmetric and B is skew symmetric. Then, A + ȷB is Hermitian, and ([ ]) A B mspec = mspec(A + ȷB) ∪ mspec(A + ȷB). BT A Related: Fact 3.24.6 and Fact 10.19.2. Fact 6.10.33. Let A, B ∈ Cn×n, and define

[

A M= −B △

] B . A

[

A B BT A

]

is symmetric,

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POLYNOMIAL MATRICES AND RATIONAL TRANSFER FUNCTIONS

Then, the following statements hold: i) χA ∈ R[s]. ii) mspec(A) is conjugate symmetric. iii) For all λ ∈ spec(A), there exist x, y ∈ Cn such that

[ x] y

and

[ ] −y x

are linearly independent

eigenvectors of M associated with λ and λ, respectively. Source: [2991, p. 106]. Related: Fact 3.24.8 n×n m×m n×m Fact 6.10.34. [ ] Let A ∈ [F ,] B ∈ F , and C ∈ F , assume that A and B are Hermitian, and △

define A0 =

A 0 0 B

△

and A =

A C C∗ B

. Furthermore, define △

η = min |λi (A) − λ j (B)|. i=1,...,n j=1,...,m

Then, for all i ∈ {1, . . . , n + m}, |λi (A) − λi (A0 )| ≤

2 2σmax (C) . √ 2 η + η + 4σmax (C)

Source: [453, pp. 142–146] and [1797]. △ Fact 6.10.35. Let A ∈ Rn×n, let b, c ∈ Rn , define p ∈ R[s] by p(s) = cT(sI − A)A b, and assume △

that p and det(sI − A) are coprime. Furthermore, for all α ∈ [0, ∞), define Aα = A − αbcT , and let λ: [0, ∞) → C be a continuous function such that, for all α ∈ [0, ∞), λ(α) ∈ spec(Aα ). Then, either limα→∞ |λ(α)| = ∞ or limα→∞ λ(α) ∈ roots(p). Remark: This result is a consequence of root locus analysis from classical control theory, which determines asymptotic pole locations under high-gain feedback. In particular, the loop transfer function is L(s) = c(sI − A)−1 b with feedback gain −α. Fact 6.10.36. Let A ∈ Fn×n, where n ≥ 2, and assume that there exist α ∈ [0, ∞) and B ∈ Fn×n such that A = αI − B and ρmax (B) ≤ α. Then, spec(A) ⊂ {0} ∪ ORHP. If, in addition, ρmax (B) < α, then spec(A) ⊂ ORHP, and thus A is nonsingular. Source: Let λ ∈ spec(A). Then, there exists µ ∈ spec(B) such that λ = α − µ. Hence, Re λ = α − Re µ. Since Re µ ≤ | Re µ| ≤ |µ| ≤ ρmax (B), it follows that Re λ ≥ α − | Re µ| ≥ α − |µ| ≥ α − ρmax (B) ≥ 0. Hence, Re λ ≥ 0. Now, suppose that Re λ = 0. Then, since α − λ = µ ∈ spec(B), it follows that α2 + |λ|2 ≤ ρ2max (B) ≤ α2. Hence, λ = 0. By a similar argument, ρmax (B) < α implies that Re λ > 0. Remark: Converses of these statements hold in the case where B is nonnegative. See Fact 6.11.13.

6.11 Facts on Graphs and Nonnegative Matrices Fact 6.11.1. Let G = (X, R) be a directed graph, where X = {x1 , . . . , xn }, and let A be the adjacency matrix of G. Then, the following statements hold: i) The number of distinct walks from xi to x j of length k ≥ 1 is (Ak )( j,i) . ii) Let k be an integer such that 1 ≤ k ≤ n − 1. Then, for distinct xi , x j ∈ X, the number of distinct walks from xi to x j whose length is either less than or equal to k is [(I + A)k ]( j,i) . Fact 6.11.2. Let G = (X, R) be a directed graph, where X = {x1 , . . . , xn }, and let A be the adjacency matrix of G. Then, every subdeterminant of A is either −1, 0, or 1. Now, assume that G is a symmetric graph. Then, the following statements hold: i) G is bipartite if and only if, for all λ ∈ spec(A), it follows that −λ ∈ spec(A) and amA (λ) = amA (−λ). ii) mspec(A) is the union of the multispectra of the adjacency matrices of the connected components of G. iii) Assume that every node of G has degree k. Then, λmax (A) = k, and amA (λ) is the number of connected components of G. In particular, if k is a simple eigenvalue of A, then G is connected.

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iv) Assume that G is connected, and let m denote the length of the longest path in G. Then, card[spec(A)] ≥ m + 1. Source: [1182, pp. 272, 325–330]. Fact 6.11.3. Let A ∈ Fn×n, and consider the directed graph G(A) = (X, R), where X = {x1 , . . . , xn }. Then, the following statements are equivalent: i) G(A) is directionally connected. ii) There exists k ≥ 1 such that (I + |A|)k−1 >> 0. iii) (I + |A|)n−1 >> 0. Source: [1448, pp. 358, 359] and [1451, p. 401]. Remark: G(A) is defined in Definition 4.2.4. Fact 6.11.4. Let G = ({x1 , . . . , xn }, R) be a directed graph, and let A ∈ Rn×n be the adjacency matrix of G. Then, the following statements are equivalent: i) G is directionally connected. ii) G has no directed cuts. iii) A is irreducible. ∑ i iv) n−1 i=0 A >> 0. v) (I + A)n−1 >> 0. If, in addition, every node has a self-arc, then the following statement is equivalent to i)–v): vi) An−1 is positive. Furthermore, the following statements are equivalent: vii) G is not directionally connected. viii) G has a directed cut. ix) A is reducible. ∑ i x) n−1 i=0 A has at least one entry that is zero. xi) (I + A)n−1 has at least one entry that is zero. If, in addition, every node has a self-arc, then the following statement is equivalent to vii)–xi): xii) An−1 has at least one zero entry. n×n Finally, suppose [ BthatC ]A is reducible and there exist k ≥ 1 and a permutation matrix S ∈ R such △ T (n−k)×(n−k) (n−k)×k k×k that SAS = 0k×(n−k) D , where B ∈ F ,C ∈F , and D ∈ F , and define [i1 · · · in ]T = S [1 · · · n]T. Then, ({xi1 , . . . , xin−k }, {xin−k+1 , . . . , xin }) is a directed cut. Source: [1448, p. 362], and [2344, p. 9-3], and [2432, pp. 238, 239]. Fact 6.11.5. Let A ∈ Rn×n, where n ≥ 2, and assume that A is nonnegative. Then, the following statements hold: i) ρmax (A) ∈ spec(A). ii) There exists a nonnegative eigenvector x ∈ Rn associated with ρmax (A). iii) If x ∈ Rn is a positive eigenvector of A associated with λ ∈ spec(A), then λ = ρmax (A). iv) If A has less than n − 1 zero entries, then A is irreducible. Furthermore, the following statements are equivalent: v) A is irreducible. vi) (I + A)n−1 >> 0. vii) For all i, j ∈ {1, . . . , n}, there exists k ≥ 1 such that (Ak )(i, j) > 0. viii) G(A) is directionally connected. ix) A has a positive eigenvector and a unique unit-length nonnegative eigenvector.

POLYNOMIAL MATRICES AND RATIONAL TRANSFER FUNCTIONS

539

If A is irreducible, then the following statements hold: x) ρmax (A) > 0. xi) ρmax (A) is a simple eigenvalue of A. xii) A has a unique positive eigenvector x ∈ Rn such that ∥x∥2 = 1. xiii) If x ∈ Rn is a positive eigenvector of A, then Ax = ρmax (A)x. xiv) Define {λ1 , . . . , λk }ms = {λ ∈ mspec(A): |λ| = ρmax (A)}ms . Then, λ1 , . . . , λk are distinct, and {λ1 , . . . , λk } = {e(2πi/k) ȷρmax (A): i = 1, . . . , k}. Furthermore, mspec(A) = e(2π/k) ȷ mspec(A). xv) If at least one diagonal entry of A is positive, then ρmax (A) is the unique eigenvalue of A whose absolute value is ρmax (A). xvi) If A has at least m positive diagonal entries, then A2n−m−1 >> 0. xvii) If x, y ∈ Rn are positive and satisfy Ax = ρmax (A)x and ATy = ρmax (A)y, then )i k ( 1 1 1∑ A = T xyT. lim k→∞ k ρmax (A) xy i=1 If A is irreducible, then the index of imprimitivity of A is card {λ ∈ spec(A) : |λ| = ρmax (A)}. In addition, the following statements are equivalent: xviii) There exists k ≥ 1 such that Ak >> 0. 2 xix) An −2n+2 >> 0. xx) A is irreducible, and {λ ∈ spec(A) : |λ| = ρmax (A)} = {ρmax (A)}. xxi) A is irreducible, and G(A) is aperiodic. xxii) A is irreducible, and the index of imprimitivity of A is 1. A is primitive if xvi)–xx) hold. The following statement holds: xxiii) If A is irreducible and tr A > 0, then A is primitive. If A is primitive, then the following statements hold: xxiv) For all k ≥ 1, Ak is primitive. xxv) If k ≥ 1 and Ak >> 0, then, for all l ≥ k, Al >> 0. xxvi) There exists a positive integer k ≤ (n − 1)nn such that Ak >> 0. xxvii) If x, y ∈ Rn are positive and satisfy Ax = ρmax (A)x and ATy = ρmax (A)y, then ( )k 1 1 lim A = T xyT. k→∞ ρmax (A) xy xxviii) If x0 ∈ Rn is nonzero and nonnegative and x, y ∈ Rn are positive and satisfy Ax = ρmax (A)x and ATy = ρmax (A)y, then Ak x0 − ρkmax (A)yTx0 x lim = 0. k→∞ ∥Ak x0 ∥2 xxix) ρmax (A) = limk→∞ (tr Ak )1/k. Source: [34, pp. 45–49], [283, p. 17], [422, pp. 26–28, 32, 55], [1040, Chapter 4], [1448, pp. 507–518, 524, 525], [2344, p. 9-3], and [2979, pp. 120–134]. For xxx), see [2441] and [2785, p. 49]. Remark: This is the Perron-Frobenius theorem. Remark: xix) is due ] [ ] [ to] H. [ Wielandt. See [2263, p. 157]. Remark: xxi) is given in [2344, p. 9-3]. Example: 10 00 , 10 10 , 10 11 , [ ] [ ] and 10 12 are reducible. 10 00 has two unit-length linearly independent nonnegative eigenvectors [ ] [ ] [ ] given by 10 and 01 , but does not have a positive eigenvector. 10 10 has two unit-length linearly [ ] [ ] 1 , and thus has a unique unit-length nonnegative independent eigenvectors given by 10 and √12 −1

540

CHAPTER 6

[ ] eigenvector, but does not have a positive eigenvector. 10 11 has one unit-length linearly independent [ ] [ ] eigenvector given by 10 , and thus has a unique unit-length nonnegative eigenvector given by 10 , [ ] but does not have a positive eigenvector. 10 12 has two unit-length linearly independent nonnegative [ ] [ ] eigenvectors given by √12 11 and 10 , and thus has a positive eigenvector but does not have a unique [ ] [ ]k unit-length nonnegative eigenvector. 01 10 is irreducible but not primitive. For all k ≥ 1, 01 10 [ ] [ ] is not positive. 11 10 is primitive. 11 11 is positive and thus primitive. Remark: For an arbitrary nonzero and nonnegative initial condition, xxvii) shows that the state xk = Ak x0 of the difference equation xk+1 = Axk approaches a distribution given by the eigenvector associated with the positive eigenvalue of maximum absolute value. In demography, this eigenvector is interpreted as the stable age distribution. See [1605, pp. 47, 63]. Example: Let x and y be positive numbers such that x + y < 1, and define    x y 1 − x − y  △   . x y A =  1 − x − y   y 1−x−y x Then, A13×1 = AT13×1 = 13×1 , and thus limk→∞ Ak = 31 13×3 . See [517, p. 213]. Related: Fact 8.11.4, Fact 15.19.20, and Fact 15.22.24. Fact 6.11.6. Let A ∈ Rn×n, where n ≥ 2, and assume that A is nonnegative. Then, there exists a permutation matrix S ∈ Rn×n such that SAS T is upper block triangular and every diagonally located block is irreducible. If, in addition, A is a permutation matrix, then there exists a permutation matrix S ∈ Rn×n such that SAS T is block diagonal and every diagonally located block is an irreducible permutation matrix. Source: If A is either zero or irreducible, then the result holds with S = I. If A is either nonzero or reducible, then there exists an n×n permutation matrix S such that SAS T is upper block triangular with square, nonnegative, diagonally located blocks B and C. If each matrix B and C is either zero or irreducible, then the result holds. If not, then either B or C can be transformed [ ] as needed. The last statement is given in [2980, p. 155] and Fact 7.18.14. Example: 10 10 and [ ] 1 1 are reducible, and every diagonally located block is irreducible. Remark: This result gives the 01 Frobenius normal form. See [587, p. 27-6]. Note that all 1 × 1 matrices are defined to be irreducible. The only 1 × 1 irreducible permutation matrix is [1]. Fact 6.11.7. Let A ∈ Rn×n. Then, the following statements hold: i) If n ≥ 2 and A2 ≤≤ 0, then A is reducible. ii) A2 has at least one nonnegative entry. Source: [1001] and [2979, pp. 130–132]. Remark: For all n ≥ 1, there exists B ∈ Rn×n such that B2 has n2 − 1 negative entries. See [1001] and [2979, pp. 131, 132]. Fact 6.11.8. Let A ∈ Rn×n, assume that A is positive, and assume that, for all i, j ∈ {1, . . . , n}, ⊙−1 A = AT . Then, ρmax (A) ≥ n. Source: [2356]. Fact 6.11.9. Let A ∈ Rn×n, and assume that A is totally nonnegative. Then, spec(A) ⊂ [0, ∞). Source: [2979, p. 139]. Fact 6.11.10. Let A ∈ Rn×n, and assume that A is totally nonnegative. Then, the following statements are equivalent: i) There exists k ≥ 1 such that Ak is totally positive. ii) A is nonsingular, and all of the entries on the subdiagonal and superdiagonal of A are positive. A is oscillatory if i) and ii) hold. If A is oscillatory, then the following statements hold: iii) A is primitive.

POLYNOMIAL MATRICES AND RATIONAL TRANSFER FUNCTIONS

541

iv) The eigenvalues of A are distinct positive numbers. v) An−1 is totally positive. Source: [2979, pp. 138, 139, 148]. Fact 6.11.11. Let A ∈ Rn×n, and assume that A is row stochastic. Then, ρmax (A) = 1. Source: Since 1n×1 is an eigenvector of A associated with the eigenvalue 1, the result follows from ii) of Fact 6.11.5. Alternatively, note that ∥A∥∞,∞ = ∥A∥row = 1. Since 1 ∈ spec(A) and ∥ · ∥∞,∞ is an induced norm, it follows from Corollary 11.4.5 that 1 ≤ ρmax (A) ≤ ∥A∥∞,∞ = 1. Remark: Fact 4.13.1 implies that, if A ∈ conv P(n), then ρmax (A) = 1. Related: Fact 15.22.12. Fact 6.11.12. Let G = (X, R) be a directed graph, where X = {x1 , . . . , xn }, and let Lout ∈ Rn×n denote the outbound Laplacian of G. Then, the following statements hold: i) 0 ∈ spec(Lout ) ⊂ {0} ∪ ORHP. ii) Lout 1n×1 = 0. iii) 0 is a semisimple eigenvalue of Lout . iv) If G is directionally connected, then 0 is a simple eigenvalue of Lout . v) The following statements are equivalent: a) There exists a node x ∈ supp(G) such that, for every node y ∈ supp(G), there exists a walk from y to x. b) 0 is a simple eigenvalue of Lout . T vi) Lout + Lout is positive semidefinite. vii) Lout is symmetric if and only if G is symmetric. Now, assume, in addition, that G is symmetric, and let L ∈ Rn×n denote the Laplacian of G. Then, the following statements hold: viii) L is positive semidefinite. ix) 0 ∈ spec(L) ⊂ {0} ∪ [0, ∞). x) G is connected if and only if rank L = n − 1. xi) If G is connected, then 0 is a simple eigenvalue of L. xii) 0 is a simple eigenvalue of L if and only if G has a spanning subgraph that is a tree. Source: [606, pp. 40, 77] and [2032, p. 27]. For xii), see [2028, p. 147]. Remark: v) means that G has at least one globally reachable node. See [606, p. 40]. Related: Fact 15.20.7. Fact 6.11.13. Let A ∈ Rn×n, where n ≥ 2, and assume that A is a Z-matrix. Then, the following statements are equivalent: i) There exist a nonnegative matrix B ∈ Rn×n and α ≥ ρmax (B) such that A = αI − B. ii) spec(A) ⊂ ORHP ∪ {0}. iii) spec(A) ⊂ CRHP. iv) If λ ∈ spec(A) is real, then λ ≥ 0. v) Every principal subdeterminant of A is nonnegative. vi) If D ∈ Rn×n is diagonal and positive definite, then A + D is nonsingular. A is an M-matrix if i)–vi) hold. The following statements are equivalent: vii) A is a nonsingular M-matrix. viii) There exist a nonnegative matrix B ∈ Rn×n and α > ρmax (B) such that A = αI − B. ix) spec(A) ⊂ ORHP. x) If λ ∈ spec(A) is real, then λ > 0. xi) A is nonsingular, and A−1 ≥≥ 0.

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xii) There exists x ∈ Rn such that x >> 0 and Ax >> 0. xiii) Every principal subdeterminant of A is positive. xiv) Every leading principal subdeterminant of A is positive. Source: i) =⇒ ii) follows from Fact 6.10.36. To prove iii) =⇒ i), let α ∈ (0, ∞) be sufficiently large △ △ that B = αI − A is nonnegative. Hence, for every µ ∈ spec(B), it follows that λ = α − µ ∈ spec(A). Since Re λ ≥ 0, it follows that every µ ∈ spec(B) satisfies Re µ ≤ α. Since B is nonnegative, it follows from i) of Fact 6.11.5 that ρmax (B) is an eigenvalue of B. Hence, setting µ = ρmax (B) implies that ρmax (B) ≤ α. iv) and v) are proved in [423, pp. 149, 150]. The [argument used ] [ to ] prove i) =⇒ ii) 1 1 is an M-matrix. shows that viii) =⇒ ix). See [2979, pp. 140–142]. Example: A = 00 −1 = I − 01 0 Related: Fact 15.20.3 and Fact 15.20.5. Fact 6.11.14. Let A ∈ Rn×n, where n ≥ 2. Then, A is a Z-matrix if and only if every principal submatrix of A is a Z-matrix. Furthermore, A is an M-matrix if and only if every principal submatrix of A is an M-matrix. Source: [1450, p. 114] and [2979, p. 139]. Fact 6.11.15. Let A ∈ Rn×n, where n ≥ 2, and assume that A is a Z-matrix. Then, the following statement holds: i) If there exists x ∈ Rn such that x >> 0 and Ax ≥≥ 0, then A is an M-matrix. Now, assume that A is an M-matrix. Then, the following statements hold: ii) There exists a nonzero vector x ∈ Rn such that x ≥≥ 0 and Ax ≥≥ 0. iii) If A is irreducible, then there exists a positive vector x ∈ Rn such that Ax is nonnegative. Now, assume, in addition, that A is singular. Then, the following statements hold: iv) rank A = n − 1. v) There exists a positive vector x ∈ Rn such that Ax = 0. vi) A is group invertible. vii) Every principal submatrix of A of size less than n and greater than 1 is a nonsingular Mmatrix. viii) If x ∈ Rn and Ax is nonnegative, then Ax = 0. Source: To prove ii), Fact 6.11.13 implies that there exist α ∈ (0, ∞) and B ∈ Rn×n such that A = αI − B, B is nonnegative, and ρmax (B) ≤ α. Consequently, ii) of Fact 6.11.5 implies that there exists a nonzero nonnegative vector x ∈ Rn such that Bx = ρmax (B)x. Therefore, Ax = [α−ρmax (B)]x is nonnegative. iv)–viii) are given in [423, p. 156]. Fact 6.11.16. Let A ∈ Rn×n, where n ≥ 2, and assume that A is a nonsingular M-matrix, B is a Z-matrix, and A ≤≤ B. Then, the following statements hold: i) tr A−1AT ≤ n. ii) tr A−1AT = n if and only if A is symmetric. iii) B is a nonsingular M-matrix. iv) 0 ≤ B−1 ≤ A−1. v) 0 < det A ≤ det B. Source: [1450, pp. 117, 370]. Fact 6.11.17. Let A ∈ Rn×n, where n ≥ 2, assume that A is a Z-matrix. Then, the following statements hold: i) αmin (A) ∈ spec(A). ∑ ii) mini∈{1,...,n} nj=1 A(i, j) ≤ αmin (A). Now, assume, in addition, that A is an M-matrix. Then, the following statements hold: iii) If A is nonsingular, then αmin (A) = ρmin (A).

POLYNOMIAL MATRICES AND RATIONAL TRANSFER FUNCTIONS

543

iv) αnmin (A) ≤ det A. v) If B ∈ Rn×n, B is an M-matrix, and B ≤≤ A, then αmin (B) ≤ αmin (A). Source: [1450, pp. 128–131]. Related: Fact 9.6.22. Fact 6.11.18. Consider the nonnegative companion matrix A ∈ Rn×n defined by   1 0 ··· 0 0   0   ..   0 . 0 1 0 0     ..   0 . 0 0 0 0    △   A =  . .. .. ..  . .. ..  . . . . . .   .     0 0 0 ··· 0 1    1/n 1/n 1/n · · · 1/n 1/n Then, A is irreducible, 1 is a simple eigenvalue of A with associated eigenvector 1n×1 , and |λ| < 1 for all λ ∈ spec(A)\{1}. Furthermore, if x ∈ Rn, then   n ∑  2  k  lim A x =  ix(i−1)  1n×1 . k→∞ n(n + 1) i=1 Source: [1304, pp. 82, 83, 263–266] and Fact 6.11.5. Fact 6.11.19. Let A ∈ Rn×m and b ∈ Rm. Then, the following statements are equivalent:

i) If x ∈ Rm and Ax ≥≥ 0, then bT x ≥ 0. ii) There exists y ∈ Rn such that y ≥≥ 0 and ATy = b. Equivalently, exactly one of the following two statements holds: iii) There exists x ∈ Rm such that Ax ≥≥ 0 and bT x < 0. iv) There exists y ∈ Rn such that y ≥≥ 0 and ATy = b. Source: [333, p. 47], [523, p. 24], and [2979, pp. 27, 28]. Remark: This is the Farkas theorem. Fact 6.11.20. Let A ∈ Rn×m. Then, the following statements are equivalent: i) There exists x ∈ Rm such that Ax >> 0. ii) If y ∈ Rn is nonzero and y ≥≥ 0, then ATy , 0. Equivalently, exactly one of the following two statements holds: iii) There exists x ∈ Rm such that Ax >> 0. iv) There exists a nonzero vector y ∈ Rn such that y ≥≥ 0 and ATy = 0. Source: [333, p. 47] and [523, p. 23]. Remark: This is Gordan’s theorem. Fact 6.11.21. Let A ∈ Cn×n, let B ∈ Rn×n , assume that B is nonnegative, and assume that |A| ≤≤ B. Then, ρmax (A) ≤ ρmax (B). Now, assume that A is irreducible, and let λ ∈ spec(A). Then, |λ| = ρmax (B) if and only if there exists a diagonal unitary matrix D ∈ Cn×n such that λDBD−1 = ρmax (B)A. Source: [2979, p. 619]. Fact 6.11.22. Let A ∈ Cn×n. Then, ρmax (A) ≤ ρmax (|A|). Source: [2036, p. 619]. Fact 6.11.23. Let A, B ∈ Rn×n, where 0 ≤≤ A ≤≤ B. Then, the following statements hold: i) ρmax (A) ≤ ρmax (B). ii) Assume that A is irreducible. Then, the following statements are equivalent: a) ρmax (A) = ρmax (B). b) A = B.

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iii) If A + B is irreducible and ρmax (A) = ρmax (B), then A = B. iv) If A , B and A + B is irreducible, then ρmax (A) < ρmax (B). Source: [423, p. 27], [970, pp. 500, 501], and [2991, p. 165]. Fact 6.11.24. Let A ∈ Rn×n and B ∈ Rm×m, and assume that A is nonnegative and B is a principal submatrix of A. Then, ρmax (B) ≤ ρmax (A). If, in addition, A is irreducible and m < n, then ρmax (B) < ρmax (A). Source: Fact 6.11.23, [2979, p. 128], and [2991, p. 170]. Fact 6.11.25. Let A ∈ Rn×n, assume that A is nonnegative, and define f : [0, 1] 7→ [0, ∞) by △ f (α) = ρmax [αA + (1 − α)AT ]. Then, f is nondecreasing on [0, 21 ] and nonincreasing on [ 12 , 1]. Hence, for all α ∈ [0, 1], ρmax (A) ≤ ρmax [αA + (1 − α)AT ] ≤ 21 ρmax (A + AT ). Source: [279] and [2979, p. 147]. Related: Fact 9.6.18. Fact 6.11.26. Let A, B ∈ Rn×n, assume that B is diagonal, assume that A and A + B are nonneg-

ative, and let α ∈ [0, 1]. Then,

ρmax [αA + (1 − α)B] ≤ αρmax (A) + (1 − α)ρmax (A + B). Source: [2344, p. 9-5]. Fact 6.11.27. Let A ∈ Rn×n, assume that A >> 0, let λ ∈ spec(A), and assume that |λ| < ρmax (A).

Then, |λ| ≤ △

α−β ρmax (A), α+β △

where β = min {A(i, j) : i, j ∈ {1, . . . , n}} and α = max {A(i, j) : i, j ∈ {1, . . . , n}}. Source: [2979, pp. 143, 144]. Remark: This is Hopf’s theorem. Remark: The equality case is considered in [1396]. Fact 6.11.28. Let A ∈ Rn×n, assume that A is nonnegative and irreducible, and let x, y ∈ Rn, where x >> 0 and y >> 0 satisfy Ax = ρmax (A)x and ATy = ρmax (A)y. Then, )k l ( 1∑ 1 lim A = xyT. l→∞ l ρ (A) max k=1 If, in addition, A is primitive, then (

)k 1 lim A = xyT. k→∞ ρmax (A) Source: [970, p. 503] and [1448, p. 516]. Fact 6.11.29. Let A ∈ Rn×n, assume that A is nonnegative, and let k ≥ 1 and m ≥ 1. Then,

(tr Ak )m ≤ nm−1 tr Akm . Source: [1723]. Remark: This is the JLL inequality.

6.12 Notes The proofs of Lemma 6.4.8 and Leverrier’s algorithm Proposition 6.4.9 are based on [2314, pp. 432, 433], where it is called the Souriau-Frame algorithm. Alternative proofs of Leverrier’s algorithm are given in [299, 1466]. The proof of Theorem 6.6.1 is based on [1448]. Polynomialbased approaches to linear algebra are given in [607, 1084], while polynomial matrices and rational transfer functions are studied in [1186, 2784]. The term normal rank is often used to refer to what we call the rank of a rational transfer function.

Chapter Seven Matrix Decompositions In this chapter we present several matrix decompositions, namely, the Smith, multicompanion, elementary multicompanion, Jordan, Schur, singular value, polar, and full-rank. The Smith, multicompanion, elementary multicompanion, Jordan, and singular value decompositions involve the transformation of a matrix into a unique canonical matrix.

7.1 Smith Decomposition For rectangular matrices under a biequivalence transformation, the following result, which follows from Theorem 6.3.2, provides a canonical matrix given by a Smith matrix. △ Theorem 7.1.1. Let A ∈ Fn×m and r = rank A. Then, there exist nonsingular matrices S 1 ∈ Fn×n m×m and S 2 ∈ F such that    Ir 0r×(m−r)   S 2 . A = S 1  (7.1.1) 0(n−r)×r 0(n−r)×(m−r) ] [ I 0r×(m−r) r The Smith matrix in (7.1.1) is the Smith form of A. Note that the Smith polyno0(n−r)×r 0(n−r)×(m−r) mials p1 , . . . , pr of a constant matrix whose rank is r are given by p1 = · · · = pr = 1. Proposition 7.1.2. Let A, B ∈ Fn×m. Then, the following statements hold: i) A and B are left equivalent if and only if N(A) = N(B). ii) A and B are right equivalent if and only if R(A) = R(B). iii) The following statements are equivalent: a) A and B are biequivalent. b) rank A = rank B. c) A and B have the same Smith form. Proof. See [2314, pp. 179–181]. 

7.2 Reduced Row Echelon Decomposition △

Definition 7.2.1. Let A ∈ Fn×m , and define r = rank A. Then, A is a reduced row echelon matrix

if the following statements hold: i) For all i ∈ {1, . . . , r}, the left-most nonzero entry of rowi (A) is 1, and all entries of A above this entry are zero. ii) For all 1 ≤ i < j ≤ r, the left-most nonzero entry of rowi (A) is to the left of the left-most nonzero entry of row j (A). iii) rowr+1 (A) = · · · = rown (A) = 0. As an example, the matrix    0 1 −2 0 4    A =  0 0 0 1 −3    0 0 0 0 0 is a reduced row echelon matrix.

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For rectangular matrices under a left equivalence transformation, the following result provides a canonical matrix given by a reduced row echelon matrix. Theorem 7.2.2. Let A ∈ Fn×m . Then, there exist a nonsingular matrix B ∈ Fn×n and a unique reduced row echelon matrix C ∈ Fn×m such that rank C = rank A and A = BC. Proof. See [1451, p. 11], where B is constructed as a product of elementary matrices.  The reduced row echelon matrix C in Theorem 7.2.2 is the reduced row echelon form of A.

7.3 Multicompanion and Elementary Multicompanion Decompositions For the monic polynomial p(s) = sn + βn−1 sn−1 + · · · + β1 s + β0 ∈ F[s] of degree n ≥ 1, the companion matrix C(p) ∈ Fn×n associated with p is defined to be   1 0 ··· 0 0   0   ..   0 . 0 1 0 0    ..  . 0 0 0 0 0     △  (7.3.1) C(p) =  . .. .. ..  . .. ..  . . . . . .   .     0 0 0 ··· 0 1    −β0 −β1 −β2 · · · −βn−2 −βn−1 If n = 1, then p(s) = s + β0 and C(p) = −β0 . Furthermore, if n = 0 and p = 1, then we define △ C(p) = 00×0 . Note that, if n ≥ 1, then tr C(p) = −βn−1 and det C(p) = (−1)n β0 = (−1)n p(0). It is easy to see that the characteristic polynomial of the companion matrix C(p) associated with p is p. For example, let n = 3 so that    0 1 0    0 1  , (7.3.2) C(p) =  0   −β0 −β1 −β2 and thus

  s  sI − C(p) =  0  β0

−1 s β1

 0   −1  .  s + β2

(7.3.3)

Adding s times the second column and s2 times the third column to the first column leaves the determinant of sI − C(p) unchanged and yields    0 −1 0    s −1  . (7.3.4)  0  p(s) β1 s + β2 △

Hence, χC(p) = p. If n = 0 and p = 1, then we define χC(p) = χ00×0 = 1. The following result shows that the characteristic polynomial of a companion matrix is also its minimal polynomial. Proposition 7.3.1. Let p ∈ F[s] be a monic polynomial having degree n. Then, there exist unimodular matrices S 1 , S 2 ∈ F[s]n×n such that    In−1 0(n−1)×1  sI − C(p) = S 1 (s)  (7.3.5)  S 2 (s). 01×(n−1) p(s) Furthermore,

χC(p) = µC(p) = p.

(7.3.6)

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(

)

Proof. Since χC(p) = p, it follows that rank[sI − C(p)] = n. Next, since det [sI − C(p)][n,1] =

(−1)n−1, it follows that ∆n−1 = 1, where ∆n−1 is the greatest common divisor (which is monic by definition) of all (n − 1) × (n − 1) subdeterminants of sI − C(p). Furthermore, since ∆i−1 divides ∆i for all i ∈ {2, . . . , n − 1}, it follows that ∆1 = · · · = ∆n−2 = 1. Consequently, the similarity invariants p1 , . . . , pn of C(p) satisfy p1 = · · · = pn−1 = 1. Furthermore, it follows from Proposition 6.6.2 that ∏ χC(p) = ni=1 pi = pn and µC(p) = pn . Therefore, µC(p) = χC(p) = p.  Next, we consider block-diagonal matrices all of whose diagonally located blocks are companion matrices. A matrix with this structure is a multicompanion matrix. Lemma 7.3.2. Let p1 , . . . , pn ∈ F[s] be monic polynomials such that pi divides pi+1 for all ∑ △ i ∈ {1, . . . , n − 1} and n = ni=1 deg pi . Furthermore, define C = diag[C(p1 ), . . . , C(pn )] ∈ Fn×n. n×n Then, there exist unimodular matrices S 1 , S 2 ∈ F[s] such that   0   p1(s)   ..  S 2 (s). (7.3.7) sI − C = S 1 (s)  .   0 pn (s) △

Proof. For all i ∈ {1, . . . , n}, define ki = deg pi , and note that

  sIk1 − C(p1 )  sI − C =   0

..

0 .

sIkn − C(pn )

    . 

For all i ∈ {1, . . . , n}, Proposition 7.3.1 implies that the Smith form of sIki − C(pi ) is 00×0 for ki = 0, △ ∑ pi for ki = 1, and diag(Iki −1, pi ) for ki ≥ 2. Note that pi = 1 if and only if i ≤ n0 = ni=1 max {0, ki −1}. By combining these Smith matrices into a block-diagonal matrix and rearranging the diagonal entries, it follows that there exist unimodular matrices S 1 , S 2 ∈ F[s]n×n such that (7.3.7) holds. Since, for all i ∈ {1, . . . , n − 1}, pi divides pi+1 , this diagonal matrix is the Smith form of sI − C.  For square matrices under a similarity transformation, the following result provides a canonical matrix given by a multicompanion matrix. Theorem 7.3.3. Let A ∈ Fn×n, and let p1 , . . . , pn ∈ F[s] denote the similarity invariants of A, where pi divides pi+1 for all i ∈ {1, . . . , n − 1}. Then, there exists a nonsingular matrix S ∈ Fn×n such that   0   C(p1 )   ..  S −1. (7.3.8) A = S  .   0 C(pn ) △

Proof. Lemma 7.3.2 implies that the n × n matrix sI − C, where C = diag[C(p1 ), . . . , C(pn )], has

the Smith form diag(p1 , . . . , pn ). Now, since sI − A has the same similarity invariants as C, Theorem 6.3.11 implies that A and C are similar.   C(p1 ) 0    ..  in (7.3.8) is the multicompanion form of A. Recall The multicompanion matrix  . 0 C(p ) n ∏ from Proposition 6.6.2 that χA = ni=1 pi and µA = pn . Corollary 7.3.4. Let A ∈ Fn×n. Then, µA = χA if and only if A and C(χA ) are similar. Proof. Suppose that µA = χA . Then, it follows from Proposition 6.6.2 that pi = 1 for all i ∈ {1, . . . , n −1} and pn = χA is the unique nonconstant similarity invariant of A. Thus, C(pi ) = 00×0 for all i ∈ {1, . . . , n − 1}, and it follows from Theorem 7.3.3 that A is similar to C(χA ). Conversely, it follows from (7.3.6) that µC(χA ) = χA . Next, since A and C(χA ) are similar, it follows from Proposition 6.6.3 that µA = µC(χA ) . Hence, µA = χA . 

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Corollary 7.3.5. Let A ∈ Fn×n be a companion matrix. Then, A = C(χA ) and µA = χA .

Note that, if A = In , then the similarity invariants of A are pi (s) = s − 1 for all i ∈ {1, . . . , n}. Thus, C(pi ) = 1 for all i ∈ {1, . . . , n}, as expected. Although the multicompanion matrix given by Theorem 7.3.3 provides a canonical matrix for all square A, in some cases it is possible to use similarity transformation to further decompose some of the companion blocks in the multicompanion matrix into multicompanion matrices. This procedure provides an alternative canonical matrix, which is also a multicompanion matrix. To show this, note that, if Ai is similar to Bi for all i ∈ {1, . . . , r}, then diag(A1 , . . . , Ar ) is similar to diag(B1 , . . . , Br ). Therefore, it follows from Corollary 7.3.9 that, if sI − Ai and sI − Bi have the same Smith form for all i ∈ {1, . . . , r}, then sI − diag(A1 , . . . , Ar ) and sI − diag(B1 , . . . , Br ) have the same Smith form. The following lemma is needed. Lemma 7.3.6. Let A = diag(A1 , A2 ), where Ai ∈ Fni ×ni for i = 1, 2. Then, µA is the least common multiple of µA1 and µA2 . In particular, if µA1 and µA2 are coprime, then µA = µA1 µA2 . Proof. Since 0 = µA(A) = diag[µA(A1 ), µA(A2 )], it follows that µA(A1 ) = 0 and µA(A2 ) = 0. Therefore, Theorem 6.6.1 implies that µA1 and µA2 both divide µA . Consequently, the least common multiple q of µA1 and µA2 also divides µA . Since q(A1 ) = 0 and q(A2 ) = 0, it follows that q(A) = 0. Therefore, µA divides q. Hence, q = µA . In the case where µA1 and µA2 are coprime, µA = µA1 µA2 .  Proposition 7.3.7. Let p ∈ F[s] be a monic polynomial of positive degree n, and let p = p1 · · · pr , where p1 , . . . , pr ∈ F[s] are monic and pairwise coprime polynomials. Then, the matrices C(p) and diag[C(p1 ), . . . , C(pr )] are similar. △ Proof. Let pˆ 2 = p2 · · · pr and Cˆ = diag[C(p1 ), C( pˆ 2 )]. Since p1 and pˆ 2 are coprime, it follows from Lemma 7.3.6 that µCˆ = µC(p1 ) µC( pˆ 2 ) . Furthermore, χCˆ = χC(p1 ) χC( pˆ 2 ) = µCˆ . Hence, Corollary 7.3.4 implies that Cˆ is similar to C(χCˆ ). However, χCˆ = p1 · · · pr = p, so that Cˆ is similar to C(p). If r > 2, then the same argument can be used to decompose C( pˆ 2 ) to show that C(p) is similar to diag[C(p1 ), . . . , C(pr )].  Proposition 7.3.7 can be used to decompose some of the companion blocks of a multicompanion matrix into multicompanion matrices. This procedure can be carried out for every companion block whose characteristic polynomial has either two or more nonconstant coprime factors. For example, suppose that A ∈ R10×10 has the similarity invariants pi(s) = 1 for all i ∈ {1, . . . , 7}, p8 (s) = (s + 1)2 , p9 (s) = (s + 1)2 (s + 2), and p10 (s) = (s + 1)2 (s + 2)(s2 + 3), so that, by Theorem 7.3.3, the multicompanion form of A is diag[C(p8 ), C(p9 ), C(p10 )], where C(p8 ) ∈ R2×2, C(p9 ) ∈ R3×3, and C(p10 ) ∈ R5×5. According to Proposition 7.3.7, the matrices C(p9 ) and C(p10 ) can be further decomposed. For example, C(p9 ) is similar to diag[C(p9,1 ), C(p9,2 )], where p9,1 (s) = (s + 1)2 and p9,2 (s) = s + 2 are coprime. Furthermore, C(p10 ) is similar to four different multicompanion matrices, three of which have two companion blocks and one of which has three companion blocks. Since p8 (s) = (s + 1)2 has no nonconstant coprime factors, however, it follows that C(p8 ) cannot be decomposed into smaller companion matrices. By using a similarity transformation, the largest number of companion blocks in a multicompanion matrix is obtained by factoring each similarity invariant into elementary divisors, which are powers of nonconstant, monic, irreducible polynomials that are pairwise coprime. In the above example, this factorization is given by p9 = p9,1 p9,2 , where p9,1(s) = (s + 1)2 and p9,2 (s) = s + 2, and by p10 = p10,1 p10,2 p10,3 , where p10,1(s) = (s + 1)2 , p10,2 (s) = s + 2, and p10,3 (s) = s2 + 3. The elementary divisors of A arising from p8 , p9 , and p10 are thus (s + 1)2 , (s + 1)2 , s + 2, (s + 1)2 , s + 2, and s2 +√3, which√yields six companion blocks. Viewing A ∈ Cn×n we can further factor p10,3 (s) = (s + 3 ȷ)(s − 3 ȷ), which yields a total of seven companion blocks. For square matrices under a similarity transformation, the following result provides a canonical matrix given by a multicompanion matrix. This result follows from Proposition 7.3.7 and Theorem 7.3.3.

MATRIX DECOMPOSITIONS

549

Theorem 7.3.8. Let A ∈ Fn×n, and let ql11 , . . . , qlhh ∈ F[s] be the elementary divisors of A, where

l1, . . . , lh are positive integers. Then, there exists a nonsingular matrix S ∈ Fn×n such that    C(ql11 ) 0   −1  ..  S . A = S  .   lh  0 C(qh )

(7.3.9)

The multicompanion matrix diag[C(ql11 ), . . . , C(qlhh )] in (7.3.9) is the elementary multicompanion form of A. The multicompanion form of A is a canonical matrix given by a multicompanion matrix with the smallest possible number of companion blocks, whereas the elementary multicompanion form of A is a canonical matrix given by a multicompanion matrix with the largest possible number of companion blocks. Of course, A may be similar to a multicompanion matrix that is neither the multicompanion form of A nor the elementary multicompanion form of A, but this matrix is not uniquely specified and thus it is not a canonical matrix. Corollary 7.3.9. Let A, B ∈ Fn×n. Then, the following statements are equivalent: i) A and B are similar. ii) A and B have the same similarity invariants. iii) A and B have the same multicompanion form. iv) A and B have the same elementary multicompanion form.

7.4 Jordan Decomposition We now present an alternative form of the multicompanion matrix in (7.3.9). To do this we define the Jordan matrix Jl (q), where l is a positive integer and q(s) = s − λ ∈ C[s], to be the l × l Toeplitz upper bidiagonal matrix     λ 1 0   0 λ 1 0   . .   .. ..   △ (7.4.1) Jl (q) = λIl + Nl =   . ..  . 1 0     0 λ 1   0 λ The following result shows that Jl (q) is similar to the companion matrix C(ql ). Lemma 7.4.1. Let l ∈ P, let λ ∈ C, and define q(s) = s − λ ∈ C[s]. Then, ql is the only elementary divisor of Jl (q). Furthermore, Jl (q) and C(ql ) are similar. Proof. Note that χJl (q) = ql and det([sI − Jl (q)][l,1] ) = (−1)l−1. Hence, as in the proof of Proposition 7.3.1, χJl (q) = µJl (q) . Corollary 7.3.4 implies that Jl (q) and C(ql ) are similar.  A block-diagonal matrix whose diagonally located blocks are Jordan matrices is a multi-Jordan matrix. For square matrices under a similarity transformation, the following result, which follows from Proposition 7.3.7 and Lemma 7.4.1, provides a canonical matrix given by a multi-Jordan matrix. Theorem 7.4.2. Let A ∈ Cn×n, and let ql11 , . . . , qlhh ∈ C[s] be the elementary divisors of A, where l1 , . . . , lh are positive integers and each polynomial q1 , . . . , qh ∈ C[s] has degree 1. Then, there exists a nonsingular matrix S ∈ Cn×n such that   0   Jl1 (q1 )   ..  S −1. A = S  (7.4.2) .   0 Jlh (qh )

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The multi-Jordan matrix diag[Jl1 (q1 ), . . . , Jlh (qh )] in (7.4.2) is the Jordan form of A. The Jordan form of A is unique up to the ordering of the Jordan matrices. Uniqueness can be ensured by specifying a total ordering on C. To illustrate the structure of the Jordan form, let li = 3 and qi (s) = s − λi , where λi ∈ C. Then, Jli (qi ) is the 3 × 3 matrix    λi 1 0    (7.4.3) Jli (qi ) = λi I3 + N3 =  0 λi 1    0 0 λi so that mspec[Jli (qi )] = {λi , λi , λi }ms . If Jli (qi ) is the unique diagonally located block of the Jordan form associated with the eigenvalue λi , then the algebraic multiplicity of λi is equal to 3, while its geometric multiplicity is equal to 1. Corollary 7.4.3. Let p ∈ F[s], let λ1 , . . . , λr denote the distinct roots of p, and, for all i ∈ △ △ {1, . . . , r}, let li = mult p (λi ) and pi (s) = s − λi . Then, C(p) is similar to diag[Jl1 (p1 ), . . . , Jlr (pr )]. For a real matrix A ∈ Rn×n , we now obtain a real decomposition that is analogous to (7.4.2). This can be done in two different ways, namely, in terms of either the coefficients of the quadratic irreducible elementary divisors of A or the real and imaginary parts of the nonreal eigenvalues of A. Note that every real elementary divisor qlii is either of the form (s − λi )li , where λi ∈ R, or of the form (s2 − β1i s − β0i )li , where β0i , β1i ∈ R. For q(s) = s2 − β1 s − β0 ∈ R[s], define the 2l × 2l real, tridiagonal matrix     0 1   β β 1 0 1   0   0 0 1   β0 β1 1  △  (7.4.4) Hl (q) =   . .. .. ..   . . .     ..  . 0 1  0   β0 β1 △

For q(s) = s − λ ∈ R[s], define Hl (q) = Jl (q). The matrix Hl (q) is a hypercompanion matrix. The following result shows that the hypercompanion matrix Hl (q) is similar to the companion matrix C(ql ). Lemma 7.4.4. Let l ∈ P, and let q(s) = s2 − β1 s − β0 ∈ R[s]. Then, ql is the only elementary divisor of Hl (q). Furthermore, Hl (q) and C(ql ) are similar. Proof. Note that χHl (q) = ql and det([sI − Hl (q)][2l,1] ) = (−1)2l−1. Hence, as in the proof of Proposition 7.3.1, χHl (q) = µHl (q) . Corollary 7.3.4 now implies that Hl (q) is similar to C(ql ).  For real square matrices under a similarity transformation, the following result provides a canonical matrix given by a multihypercompanion matrix, which is a block-diagonal matrix whose diagonally located blocks are hypercompanion matrices. Theorem 7.4.5. Let A ∈ Rn×n, and let ql11 , . . . , qlhh ∈ R[s] be the real elementary divisors of A, where l1, . . . , lh are positive integers. Then, there exists a nonsingular matrix S ∈ Rn×n such that   0   Hl1 (q1 )   ..  S −1. A = S  (7.4.5) .   0 Hlh (qh ) The multihypercompanion matrix diag[Hl1 (q1 ), . . . , Hlh (qh )] in (7.4.5) is the multihypercompanion form of A. Applying an additional real similarity transformation to each diagonally located block of a multihypercompanion matrix yields a real Jordan matrix. To do this, define the real Jordan ma-

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trix Jl (q) for the positive integer l as follows. For λ ∈ R and q(s) = s−λ ∈ R[s], define Jl (q) = Hl (q), whereas, for irreducible q(s) = s2 − β1 s − β0 ∈ R[s] with a nonreal root λ = ν + ω ȷ, define the 2l × 2l upper Hessenberg matrix   0   ν ω 1   ..  −ω ν  . 0 1 0   ..   .   0 ν ω 1   .. ..   . . −ω ν 0  △  (7.4.6) Jl (q) =   . .. ..  . . 1 0     .. . 0  1      0 ν ω     −ω ν For real square matrices under a similarity transformation, the following result provides a canonical matrix given by a multi-real-Jordan matrix, which is a block-diagonal matrix whose diagonally located blocks are real Jordan matrices. Theorem 7.4.6. Let A ∈ Rn×n, and let ql11 , . . . , qlhh ∈ R[s] be the elementary divisors of A, where l1 , . . . , lh are positive integers. Then, there exists a nonsingular matrix S ∈ Rn×n such that   0   Jl1 (q1 )   ..  S −1. (7.4.7) A = S  .   0 Jlh (qh ) Proof. For the irreducible quadratic q(s) = s2 − β1 s − β0 ∈ R[s], we show that Jl (q) and Hl (q)

are similar. Writing q(s) = (s − λ)(s − λ), Theorem 7.4.2 implies that Hl (q) ∈ R2l×2l is similar to diag(λIl + Nl , λIl + Nl ). Next, by using a permutation similarity transformation, Hl (q) is similar to    λ 0 1 0     0 λ 0 1 0  0     0 λ 0 1 0     0 λ 0 1     . . . . . .   . . . .     .. .. . . 1 0      ..  . 0 1     0 λ 0     0 λ

△ △ ˆ . . . , Sˆ ) to the above matrix, where Sˆ = Finally, transformation S = diag(S, [ − ȷ − ȷ ] applying the [ ȷ similarity ] 1 1 −1 ˆ = ȷ −1 , yields Jl (q).  1 −1 and S 2 The multi-real-Jordan matrix [Jl1 (q1 ), . . . , Jlh (qh )] in (7.4.7) is the multi-real-Jordan form of A.

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Example 7.4.7. Let A, B ∈ R4×4 and C ∈ C4×4 be given by

 1 0  0  0 0 1  A =  0 0  0 −16 0 −8

 0   0  , 1  0

  0  −4 B =   0 0

1 0 0 0

0 1 0 −4

 0   0  , 1  0

  0 0   2 ȷ 1   0 2 ȷ 0 0  . C =   0 0 −2 ȷ 1  0 0 0 −2 ȷ

Then, A is a companion matrix, B is a hypercompanion matrix, and C is Furthermore, A, B, and C are similar. Example 7.4.8. Let A, B, D ∈ R6×6 and C ∈ C6×6 be given by    1 0 0 0 0   0 1 0 0  0   3 2 0 0   0 0 1 0 0 0      0 0 0 1  0 0 0 1 0 0  A =   , B =  0 0 0 1 0   0 0 −3 2  0   0 0 0 0  0 0 0 0 0 1     0 0 0 0 27 −18 −3 20 −15 6   −1  0   0 C =   0  0  0

0 3 0 0 0 0

  0 0 0 0   −1   0 0√ 0 0 0     0 1√ 0 0  1− 2 ȷ  , D =   0 0 1− 2 ȷ 0√ 0     0 0 0 1+ 2 ȷ 1√    0 0 0 1+ 2 ȷ 0

0 3 0 0 0 0

a multi-Jordan matrix. ^ 0 0 0 1 0 −3

0 0 1 √ − 2 0 0

 0   0  0  , 0   1   2 0 0 0 0 √ 2 0 1 1 0 1 √ 0 − 2

 0   0   0   .  0 √  2   1

Then, A is a companion matrix, B is a multihypercompanion matrix, C is a multi-Jordan matrix, and D is a multi-real-Jordan matrix. Furthermore, A, B, C, and D are similar. ^ The next result shows that every matrix is similar to its transpose by means of a symmetric similarity transformation. This result, which improves Corollary 6.3.12, is due to F. G. Frobenius. Corollary 7.4.9. Let A ∈ Fn×n. Then, there exists a symmetric, nonsingular matrix S ∈ Fn×n such that A = SATS −1. Proof. It follows from Theorem 7.4.2 that there exists a nonsingular matrix Sˆ ∈ Cn×n such that ˆ Sˆ −1, where B = diag(B1 , . . . , Br ) is the Jordan form of A, and Bi ∈ Cni ×ni for all i ∈ {1, . . . , r}. A = SB △ △ Now, define the symmetric nonsingular matrix S = Sˆ I˜Sˆ T, where I˜ = diag(Iˆn1 , . . . , Iˆnr ) is symmetric T ˜ I˜ = BT, and thus and involutory. Furthermore, note that Iˆni Bi Iˆni = Bi for all i ∈ {1, . . . , r} so that IB T ˜ I˜ = B. Hence, it follows that IB ˜ TI˜Sˆ −1 = SB ˆ Sˆ −1 = A. SATS −1 = S Sˆ −TBTSˆ TS −1 = Sˆ I˜Sˆ TSˆ −TBTSˆ TSˆ −TI˜Sˆ −1 = Sˆ IB If A is real, then the real Jordan form can be used to show that S can be chosen to be real.  An extension of Corollary 7.4.9 to the case where A is normal is given by Fact 7.10.10. Corollary 7.4.10. Let A ∈ Fn×n. Then, there exist symmetric matrices S 1 , S 2 ∈ Fn×n such that S 2 is nonsingular and A = S 1 S 2 . Proof. From Corollary 7.4.9 it follows that there exists a symmetric, nonsingular matrix S ∈ △ △ Fn×n such that A = SATS −1. Now, let S 1 = SAT and S 2 = S −1. Note that S 2 is symmetric and nonsingular. Furthermore, S 1T = AS = SAT = S 1 , which shows that S 1 is symmetric.  Corollary 7.4.10 implies Corollary 7.4.9. To show this, note that, if A = S 1 S 2 , where S 1 , S 2 are symmetric and S 2 is nonsingular, then A = S 2−1S 2 S 1 S 2 = S 2−1ATS 2 .

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7.5 Schur Decomposition The Schur decomposition uses a unitary similarity transformation to transform an arbitrary square matrix into an upper triangular matrix. Theorem 7.5.1. Let A ∈ Cn×n. Then, there exist a unitary matrix S ∈ Cn×n and an upper triangular matrix B ∈ Cn×n such that A = SBS ∗.

(7.5.1)

Proof. Let λ1 ∈ C be an eigenvalue of A with associated eigenvector x ∈ Cn chosen such that

△ x x = 1. Furthermore, let S 1 = [x Sˆ1 ] ∈ Cn×n be unitary, where Sˆ1 ∈ Cn×(n−1) satisfies Sˆ1∗S 1 = In−1 and x∗Sˆ1 = 01×(n−1) . Then, S 1 e1 = x, and col1 (S 1−1AS 1 ) = S 1−1Ax = λ1 S 1−1x = λ1 e1. Consequently,    λ1 C1  −1  S 1 ,  A = S 1  0(n−1)×1 A1

∗

where C1 ∈ C1×(n−1) and A1 ∈ C(n−1)×(n−1). Next, let S 20 ∈ C(n−1)×(n−1) be a unitary matrix such that    λ2 C2  −1  S 20 ,  A1 = S 20  0(n−2)×1 A2 where C2 ∈ C1×(n−2) and A2 ∈ C(n−2)×(n−2). Hence,    λ1 C11 C12    A = S 1 S 2  0 λ2 C2  S 2−1S 1 ,   0 0 A2 [ ] △ where C1 = [C11 C12 ], C11 ∈ C, and S 2 = 10 S020 is unitary. Proceeding in a similar manner yields △

(7.5.1) with S = S 1 S 2 · · · S n−1 , where S 1 , . . . , S n−1 ∈ Cn×n are unitary.  Since A and B in (7.5.1) are similar and B is upper triangular, Fact 6.10.14 implies that A and B have the same eigenvalues with the same algebraic multiplicities. The real Schur decomposition uses a real orthogonal similarity transformation to transform a real matrix into an upper Hessenberg matrix with real 1 × 1 and 2 × 2 diagonally located blocks. Corollary 7.5.2. Let A ∈ Rn×n, and let mspec(A) = {λ1 , . . . , λr }ms ∪ {ν1 + ω1 ȷ, ν1 − ω1 ȷ, . . . , νl + ωl ȷ, νl − ωl ȷ}ms , where λ1 , . . . , λr ∈ R and, for all i ∈ {1, . . . , l}, νi , ωi ∈ R and ωi , 0. Then, there exists an orthogonal matrix S ∈ Rn×n such that A = SBS T ,

(7.5.2)

where B is upper block triangular and the diagonally located blocks B1 , . . . , Br ∈ R and Bˆ 1 , . . . , Bˆ l ∈ △ R2×2 of B satisfy Bi = [λi ] for all i ∈ {1, . . . , r} and spec( Bˆ i ) = {νi + ωi ȷ, νi − ωi ȷ} for all i ∈ {1, . . . , l}. Proof. The proof is analogous to the proof of Theorem 7.4.6. See also [1448, p. 82].  n×n Corollary 7.5.3. Let A ∈ R , and assume that the spectrum of A is real. Then, there exist an orthogonal matrix S ∈ Rn×n and an upper triangular matrix B ∈ Rn×n such that A = SBS T.

(7.5.3)

The Schur decomposition reveals the structure of range-Hermitian matrices and thus, as a special case, normal matrices. △ Corollary 7.5.4. Let A ∈ Fn×n, and define r = rank A. Then, A is range Hermitian if and only if there exist a unitary matrix S ∈ Fn×n and a nonsingular matrix B ∈ Fr×r such that [ ] B 0 ∗ A=S S. (7.5.4) 0 0

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In addition, A is normal if and only if there exist a unitary matrix S ∈ Cn×n and a diagonal matrix B ∈ Cr×r such that (7.5.4) holds. ˆ ∗, where Bˆ is upper triangular and Proof. Suppose that A is range Hermitian, and let A = SBS n×n S ∈F is unitary. Assume that A is singular, and choose S such that Bˆ ( j, j) = Bˆ ( j+1, j+1) = · · · = ˆ ˆ = 0, which implies B(n,n) = 0 and such that all other diagonal entries of Bˆ are nonzero. Thus, rown ( B) ⊥ ∗ ˆ ˆ ˆ that en ∈ R( B) . Since A is range Hermitian, it follows that R( B) = R( B ), and thus en ∈ R( Bˆ ∗ )⊥. ˆ which implies that coln ( B) ˆ = 0. In the case where Therefore, it follows from (3.5.13) that en ∈ N( B), [ ] ˆ ˆ B(n−1,n−1) = 0, it follows that coln−1 ( B) = 0. Repeating this argument shows that Bˆ has the form B0 00 , where B ∈ Fr×r is nonsingular. The converse result is immediate. ˆ ∗, where Bˆ ∈ Cn×n is upper triangular and S ∈ Now, suppose that A is normal, and let A = SBS n×n ˆ Since Bˆ C is unitary. Since A is normal, it follows that AA∗ = A∗A, which implies that Bˆ Bˆ ∗ = Bˆ ∗B. ∗ ∗ˆ ˆ ˆ ˆ ˆ ˆ ∗= ˆ ˆ is upper triangular, it follows that ( B B)(1,1) = B(1,1) B(1,1) , whereas ( BB )(1,1) = row1( B)[row1( B)] ∑n ˆ ˆ ˆ∗ ˆ ˆ ˆ∗ ˆ i=1 B(1,i) B(1,i) . Since ( B B)(1,1) = ( BB )(1,1) , it follows that B(1,i) = 0 for all i ∈ {2, . . . , n}. Continuing ˆ in a similar fashion row by row, it follows that B is diagonal. The converse result is immediate.  △ Corollary 7.5.5. Let A ∈ Fn×n, assume that A is Hermitian, and define r = rank A. Then, there n×n r×r exist a unitary matrix S ∈ F and a diagonal matrix B ∈ R such that (7.5.4) holds. In addition, A is positive semidefinite if and only if the diagonal entries of B are positive, and A is positive definite if and only if A is positive semidefinite and r = n. Proof. Corollary 7.5.4 and x), xi) of Proposition 6.4.5 imply that there exist a unitary matrix S ∈ Fn×n and a diagonal matrix B ∈ Rr×r such that (7.5.4) holds. If A is positive semidefinite, then x∗Ax ≥ 0 for all x ∈ Fn. Choosing x = Sei , it follows that B(i,i) = eTi S ∗ASei ≥ 0 for all i ∈ {1, . . . , r}. If A is positive definite, then r = n and B(i,i) > 0 for all i ∈ {1, . . . , n}.  n×n Proposition 7.5.6. Let A ∈ F be Hermitian. Then, there exists a nonsingular matrix S ∈ Fn×n such that   0 0   −Iν− (A)   0ν0 (A)×ν0 (A) 0  S ∗. (7.5.5) A = S  0   0 0 Iν+ (A) Furthermore,

rank A = ν+ (A) + ν− (A),

(7.5.6)

def A = ν0 (A).

(7.5.7)

Proof. Since A is Hermitian, it follows from Corollary 7.5.5 that there exist a unitary matrix ˆ Sˆ ∗. Choose Sˆ to arrange the diagonal Sˆ ∈ Fn×n and a diagonal matrix B ∈ Rn×n such that A = SB entries of B such that B = diag(−B1 , 0ν0 (A)×ν0 (A) , B2 ), where the diagonal matrices B1 , B2 are both positive definite. Since the diagonal entries of B are the eigenvalues of A, it follows that B1 ∈ △ Rν−(A)×ν−(A) and B2 ∈ Rν+(A)×ν+(A) . Now, define Bˆ = diag(B1 , Iν0 (A) , B2 ). Then, B = Bˆ 1/2DBˆ 1/2, where △ △ D = diag(−Iν−(A) , 0ν0 (A)×ν0 (A) , Iν+(A) ). Hence, A = SˆBˆ 1/2DBˆ 1/2Sˆ ∗ = SDS ∗, where S = SˆBˆ 1/2.  The following result is Sylvester’s law of inertia. Theorem 7.5.7. Let A, B ∈ Fn×n be Hermitian. Then, A and B are congruent if and only if In A = In B. Proof. To prove sufficiency, assume that In A = In B = [n1 n2 n3 ]T . It thus follows from Proposition 7.5.6 that there exist nonsingular matrices S 1 ∈ Fn×n and S 2 ∈ Fn×n     0 0  0 0   −In1  −In1  ∗    0n2 ×n2 0  S 1 , B = S 2  0 0n2 ×n2 0  S 2∗ . A = S 1  0       0 0 In3 0 0 In3

555

MATRIX DECOMPOSITIONS

Therefore, A = S 1 S 2−1B(S 1 S 2−1 )∗ , which shows that A and B are congruent. Conversely, assume that A and B are congruent, and let Sˆ ∈ Fn×n be a nonsingular matrix such ˆ Sˆ ∗ . Proposition 7.5.6 implies that there exists a nonsingular matrix S ∈ Fn×n such that that A = SB   0 0   −Iν− (B)   0ν0 (B)×ν0 (B) 0  S ∗. B = S  0   0 0 Iν+ (B) Therefore,

 −I  ν− (B) Since A and  0 0

  0 0   −Iν− (B)  ˆ )∗. ˆ  0 0ν0 (B)×ν0 (B) 0  (SS A = SS   0 0 Iν+ (B)  0 0   0ν0 (B)×ν0 (B) 0   are congruent, Fact 7.9.17 implies that In A = In B. 0



Iν+ (B)

Proposition 6.5.4 shows that eigenvectors associated with distinct eigenvalues of a normal matrix are mutually orthogonal. Thus, a normal matrix has at least as many mutually orthogonal eigenvectors as it has distinct eigenvalues. The next result, which follows from Corollary 7.5.4, shows that every n × n normal matrix has n mutually orthogonal eigenvectors. In fact, the converse is also true. Corollary 7.5.8. Let A ∈ Cn×n. Then, A is normal if and only if A has n mutually orthogonal eigenvectors. The following result concerns the real normal form. Corollary 7.5.9. Let A ∈ Rn×n be range symmetric. Then, there exist an orthogonal matrix △ S ∈ Rn×n and a nonsingular matrix B ∈ Rr×r, where r = rank A, such that [ ] B 0 T A=S S . (7.5.8) 0 0 In addition, assume that A is normal, and let mspec(A) = {λ1 , . . . , λr }ms ∪ {ν1 + ω1 ȷ, ν1 − ω1 ȷ, . . . , νl + ωl ȷ, νl − ωl ȷ}ms , where λ1 , . . . , λr ∈ R and, for all i ∈ {1, . . . , l, } νi , ωi ∈ R and ωi , 0. Then, there exists an orthogonal matrix S ∈ Rn×n such that A = SBS T, △

△

△

where B = diag(B1 , . . . , Br , Bˆ 1 , . . . , Bˆ l ), Bi = [λi ] for all i ∈ {1, . . . , r}, and Bˆ i = i ∈ {1, . . . , l}.

[

νi ωi ] −ωi νi

(7.5.9) for all

7.6 Singular Value Decomposition, Polar Decomposition, and Full-Rank Factorization We now consider the singular value decomposition, which, like the Smith decomposition but unlike the Jordan and Schur decompositions, applies to rectangular matrices. Let A ∈ Fn×m, where A , 0, and consider the positive-semidefinite matrices AA∗ ∈ Fn×n and A∗A ∈ Fm×m. It follows from Proposition 6.4.10 that AA∗ and A∗A have the same nonzero eigenvalues with the same algebraic multiplicities. Since AA∗ and A∗A are positive semidefinite, it follows that they have the same positive eigenvalues with the same algebraic multiplicities. Furthermore, since AA∗ is Hermitian, it follows that the number of positive eigenvalues of AA∗ (or A∗A), counting algebraic multiplicity, is equal to the rank of AA∗ (or A∗A). Since rank A = rank AA∗ = rank A∗A, it thus follows that AA∗ and A∗A both △ have r positive eigenvalues, where r = rank A. Definition 7.6.1. Let A ∈ Fn×m. Then, the singular values of A are the min {n, m} nonnegative

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numbers σ1 (A), . . . , σmin{n,m} (A), where, for all i ∈ {1, . . . , min {n, m}}, △

1/2 ∗ ∗ σi (A) = λ1/2 i (AA ) = λi (A A).

(7.6.1)

For A ∈ Fn×m, define the singular value vector of A by    σ1 (A)    △  ..  σ(A) =  .   σmin {n,m} (A)

(7.6.2)

and the multiset of singular values of A by △

msval(A) = {σ1 (A), . . . , σmin {n,m} (A)}ms .

(7.6.3)

In terms of the notation ⟨A⟩ defined by (10.5.6), it follows that, if m ≤ n, then σ(A) = λ(⟨A⟩), whereas, if n ≤ m, then σ(A) = λ(⟨A∗ ⟩). Let A ∈ Fn×m. Since AA∗ is positive semidefinite, it follows that 0 ≤ σmin {n,m}(A) ≤ · · · ≤ σ1 (A).

(7.6.4)

△

Define r = rank A. If 1 ≤ r < min {n, m}, then 0 = σmin {n,m}(A) = · · · = σr+1 (A) < σr (A) ≤ · · · ≤ σ1 (A),

(7.6.5)

whereas, if r = min {m, n}, then 0 < σmin {n,m}(A) = σr (A) ≤ · · · ≤ σ1 (A).

(7.6.6)

Consequently, rank A is the number of positive singular values of A. For convenience, define △

σmax (A) = σ1 (A) and, if n = m,

△

σmin (A) = σn (A).

(7.6.7) (7.6.8)

If n , m, then σmin (A) is not defined. By convention, we define σmax (0n×m ) = σmin (0n×n ) = 0,

(7.6.9)

σi (A) = σi (A∗ ) = σi (A) = σi (AT ).

(7.6.10)

and, for all i ∈ {1, . . . , min {n, m}}, Now, suppose that n = m. If A is Hermitian, then, for all i ∈ {1, . . . , n}, σi (A) = ρi (A) and

{σ1 (A), . . . , σn (A)}ms = {ρ1 (A), . . . , ρn (A)}ms = {|λ1 (A)|, . . . , |λn (A)|}ms .

(7.6.11) (7.6.12)

Finally, if A is positive semidefinite, then, for all i ∈ {1, . . . , n}, σi (A) = ρi (A) = λi (A).

(7.6.13)

Proposition 7.6.2. Let A ∈ Fn×m. If n ≤ m, then the following statements are equivalent:

i) rank A = n. ii) σn (A) > 0. If m ≤ n, then the following statements are equivalent: iii) rank A = m. iv) σm(A) > 0.

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MATRIX DECOMPOSITIONS

If n = m, then the following statements are equivalent: v) A is nonsingular. vi) σmin (A) > 0. We now state the singular value decomposition. △ △ Theorem 7.6.3. Let A ∈ Fn×m, assume that A is nonzero, let r = rank A, and define B = n×n m×m diag[σ1 (A), . . . , σr (A)]. Then, there exist unitary matrices S 1 ∈ F and S 2 ∈ F such that    B 0r×(m−r)   S 2 . (7.6.14) A = S 1  0(n−r)×r 0(n−r)×(m−r) Furthermore, each column of S 1 is an eigenvector of AA∗ , while each column of S 2∗ is an eigenvector of A∗A. Proof. For convenience, assume that r < min {n, m}, since otherwise some of the zero matrices in (7.6.14) become empty matrices. By Corollary 7.5.5 there exists a unitary matrix U ∈ Fn×n such [ 2 ] that B 0 ∗ ∗ AA = U U. 0 0 Partition U = [U1 U2 ], where U1 ∈ Fn×r and U2 ∈ Fn×(n−r). Since U ∗ U = In, it follows that U1∗U1 = Ir △ and U1∗U = [Ir 0r×(n−r) ]. Now, define V1 = A∗U1B−1 ∈ Fm×r, and note that [ 2 ] B 0 ∗ ∗ −1 ∗ ∗ −1 −1 ∗ V1 V1 = B U1 AA U1 B = B U1 U U U1 B−1 = Ir . 0 0 Next, since U2∗U = [0(n−r)×r In−r ], it follows that U2∗ AA∗

[

B2 = [0 I] 0

] 0 ∗ U = 0. 0

However, since R(A) = R(AA∗ ), it follows that U2∗ A = 0. Finally, let V2 ∈ Fm×(m−r) be such that △ V = [V1 V2 ] ∈ Fm×m is unitary. Hence, [ ] [ ][ ] B 0 ∗ B 0 V1∗ U V = [U1 U2 ] = U1 BV1∗ = U1BB−1 U1∗A 0 0 0 0 V2∗ = U1U1∗A = (U1U1∗ + U2 U2∗ )A = UU ∗A = A, which yields (7.6.14) with S 1 = U and S 2 = V ∗.  A corollary of the singular value decomposition is the polar decomposition. Corollary 7.6.4. Let A ∈ Fn×n. Then, there exists a positive-semidefinite matrix M ∈ Fn×n and a unitary matrix S ∈ Fn×n such that A = MS.

(7.6.15)

Proof. It follows from the singular value decomposition that there exist unitary matrices S[ 1 , S]2 ∈ △ Fn×n and a diagonal positive-definite matrix B ∈ Fr×r, where r = rank A, such that A = S 1 B0 00 S 2 . Hence, [ ] B 0 ∗ A = S1 S S S = MS, 0 0 1 1 2 [ ] △ △ where M = S 1 B0 00 S 1∗ is positive semidefinite and S = S 1 S 2 is unitary.  [ ] △ △ n×m Proposition 7.6.5. Let A ∈ F , let r = rank A, and define the Hermitian matrix A = A0∗ A0 ∈ (n+m)×(n+m) F . Then, In A = [r 0 r]T, and the 2r nonzero eigenvalues of A are the r positive singular

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values of A and their negatives. Proof. Since χA(s) = det(s2I − A∗A), it follows that mspec(A)\{0, . . . , 0}ms = {σ1 (A), −σ1 (A), . . . , σr (A), −σr (A)}ms .



The following result shows that every nonzero matrix has a full-rank factorization A = BC, which is unique up to a nonsingular factor. Note that B is left invertible and C is right invertible. △ Proposition 7.6.6. Let A ∈ Fn×m, assume that A is nonzero, and let r = rank A. Then, the following statements hold. i) There exist B ∈ Fn×r and C ∈ Fr×m such that rank B = rank C = r and A = BC. ii) Let B, Bˆ ∈ Fn×r and C, Cˆ ∈ Fr×m , and assume that A = BC = Bˆ Cˆ and rank B = rank Bˆ = ˆ rank C = rank Cˆ = r. Then, there exists a nonsingular matrix S ∈ Fr×r such that B = BS −1 ˆ and C = S C. [ ] △ ˆ T (CC T )−1 Proof. i) follows from (7.6.14) with B = S 1 B0 and C = [I 0]S 2 . ii) holds with S = CC ˆ and S −1 = (BTB)−1BTB.  Corollary 7.6.7. Let A ∈ Fn×m, and assume that rank A = 1. Then, there exist nonzero x ∈ Fn and nonzero y ∈ Fm such that A = xyT . If, in addition, xˆ ∈ Fn and yˆ ∈ Fm satisfy A = xˆyˆ T , then there exists nonzero α ∈ F such that xˆ = αx and yˆ = (1/α)y.

7.7 Eigenstructure Properties Definition 7.7.1. Let A ∈ Fn×n and λ ∈ C. Then, the index of λ with respect to A, denoted by

indA (λ), is the smallest nonnegative integer k such that R[(λI − A)k ] = R[(λI − A)k+1 ].

(7.7.1)

indA (λ) = ind(λI − A).

(7.7.2)

That is,

Note that λ < spec(A) if and only if indA (λ) = 0. Hence, 0 < spec(A) if and only if ind A = indA (0) = 0. Finally, note that indA (0) = ind A. Proposition 7.7.2. Let A ∈ Fn×n and λ ∈ C. Then, indA (λ) is the smallest nonnegative integer k such that rank[(λI − A)k ] = rank[(λI − A)k+1 ].

(7.7.3)

Furthermore, ind A is the smallest nonnegative integer k such that rank(Ak ) = rank(Ak+1 ).

(7.7.4)

Proof. Corollary 3.5.2 implies that R[(λI − A)k ] ⊆ R[(λI − A)k+1 ]. Consequently, Lemma 3.1.5 implies that R[(λI − A)k ] = R[(λI − A)k+1 ] if and only if rank[(λI − A)k ] = rank[(λI − A)k+1 ].  n×n Proposition 7.7.3. Let A ∈ F and λ ∈ spec(A). Then, the following statements hold: i) The largest size of all of the Jordan blocks of A associated with λ is indA (λ). ii) multµA (λ) = indA (λ). iii) The number of Jordan blocks of A associated with λ is gmultA (λ). iv) The number of linearly independent eigenvectors of A associated with λ is gmultA (λ). v) indA (λ) ≤ amultA (λ). vi) indA (λ) = amultA (λ) if and only if exactly one block is associated with λ. vii) gmultA (λ) ≤ amultA (λ). viii) gmultA (λ) = amultA (λ) if and only if every block associated with λ is of size equal to 1.

MATRIX DECOMPOSITIONS

559

ix) indA (λ) + gmultA (λ) ≤ amultA (λ) + 1. x) indA (λ) + gmultA (λ) = amultA (λ) + 1 if and only if at most one block associated with λ is of size greater than 1. Definition 7.7.4. Let A ∈ Fn×n and λ ∈ spec(A). Then, the following terminology is defined: i) λ is simple if amultA (λ) = 1. ii) A is simple if every eigenvalue of A is simple. iii) λ is cyclic (or nonderogatory) if gmultA (λ) = 1. iv) A is cyclic (or nonderogatory) if every eigenvalue of A is cyclic. v) λ is derogatory if gmultA (λ) > 1. vi) A is derogatory if A has at least one derogatory eigenvalue. vii) λ is semisimple if gmultA (λ) = amultA (λ). viii) A is semisimple if every eigenvalue of A is semisimple. ix) λ is defective if gmultA (λ) < amultA (λ). x) A is defective if A has at least one defective eigenvalue. Proposition 7.7.5. Let A ∈ Fn×n and λ ∈ spec(A). Then, λ is simple if and only if λ is cyclic and semisimple. Proposition 7.7.6. Let A ∈ Fn×n and λ ∈ spec(A). Then, gmultA (λ) = def(λI − A) ≤ def (λI − A)indA (λ) = amultA (λ).

(7.7.5)

Theorem 7.4.2 yields the following result, which shows that the subspaces N[(λI − A)k ], where λ ∈ spec(A) and k = indA (λ), provide a decomposition of Fn. Proposition 7.7.7. Let A ∈ Fn×n, let spec(A) = {λ1 , . . . , λr }, and, for all i ∈ {1, . . . , r}, let △ ki = indA (λi ). Then, the following statements hold: i) N[(λi I − A)ki ] ∩ N[(λ j I − A)k j ] = {0} for all i, j ∈ {1, . . . , r} such that i , j. ∑ ii) ri=1 N[(λi I − A)ki ] = Fn. Proposition 7.7.8. Let A ∈ Fn×n and λ ∈ spec(A). Then, the following statements are equivalent: i) λ is semisimple. ii) def(λI − A) = def[(λI − A)2 ]. iii) N(λI − A) = N[(λI − A)2 ]. iv) indA (λ) = 1. Proof. To prove i) =⇒ ii), suppose that λ is semisimple so that gmultA (λ) = amultA (λ), and thus def(λI − A) = amultA (λ). Then, it follows from Proposition 7.7.6 that def[(λI − A)k ] = amultA (λ), △ where k = indA (λ). Therefore, Corollary 3.6.6 implies that amultA (λ) = def(λI − A) ≤ def[(λI − A)2 ] ≤ def[(λI − A)k ] = amultA (λ), which implies that def(λI − A) = def[(λI − A)2 ]. To prove ii) =⇒ iii), note that Corollary 3.6.6 implies that N(λI − A) ⊆ N[(λI − A)2 ]. Since, by ii), these subspaces have equal dimension, Lemma 3.1.5 implies that these subspaces are equal. To prove iii) =⇒ iv), note that iii) implies ii), and thus rank(λI − A) = n − def(λI − A) = n − def[(λI − A)2 ] = rank[(λI − A)2 ]. Therefore, since R(λI − A) ⊆ R[(λI − A)2 ], Corollary 3.6.6 implies that R(λI − A) = R[(λI − A)2 ]. Finally, since λ ∈ spec(A), it follows from Definition 7.7.1 that indA (λ) = 1. Finally, to prove iv) =⇒ i), note that iv) is equivalent to the fact that every Jordan block of A associated with λ has size 1, which is equivalent to the fact that the geometric multiplicity of λ is equal to the algebraic multiplicity of λ; that is, λ is semisimple. 

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Corollary 7.7.9. Let A ∈ Fn×n. Then, A is group invertible if and only if ind A ≤ 1. Proposition 7.7.10. Let A, B ∈ Fn×n . If A and B are similar, then the following statements hold:

i) mspec(A) = mspec(B). ii) For all λ ∈ spec(A), gmultA (λ) = gmultB (λ). Furthermore, if n ≤ 3 and i) and ii) hold, then A and B are similar. The example     0 1 0  0 1 0 0    0 0 0 0   , B =  0 0 1 A =   0 0 0   0 0 0 1   0 0 0 0 0 0 0

 0   0   0  0

(7.7.6)

shows that, for all n ≥ 4, the converse of Proposition 7.7.10 does not hold. Proposition 7.7.11. Let A ∈ Cn×n . Then, the following statements are equivalent: i) A is semisimple. ii) There exists a nonsingular matrix S ∈ Cn×n such that SAS −1 is diagonal. Now, assume that A ∈ Rn×n . Then, the following statements are equivalent: iii) A is semisimple and spec(A) ⊂ R. iv) There exists a nonsingular matrix S ∈ Rn×n such that SAS −1 is diagonal. In view of Proposition 7.7.11, a complex matrix A is diagonalizable over C if and only if A is semisimple. If, in addition, A is real, then A is diagonalizable over R if and only if A is semisimple and every eigenvalue of A is real. In the following result, “similar over F” means that the entries of the similarity transformation are elements of F. Proposition 7.7.12. Let A ∈ Cn×n. Then, A is semisimple if and only if A is similar over C to a normal matrix. Now, assume that A ∈ Rn×n. Then, A is semisimple and spec(A) ⊂ R if and only if A is similar over R to a real symmetric matrix. The following result is an extension of Corollary 7.4.10. Proposition 7.7.13. Let A ∈ Fn×n. Then, the following statements are equivalent: i) A is semisimple, and spec(A) ⊂ R. ii) There exists a positive-definite matrix S ∈ Fn×n such that A = SA∗S −1. iii) There exist a Hermitian matrix S 1 ∈ Fn×n and a positive-definite matrix S 2 ∈ Fn×n such that A = S1 S 2 . ˆ Sˆ −1 , where Proof. To prove i) =⇒ ii), let Sˆ ∈ Fn×n be a nonsingular matrix such that A = SB ˆ Sˆ −1 = Sˆ (Sˆ ∗A∗Sˆ −∗ )Sˆ −1 = B ∈ Rn×n is diagonal. Then, B = Sˆ −1ASˆ = Sˆ ∗A∗Sˆ −∗. Hence, A = SB △ (Sˆ Sˆ ∗ )A∗ (Sˆ Sˆ ∗ )−1 = SA∗S −1, where S = Sˆ Sˆ ∗ is positive definite. To show that ii) implies iii), note that △ A = SA∗S −1 = S 1 S 2 , where S 1 = SA∗ and S 2 = S −1. Since S ∗1 = (SA∗ )∗ = AS ∗ = AS = SA∗ = S 1 , it follows that S 1 is Hermitian. Furthermore, since S is positive definite, it follows that S −1, and hence S 2 , is also positive definite. Finally, to prove iii) =⇒ i), note that A = S 1 S 2 = S 2−1/2 (S 21/2S 1 S 21/2 )S 21/2. Since S 21/2 S 1 S 21/2 is Hermitian, Corollary 7.5.5 implies that S 21/2S 1 S 21/2 is unitarily similar to a real diagonal matrix. Thus, A is semisimple and spec(A) ⊂ R.  If a matrix is block triangular, then the following result shows that its eigenvalues and their algebraic multiplicity are determined by the diagonally located blocks. If, in addition, the matrix is block diagonal, then the geometric multiplicities of its eigenvalues are determined by the diagonally located blocks.  A11 ··· A1k    . . .  Proposition 7.7.14. Let A ∈ Fn×n, assume that A is partitioned as A =  .. · .· · ..  , where, for   Ak1 ···

Akk

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MATRIX DECOMPOSITIONS

all i, j ∈ {1, . . . , k}, Ai j ∈ Fni ×n j , and let λ ∈ spec(A). Then, the following statements hold: i) If Aii is the unique nonzero block in the ith column of blocks, then amultAii (λ) ≤ amultA(λ).

(7.7.7)

ii) If A is either upper block triangular or lower block triangular, then amultA(λ) =

r ∑

mspec(A) =

amultAii (λ),

i=1

k ∪

mspec(Aii ).

(7.7.8)

i=1

iii) If Aii is the unique nonzero block in the ith column of blocks, then gmultAii (λ) ≤ gmultA(λ).

(7.7.9)

iv) If A is upper block triangular, then gmultA11 (λ) ≤ gmultA(λ).

(7.7.10)

v) If A is lower block triangular, then gmultAkk (λ) ≤ gmultA(λ).

(7.7.11)

vi) If A is block diagonal, then gmultA(λ) =

r ∑

gmultAii (λ).

(7.7.12)

i=1 △

Proposition 7.7.15. Let A ∈ Fn×n, let spec(A) = {λ1, . . . , λr }, and, for all i ∈ {1, . . . , r}, let

ki = indA (λi ). Then,

µA(s) =

r ∏

(s − λi )ki ,

deg µA =

i=1

r ∑

ki .

(7.7.13)

i=1

Furthermore, the following statements are equivalent: i) µA = χA . ii) A is cyclic. iii) For all λ ∈ spec(A), the Jordan form of A contains exactly one block associated with λ. iv) A is similar to C(χA ). Proof. Let A = SBS −1, where B = diag(B1 , . . . , Bnh ) denotes the Jordan form of A given by (7.4.2). Let λi ∈ spec(A), and let B j be a Jordan block associated with λi . Then, the size of B j is either less than or equal to ki . Consequently, (B j − λi I)ki = 0. Next, let p(s) denote the right-hand side of the first equality in (7.7.13). Then,  r  r ∏ ∏  ki k p(A) = (A − λi I) = S  (B − λi I) i  S −1 i=1

i=1

 r  r ∏ ∏  = S diag (B1 − λi I)ki , . . . , (Bnh − λi I)ki  S −1 = 0. i=1

i=1

Therefore, it follows from Theorem 6.6.1 that µA divides p. Furthermore, note that, replacing ki by kˆ i < ki yields p(A) , 0. Hence, p is the minimal polynomial of A. The equivalence of i) and ii) is now immediate, while the equivalence of ii) and iii) follows from Theorem 7.4.6. The equivalence of i) and iv) is given by Corollary 7.3.4.  Example 7.7.16. The standard nilpotent matrix Nn is in companion form, and thus is cyclic. In ^ fact, Nn consists of a single Jordan block, and χNn (s) = µNn (s) = sn.

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Example 7.7.17. The matrix

[

]

[

1 1 −1 1

]

is normal but is neither symmetric nor skew symmetric,

is normal but is neither symmetric nor semisimple with real eigenvalues. ^ [ ] [ ] 0 and 1 1 are diagonalizable over R but not normal, while Example 7.7.18. The matrices 12 −1 02 [ ] −1 1 is diagonalizable but is neither normal nor diagonalizable over R. the matrix −2 ^ 1 [ ] [ ] 1 has no real eigenvalues. Example 7.7.19. The product of the Hermitian matrices 12 21 and 21 −2 ^ [ ] [ ] [ ] [ ] 0 1 are similar, whereas 1 0 and 0 1 have the Example 7.7.20. The matrices 10 02 and −2 3 01 −1 2 same spectrum but are not similar. ^ Proposition 7.7.21. Let A ∈ Fn×n. Then, the following statements hold: i) A is singular if and only if 0 ∈ spec(A). ii) A is group invertible if and only if either A is nonsingular or 0 ∈ spec(A) is semisimple. iii) A is Hermitian if and only if A is normal and spec(A) ⊂ R. iv) A is skew Hermitian if and only if A is normal and spec(A) ⊂ IA . v) A is positive semidefinite if and only if A is normal and spec(A) ⊂ [0, ∞). vi) A is positive definite if and only if A is normal and spec(A) ⊂ (0, ∞). vii) A is unitary if and only if A is normal and spec(A) ⊂ UC. viii) A is shifted unitary if and only if A is normal and while the matrix

0 1 −1 0

spec(A) ⊂ {λ ∈ C: |λ − 12 | = 21 }.

(7.7.14)

ix) A is involutory if and only if A is semisimple and spec(A) ⊆ {−1, 1}. x) A is skew involutory if and only if A is semisimple and spec(A) ⊆ {− ȷ, ȷ}. xi) A is idempotent if and only if A is semisimple and spec(A) ⊆ {0, 1}. xii) A is skew idempotent if and only if A is semisimple and spec(A) ⊆ {0, −1}. xiii) A is tripotent if and only if A is semisimple and spec(A) ⊆ {−1, 0, 1}. xiv) A is nilpotent if and only if spec(A) = {0}. xv) A is unipotent if and only if spec(A) = {1}. xvi) A is a projector if and only if A is normal and spec(A) ⊆ {0, 1}. xvii) A is a reflector if and only if A is normal and spec(A) ⊆ {−1, 1}. xviii) A is a skew reflector if and only if A is normal and spec(A) ⊆ {−ȷ, ȷ}. xix) A is an elementary projector if and only if A is normal and mspec(A) = {0, 1, . . . , 1}ms . xx) A is an elementary reflector if and only if A is normal and mspec(A) = {−1, 1, . . . , 1}ms . If, furthermore, A ∈ F2n×2n, then the following statements hold: xxi) If A is Hamiltonian, then mspec(A) = mspec(−A). xxii) If A is symplectic, then mspec(A) = mspec(A−1 ). The following result is a consequence of Proposition 7.7.13 and Proposition 7.7.21. Corollary 7.7.22. Let A ∈ Fn×n, and assume that A is either involutory, idempotent, skew idempotent, tripotent, a projector, or a reflector. Then, the following statements hold: i) There exists a positive-definite matrix S ∈ Fn×n such that A = SA∗S −1. ii) There exist a Hermitian matrix S 1 ∈ Fn×n and a positive-definite matrix S 2 ∈ Fn×n such that A = S1 S 2 . Proposition 7.7.23. Let A, B ∈ Fn×n. Then, the following statements hold: i) Assume that A and B are normal. Then, A and B are unitarily similar if and only if

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MATRIX DECOMPOSITIONS

ii) iii) iv) v) vi) vii)

mspec(A) = mspec(B). Assume that A and B are projectors. Then, A and B are unitarily similar if and only if rank A = rank B. Assume that A and B are (projectors, reflectors). Then, A and B are unitarily similar if and only if tr A = tr B. Assume that A and B are semisimple. Then, A and B are similar if and only if mspec(A) = mspec(B). Assume that A and B are (involutory, skew involutory, idempotent). Then, A and B are similar if and only if tr A = tr B. Assume that A and B are idempotent. Then, A and B are similar if and only if rank A = rank B. Assume that A and B are tripotent. Then, A and B are similar if and only if rank A = rank B and tr A = tr B.

7.8 Pencils and the Kronecker Canonical Form Let A, B ∈ Fn×m, and define the polynomial matrix PA,B ∈ F[s]n×m , called a pencil, by △

PA,B (s) = sB − A. The pencil PA,B is regular if rank PA,B = min {n, m} (see Definition 6.2.4). Otherwise, PA,B is singular. Let A, B ∈ Fn×m. Since PA,B ∈ Fn×m we define the generalized spectrum of PA,B by △

spec(A, B) = Szeros(PA,B )

(7.8.1)

and the generalized multispectrum of PA,B by △

mspec(A, B) = mSzeros(PA,B ).

(7.8.2)

Furthermore, the elements of spec(A, B) are the generalized eigenvalues of PA,B . The structure of a pencil is illuminated by the following result called the Kronecker canonical form. Theorem 7.8.1. Let A, B ∈ Cn×m. Then, there exist nonsingular matrices S 1 ∈ Cn×n and S 2 ∈ m×m C such that, for all s ∈ C, PA,B (s) = S 1 diag(sIr1 − A1 , sB2 − Ir2 , [sIk1−Nk1 − ek1 ], . . . , [sIk p−Nk p − ek p ], [sIl1−Nl1 − el1 ]T, . . . , [sIlq−Nlq − elq ]T, 0t×u )S 2 ,

(7.8.3)

where A1 ∈ Cr1 ×r1 is in Jordan form, B2 ∈ Rr2 ×r2 is nilpotent and in Jordan form, k1, . . . , k p , l1, . . . , lq are positive integers, and [sIl − Nl − el ] ∈ Cl×(l+1). Furthermore, rank PA,B = r1 + r2 +

p ∑

ki +

i=1

q ∑

li .

(7.8.4)

i=1

Proof. See [151, Chapter 2], [1140, Chapter XII], [1573, pp. 395–398], [1737], [1762, pp. 128, 129], and [2539, Chapter VI].  In Theorem 7.8.1, note that

n = r1 + r2 +

p ∑ i=1

ki +

q ∑

li + q + t,

m = r1 + r2 +

i=1

p ∑ i=1

ki +

q ∑

li + p + u.

(7.8.5)

i=1

Proposition 7.8.2. Let A, B ∈ Cn×m, and consider the notation of Theorem 7.8.1. Then, PA,B is

regular if and only if t = u = 0 and either p = 0 or q = 0.

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Let A, B ∈ Fn×m, and let λ ∈ C. Then, rank PA,B (λ) = rank(λI − A1 ) + r2 +

p ∑

ki +

i=1

q ∑

li .

(7.8.6)

i=1

Note that λ is a generalized eigenvalue of PA,B if and only if rank PA,B (λ) < rank PA,B . Consequently, λ is a generalized eigenvalue of PA,B if and only if λ is an eigenvalue of A1 ; that is, spec(A, B) = spec(A1 ).

(7.8.7)

mspec(A, B) = mspec(A1 ).

(7.8.8)

Furthermore, The generalized algebraic multiplicity amultA,B (λ) of λ ∈ spec(A, B) is defined by △

amultA,B (λ) = amultA1(λ).

(7.8.9)

The generalized geometric multiplicity gmultA,B (λ) of λ ∈ spec(A, B) is defined by △

gmultA,B (λ) = gmultA1(λ). For all λ ∈ spec(A, B),

(7.8.10)

gmultA1(λ) = rank PA,B − rank PA,B (λ).

Now, assume that A, B ∈ Fn×n, and thus A and B are square, which, from (7.8.5), is equivalent to q + t = p + u. Then, the characteristic polynomial χA,B ∈ F[s] of (A, B) is defined by △

χA,B (s) = det PA,B (s) = det(sB − A). Proposition 7.8.3. Let A, B ∈ Fn×n. Then, the following statements hold:

i) ii) iii) iv) v) vi) vii) viii) ix) x) xi) xii) xiii)

PA,B is singular if and only if χA,B = 0. PA,B is singular if and only if deg χA,B = −∞. PA,B is regular if and only if χA,B is not the zero polynomial. PA,B is regular if and only if 0 ≤ deg χA,B ≤ n. If PA,B is regular, then multχA,B (0) = n − deg χB,A . deg χA,B = n if and only if B is nonsingular. If B is nonsingular, then χA,B = (det B)χB−1A , spec(A, B) = spec(B−1A), and mspec(A, B) = mspec(B−1A). roots(χA,B ) = spec(A, B). mroots(χA,B ) = mspec(A, B). If either A or B is nonsingular, then PA,B is regular. If all of the generalized eigenvalues of (A, B) are real, then PA,B is regular. If PA,B is regular, then N(A) ∩ N(B) = {0}. If PA,B is regular, then there exist nonsingular matrices S 1 , S 2 ∈ Cn×n such that, for all s ∈ C, ( [ ] [ ]) Ir 0 A1 0 PA,B (s) = S 1 s − S 2, (7.8.11) 0 B2 0 In−r △

where r = deg χA,B , A1 ∈ Cr×r is in Jordan form, and B2 ∈ R(n−r)×(n−r) is nilpotent and in Jordan form. Furthermore, χA,B = χA1 ,

roots(χA,B ) = spec(A1 ),

mroots(χA,B ) = mspec(A1 ).

(7.8.12)

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MATRIX DECOMPOSITIONS

Proof. See [1762, p. 128] and [2539, Chapter VI].



xiii) is the Weierstrass canonical form for a square, regular pencil. Proposition 7.8.4. Let A, B ∈ Fn×n, and assume that A is positive semidefinite and B is Hermitian. Then, the following statements are equivalent: i) PA,B is regular. ii) There exists α ∈ F such that A + αB is nonsingular. iii) N(A) ∩ N(B) = {0}. ([ ]) iv) N AB = {0}. v) There exists nonzero α ∈ F such that N(A) ∩ N(B + αA) = {0}. vi) For all nonzero α ∈ F, N(A) ∩ N(B + αA) = {0}. vii) All generalized eigenvalues of (A, B) are real. If, in addition, B is positive semidefinite, then the following statement is equivalent to i)–vii): viii) There exists β ∈ R such that βB < A. Proof. i) =⇒ ii) and ii) =⇒ iii) are immediate. Next, Fact 3.13.17 and Fact 3.14.10 imply that iii), iv), v), and vi) are equivalent. Next, to prove iii) =⇒ vii), let λ ∈ C be a generalized eigenvalue of (A, B). Since λ = 0 is real, suppose λ , 0. Since det(λB − A) = 0, let nonzero θ ∈ Cn satisfy (λB − A)θ = 0, and thus it follows that θ∗Aθ = λθ∗Bθ. Furthermore, note that θ∗Aθ and θ∗Bθ are real. Now, suppose θ ∈ N(A). Then, it follows from (λB − A)θ = 0 that θ ∈ N(B), which contradicts N(A) ∩ N(B) = {0}. Hence, θ < N(A), and thus θ∗Aθ > 0 and, consequently, θ∗Bθ , 0. Hence, it follows that λ = θ∗Aθ/θ∗Bθ, and thus λ is real. Hence, all generalized eigenvalues of (A, B) are real. To prove vii) =⇒ i), let λ ∈ C\R so that λ is not a generalized eigenvalue of (A, B). Consequently, χA,B (s) is not the zero polynomial, and thus (A, B) is regular. To prove i)–vii) =⇒ viii), let θ ∈ Rn be nonzero, and note that N(A) ∩ N(B) = {0} implies that either Aθ , 0 or Bθ , 0. Hence, either θTAθ > 0 or θTBθ > 0. Thus, θT(A + B)θ > 0, which implies that A + B > 0 and hence −B < A. Finally, to prove viii) =⇒ i)–vii), let β ∈ R satisfy βB < A, so that, for all nonzero θ ∈ Rn, T βθ Bθ < θTAθ. Next, suppose θˆ ∈ N(A) ∩ N(B) is nonzero. Then, Aθˆ = 0 and Bθˆ = 0, and thus ˆ Thus, N(A) ∩ N(B) = {0}. θˆ TBθˆ = 0 and θˆ TAθˆ = 0, which contradicts βθˆ TBθˆ < θˆ TAθ. 

7.9 Facts on the Inertia Fact 7.9.1. Let A ∈ Fn×n, and assume that A is idempotent. Then,

rank A = sig A = tr A,

   0    In A =  n − tr A  .   tr A

Fact 7.9.2. Let A ∈ Fn×n, and assume that A is involutory. Then,

rank A = n,

sig A = tr A,

1   2 (n − tr A)     . 0 In A =   1  2 (n + tr A)

Fact 7.9.3. Let A ∈ Fn×n, and assume that A is tripotent. Then,

rank A = tr A2,

sig A = tr A,

1   2 (tr A2 − tr A)    In A =  n − tr A2  .   1 2 (tr A + tr A) 2

Fact 7.9.4. Let A ∈ Fn×n, and assume that A is either skew Hermitian, skew involutory, or

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nilpotent. Then, sig A = ν− (A) = ν+ (A) = 0,

   0    In A =  n  .   0

Fact 7.9.5. Let A ∈ Fn×n, assume that A is group invertible, and assume that spec(A) ∩ IA ⊆ {0}.

Then,

rank A = ν− (A) + ν+ (A),

def A = ν0 (A) = amultA (0).

Fact 7.9.6. Let A ∈ Fn×n, and assume that A is Hermitian. Then,

rank A = ν− (A) + ν+ (A),

     ν− (A)   21 (rank A − sig A)       . def A In A =  ν0 (A)  =      1 ν+ (A) (rank A + sig A) 2

Fact 7.9.7. Let A, B ∈ Fn×n, and assume that A and B are Hermitian. Then, In A = In B if and

only if rank A = rank B and sig A = sig B. Fact 7.9.8. Let A ∈ Fn×n, assume that A is Hermitian, and let A0 be a principal submatrix of A. Then, ν− (A0 ) ≤ ν− (A) and ν+ (A0 ) ≤ ν+ (A). Source: [1546] and [2991, p. 259]. Fact 7.9.9. Let A ∈ Fn×n, and assume that A is positive semidefinite. Then,    0    rank A = sig A = ν+ (A), In A =  def A .   rank A If, in addition, A is positive definite, then In A = [0 0 n]T . Fact 7.9.10. Let A ∈ Fn×n. Then, the following statements are equivalent: i) A is an elementary projector. ii) A is a projector, and tr A = n − 1. [ 0 ] iii) A is a projector, and In A = 1 . n−1

Furthermore, the following statements are equivalent: iv) A is an elementary reflector. v) A is a reflector, and tr A = n − 2. [ 1 ] vi) A is a reflector, and In A = 0 . n−1

Source: Proposition 7.7.21. Fact 7.9.11. Let A, B ∈ Fn×n , and assume that A and B are projectors. Then,

        rank [A B] − rank A rank(B − AB)     In(A − B) =  rank A + rank B + n − 2 rank [A B]  =  n − rank(B − AB) − rank(A − AB)  .     rank [A B] − rank B rank(A − AB)

Furthermore, the following statements are equivalent: i) rank(A − B) = rank A + rank B. ii) ν+ (A − B) = rank A. iii) ν− (A − B) = rank B. iv) R(A) ∩ R(B) = {0}. In addition, the following statements are equivalent: v) rank(A − B) = rank A − rank B.

567

MATRIX DECOMPOSITIONS

vi) ν+ (A − B) = rank A − rank B. vii) ν− (A − B) = 0. viii) R(B) ⊆ R(A). Source: [2659]. Fact 7.9.12. Let A, B ∈ Fn×n , and assume that A and B are projectors. Then,     rank AB   In(I − A − B) =  rank A + rank B − 2 rank AB  .   n − rank A − rank B + rank AB Furthermore, the following statements hold: i) I − A − B is nonsingular if and only if rank A = rank B = rank AB. ii) A + B = I if and only if rank A + rank B = n and AB = 0. iii) A + B < I if and only if A = B = 0. iv) I < A + B if and only if A = B = I. v) A + B ≤ I if and only if AB = 0. vi) I ≤ A + B if and only if rank A + rank B = n + rank AB. Source: [2659]. Fact 7.9.13. Let A ∈ Fn×n. Then, the following statements are equivalent: i) A + A∗ is positive definite. ii) For all Hermitian matrices B ∈ Fn×n, In B = In AB. Source: [613]. Fact 7.9.14. Let A, B ∈ Fn×n, assume that AB and B are Hermitian, and assume that spec(A) ∩ [0, ∞) = ∅. Then, In(−AB) = In B. Source: [613]. Fact 7.9.15. Let A, B ∈ Fn×n, assume that A and B are Hermitian and nonsingular, and assume that spec(AB) ∩ [0, ∞) = ∅. Then, ν+ (A) + ν+ (B) = n. Source: Fact 7.9.14 and [613]. Remark: Weaker versions are given in [1538, 2125]. Fact 7.9.16. Let A ∈ Fn×n, assume that A is Hermitian, and let S ∈ Fm×n. Then, ν− (SAS ∗ ) + ν+ (SAS ∗ ) = rank SAS ∗ ≤ min {rank A, rank S }, ν− (A) + rank S − n ≤ ν− (SAS ∗ ) ≤ ν− (A), ν+ (A) + rank S − n ≤ ν+ (SAS ∗ ) ≤ ν+ (A), m + ν0 (A) ≤ n + ν0 (SAS ∗ ), rank SAS ∗ ≤ rank A ≤ 2(n − rank S ) + rank SAS ∗ ≤ 2n − rank S . Furthermore, consider the following statements: i) rank S = n. ii) rank SAS ∗ = rank A. iii) ν− (SAS ∗ ) = ν− (A) and ν+ (SAS ∗ ) = ν+ (A). iv) m + ν0 (A) = n + ν0 (SAS ∗ ). Then, i) =⇒ ii) ⇐⇒ iii) =⇒ iv). Finally, the following statements hold: v) If rank S = m, then rank SAS ∗ ≤ rank A ≤ 2(n − m) + rank SAS ∗ ≤ 2n − m, ν0 (SAS ∗ ) ≤ n − m + ν0 (A) ≤ 2(n − m) + ν0 (SAS ∗ ).

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vi) If SAS ∗ is positive semidefinite, then rank S ≤ ν0 (A) + ν+ (A). vii) If SAS ∗ is positive definite, then rank S = m = rank SAS ∗ = ν+ (SAS ∗ ) ≤ ν+ (A). Source: [970, pp. 430, 431] and [1084, p. 194]. The first three strings are given in [2184]. vi) follows from the second string. Adding the second and third strings yields the fifth string. Remark: vi) and vii) are given in [1448, p. 192]. Fact 7.9.17. Let S ∈ Fn×n, assume that S is nonsingular, let n1 , n2 , and n3 be nonnegative △ integers such that n1 + n2 + n3 = n, define D = diag(−In1 , 0n2 ×n2 , In3 ), and define A ∈ Fn×n by △ A = SDS ∗ . Then, In(A) = [n1 n2 n3 ]T . Source: Note that ν0 (A) = def A = def D = n2 . Next, △ ˆ Sˆ ∗ = In3 . Since SA ˆ Sˆ ∗ is positive definite, it follows from define Sˆ = [0n3 ×(n−n3 ) In3 ]S −1 so that SA ˆ Sˆ ∗ = vii) and the second inequality in the third string of Fact 7.9.16 that rank Sˆ = n3 = rank SA ∗ ∗ ˆ ˆ ν+ (SAS ) ≤ ν+ (A) = ν+ (SDS ) ≤ ν+ (D) = n3 . Hence, ν+ (A) = n3 . Likewise, ν− (A) = n1 . Remark: This result completes the proof of Theorem 7.5.7. Fact 7.9.18. Let A, S ∈ Fn×n, and assume that A is Hermitian and S is nonsingular. Then, there exist α1 , . . . , αn ∈ [λmin (SS ∗ ), λmax (SS ∗ )] such that, for all i ∈ {1, . . . , n}, λi (SAS ∗ ) = αi λi (A). Source: [2893]. Remark: This is a quantitative version of Sylvester’s law of inertia given by Theorem 7.5.7. Credit: A. Ostrowski. Fact 7.9.19. Let A, S ∈ Fn×n, assume that A is Hermitian. Then, the following statements are equivalent: i) In(SAS ∗ ) = In A. ii) rank(SAS ∗ ) = rank A. iii) R(A) ∩ N(A) = {0}. Source: [228]. Fact 7.9.20. Let A, B ∈ Rn×n, and assume that A and B are projectors. Then, ν+ (A− B) = rank(A+ B)−rank B,

ν− (A− B) = rank(A+ B)−rank A,

Source: [2661]. Fact 7.9.21. Let A ∈ Rn×m. Then,

[

0 In ∗ A

] [ ∗ A AA = In 0 0

] [ + 0 AA = In −A∗A 0

sig(A− B) = rank A−rank B.

  ]   rank A  0   n + m − 2 rank A =  .   −A+A rank A

Source: [970, pp. 432, 434]. Fact 7.9.22. Let A ∈ Cn×n, assume that A is Hermitian, and let B ∈ Cn×m. Then,

[

A In ∗ B Furthermore, if R(A) ⊆ R(B), then

  ]   rank B B   =  n + m − 2 rank B  . 0   rank B

[

A In ∗ B Finally, if rank B = n, then

  ]  rank B  B   ≥≥  m − rank B  . 0   rank B

[

A In ∗ B

  ]  n    B =  m − n  . 0   n

Source: [970, pp. 433, 434] and [1921]. Related: Fact 10.19.21.

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MATRIX DECOMPOSITIONS

Fact 7.9.23. Let A ∈ Cn×n, let C ∈ Cm×m, assume that A and C are Hermitian, and let B ∈ Cn×m.

Then,

  ]   rank B  B  =  n + m − 2 rank B  , 0   rank B ([ ]) ([ ]) A B A B max {ν− (A), ν− (C)} ≤ ν− , max {ν (A), ν (C)} ≤ ν . + + + B∗ C B∗ C ([ ]) ([ ]) A B A B max {ν− (A) + ν0 (A), ν− (C) + ν0 (C)} ≤ ν− + ν , 0 B∗ C B∗ C ([ ]) ([ ]) A B A B max {ν+ (A) + ν0 (A), ν+ (C) + ν0 (C)} ≤ ν+ + ν0 . B∗ C B∗ C [

] A 0 In = In A + In C, 0 C

[

0 In ∗ B

Source: [2656]. Remark: The last four inequalities are Poincar´e’s inequalities. See [824]. Fact 7.9.24. Let A ∈ [Cn×n, ]let C ∈ Cm×m, assume that A and C are positive semidefinite, let △

B ∈ Cn×m , and define A =

. Then,     rank [B∗ C]   In A =  n + m − rank [A B] − rank [B∗ C]  .   rank [A B] A B B∗ −C

Source: [2662]. Fact 7.9.25. Let A, B ∈ Fn×n, and assume that A and B are projectors. Then,

[

A In B

  ]   rank [A B] − rank A  B  =  2n − 3 rank [A B] + rank A + rank B  . A   2 rank [A B] − rank B

Source: [2667]. Related: Fact 4.18.22. Fact 7.9.26. Let A ∈ Fn×n. Then, there exist a nonsingular matrix S ∈ Fn×n and a skew-Her-

mitian matrix B ∈ Fn×n such that   Iν− (A+A∗ )  A = S  0  0

0

0

0ν0 (A+A∗ )×ν0 (A+A∗ )

0

0

−Iν+ (A+A∗ )

       + B S ∗.  

Source: Write A = 12 (A + A∗ ) + 21 (A − A∗ ), and apply Proposition 7.5.6 to 21 (A + A∗ ). Fact 7.9.27. Let A ∈ Fn×n, B ∈ Fn×m, and C ∈ Fm×m, and assume that A is positive definite and

C is negative definite. Then,

  A  In  B∗  0

B C 0

   0   m     0  =  l  .    0l×l n

Source: Fact 7.9.6 and [1546].

7.10 Facts on Matrix Transformations for One Matrix Fact 7.10.1. Define S ∈ C3×3 by

  1 1   0  1  S = √  √0 − ȷ ȷ  .   2 2 0 0 △

Then, S is unitary, and K(e3 ) = S diag(0, ȷ, − ȷ)S −1. Related: Fact 7.10.2.

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CHAPTER 7

Fact 7.10.2. Let x ∈ R3 , assume that either x(1) , 0 or x(2) , 0, and define

   a    △ 1 x,  b  = ∥x∥2 c

△

α=

√ a2 + b2 + a2 c2 + b2 c2 + (1 − c2 )2 ,

  a  △  S =  b  c

    .   (1−c2 ) ȷ 

−b+ac ȷ α a+bc ȷ α

−b−ac ȷ α a−bc ȷ α

−(1−c2 ) ȷ α

α

√ Then, α ≥ a2 + b2 > 0, S is unitary, and K(x) = S diag(0, ∥x∥2 ȷ, −∥x∥2 ȷ)S −1. Source: [1656, p. 154]. Related: If x(1) = x(2) = 0, then a = b = 0, c = 1, and α = 0. See Fact 7.10.1. Problem: Find a decomposition of K(x) that applies to all x ∈ R3 . Fact 7.10.3. Let A ∈ Fn×n, and assume that there exists (a nonsingular matrix S ∈ Fn×n such [ ]) Ir −1 that S AS is upper triangular. Then, for all r ∈ {1, . . . , n}, R S 0 is an invariant subspace of A. Remark: Analogous results hold for lower triangular matrices and block-triangular matrices. Fact 7.10.4. Let A ∈ Fn×n. Then, there exist unique matrices B, C ∈ Fn×n such that the following statements hold: i) B is semisimple. ii) C is nilpotent. iii) A = B + C. iv) BC = CB. Furthermore, the following statements hold: v) B = (AD )D = (AD )# = A2AD . vi) C = A − A2AD . vii) mspec(A) = mspec(B). Source: [1401, p. 112], [1451, pp. 181, 189], and [1474, p. 74]. v) is given by xiv) of Proposition 8.2.2; vi) follows from Fact 7.19.5. Remark: This is the S-N decomposition (also called the JordanChevalley decomposition). Fact 7.10.5. Let A ∈ Fn×n. Then, the following statements are equivalent: i) A is similar to a skew-Hermitian matrix. ii) A is semisimple, and spec(A) ⊂ IA . Related: Fact 15.19.12. △ Fact 7.10.6. Let A ∈ Fn×n, and let r = rank A. Then, A is group invertible if and only if there r×r exist a nonsingular matrix B ∈ F and a nonsingular matrix S ∈ Fn×n such that [ ] B 0 −1 A=S S . 0 0 △

Fact 7.10.7. Let A ∈ Fn×n, and let r = rank A. Then, A is range Hermitian if and only if there

exist a nonsingular matrix S ∈ F

n×n

and a nonsingular matrix B ∈ Fr×r such that [ ] B 0 ∗ A=S S. 0 0

Remark: S need not be unitary for sufficiency. See Corollary 7.5.4. Source: Use the QR decom-

△ ˆ where Sˆ is unitary and R is upper triangular. See [2658]. position Fact 7.17.11 to let S = SR, Fact 7.10.8. Let A ∈ Fn×n. Then, there exists involutory S ∈ Fn×n such that AT = SAS T. Source: [920] and [1215]. Remark: Note AT rather than A∗. Remark: Unitary similarity between a matrix and its transpose is discussed in [1144]. Fact 7.10.9. Let A ∈ Fn×n. Then, there exists a nonsingular matrix S ∈ Fn×n such that A =

571

MATRIX DECOMPOSITIONS

SA∗S −1 if and only if there exist Hermitian matrices S 1 , S 2 ∈ Fn×n such that A = S 1 S 2 . Source: [2983, pp. 215, 216] and [2991, pp. 262, 263]. Remark: For normal A, an analogous result in Hilbert space is given in [1531]. Related: Corollary 6.3.12, Corollary 7.4.9, Proposition 7.7.13, and Fact 7.17.26. Fact 7.10.10. Let A ∈ Fn×n, and assume that A is normal. Then, there exists a symmetric, nonsingular matrix S ∈ Fn×n such that AT = SAS −1 and such that S −1 = S. Source: For F = C, let △ A = UBU ∗, where U is unitary and B is diagonal. Then, AT = SAS = SAS −1, where S = UU −1. For △ △ ˜ T, where U is orthogonal and I˜ = diag(I,ˆ . . . , I). ˆ F = R, use the real normal form and let S = UIU Related: Corollary 7.4.9. Fact 7.10.11. Let A ∈ Rn×n, and assume that A is normal. Then, there exists a reflector S ∈ Rn×n such that AT = SAS −1. Consequently, A and AT are orthogonally similar. Finally, if A is skew symmetric, then A and −A are orthogonally similar. Source: Specialize Fact 7.10.10 to the case F = R. Fact 7.10.12. Let A ∈ Fn×n. Then, there exists a reverse-symmetric, nonsingular matrix S ∈ ˆ Fn×n such that AT = SAS −1. Source: Corollary 7.4.9 and [1775]. Fact 7.10.13. Let A ∈ Fn×n. Then, there exist reverse-symmetric matrices S 1 , S 2 ∈ Fn×n such that S 2 is nonsingular and A = S 1 S 2 . Source: Corollary 7.4.10 and [1775]. Fact 7.10.14. Let A ∈ Rn×n, and assume that A is not of the form aI, where a ∈ R. Then, A is similar to a matrix with diagonal entries 0, . . . , 0, tr A. Source: [2263, p. 77]. Credit: P. M. Gibson. Fact 7.10.15. Let A ∈ Rn×n, and assume that A is not zero. Then, A is similar to a matrix whose diagonal entries are all nonzero. Source: [2263, p. 79]. Credit: M. Marcus and R. Purves. Fact 7.10.16. Let A ∈ Rn×n, and assume that A is symmetric. Then, there exists an orthogonal matrix S ∈ Rn×n such that −1 < spec(S ) and SAS T is diagonal. Source: [2263, p. 101]. Credit: P. L. Hsu. Fact 7.10.17. Let A ∈ Rn×n, and assume that A is symmetric. Then, there exist a diagonal matrix B ∈ Rn×n and a skew-symmetric matrix C ∈ Rn×n such that A = [2(I + C)−1 − I]B[2(I + C)−1 − I]T. Source: Fact 7.10.16 and [2263, p. 101]. Fact 7.10.18. Let A ∈ Fn×n. Then, there exists a unitary matrix S ∈ Fn×n such that S ∗AS has equal diagonal entries. In particular, I⊙(S ∗AS ) = n1 (tr A)I. Source: Fact 7.10.19, [1048], and [2263, p. 78]. Credit: W. V. Parker. See [1130]. Fact 7.10.19. Let A ∈ Fn×n. Then, the following statements are equivalent:

i) tr A = 0. ii) There exist B, C ∈ Fn×n such that A = [B, C]. iii) A is unitarily similar to a matrix whose diagonal entries are zero. Source: [29, 1130], [1301, p. 146], and [1596, 1625]. Remark: This is Shoda’s theorem. Related: Fact 7.10.20. Fact 7.10.20. Let R ∈ Fn×n, and assume that R is Hermitian. Then, the following statements are equivalent: i) tr R < 0. ii) R is unitarily similar to a matrix all of whose diagonal entries are negative. iii) There exists an asymptotically stable matrix A ∈ Fn×n such that R = A + A∗. Source: [266]. Related: Fact 7.10.19. Fact 7.10.21. Let A ∈ Fn×m. Then, there exists S ∈ Fn×m such that S ∗S = Im and such that AA∗ =

572

CHAPTER 7

SA∗AS ∗ . Now, assume that n = m. Then, AA∗ and A∗A are unitarily similar. Source: Fact 10.8.4. Fact 7.10.22. Let A ∈ Fn×n. Then, A is symmetric if and only if there exists a unitary matrix △ S ∈ Fn×n such that A = SBS T, where B = diag[σ1 (A), . . . , σn (A)]. Source: [1451, pp. 153, 263]. Remark: A is symmetric, complex, and T-congruent to B. Remark: This is the Autonne-Takagi factorization. Fact 7.10.23. Let A ∈ Fn×n. Then, [ ] [ ][ ] [ ] 1 I I 0 A 1 I −I A 0 = √ . √ 0 −A 2 −I I A 0 2 I I [ ] [ ] 0 and 0 A are unitarily similar. Related: Fact 4.13.31. Hence, A0 −A A 0 Fact 7.10.24. Let n be a positive integer. Then,      I  ȷIn I  0  ∗ 1 △ J2n = S   .  S , S = √  0 − ȷIn 2 ȷI − ȷI Hence, mspec(J2n ) = { ȷ, − ȷ, . . . , ȷ, − ȷ}ms , and det J2n = 1. Source: Fact 3.24.6. Remark: By Fact 4.28.3, J2n is Hamiltonian, and thus, by Fact 6.9.20, mspec(J2n ) = − mspec(J2n ). Remark: S is unitary. See Fact 4.13.31. Fact 7.10.25. Let n be a positive integer. Then,        In/2   0  T 1  In/2 −Iˆn/2         n even, S  n even, S ,  ,   √  ˆ       0 −In/2  2 In/2 In/2           △     S = Iˆn =    I(n−1)/2 0  I(n−1)/2 0 −Iˆ(n−1)/2  0       √       1          S  0 1 0  S T, n odd, 2 0  , n odd. √  0              2  Iˆ   0 0 −I 0 I (n−1)/2

Therefore,

(n−1)/2

(n−1)/2

   n even,  {−1, 1, . . . , −1, 1}ms , mspec(Iˆn ) =    {1, −1, 1, . . . , −1, 1}ms , n odd.

Remark: For even n, Fact 4.28.3 implies that Iˆn is Hamiltonian, and thus, by Fact 6.9.20, mspec(Iˆn ) = − mspec(Iˆn ). See [2844]. Remark: S is orthogonal. See Fact 4.13.31. △ Fact 7.10.26. Let A ∈ Fn×n, assume that A is idempotent, and let r = rank A. Then, there exist

a unitary matrix S ∈ Fn×n and positive numbers a1 , . . . , ak such that ([ ] [ ] ) 1 a1 1 ak A = S diag ,..., , Ir−k , 0(n−r−k)×(n−r−k) S ∗. 0 0 0 0

Source: [919]. Remark: This result provides a canonical matrix for idempotent matrices under unitary similarity. Remark: See [1134]. △ △ Fact 7.10.27. Let A ∈ Fn×n be nonzero, define r = rank A, define B = diag[σ1 (A), . . . , σr (A)],

let S, T ∈ Fn×n be unitary matrices such that   B A = S  0(n−r)×r

 0r×(n−r)   T,

0(n−r)×(n−r)

573

MATRIX DECOMPOSITIONS △

and define K ∈ Fr×r and L ∈ Fr×(n−r) by [K L] = [Ir 0r×(n−r) ]TS. Then,    ∗  BK BL ∗ ∗  S .  KK + LL = Ir , A = S  0(n−r)×r 0(n−r)×(n−r) Furthermore, the following statements hold: i) A is group invertible if and only if K is nonsingular. ii) A is idempotent if and only if BK = Ir . iii) A is tripotent if and only if BK is involutory. iv) A2 = 0 if and only if K = 0. v) A is range Hermitian if and only if L = 0. vi) A is range disjoint if and only if rank L = r. vii) A is range spanning if and only if rank L = n − r. viii) A is Hermitian if and only if L = 0 and BK = K ∗B. ix) A is normal if and only if L = 0 and BK = KB. x) A is a projector if and only if B = K = Ir . xi) A is a partial isometry if and only if B = Ir . xii) Let k ≥ 1. Then, Ak = A if and only if (BK)k−1 = Ir . xiii) A2 = 0 if and only if K = 0. xiv) A2 = A∗ if and only if L = 0 and B = K 3 = Ir . xv) A4 = A if and only if (BK)3 = Ir . xvi) A2 = A+ if and only if L = 0 and (BK)3 = Ir . xvii) A is semicontractive if and only if B is semicontractive. xviii) A is contractive if and only if B is contractive. xix) A is Hermitian and tripotent if and only if L = 0, B = Ir , and K is Hermitian. xx) N(A) ⊆ R(A) if and only if L∗ L = In−r . xxi) R(A) ⊆ N(A) if and only if LL∗ = Ir . xxii) The following statements are equivalent: a) R(A) ∩ N(A) = {0}; b) R(L) ⊆ R(K); c) rank K = r. Source: [234, 236, 240, 247, 257, 1338]. Remark: L = 0 if and only if K is unitary. Related: Fact 8.3.23, Fact 8.5.13, and Fact 8.5.5. △ Fact 7.10.28. Let A ∈ Fn×n, assume that A is idempotent, and let r = rank A. Then, there exists r×(n−r) n×n B∈F and a unitary matrix S ∈ F such that [ ] Ir B A=S S ∗. 0 0(n−r)×(n−r) Source: Use Fact 7.10.27. See [1133, p. 46]. Fact 7.10.29. Let A ∈ Fn×n, assume that A is unitary, and partition A as

[

A=

A11 A21

] A12 , A22

where A11 ∈ Fm×k, A12 ∈ Fm×q, A21 ∈ F p×k, A22 ∈ F p×q, and m + p = k + q = n. Then, there exist

574

CHAPTER 7

unitary matrices U, V ∈ Fn×n and nonnegative integers l, r such that   0 0 0 0   Ir 0     0 Γ 0 0 Σ 0     0 0 0 0 0 Im−r−l    V, A = U   0 0 0 Iq−m+r 0 0      0 Σ 0 0 −Γ 0     0 0 Ik−r−l 0 0 0 where Γ, Σ ∈ Rl×l are diagonal and satisfy 0 < Γ(l,l) ≤ · · · ≤ Γ(1,1) < 1,

0 < Σ(1,1) ≤ · · · ≤ Σ(l,l) < 1,

Γ 2 + Σ 2 = Im .

Source: [1133, p. 12] and [2539, p. 37]. Remark: This is the CS decomposition. See [2183, 2185].

The diagonal entries of Σ and Γ can be interpreted as the sines and cosines, respectively, of the principal angles between a pair of subspaces S1 = R(X1 ) and S2 = R(Y1 ) such that [X1 X2 ] and [Y1 Y2 ] are unitary and A = [X1 X2 ]∗ [Y1 Y2 ]; see [1133, pp. 25–29], [1134, 1595], [2539, pp. 40–43], [2847], and Fact 3.12.21. Related: Fact 7.10.30 and Fact 7.13.27. △ Fact 7.10.30. Let A ∈ Fn×n , define r = rank A, let P ∈ Fn×n and Q ∈ Fn×n be the projectors ∗ onto R(A) and R(A ), respectively, let k ≥ 0 be the multiplicity of the singular value 1 of PQ, define △ △ l = r − rank PQ ≥ 0, and, for all i ∈ {1, . . . , r}, define θi = acos σi (PQ). Then, the following statements hold: i) dim[R(A) ∩ R(A∗ )] = k. ii) dim[R(A) ∩ N(A)] = l. iii) def PQ = n + l − r. iv) k ≤ rank PQ = r − l ≤ r. v) If k = 0, then 1 > σ1 (PQ) ≥ · · · ≥ σr−l (PQ) > σr−l+1 (PQ) = · · · = σn (PQ) = 0. vi) If 1 ≤ k < r, then 1 = σ1 (PQ) = · · · = σk (PQ) > σk+1 (PQ) ≥ · · · ≥ σr−l (PQ) > σr−l+1 (PQ) = · · · = σn (PQ) = 0. vii) If k = r, then l = 0 and 1 = σ1 (PQ) = · · · = σr (PQ) > σr+1 (PQ) = · · · = σn (PQ) = 0. viii) θ1 = · · · = θk = 0, θk+1 , . . . , θr−l ∈ (0, π2 ), and θr−l+1 = · · · = θr = π2 . ix) σmax (P − Q) = max {sin θi : i ∈ {1, . . . , r}}. x) There exist a unitary matrix U ∈ Fn×n and a nonsingular matrix M ∈ Fr×r such that [ ] MC MS A=U U ∗, 0(n−r)×r 0(n−r)×(n−r) △

xi) xii) xiii) xiv) xv) xvi)

△

where C = diag(0l×l , cos θk+1 , . . . , cos θr−l , Ik ) ∈ Rr×r and S = diag(sin θk+1 , . . . , sin θr−l , Il , 0k×(n+k−2r) ) ∈ Rr×(n−r) . Furthermore, C 2 + S S ∗ = Ir . A is a partial isometry if and only if M is unitary. A is idempotent if and only if CM = Ir . A is a projector if and only if θ1 = · · · = θr = 0 and M = Ir . A is range Hermitian if and only if θ1 = · · · = θr = 0. A is normal if and only if θ1 = · · · = θr = 0 and M is normal. The following conditions are equivalent: a) A is group invertible.

575

MATRIX DECOMPOSITIONS

b) l = 0. c) C is nonsingular. If these statements hold, then

[

(MC)−1 A =U 0(n−r)×r #

] (CMC)−1S U ∗. 0(n−r)×(n−r)

xvii) A is (contractive, semicontractive) if and only if M is. Source: [402, 404]. Remark: θ1 , . . . , θr are the principal angles between the subspaces R(A) and R(A∗ ). Related: Fact 7.12.42. △ △ ∏ Fact 7.10.31. Let A ∈ Fn×n, and define r = rank A and α = [ ri=1 σi (A)]1/r . Then, there exist

S 1 ∈ Fn×r, B ∈ Rr×r, and S 2 ∈ Fr×n such that S 1 is left inner, S 2 is right inner, B is upper triangular, I ⊙ B = αI, and A = S 1BS 2 . Source: [1534]. Remark: B is real. Remark: This is the geometric mean decomposition. Fact 7.10.32. Let A ∈ Fn×n. Then, there exist upper triangular matrices S 1 ∈ Fn×n and S 2 ∈ Fn×n and a permutation matrix B ∈ Rn×n such that A = S 1BS 2 . Source: [2336, pp. 54, 55]. Remark: This is the Bruhat decomposition. Fact 7.10.33. Let A ∈ Cn×n. Then, there exists B ∈ Rn×n such that AA and B2 are similar. Source: [913].

7.11 Facts on Matrix Transformations for Two or More Matrices Fact 7.11.1. Let A ∈ Rn×n , let λ ∈ spec(A), and assume that λ < R. Then, the number and size

of the Jordan blocks associated with λ in the Jordan form of A are equal to the number and size of the Jordan blocks associated with λ¯ in the Jordan form of A. △ Fact 7.11.2. Let q(s) = s2 − β1 s − β0 ∈ R[s] be irreducible, and let λ = ν + ω ȷ denote a root of q so that β1 = 2ν and β0 = −(ν2 + ω2 ). Then, [ ] [ ][ ][ ] 0 1 1 0 ν ω 1 0 H1 (q) = = = S J1(q)S −1. β0 β1 ν ω −ω ν −ν/ω 1/ω [ ] [ ν ] The transformation matrix S = 1ν ω0 is not unique; an alternative choice is S = ω0 ν2 +ω 2 . Furthermore,     0   ν ω 1  0 1 0 0    −ω ν  β β 0 1  −1 1 0  1  S = S J2 (q)S −1,  = S  H2 (q) =  0 0 ν ω   0  0 0 0 1  0 0 −ω ν 0 0 β0 β1 where

  ω  0 △  S =   0 0

ν ν 2 + ω2 0 0

 ω ν  ω ν2 + ω2 + ν   . −2ων 2ω2  −2ω(ν2 + ω2 ) 0

△

Fact 7.11.3. Let q(s) = s2 − 2νs + ν2 + ω2 ∈ R[s] with roots λ = ν + ω ȷ and λ = ν − ω ȷ. Then,

[

] ν ω H1(q) = = −ω ν   ν ω  −ω ν H2 (q) =  0  0 0 0

] [ ] λ 0 1 1 −ȷ , √ 0 λ 2 1 ȷ    1 0   λ 1 0 0     0 λ 0 0  −1 0 1    = S   0 0 λ 1  S , ν ω    −ω ν 0 0 0 λ [ 1 1 √ 2 ȷ

1 −ȷ

][

576

where

CHAPTER 7

  1 0 1  ȷ 0 △ S = √  2  0 1 0 ȷ

 1 0   − ȷ 0  , 0 1  0 −ȷ

S −1

  1 − ȷ 1  0 0 = √  2  1 ȷ 0 0

0 1 0 1

 0   − ȷ  . 0  ȷ

Fact 7.11.4. Left equivalence, right equivalence, biequivalence, unitary left equivalence, unitary right equivalence, and unitary biequivalence are equivalence relations on Fn×m. Similarity, congruence, and unitary similarity are equivalence relations on Fn×n. n×m Fact 7.11.5. Let A, B ∈ Fn×m. Then, A and B are in the same equivalence[ class induced ] of F I 0 by biequivalent transformations if and only if A and B are biequivalent to 0 0 . Now, let n = m. Then, A and B are in the same equivalence class of Fn×n induced by similarity transformations if and only if A and B have the same Jordan form. Fact 7.11.6. Let A, B ∈ Cn×n . Then, the following statements are equivalent: i) A and B are similar. ii) spec(A) = spec(B), and, for all λ ∈ spec(A) and i ∈ {1, . . . , n}, def[(λI−A)i ] = def[(λI−B)i ]. Source: [2166, pp. 63–65]. Fact 7.11.7. Let A, B ∈ Fn×n, and assume that A and B are similar. Then, A is semisimple if and only if B is. Fact 7.11.8. Let A ∈ Fn×n, and assume that A is normal. Then, A is unitarily similar to its Jordan form. Related: Fact 7.11.10. Fact 7.11.9. Let A, B ∈ Rn×n. Then, the following statements are equivalent: i) There exists a nonsingular matrix S ∈ Cn×n such that A = SBS −1 . ii) There exists a nonsingular matrix S ∈ Rn×n such that A = SBS −1 . Furthermore, the following statements are equivalent: iii) There exists an orthogonal matrix S ∈ Cn×n such that A = SBS −1 . iv) There exists a unitary matrix S ∈ Rn×n such that A = SBS −1 . Source: [2979, pp. 181, 182]. Fact 7.11.10. Let A, B ∈ Fn×n, and assume that A and B are normal. Then, the following statements are equivalent: i) mspec(A) = mspec(B). ii) A and B are similar. iii) A and B are unitarily similar. Source: Since A and B are similar, it follows that mspec(A) = mspec(B). Since A and B are normal, it follows that they are unitarily similar to the same diagonal matrix. See [1140, p. 8], [1302, p. 104], and [2991, p. 303]. Related: Fact 7.11.8. Fact 7.11.11. Let A, B ∈ Fn×n. Then, the following statements are equivalent: i) A and B are unitarily similar. √ ∑ △ n ii) For all k1, . . . , kr , l1, . . . , lr ∈ N such that ri, j=1 (ki + l j ) ≤ r, where r = 2n2 /(n − 1) + 1/4 + n/2 − 2, it follows that

tr Ak1Al1 ∗ · · · AkrAlr ∗ = tr Bk1Bl1 ∗ · · · BkrBlr ∗. Source: [481], [1451, pp. 97, 98], [1580, pp. 71, 72], and [2209, 2429]. Remark: This is Specht’s theorem. Remark: In the case n = 2, it suffices to check three equalities, specifically, tr A = tr B, tr A2 = tr B2, and tr A∗A = tr B∗B. In the case n = 3, it suffices to check 7 equalities. See [481], [1451, p. 98], and [2429]. Related: Fact 7.11.15.

577

MATRIX DECOMPOSITIONS

Fact 7.11.12. Let A, B ∈ Fn×n, assume that A and B are idempotent, assume that ρmax (A−B) < 1,

and define

△

S = (AB + A⊥B⊥ )[I − (A − B)2 ]−1/2.

Then, the following statements hold: i) S is nonsingular. ii) If A = B, then S = I. iii) S −1 = (BA + B⊥A⊥ )[I − (B − A)2 ]−1/2. iv) A and B are similar. In fact, A = SBS −1. v) If A and B are projectors, then S is unitary and A and B are unitarily similar. Source: [1399, p. 412]. Remark: [I − (A − B)2 ]−1/2 is defined by ix) of Fact 13.4.15. Fact 7.11.13. Let A, B ∈ Fn×n, and assume that A and B are idempotent. Then, the following statements are equivalent: i) A and B are unitarily similar. ii) tr A = tr B and, for all i ∈ {1, . . . , ⌊n/2⌋}, tr (AA∗ )i = tr (BB∗ )i. iii) χAA∗ = χBB∗. Source: Fact 7.10.26 and [919]. Fact 7.11.14. Let A, B ∈ Fn×n, and assume that either A or B is nonsingular. Then, AB and BA are similar. Source: If A is nonsingular, then AB = A(BA)A−1, whereas, if B is nonsingular, then BA = B(AB)B−1. Fact 7.11.15. Let A, B ∈ Fn×n, and assume that A and B are projectors. Then, AB and BA are unitarily similar. Source: [2291]. The result follows from Fact 7.11.11. ∗ Fact 7.11.16. Let A ∈ Fn×n, and assume that A[ is ]idempotent. [ ] Then, A and A are unitarily 1 0 1 a similar. Source: Use Fact 7.10.26 and the fact that 0 0 and a 0 are unitarily similar. See [919]. Alternatively, the result follows from Fact 7.11.15 and Fact 8.7.2. Fact 7.11.17. Let A, B ∈ Fn×n, where A and B are Hermitian. Then, AB and BA are similar. Source: [2991, p. 264]. Fact 7.11.18. Let A ∈ Fn×n. Then, A is idempotent if and only if there exists a projector B ∈ Fn×n such that A and B are similar. Fact 7.11.19. Let A, B ∈ Fn×n, assume that A and B are idempotent, and assume that A + B − I is nonsingular. Then, A and B are similar. In particular, A = (A + B − I)−1B(A + B − I). Fact 7.11.20. Let A1 , . . . , Ar ∈ Fn×n, and assume that, for all i, j ∈ {1, . . . , r}, Ai Aj = Aj Ai . Then,  r      ∏ ni  dim span  Ai : 0 ≤ ni ≤ n − 1 for all i ∈ {1, . . . , r} ≤ ⌊n2 /4⌋ + 1.     i=1

Source: [1582, 1722] and [2166, p. 114]. Remark: This result bounds the dimension of a commutative subalgebra. Credit: I. Schur. Fact 7.11.21. Let A, B ∈ Fn×n, and assume that AB = BA. Then,

dim span {AiB j : 0 ≤ i ≤ n − 1, 0 ≤ j ≤ n − 1} ≤ n. Source: [306, 1432, 1722] and [2166, p. 219]. Remark: This result bounds the dimension of a commutative subalgebra. Credit: M. Gerstenhaber. △ Fact 7.11.22. Let n ≥ 2, let A, B ∈ Cn×n, define S0 (A, B) = {αI : α ∈ C}, for all k ≥ 1, define △

△

Sk (A, B) = span {X1 · · · Xk : X1 , . . . , Xk ∈ {A, B, I}}, and define ℓ(A, B) = maxk≥0 dim Sk (A, B). Then, the following statements hold: i) There exist C, D ∈ Cn×n such that ℓ(C, D) = 2n − 2.

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CHAPTER 7

ℓ(A, B) ≤ ⌈(n2 + 2)/3⌉. If AB = BA, then ℓ(A, B) ≤ n − 1. If either A or B is cyclic, then ℓ(A, B) ≤ 3n − 3. If either A or B is simple, then ℓ(A, B) ≤ 2n − 2. If n ≤ 6, then ℓ(A, B) ≤ 2n − 2. Source: [1727, 2197, 2208]. Remark: ∪∞ i=0 R[Wi (A, B)] is the unital algebra generated by A, B. Remark: The bound 2n − 2 is conjectured to hold for all n ≥ 2 and all A, B ∈ Cn×n. See [1727]. Fact 7.11.23. Let A, B ∈ Fn×m. Then, the following statements hold: i) The matrices A and B are unitarily left equivalent if and only if A∗A = B∗B. ii) The matrices A and B are unitarily right equivalent if and only if AA∗ = BB∗. iii) The matrices A and B are unitarily biequivalent if and only if A and B have the same singular values with the same multiplicity. Source: [1455] and [2314, pp. 372, 373]. Remark: In [1455] A and B need not be the same size. Remark: The singular value decomposition provides a canonical matrix under unitary biequivalence in analogy with the Smith form under biequivalence. Remark: Note that AA∗ = BB∗ implies that R(A) = R(B), which implies right equivalence. This is an alternative proof of the fact that unitary right equivalence implies right equivalence. Fact 7.11.24. Let A, B ∈ Fn×n. Then, the following statements hold: i) A∗A = B∗B if and only if there exists a unitary matrix S ∈ Fn×n such that A = SB. ii) A∗A ≤ B∗B if and only if there exists S ∈ Fn×n such that A = SB and S ∗S ≤ I. iii) A∗B + B∗A = 0 if and only if there exists a unitary matrix S ∈ Fn×n such that (I − S )A = (I + S )B. iv) A∗B + B∗A ≥ 0 if and only if there exists S ∈ Fn×n such that (I − S )A = (I + S )B and S ∗S ≤ I. Source: [1448, p. 406] and [2298]. Remark: iii) and iv) follow from i) and ii) by replacing A and B with A − B and A + B, respectively. Fact 7.11.25. Let A ∈ Fn×n, B ∈ Fm×m, and C ∈ Fn×m. Then, there exist X, Y ∈ Fn×m satisfying ii) iii) iv) v) vi)

AX + YB + C = 0 [

if and only if rank

A 0

] [ ] 0 A C = rank . B 0 B

Source: [2263, pp. 194, 195] and [2835]. Remark: AX + YB + C = 0 is a generalization of Sylvester’s equation. See Fact 7.11.26. Remark: An explicit expression for all solutions is given

by Fact 8.9.10, which applies to the case where A and B are not necessarily square and thus X and Y are not necessarily the same size. Credit: W. E. Roth. Fact 7.11.26. Let A ∈ Fn×n, B ∈ Fm×m, and C ∈ Fn×m. Then, there exists X ∈ Fn×m satisfying

if and only if

[

A 0 0 −B

]

and

[

A C 0 −B

]

[

AX + XB + C = 0

are similar. If these conditions hold, then ] [ ][ ][ ] A C I X A 0 I −X = . 0 −B 0 I 0 −B 0 I

Source: [2835] and [2979, pp. 41, 42]. For sufficiency, see [1738, pp. 422–424] and [2263, pp. 194, 195]. Remark: AX + XB + C = 0 is Sylvester’s equation. See Proposition 9.2.5, Corollary 9.2.6, and Proposition 15.10.3 for the case where X is unique. Credit: W. E. Roth. See [478].

579

MATRIX DECOMPOSITIONS

[

n×n Fact 7.11.27. ] [ Let A,] B ∈ F , and assume that A and B are idempotent. Then, the matrices

A+B A 0 −A−B

and

A+B 0 0 −A−B

[

are similar and satisfy ] [ ][ ][ ] A+B −A I X A+B 0 I −X = , 0 −A − B 0 I 0 −A − B 0 I

△

where X = 14 (I + A − B). Credit: Y. Tian. Related: Fact 7.11.26. Fact 7.11.28. [Let ]A ∈ F[n×n, ]B ∈ Fm×m, and C ∈ Fn×m, and assume that A and B are nilpotent. Then, the matrices A0 CB and A0 B0 are similar if and only if [ ] A C rank = rank A + rank B, AC + CB = 0. 0 B Source: [2643]. [ ] [ 0 0 Fact 7.11.29. Let A ∈ Fn×m and B ∈ Fm×n . Then, AB B 0 and B

0 BA

]

are similar. Source: [2979,

p. 32]. Fact 7.11.30. Let A, B, C ∈ Fn×n, assume that spec(A) ∩ spec(B) = ∅, and assume that [A + B, C] = 0 and [AB, C] = 0. Then, [A, C] = [B, C] = 0. Source: This result follows from Corollary 9.2.6. Credit: M. R. Embry. See [478].

7.12 Facts on Eigenvalues and Singular Values for One Matrix Fact 7.12.1. Let A ∈ Fn×n, and assume that A is singular. If A is either simple or cyclic, then

rank A = n − 1.

Fact 7.12.2. Let A ∈ Rn×n, and assume that A ∈ SO(n). Then, amultA (−1) is even. Now, assume

that n = 3. Then, the following statements hold: i) Either amultA (1) = 1 or amultA (1) = 3. ii) tr A ≥ −1. iii) tr A = −1 if and only if mspec(A) = {1, −1, −1}ms . Fact 7.12.3. Let A ∈ Fn×n, let α ∈ F, and assume that A2 = αA. Then, spec(A) ⊆ {0, α}. Fact 7.12.4. Let A ∈ Fn×n, assume that A is Hermitian, and let α ∈ R. Then, A2 = αA if and only if spec(A) ⊆ {0, α}. Related: Fact 4.10.22. Fact 7.12.5. Let A ∈ Fn×n . Then, the following statements are equivalent: i) A2 = −A. ii) rank A = − tr A and rank(A + I) = n + tr A. Source: [2238, p. 530] and [2714]. Fact 7.12.6. Let A ∈ Fn×n, assume that A is Hermitian, let x1 , . . . , xn ∈ Fn be eigenvectors associated with λ1 (A), . . . , λn (A), respectively, let k, l ∈ {1, . . . , n}, where k ≤ l, and let x ∈ span {xk , . . . , xl }, where x , 0. Then, x∗Ax ≤ λk (x). x∗ x Source: [2991, p. 266]. Related: This result implies Lemma 10.4.3. Fact 7.12.7. Let A ∈ Fn×n, assume that A is Hermitian, let k ∈ {1, . . . , n}, and let W denote the set of subspaces of Fn whose dimension is k. Then, λl (A) ≤

λk (A) = max

min

S∈W {x∈S: x,0}

x∗Ax x∗Ax = max min . ∗ ⊥ S ∈W {x∈S: x,0} x∗ x xx

Source: [2991, pp. 268, 269]. Remark: This is the min-max theorem. Credit: R. Courant and E.

S. Fischer.

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CHAPTER 7

Fact 7.12.8. Let A ∈ Fn×n, and assume that A is Hermitian. Then,

|x∗Ax| = σmax (A). x∈F \{0} x∗x max n

Furthermore, if A is either positive definite or negative definite, then |x∗Ax| = σmin (A). x∈F \{0} x∗x min n

Finally, A is neither positive definite nor negative definite, then |x∗Ax| = 0. x∈F \{0} x∗x min n

Related: Fact 11.15.1. Fact 7.12.9. Let A ∈ Fn×n, and assume that A is Hermitian. Then,

αmax (A) = λmax (A),

ρmax (A) = σmax (A) = max {|λmin (A)|, λmax (A)}.

If, in addition, A is positive semidefinite, then ρmax (A) = σmax (A) = αmax (A) = λmax (A). Related: Fact 7.13.8. Fact 7.12.10. Let A ∈ Fn×n, and assume that A is skew Hermitian. Then, the eigenvalues of A are imaginary. Source: Let λ ∈ spec(A). Since 0 ≤ AA∗ = −A2, it follows that −λ2 ≥ 0, and thus

λ2 ≤ 0.

Fact 7.12.11. Let A ∈ Fn×n. Then, the following statements are equivalent:

i) ii) iii) iv)

A is idempotent. rank(I − A) ≤ tr(I − A), A is group invertible, and every eigenvalue of A is nonnegative. A and I − A are group invertible, and every eigenvalue of A is nonnegative. A is group invertible, and there exists a positive integer k such that Ak = Ak+1 . Source: [1336]. △ Fact 7.12.12. Let A ∈ Fn×n, and define k = n − amultA (0). Then, 2  k k ∑  ∑ 2     ρ2i (A). | tr A| ≤  ρi (A) ≤ k i=1

i=1

Source: Fact 2.11.17. Fact 7.12.13. Let A ∈ Fn×n, and assume that A has exactly k nonzero eigenvalues. Then,

 | tr A|2    ≤ k tr A∗A ≤ (rank A) tr A∗A. 2 2∗ 2 1/2   k| tr A | ≤ k tr (A A ) 

Furthermore, the upper left-most inequality is an equality if and only if A is normal and all of the nonzero eigenvalues of A have the same absolute value. Moreover, the right-most inequality is an equality if and only if A is group invertible. If, in addition, spec(A) ⊂ R, then { } (rank A) tr A2 (tr A)2 ≤ k tr A2 ≤ ≤ (rank A) tr A∗A. k tr A∗A Source: The upper left-hand inequality in the first string is given in [2905]. The lower left-hand inequality in the first string is given by Fact 11.13.2. In the case where all of the eigenvalues of A are real, the inequality (tr A)2 ≤ k tr A2 follows from Fact 7.12.12. See [2991, p. 264]. Remark:

581

MATRIX DECOMPOSITIONS

| tr A|2 ≤ k| tr A2 | does not necessarily hold. Consider mspec(A) = {1, 1, ȷ, −ȷ}ms . Related: Fact 4.10.22, Fact 10.13.3, Fact 10.13.5, and Fact 10.13.6. Fact 7.12.14. Let A ∈ Rn×n and mspec(A) = {λ1 , . . . , λn }ms . Then, n ∑ (Re λi )(Im λi ) = 0,

tr A2 =

i=1

n n ∑ ∑ (Re λi )2 − (Im λi )2. i=1

i=1

Fact 7.12.15. Let n ≥ 2, let a1 , . . . , an > 0, and define the symmetric matrix A ∈ Rn×n by △

A(i, j) = ai + a j for all i, j ∈ {1, . . . , n}. Then, rank A ≤ 2 and mspec(A) = {λ, µ, 0, . . . , 0}ms , where v v t n t n n n ∑ ∑ ∑ ∑ △ △ 2 λ= ai + n ai , µ = ai − n a2i . i=1

i=1

i=1

i=1

Furthermore, the following statements hold: i) λ > 0. ii) µ ≤ 0. Moreover, the following statements are equivalent: iii) µ < 0. iv) At least two of the numbers a1 , . . . , an > 0 are distinct. v) rank A = 2. ∑ If iii)–v) hold, then λmin (A) = µ < 0 < tr A = 2 ni=1 ai < λmax (A) = λ. Source: A = a11×n + 1n×1 aT, △ where a = [a1 · · · an ]T. Then, Fact 3.14.18 implies that rank A ≤ rank(a11×n ) + rank(1n×1 aT ) = 2. Furthermore, mspec(A) follows from Fact 7.12.16, while Fact 2.11.81 implies that µ ≤ 0. Related: Fact 10.9.11. Fact 7.12.16. Let x, y ∈ Rn. Then, } { √ √ mspec(xyT + yxT ) = xTy + xTxyTy, xTy − xTxyTy, 0, . . . , 0 , ms

αmax (xyT + yxT ) = xTy +

√ xTxyTy,

√  T   xTxyTy, x y +   ρmax (xyT + yxT ) =  √    xTy − xTxyTy ,

xTy ≥ 0, xTy ≤ 0.

Now, assume that x and y are nonzero, and define v1, v2 ∈ Rn by △

v1 =

1 ∥x∥ x

+

1 ∥y∥ y,

△

v2 =

1 ∥x∥ x

−

1 ∥y∥ y.

√ √ T Then, v1 and v2 are eigenvectors of xyT + yxT corresponding xTxyTy and xTy − xTxyTy, ([ 0 1to x])y + √ √ n×1 respectively. Source: [835, p. 539]. Example: mspec 1n×n = {− n, 0, . . . , 0, n}ms . Prob0 1×n lem: Extend this result to C and xyT + zwT. See Fact 6.9.15. Fact 7.12.17. Let A ∈ Fn×n, and let mspec(A) = {λ1 , . . . , λn }ms . Then, mspec[(I + A)2 ] = {(1 + λ1 )2, . . . , (1 + λn )2 }ms . If A is nonsingular, then

−1 mspec(A−1 ) = {λ−1 1 , . . . , λn }ms .

Finally, if I + A is nonsingular, then mspec[(I + A)−1 ] = {(1 + λ1 )−1, . . . , (1 + λn )−1 }ms , mspec[A(I + A)−1 ] = {λ1 (1 + λ1 )−1, . . . , λn (1 + λn )−1 }ms .

582

CHAPTER 7

Source: Fact 7.12.18. △ Fact 7.12.18. Let p, q ∈ F[s], assume that p and q are coprime, and define g = p/q ∈ F(s).

Furthermore, let A ∈ Fn×n, let mspec(A) = {λ1 , . . . , λn }ms , assume that roots(q) ∩ spec(A) = ∅, and △ define g(A) = p(A)[q(A)]−1. Then, mspec[g(A)] = {g(λ1 ), . . . , g(λn )}ms . Source: ii) of Fact 6.9.36 implies that q(A) is nonsingular. Fact 7.12.19. Let x ∈ Fn and y ∈ Fm. Then, √ σmax (xy∗ ) = x∗xy∗ y.

If, in addition, m = n, then mspec(xy∗ ) = {x∗y, 0, . . . , 0}ms , ρmax (xy∗ ) = |x∗y|,

mspec(I + xy∗ ) = {1 + x∗y, 1, . . . , 1}ms , αmax (xy∗ ) = max {0, Re x∗ y}.

Related: Fact 11.8.25. Fact 7.12.20. Let A ∈ Fn×n, and assume that rank A = 1. Then, σmax (A) = (tr AA∗ )1/2. Fact 7.12.21. Let x, y ∈ Fn, and assume that x∗y , 0. Then, σmax [(x∗y)−1xy∗ ] ≥ 1. Fact 7.12.22. Let A ∈ Fn×m and α ∈ F. Then, for all i ∈ {1, . . . , min {n, m}}, σi (αA) = |α|σi (A). Fact 7.12.23. Let A ∈ Fn×m. Then, for all i ∈ {1, . . . , rank A}, σi (A) = σi (A∗ ). Fact 7.12.24. Let A ∈ Fn×n, and let λ ∈ spec(A). Then, the following statements hold:

i) σmin (A) ≤ |λ| ≤ σmax (A). ii) λmin [ 12 (A + A∗ )] ≤ Re λ ≤ λmax [ 12 (A + A∗ )]. iii) λmin [ 21ȷ (A − A∗ )] ≤ Im λ ≤ λmax [ 21ȷ (A − A∗ )]. iv) σn [ 21ȷ (A − A∗ )] ≤ | Im λi | ≤ σ1 [ 21ȷ (A − A∗ )]. Remark: i) is Browne’s theorem, ii) is Bendixson’s theorem, and iii) is Hirsch’s theorem. See [688, p. 17] and [1952, pp. 140–144]. iv) is given in [1173]. Related: Fact 7.12.29, Fact 7.13.9, Fact 11.13.6, and Fact 11.13.7. Fact 7.12.25. Let A ∈ Fn×n. Then, λ[ 12 (A + A∗ )] ≤≤ σ(A). Hence, for all i ∈ {1, . . . , n}, λi [ 21 (A + A∗ )] ≤ σi (A),

−σmax (A) ≤ λi [ 21 (A + A∗ )] ≤ σmax (A).

In particular, λmin [ 12 (A + A∗ )] ≤ σmin (A), Furthermore,

−σmax (A) ≤ λmin [ 12 (A + A∗ )] ≤ λmax [ 12 (A + A∗ )] ≤ σmax (A). w

λ[ 12 (A + A∗ )] ≺ σ(A),

Re tr A = tr[ 21 (A + A∗ )] ≤ tr ⟨A⟩.

Source: [1399, p. 447], [1450, p. 151], [2479], and [2991, pp. 288, 289, 361]. Remark: This result generalizes Re z ≤ |z|, where z ∈ C. Remark: For all i ∈ {1, . . . , n}, |λi [ 12 (A + A∗ )]| ≤ σmax (A).

However, |λi [ 21 (A + A∗ )]| ≤ σi (A) given in [1969, p. 240] and [1971, p. 327] is erroneous. Consider [ ] 0 A = −1 0 0 . Related: Fact 7.12.26, Fact 10.13.18, and Fact 10.21.9. Fact 7.12.26. Let A ∈ Fn×n. Then, w

d(|A|) ≺ σ(A), If, in addition, A is Hermitian, then

w

ρ(A) ≺ σ(A), s

w

ρ[ 21 (A + A∗ )] ≺ σ(A).

d(A) ≺ λ(A).

583

MATRIX DECOMPOSITIONS

Source: Fact 7.12.25, [1969, pp. 228, 240], [1971, pp. 314, 327], [2979, p. 82], and [2991, pp. 349, 351, 361]. Related: Fact 7.12.25, Fact 10.21.10, and Fact 10.21.11. Fact 7.12.27. Let A ∈ Fn×n. Then, s

α(A) ≺ λ[ 21 (A + A∗ )]. In particular, for all i ∈ {1, . . . , n}, λmin [ 12 (A + A∗ )] ≤ αn (A) ≤ αi (A) ≤ αmax (A) = α1 (A) ≤ λmax [ 21 (A + A∗ )]. Furthermore, n n n n ∑ ∑ ∑ ∑ αi (A) = Re tr A = tr 12 (A + A∗ ) = λi [ 12 (A + A∗ )] ≤ | tr A| ≤ ρi (A) ≤ σi (A). i=1

i=1

i=1

i=1

Source: [449, p. 74] and [2991, p. 360]. Credit: K. Fan. Related: Fact 7.12.26, Fact 10.13.2, and

vi) and xvii) of Fact 15.16.7. Fact 7.12.28. Let A ∈ Fn×n. Then, −σmax (A) ≤

λmin [ 12 (A

{

} |αmax (A)| ≤ ρmax (A) + A )] ≤ αmax (A) ≤ ≤ σmax (A). λmax [ 12 (A + A∗ )] ∗

Source: Fact 7.12.25 and Fact 7.12.27. Fact 7.12.29. Let A ∈ Fn×n, and let mspec(A) = {λ1 , . . . , λn }ms . Then, for all k ∈ {1, . . . , n}, k k ∑ ∑ [σ2n−i+1 (A) − |λi |2 ] ≤ 2 (σ2i [ 21ȷ (A − A∗ )] − | Im λi |2 ), i=1

2

i=1

k ∑

k ∑ [σ2i (A) − |λi |2 ],

i=1

i=1

(σ2n−i+1 [ 21ȷ (A − A∗ )] − | Im λi |2 ) ≤

n n ∑ ∑ 2 2 [σi (A) − |λi | ] = 2 (σ2i [ 21ȷ (A − A∗ )] − | Im λi |2 ). i=1

i=1

Source: [1173]. Related: Fact 11.13.6. Fact 7.12.30. Let A ∈ Fn×n. Then, the following statements are equivalent:

i) ii) iii) iv) v)

A is normal. {σ1 (A), . . . , σn (A)}ms = {ρ1 (A), . . . , ρn (A)}ms . ∏ ∏ For all k ∈ {1, . . . , n}, ki=1 σi (A) = ki=1 ρi (A). ∑n ∑n 2 ρi (A) = i=1 σ2i (A). ∑i=1 n 1 2 ∗ 2 i=1 αi (A) = 4 tr (A + A ) . ∑n 2 1 vi) i=1 βi (A) = − 4 tr (A − A∗ )2 . vii) {ρ21 (A), . . . , ρ2n (A)}ms = mspec(A∗A). viii) {α1 (A), . . . , αn (A)}ms = mspec[ 12 (A + A∗ )]. ix) {β1 (A), . . . , βn (A)}ms = mspec[ 21ȷ (A + A∗ )]. x) There exists a permutation σ of (1, . . . , n) such that mspec(AA∗ ) = {λ1 λσ(1) , . . . , λn λσ(n) }ms . s

xi) ρ(A) ≺ σ(A). s

xii) [σ(A)]⊙2 ≺ σ(A2 ). xiii) For all x ∈ Fn , ∥Ax∥2 = ∥A∗ x∥2 .

584

CHAPTER 7

If these statements hold, then the following statements hold: s

xiv) [Re A(1,1) · · · Re A(n,n) ] ≺ [α1 (A) · · · αn (A)]. xv) For all x ∈ Fn such that ∥x∥2 = 1, |x∗Ax|2 ≤ 12 (∥Ax∥22 + |x∗A2 x|) ≤ ∥Ax∥22 . Source: [940, 987, 1242] and [2991, pp. 294, 355, 361]. Related: Fact 4.10.12, Fact 4.13.6, Fact 7.15.16, Fact 10.21.10, and Fact 11.13.6. Fact 7.12.31. Let A ∈ Fn×n, assume that A is normal, let α, β ∈ F, and let p ∈ [1, ∞). If p ∈ [1, 2], then p p p [(|α| + |β|) p + ||α| − |β|| p ]σmax (A) ≤ σmax (αA + βA∗ ) + σmax (αA − βA∗ ).

If p ∈ [2, ∞), then p p p 2(|α| p + |β| p )σmax (A) ≤ σmax (αA + βA∗ ) + σmax (αA − βA∗ ).

Source: [940]. slog

Fact 7.12.32. Let A ∈ Fn×n. Then, ρ(A) ≺ σ(A). Therefore,

| det A| =

n ∏

ρi (A) =

i=1

and, for all k ∈ {1, . . . , n},

n ∏

n ∏

σi (A),

i=1

σi (A) ≤

i=k

n ∏

ρi (A).

i=k

In particular, σmin (A) ≤ ρmin (A). Source: Fact 3.25.15, [449, p. 43], [1399, p. 445], [1450, p. 171], [2977, p. 19], and [2991, p. 353]. Credit: H. Weyl. Related: Fact 10.22.28 and Fact 11.15.20. Fact 7.12.33. Let A ∈ Fn×n. Then, (n−1)/n (n−1)/n 1/n 1/n σmin (A) ≤ σmax (A)σmin (A) ≤ ρmin (A) ≤ ρmax (A) ≤ σmin (A)σmax (A) ≤ σmax (A), n n−1 n−1 n σmin (A) ≤ σmax (A)σmin (A) ≤ | det A| ≤ σmin (A)σmax (A) ≤ σmax (A).

Source: Fact 7.12.32 and [1399, p. 445]. Related: Fact 10.15.1 and Fact 15.21.19. Fact 7.12.34. Let β0 , . . . , βn−1 ∈ F, define A ∈ Fn×n by

        △  A =       

△

and define α = 1 +

0

0

0

0

0

··· .. . .. .

0

0

.. .

.. .

..

.

..

.

.. .

0

0

0

···

0

1

−β0

−β1

−β2

···

−βn−2

−βn−1

0

1

0

0

0

1

0

0

.. .

         ,     

∑n−1

|βi |2. Then, σ2 (A) = · · · = σn−1(A) = 1 and √ ( √ ( ) ) √ √ 1 1 2 2 2 2 α + α − 4|β0 | , σn (A) = α − α − 4|β0 | . σ1 (A) = 2 2 i=0

In particular, σ1(Nn ) = · · · = σn−1 (Nn ) = 1 and σmin (Nn ) = 0. Source: [1389, p. 523] and [1601, 1628]. Related: Fact 8.3.31 and Fact 15.21.19.

585

MATRIX DECOMPOSITIONS

Fact 7.12.35. Let β ∈ C. Then,

([

σmax

1 0

2β 1

])

= |β| +

√

([ 1 + |β|2 ,

σmin

1 2β 0 1

]) =

√

1 + |β|2 − |β|.

Source: [1801]. Remark: Singular-value inequalities for block-triangular matrices are given in

[1801]. Fact 7.12.36. Let A ∈ Fn×m. Then,

([

σmax

I 2A 0 I

]) = σmax (A) +

√

2 1 + σmax (A).

Source: [1389, p. 116]. Fact 7.12.37. For all i ∈ {1, . . . , l}, let Ai ∈ Fni ×mi. Then,

σmax [diag(A1 , . . . , Al )] = max {σmax (A1 ), . . . , σmax (Al )}. △

Fact 7.12.38. Let A ∈ Fn×m, and define r = rank A. Then, for all i ∈ {1, . . . , r},

λi (AA∗ ) = λi (A∗A) = σi (AA∗ ) = σi (A∗A) = σ2i (A). In particular,

2 σmax (AA∗ ) = σmax (A),

and, if n = m, then

2 σmin (AA∗ ) = σmin (A).

Furthermore, for all i ∈ {1, . . . , r},

σi (AA∗A) = σ3i (A).

Related: Fact 10.10.24. Fact 7.12.39. Let A ∈ Fn×n. Then, for all i ∈ {1, . . . , n},

σi (AA ) =

n ∏

σj (A).

j=1 j,n+1−i

Source: Fact 6.10.13 and [2263, p. 149]. Fact 7.12.40. Let A ∈ Fn×n. Then, σ1 (A) = σn (A) if and only if there exist λ ∈ F and a unitary matrix B ∈ Fn×n such that A = λB. Source: [2263, pp. 149, 165]. Fact 7.12.41. Let A ∈ Fn×n, and assume that A is idempotent. Then, the following statements

hold: i) ii) iii) iv) v)

If σ is a singular value of A, then either σ = 0 or σ ≥ 1. If A , 0, then σmax (A) ≥ 1. σmax (A) = 1 if and only if A is a projector. If 1 ≤ rank A ≤ n − 1, then σmax (A) = σmax (A⊥ ). If A , 0, then σmax (A) = σmax (A + A∗ − I) = σmax (A + A∗ ) − 1, 2 σmax (I − 2A) = σmax (A) + [σmax (A) − 1]1/2.

Source: [1134, 1469, 1504]. iv) is given in [1133, p. 61] and follows from Fact 7.12.42. Problem:

Use Fact 7.10.28 to prove iv).

586

CHAPTER 7

Fact 7.12.42. Let A ∈ Fn×n, assume that A is idempotent, and assume that 1 ≤ rank A ≤ n − 1.

Then, σmax (A) = σmax (A + A∗ − I) =

1 , sin θ

where θ ∈ (0, π2 ] is defined by cos θ = max {|x∗ y| : (x, y) ∈ R(A) × N(A) and x∗ x = y∗ y = 1}. Source: [1134, 1504]. Remark: θ is the minimal principal angle. See Fact 3.12.21 and Fact 7.13.27. Remark: Note that N(A) = R(A⊥ ). See Fact 4.15.5. Remark: This result yields iii) of Fact 7.12.41. Credit: V. E. Ljance. Related: Fact 7.10.30 and Fact 12.12.28. Fact 7.12.43. Let A ∈ Rn×n, where n ≥ 2, be the tridiagonal matrix   0 0   b1 c1 0 · · ·   0 0   a1 b2 c2 · · ·   ..   0 a . b 0 0 2 3  △    A =  . . .. , .. ... ... ...  .. .    ..  . bn−1 cn−1  0  0 0   0 0 0 · · · an−1 bn

and assume that, for all i ∈ {1, . . . , n − 1}, ai ci > 0. Then, A is simple, and every eigenvalue of A △ △ is real. Hence, rank A ≥ n − 1. Source: SAS −1 is symmetric, where S = diag(d1, . . . , dn ), d1 = 1, △ 1/2 and, for all i ∈ {1, . . . , n − 1}, di+1 = (ci /ai ) di . For a proof that A is simple, see [1040, p. 198]. Related: Fact 7.12.44. Fact 7.12.44. Let A ∈ Rn×n, where n ≥ 2, be the tridiagonal matrix   0 0   b1 c1 0 · · ·   0 0   a1 b2 c2 · · ·   ..   0 a . b3 0 0  2 △     A =  . . .. , .. ... ... ...   .. .   ..   . bn−1 cn−1  0  0 0   0 0 0 · · · an−1 bn and assume that, for all i ∈ {1, . . . , n − 1}, ai ci , 0. Then, A is irreducible. Furthermore, let k+ and k− denote, respectively, the number of positive and negative numbers in the n-tuple (1, a1 c1, a1 a2 c1 c2 , . . . , a1 a2 · · · an−1 c1 c2 · · · cn−1 ). Then, A has at least |k+ − k− | distinct real eigenvalues, at least max {0, n − 3 min {k+ , k− }} of which are simple. Source: [2794]. Remark: Note that k+ + k− = n and |k+ − k− | = n − 2 min {k+ , k− }. Remark: This result implies Fact 7.12.43.

587

MATRIX DECOMPOSITIONS

Fact 7.12.45. Let A ∈ Rn×n be the tridiagonal matrix

 1  0   n − 1 0    0 n−2   △ .. A =  .      0  

Then, χA (s) =

0 2

0

0

..

.

..

..

.

..

.

0

n−2

.

2

0

0

1

.

..

.

.. ..

n ∏

.

          .   0    n − 1   0

[s − (n + 1 − 2i)].

i=1

Hence,

   n even,  {n − 1, −(n − 1), . . . , 1, −1}, spec(A) =    {n − 1, −(n − 1), . . . , 2, −2, 0}, n odd.

Source: [2594]. Fact 7.12.46. Let A ∈ Rn×n, where n ≥ 1, be the tridiagonal, Toeplitz matrix

       △  A =      

c

0

···

0

a b

c

0

0 a .. .. . .

b .. .

0 .. .

0 0

0

··· .. . .. . .. .

0 0

0

···

a

b

b

 0   0    0   ..  , .   c   b

and assume that ac ≥ 0. Then,

{ } √ iπ spec(A) = b + 2 ac cos : i ∈ {1, . . . , n} . n+1 Remark: See [1389, p. 522]. Related: Fact 4.23.9. Fact 7.12.47. Let A ∈ Rn×n, where n ≥ 1, be the tridiagonal, Toeplitz matrix   0 0   0 1/2 0 · · ·   0 0   1/2 0 1/2 · · ·   ..   0 1/2 0 . 0 0  △    A =  . .. ..  . .. .. ..  .. . . . . .    ..  . 0 0 0 1/2   0   0 0 0 · · · 1/2 0 Then,

} { iπ : i ∈ {1, . . . , n} . spec(A) = cos n+1

588

CHAPTER 7

Furthermore, the associated eigenvectors v1, . . . , vn are given by  iπ   sin n+1  √   2iπ   2  sin n+1 vi =  .  , n + 1  ..    niπ  sin n+1 which are mutually orthogonal and satisfy ∥vi ∥2 = 1 for all i ∈ {1, . . . , n}. Source: [1633]. Fact 7.12.48. Let A ∈ Fn×n, and assume that A has real eigenvalues. Then, √ [ √ ] [ ] 1 1 n−1 1 2 − 1 (tr A)2 ≤ λ 2 − 1(tr A)2 tr A − tr A (A) ≤ tr A − tr A min 2 n n n n n n −n √ ] [ 1 1 1 ≤ n tr A + n2 −n tr A2 − n (tr A)2 √ [ ] 1 2 2 ≤ λmax (A) ≤ 1n tr A + n−1 n tr A − n (tr A) . Furthermore, for all i ∈ {1, . . . , n}, ] √ [ 1 2 2 λi (A) − 1n tr A ≤ n−1 n tr A − n (tr A) . Finally, if n = 2, then √ 1 1 2 tr A − n n tr A −

1 (tr A)2 n2

= λmin (A) ≤ λmax (A) = 1n tr A +

√

1 2 n tr A

−

1 (tr A)2 . n2

Source: [2905, 2906]. Remark: See [2825]. Related: Fact 2.11.37 and Fact 7.12.49. Fact 7.12.49. Let A ∈ Fn×n, assume that A has real eigenvalues, and define △

a= Then,

( )2 1 1 tr A2 − tr A , n n

△

b=

1 3 2 tr A3 − 2 (tr A) tr A2 + 3 (tr A)3 . n n n

√ √ b + b2 + 4a3 b2 + 4a3 1 ≤ tr A + ≤ λmax (A), 2a n 2a √ √ b2 + 4a3 λmax (A) 2 b2 + 4a3 ≤ ≤ λmax (A) − λmin (A), 1 + . √ 2 3 a λmin (A) b + 2a(tr A)/n − b + 4a λmin (A) ≤

1 b− tr A + n

Source: [2433]. Related: Fact 7.12.48. Fact 7.12.50. Let A ∈ Fn×n. Then,

√ − 1n | tr A|2 ) ≤ ρmin (A) ≤ 1n tr AA∗ , √ 1 1 n−1 1 ∗ 2 n | tr A| ≤ ρmax (A) ≤ n | tr A| + n (tr AA − n | tr A| ).

1 n | tr A|

−

√

Source: Theorem 3.1 of [2905].

n−1 ∗ n (tr AA

589

MATRIX DECOMPOSITIONS

Fact 7.12.51. Let A ∈ Fn×n, where n ≥ 2, be the bidiagonal matrix

       △  A =      

a1

b1

0

···

0

0

0

a2

b2

0

0

0 .. .

0 .. .

a3 .. .

0 .. .

0 .. .

0

0

0

··· .. . .. . .. .

an−1

bn−1

0

0

0

···

0

an

        ,     

and assume that a1 , . . . , an , b1 , . . . , bn−1 are nonzero. Then, the following statements hold: i) The singular values of A are distinct. ii) If B ∈ Fn×n is bidiagonal and |B| = |A|, then A and B have the same singular values. iii) If B ∈ Fn×n is bidiagonal, |A| ≤ |B|, and |A| , |B|, then σmax (A) < σmax (B). iv) If B ∈ Fn×n is bidiagonal, |I ⊙ A| ≤ |I ⊙ B|, and |I ⊙ A| , |I ⊙ B|, then σmin (A) < σmin (B). v) If B ∈ Fn×n is bidiagonal, |Nn ⊙A| ≤ |Nn ⊙B|, and |Nn ⊙A| , |Nn ⊙B|, then σmin (B) < σmin (A). Source: [1990, p. 17-5].

7.13 Facts on Eigenvalues and Singular Values for Two or More Matrices Fact 7.13.1. Let A, B ∈ Fn×n, and assume that A and B are idempotent. Then, the following

statements are equivalent: i) mspec(A) = mspec(B). ii) rank A = rank B. iii) tr A = tr B. Fact 7.13.2. Let A, B ∈ Fn×n, and assume that A and B are projectors. Then, the following statements hold: i) amultAB (1) = def(I − AB) ≤ tr AB. ii) rank(I − AB) = def AB + card[mspec(AB) ∩ (0, 1)] = n − amultAB (1). iii) rank(AB + BA) = rank AB + card[mspec(AB) ∩ (0, 1)]. iv) rank[AB − (AB)2 ] = card[mspec(AB) ∩ (0, 1)]. v) rank(I − ABA) = def AB + card[mspec(AB) ∩ (0, 1)]. vi) rank(AB)2 = rank ABA = rank AB. vii) R(AB) ⊆ R(AB + BA) = R(AB) + R(BA). viii) If AB + BA is idempotent, then AB = BA = 0. ix) rank(AB − BA) = 2 card[mspec(AB) ∩ (0, 1)]. x) If AB + BA and AB − BA are nonsingular, then rank AB = n/2. Source: [245]. Related: Fact 8.8.3. △ Fact 7.13.3. Let A, B ∈ Fn×n, assume that A and B are projectors, and define r = rank A. Then, n×n there exists a unitary matrix S ∈ F such that [ ] [ ] I 0 ∗ B B12 ∗ A=S r S , B = S 11 S , 0 0 B∗12 B22 where B11 ∈ Fr×r , B12 ∈ Fr×(n−r) , and B22 ∈ F(n−r)×(n−r) . Furthermore, the following statements hold: i) rank B = rank B11 − rank B12 + rank B22 .

590

CHAPTER 7

rank AB = rank BA = rank B11 . rank(I − AB) = n − rank B11 + rank B12 . rank(A + B) = r + rank B22 . rank(A − B) = r − rank B11 + rank B12 + rank B22 . rank(AB + BA) = rank B11 + rank B12 . rank(AB − BA) = 2 rank B12 . A + A⊥ (A⊥ B)+ is the projector onto R(A) + R(B). A − A(AB⊥ )+ is the projector onto R(A) ∩ R(B). R(A) ∩ R(B) = {0} if and only if rank B11 = rank B12 . R(A) + R(B) = Fn if and only if rank B22 = n − r. R(A) ⊥ R(B) if and only if B11 = 0. R(A) and R(B) are complementary subspaces if and only if rank B11 = rank B12 and rank B22 = n − r. xiv) R(A)⊥ = R(B) if and only if B11 = 0 and rank B22 = n − r. xv) amultB11 (1) = rank B11 − rank B12 and amultB22 (1) = rank B22 − rank B12 xvi) spec(AB) ⊂ [0, 1]. xvii) def(AB) = n − rank B11 = dim(N(A) + [R(A) ∩ N(B)]). xviii) amultAB (1) = rank B11 − rank B12 = dim[R(A) ∩ R(B)]. xix) card[mspec(AB) ∩ (0, 1)] = rank B12 . xx) rank(AB − BA) = card[mspec[(A − B)2 ] ∩ (0, 1)]. xxi) spec(A − B) ⊂ [−1, 1]. xxii) amultA−B (1) = def(I − A + B) = r − rank B11 = dim[R(A) ∩ N(B)]. xxiii) amultA−B (−1) = def(I + A − B) = rank B22 − rank B12 = dim[N(A) ∩ R(B)]. xxiv) def(A − B) = n − r + rank B11 − rank B12 − rank B22 . xxv) card(mspec(A − B) ∩ [(−1, 0) ∪ (0, 1)]) = 2 rank B12 . xxvi) spec(A + B) ⊂ [0, 2]. xxvii) amultA+B (2) = def(2I − A − B) = rank B11 − rank B22 . xxviii) amultA+B (1) = def(I − A − B) = r − rank B11 − rank B12 + rank B22 . xxix) def(A + B) = n − r − rank B22 = dim[N(A) ∩ N(B)]. xxx) card(mspec(A + B) ∩ [(0, 1) ∪ (1, 2)]) = 2 rank B12 . xxxi) spec(AB − BA) ⊂ ȷ[−1, 1]. xxxii) def(AB − BA) = n − 2 rank B12 . xxxiii) card(mspec(AB − BA) ∩ ȷ[[−1, 0) ∪ (0, 1)]) = 2 rank B12 . xxxiv) (A − B)2 + (I − A − B)2 = I. xxxv) AB − BA = (I − A − B)(A − B) = (B − A)(I − A − B). Source: [243]. Related: Fact 7.13.4. Fact 7.13.4. Let A, B ∈ Fn×n, and assume that A and B are projectors. Then, ii) iii) iv) v) vi) vii) viii) ix) x) xi) xii) xiii)

spec(A + B) ⊂ [0, 2], spec(AB) ⊂ [0, 1],

spec(A − B) ⊂ [−1, 1],

spec(AB + BA) ⊂ [− 41 , 2],

spec([A, B]) ⊂ ȷ[−1, 1].

Source: [83], [246, 251], [1133, p. 53], [2081], and [2263, p. 147]. Remark: Let A =

[

10 00

]

591

MATRIX DECOMPOSITIONS

and B =

1 4

[

√ ] 3 √1 . 3 3

Then, − 14 ∈ spec(AB + BA). Credit: The first inclusion is due to S. N. Afriat.

Related: Fact 7.13.3, Fact 8.5.14, and Fact 10.11.10 . Fact 7.13.5. Let A, B ∈ Fn×n, and assume that A and B are projectors. Then, the following

statements are equivalent: i) AB is a projector. ii) spec(AB) ⊆ {0, 1}. iii) spec(A + B) ⊂ {0} ∪ [1, 2]. iv) spec(A − B) ⊆ {−1, 0, 1}. Source: [245, 1134, 1252] and [2238, p. 336]. Related: Fact 4.18.7, Fact 7.13.6, Fact 8.8.5, and Fact 8.11.13. Fact 7.13.6. Let A, B ∈ Fn×n, and assume that A and B are projectors. Then, the following statements are equivalent: i) AB is a projector. ii) AB is Hermitian. iii) AB is normal. iv) AB is range Hermitian. v) 2AB − I is involutory. vi) 2AB − I is unitary. vii) [A, B] is idempotent. viii) [A, B] is nilpotent. ix) [A, B]2 = 0. x) [ABA, BAB] = 0. xi) R(BABAB) ⊆ R(AB). xii) amultAB (1) = tr AB. xiii) def(I − AB) = tr AB. xiv) rank(AB + BA) = rank AB. xv) spec(AB) ∩ (0, 1) = ∅. xvi) rank(A + B) = rank AB + rank(A − B). xvii) rank(A + B) + rank AB = rank A + rank B. xviii) rank(2I − A − B) = def AB. xix) rank(I − BAB) = def AB. xx) rank [AB BA] = rank AB. xxi) tr (AB)2 = tr AB. xxii) tr (I − AB)2 = tr(I − AB). xxiii) tr(I − AB) = rank(I − AB). Source: [245, 2705, 2706]. Related: Fact 7.13.5 and Fact 8.8.5. [ ] △ △ Fact 7.13.7. Let A ∈ Fn×n and B ∈ Fn×m, and define r = rank B and A = BA∗ B0 . Then, ν− (A) ≥ r, [n] ν0 (A) ≥ 0, and ν+ (A) ≥ r. If, in addition, n = m and B is nonsingular, then In A = 0 . Source: n [1458]. Related: Proposition 7.6.5. Fact 7.13.8. Let A, B ∈ Fn×n. Then, ρmax (A + B) ≤ σmax (A + B) ≤ σmax (A) + σmax (B).

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If, in addition, A and B are Hermitian, then ρmax (A + B) = σmax (A + B) ≤ σmax (A) + σmax (B) = ρmax (A) + ρmax (B), λmin (A) + λmin (B) ≤ λmin (A + B) ≤ λmax (A + B) ≤ λmax (A) + λmax (B). Source: Use Lemma 10.4.3 for the last string of inequalities. Related: Fact 7.12.9. Fact 7.13.9. Let A, B ∈ Fn×n and λ ∈ spec(A + B). Then, ∗ 1 2 λmin (A

+ A) + 21 λmin (B∗ + B) ≤ Re λ ≤ 12 λmax (A∗ + A) + 21 λmax (B∗ + B).

Source: [688, p. 18]. Related: Fact 7.12.24. Fact 7.13.10. Let A ∈ Fn×n . Then, the following statements hold:

i) tr AX = 0 for all Hermitian X ∈ Fn×n if and only if A = 0. ii) Im tr AX = 0 for all Hermitian X ∈ Fn×n if and only if A is Hermitian. iii) If A is Hermitian and Re tr AX ≤ tr A for all unitary X ∈ Fn×n , then A is positive semidefinite. Source: [2991, p. 258]. Fact 7.13.11. Let A, B ∈ Fn×n, assume that either A and B are Hermitian or A and B are skew Hermitian, and let k ≥ 1. Then, tr AkBk and tr (AB)k are real. Source: tr AB = tr A∗B∗ = tr (BA)∗ = tr BA = tr AB. See [103, 2954] and [2991, pp. 260, 264]. Fact 7.13.12. Let A, B ∈ Fn×n be normal, and let mspec(A) = {λ1 , . . . , λn }ms and mspec(B) = {µ1 , . . . , µn }ms . Then, min Re

n ∑

λi µσ(i) ≤ Re tr AB ≤ max Re

i=1

n ∑

λi µσ(i) ,

i=1

where “max” and “min” are taken over all permutations σ of the eigenvalues of B. Now, assume that A and B are Hermitian. Then, tr AB is real, and n n ∑ ∑ λi (A)λn−i+1 (B) ≤ tr AB ≤ λi (A)λi (B). i=1

i=1

Furthermore, the last inequality is an equality if and only if there exists a unitary matrix S ∈ Fn×n such that A = S diag[λ1 (A), . . . , λn (A)]S ∗ and B = S diag[λ1 (B), . . . , λn (B)]S ∗. Source: [1946]. For the second string of inequalities, use Fact 2.12.8. For the last statement, see [523, p. 10] and [1792]. Credit: The upper bound for tr AB is due to K. Fan. Related: Fact 7.13.13, Fact 7.13.16, Proposition 10.4.13, Fact 10.14.35, and Fact 10.22.25. Fact 7.13.13. Let A, B ∈ Fn×n, and assume that B is Hermitian. Then, n n ∑ ∑ λi [ 12 (A + A∗ )]λn−i+1(B) ≤ Re tr AB ≤ λi [ 12 (A + A∗ )]λi (B). i=1

i=1

Source: Apply the second string of inequalities in Fact 7.13.12. Remark: For A, B real, these inequalities are given in [1663]. For the complex case, see [1760]. Related: Proposition 10.4.13 in

the case where B is positive semidefinite. △ Fact 7.13.14. Let A ∈ Fn×m and B ∈ Fm×n, and define r = min {m, n}. Then, r ∑ | tr AB| ≤ σi (A)σi (B). i=1

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MATRIX DECOMPOSITIONS

Now, assume that n = m. Then, max Re tr UA∗ V B = max | tr UA∗ V B| =

r ∑

σi (A)σi (B),

i=1

where “max” is taken over all unitary matrices U, V ∈ Fn×n . Source: [541], [1969, [ ]pp. 514, [ ∗515], ] [1971, pp. 789, 790], and [2263, p. 148]. Remark: Applying Fact 7.13.12 to A0∗ A0 and B0 B0 and ∑r using Proposition 7.6.5 yields the weaker result |Re tr AB| ≤ i=1 σi (A)σi (B). Remark: See [523, p. 14] and [1665]. Credit: The equalities are due to J. von Neumann and K. Fan. Related: Fact 7.13.15, Fact 11.14.1, and Fact 11.16.2. Fact 7.13.15. Let A, B ∈ Fn×n, and assume that B is positive semidefinite. Then, | tr AB| ≤

n ∑

σi (A)λi (B) ≤ σmax (A) tr B.

i=1

Source: Fact 7.13.14. Related: An extension of this result is given by Fact 11.16.2. △ Fact 7.13.16. Let A, B ∈ Rn×n, assume that B is symmetric, and define C = 21 (A + AT ). Then,

λmin (C)tr B − λmin (B)[nλmin (C) − tr A] ≤ tr AB ≤ λmax (C)tr B − λmin (B)[nλmax (C) − tr A]. Source: [1014]. Remark: Extensions are given in [2201]. Related: Fact 7.13.12, Proposition

10.4.13, and Fact 10.14.35. Fact 7.13.17. Let A, B ∈ Fn×n, assume that A and B are Hermitian, and assume that tr A and tr B are positive. Then, tr (A + B)2 tr A2 tr B2 ≤ + . tr(A + B) tr A tr B Furthermore, equality holds if and only if there exists α ∈ (0, ∞) such that A = αB. Source: Note that √ √ 2 ( 2 )  tr B tr A tr B2 tr A  2  tr (A + B) + tr  A− B = (tr A + tr B) + . tr A tr B  tr A tr B See [2991, p. 264]. Fact 7.13.18. Let A, B, Q, S 1 , S 2 ∈ Rn×n, assume that A and B are symmetric, and assume that

Q, S 1 , and S 2 are orthogonal. Furthermore, assume that S 1TAS 1 and S T2 BS 2 are diagonal with the diagonal entries arranged downward in nonincreasing order, and define the orthogonal matrices △ △ Q1, Q2 ∈ Rn×n by Q1 = S 1 Iˆn S 1T and Q2 = S 1 S T2. Then, tr AQ1BQT1 ≤ tr AQBQT ≤ tr AQ2BQT2 . Source: [330, 1792]. Related: Fact 7.13.16 and Fact 7.13.19. Fact 7.13.19. Let A ∈ Fn×n, assume that A is Hermitian, let k ≤ rank A, let S , S 1 , S 2 ∈ Fn×k ,

assume that S, S 1 , and S 2 are left inner, and assume that AS 1 = S 1 diag[λn−k+1 (A), . . . , λn (A)] and AS 2 = S 2 diag[λ1 (A), . . . , λk (A)]. Then, tr S1∗AS 1 ≤ tr S ∗AS ≤ tr S 2∗ AS 2 .

Source: [775]. Remark: This is the Fan trace minimization principle. Related: Fact 7.13.18 and Fact 11.17.16. Fact 7.13.20. Let A, B ∈ Fn×n , and assume that A and B are Hermitian. Then, n ∑ max tr AS ∗BS = λi (A)λi (B). S ∈U(n)

i=1

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CHAPTER 7

Source: [775]. Related: Fact 7.13.14, Fact 7.13.19, and Fact 11.17.15. Fact 7.13.21. Let A1 , . . . , Am , B1 , . . . , Bm ∈ Fn×n, and assume that A1 , . . . , Am are unitary. Then,

| tr A1B1 · · · Am Bm | ≤

n ∑

σi (B1 ) · · · σi (Bm ).

i=1

Source: [1969, p. 516] and [1971, p. 791]. Credit: K. Fan. Related: Fact 7.13.18. Fact 7.13.22. Let A1 , . . . , Am ∈ Fn×n. Then, for all k ∈ {1, . . . , n}, k ∑

σi (A1 · · · Am ) ≤

i=1

k ∑

σi (A1 ) · · · σi (Am ).

i=1

Source: [1969, p. 250] and [1971, p. 342]. Fact 7.13.23. Let A, B ∈ Rn×n, and assume that AB = BA. Then,

ρmax (AB) ≤ ρmax (A)ρmax (B),

ρmax (A + B) ≤ ρmax (A) + ρmax (B).

Source: Fact[ 7.19.5. Remark: ] [ ] If AB , BA, then both of these inequalities may be violated.

and B = 01 00 . Fact 7.13.24. Let A, B ∈ Cn×n, assume that A and B are normal, and let mspec(A) = {λ1 , . . . , λn }ms and mspec(B) = {µ1 , . . . , µn }ms . Then,     n n   ∏    ∏ . max |λ + µ |, max |λ + µ | | det(A + B)| ≤ min  i j i j     i∈{1,...,n}   i=1 j∈{1,...,n} j=1

Consider A =

01 00

Source: [2286]. Remark: Equality is discussed in [339]. Related: Fact 11.16.22. Fact 7.13.25. Let A, B ∈ Fn×n, assume that B is nonsingular, assume that spec(B−1A) ⊂ ORHP,

and let p ≥ 2/n. Then,

| det A| p + | det B| p ≤ | det(A + B)| p .

Source: [2972]. Related: Corollary 10.4.15 and Fact 10.16.8. Fact 7.13.26. Let A ∈ Fn×m and B ∈ Fm×m, and assume that n ≤ m. Then,

det(ABB∗A∗ ) ≤ det(AA∗ )

n ∏

σ2i (B) =

i=1

n ∏

σ2i (A)σ2i (B).

i=1

Source: [970, p. 218]. Fact 7.13.27. Let A, B ∈ Fn×n, assume that A and B are nonzero projectors, and define the

minimal principal angle θ ∈ [0, π2 ] by

cos θ = max {|x∗ y| : (x, y) ∈ R(A) × R(B) and x∗ x = y∗ y = 1}. Then, the following statements hold: i) σmax (AB) = σmax (BA) = cos θ. ii) σmax (A + B) = 1 + σmax (AB) = 1 + cos θ. iii) 1 ≤ σmax (AB) + σmax (A − B). iv) σmax (A − B) = max {σmax (AB⊥ ), σmax (BA⊥ )} ≤ 1. v) θ > 0 if and only if R(A) ∩ R(B) = {0}. Furthermore, the following statements are equivalent: vi) σmax (A − B) < 1. vii) rank A = rank B = rank AB.

595

MATRIX DECOMPOSITIONS

viii) R(A) ∩ N(B) = {0} and N(A) ∩ R(B) = {0}. ix) R(A) = R(AB) and R(B) = R(BA). x) rank(A + B) = rank(AB + BA). xi) 1 < spec(A + B). If these statements hold, then A and B are unitarily similar. Furthermore, the following statements are equivalent: xii) A − B is nonsingular. xiii) R(A) and R(B) are complementary subspaces. xiv) σmax (A + B − I) < 1. Now, assume that A − B is nonsingular. Then, the following statements hold: xv) θ > 0. xvi) σmax (AB) < 1. xvii) σmax [(A − B)−1 ] = √ 12 = 1/sin θ. 1−σmax (AB)

σmin (A − B) = sin θ. 2 2 σmin (A − B) + σmax (AB) = 1. I − AB is nonsingular. If rank A = rank B, then σmax (A − B) = sin θmax , where θmax is the maximum principal angle defined in Fact 7.10.29. √ √ xxii) tr (A − B)2 = rank A + rank B − 2 tr AB ≤ rank(A − B). Source: i) is given in [1504]; ii) is given in [1134]; iii) follows from the first inequality in Fact 10.22.17; iv) is given in [779] and [1590, p. 56]; v) is given in [1187, p. 393]; vi)–xi) are given in [249], see also [970, p. 195] and [1187, p. 389]; Fact 4.18.19 implies that xi) and xii) are equivalent; xiv) is given in [602], see also [1133, p. 236]; xxi) follows from [2539, pp. 92, 93]; xxii) is given in [249]. Remark: The equality equality. See [779]. [ ]in iv) is the [ Krein-Krasnoselskii-Milman ] Remark: The projectors A = 10 00 and B = 00 01 satisfy A = Iˆ2 BIˆ2 and thus are unitarily similar, whereas σmax (A − B) = 1. Hence, the converse of the statement after xi) does not hold. Remark: Additional results for A − B nonsingular are given in Fact 4.18.19. Related: Fact 3.12.21, Fact 7.12.42, Fact 7.13.28, and Fact 8.8.17. Fact 7.13.28. Let A ∈ Fn×n, assume that A is idempotent, let P, Q ∈ Fn×n, where P is the projector onto R(A) and Q is the projector onto N(A), and define the minimal principal angle θ ∈ [0, π2 ] by cos θ = max {|x∗ y| : (x, y) ∈ R(P) × R(Q) and x∗ x = y∗ y = 1}. xviii) xix) xx) xxi)

Then, the following statements hold: i) P − Q is nonsingular. ii) (P − Q)−1 = A + A∗ − I = A − A∗⊥ . iii) σmax (A) = √ 12 = σmax [(P − Q)−1 ] = σmax (A + A∗ − I) = 1/sin θ. 2 σmin (P

1−σmax (PQ) 2 = 1 − σmax (PQ).

iv) − Q) v) σmax (PQ) = σmax (QP) = σmax (P + Q − I) < 1. vi) A = (I − PQ)−1 P(I − PQ). Source: [2296] and Fact 7.13.27. The nonsingularity of P − Q follows from Fact 4.18.19; ii) is given by Fact 4.18.19 and Fact 8.7.3; the first equality in iii) is given in [602], see also [1134]; vi) is given in [15, 531]. Remark: A⊥∗ is the idempotent matrix onto R(A)⊥ along N(A)⊥ . Remark: P = AA+ and Q = (A+A)⊥ . Related: Fact 4.15.5, Fact 4.18.19, Fact 7.13.27, and Fact 8.8.17.

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Fact 7.13.29. Let A, B ∈ Fn×n, and assume that A and B are idempotent. Then, A − B is

idempotent if and only if A − B is group invertible and every eigenvalue of A − B is nonnegative. Source: [1336]. Remark: Conditions under which a matrix is a difference of idempotents are given in [1336]. Credit: T. Makelainen and G. P. H. Styan. Related: Fact 4.16.13. [ ] △ Fact 7.13.30. Let A ∈ Fn×n, B ∈ Fn×m, and C ∈ Fm×m, define A = BA∗ CB ∈ F(n+m)×(n+m), and assume that A is Hermitian. Then, for all i ∈ {1, . . . , n} and j ∈ {1, . . . , n} such that i + j ≤ n + 1, λi+ j−1 (A) + λmin (A) ≤ λi (A) + λ j (C). In particular,

λmax (A) + λmin (A) ≤ λmax (A) + λmax (C).

Source: [486, p. 56] and [2979, p. 74]. Fact 7.13.31. Let M ∈ Rr×r, assume that M is positive definite, let C, K ∈ Rr×r, assume that C

and K are positive semidefinite, and consider the equation △

Then, x(t) =

[ q(t) ] q(t) ˙

M¨q + Cq˙ + Kq = 0. satisfies x˙(t) = Ax(t), where A is the 2r × 2r matrix    0 I  △  A =  . −M−1K −M−1C

Furthermore, the following statements hold: i) A, K, and M satisfy det K . det A = det M ii) A and K satisfy rank A = r + rank K. iii) A is nonsingular if and only if K is positive definite. If these conditions hold, then   −1  −K C −K−1M  −1 A =   . I 0 iv) If K is singular, then A is not group invertible. In particular, every Jordan block of A associated with the zero eigenvalue is of size 2. v) χA (s) = det(s2I + sM−1C + M−1K). vi) Let λ ∈ C. Then, λ ∈ spec(A) if and only if det(λ2 M + λC + K) = 0. vii) If λ ∈ spec(A), Re λ = 0, and Im λ , 0, then λ is semisimple. viii) mspec(A) ⊂ CLHP. ix) If C = 0, then spec(A) ⊂ IA . x) If C and K are positive definite, then spec(A) ⊂ OLHP. [ ] △ K 1/2 q(t) ˆ satisfies x˙(t) = Ax(t), where xi) xˆ(t) = M 1/2 q(t) ˙   Aˆ =  △

0

K 1/2M −1/2

−M−1/2K 1/2

−M−1/2CM−1/2

If, in addition, C = 0, then Aˆ is skew symmetric.

   .

597

MATRIX DECOMPOSITIONS △

xii) xˆ(t) =

[

M 1/2 q(t) M 1/2 q(t) ˙

]

ˆ satisfies x˙(t) = Ax(t), where   △  Aˆ = 

0

I

−M−1/2KM−1/2

−M−1/2CM−1/2

   .

If, in addition, C = 0, then Aˆ is Hamiltonian. Remark: M, C, and K are mass, damping, and stiffness matrices, respectively. See [431]. iv) shows that the absence of a stiffness implies that the structure has a rigid-body mode. vii) shows that the only type of instability is a rigid-body mode. Related: Fact 7.13.32, Fact 7.15.32, and Fact 15.19.38. Problem: Prove vii). n×n Fact 7.13.32. [ Let] A, B ∈ R , and assume that A and B are positive semidefinite. [Then,] every 0 A 0 A is not eigenvalue λ of −B 0 satisfies Re λ = 0. If, in addition, rank A + rank B is odd, then −B 0 n×n group invertible. Now, let C[ ∈ R ] , and assume that C is (positive semidefinite, positive definite). 0 A satisfies (Re λ ≤ 0, Re λ < 0). Source: To prove the second Then, every eigenvalue of −B −C [ ]2 [ ] [ ] △ △ 0 A 0 A > statement, note that rank −B 0 is even. Example: If A = 10 00 and B = I2 , then 3 = rank −B 0 ]2 [ 0 A = 2. Problem: Determine the structure of the Jordan blocks associated with the zero rank −B 0 eigenvalue. Problem: Consider the case where A and B are of different size as well as the matrix [ ] −C A . −B −C Fact 7.13.33. Let A0 , . . . , Ar−1 ∈ Fn×n, and define A ∈ Frn×rn by   In 0 ··· 0 0   0   ..   0 . 0 I 0 0 n   ..   . 0 0 0 0   0  △  A =  . .. .. ..  . .. ..  .. . . . . .       0 0 0 · · · 0 I  n   −A0 −A1 −A2 · · · −Ar−2 −Ar−1 Then,

χA (s) = det(sr In + sr−1Ar−1 + · · · + sA1 + A0 ).

7.14 Facts on Matrix Pencils Fact 7.14.1. Let A, B ∈ Fn×n, assume that PA,B is a regular pencil, let S, T ⊆ Fn , assume that △

S, T are subspaces of the same dimension, define k = dim S = dim T, let S , T ∈ Fn×k , and assume that R(S ) = S and R(T ) = T. Then, there exist M1 , M2 ∈ Fk×k such that AS = T M1 and BS = T M2 . If, in addition, B is nonsingular, then S is an invariant subspace of B−1A. Source: [1705, p. 68]. Remark: S is a right deflating subspace of PA,B , and T is a left deflating subspace of PA,B . See also [2539, pp. 303, 304]. Related: Fact 3.11.21.

7.15 Facts on Eigenstructure for One Matrix Fact 7.15.1. Let A ∈ Fn×n. Then, def A = gmultA (0) ≤ amultA (0). If, in addition, A is group invertible, then def A = gmultA (0) = amultA (0). Fact 7.15.2. Let A ∈ Fn×n. Then, amultA∗A (0) = gmultA∗A (0) = gmultA (0). Source: [2991, p. 85]. Fact 7.15.3. Let A ∈ Fn×n, and let p ∈ F[s]. Then, the following statements are equivalent: i) µA divides p.

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CHAPTER 7

ii) spec(A) ⊆ roots(p), and, for all λ ∈ spec(A), indA(λ) ≤ mult p (λ). Source: Proposition 7.7.3. Fact 7.15.4. Let A ∈ Fn×n, let λ ∈ spec(A), assume that λ is cyclic, let i ∈ {1, . . . , n} satisfy rank (A − λI)[i,·] = n − 1, and define x ∈ Cn by   det (A − λI)[i,1]      − det (A − λI)[i,2]  △  . x =  ..   .    (−1)n+1 det (A − λI)[i,n] Then, x is an eigenvector of A associated with λ. Source: [2737]. Fact 7.15.5. Let A ∈ Fn×n, where n ≥ 2. Then, the following statements are equivalent: i) rank A = 1. ii) gmultA (0) = n − 1. Now, assume that these statements hold, and let x, y ∈ Fn satisfy A = xyT. Then, the following statements are equivalent: iii) tr A = 0. iv) amultA (0) = n. v) spec(A) = {0}. vi) A2 = 0. vii) A is nilpotent. viii) A is not group invertible. ix) A is defective. x) ind A = 2. xi) yTx = 0. xii) A is similar to diag(N2 , 0(n−2)×(n−2) ). In addition, the following statements are equivalent: xiii) tr A , 0. xiv) amultA (0) = n − 1. xv) {0} ⊂ spec(A). xvi) A2 , 0. xvii) A is group invertible. xviii) A is semisimple. xix) ind A = 1. xx) yTx , 0. xxi) A is similar to diag(tr A, 0(n−1)×(n−1) ). Fact 7.15.6. Let A ∈ Fn×n. Then, the following statements are equivalent: i) A is group invertible. ii) R(A) = R(A2 ). iii) rank A = rank A2 . iv) ind A ≤ 1. ∑ v) rank A = ri=1 amultA(λi ), where {λ1 , . . . , λr } = spec(A)\{0}.

MATRIX DECOMPOSITIONS

599

vi) R(A) + N(A) = Fn . vii) R(A) ∩ N(A) = {0}. Related: Corollary 4.8.10. Fact 7.15.7. Let A ∈ Fn×n, assume that A is diagonalizable over F with eigenvalues λ1 , . . . , λn , △ and let B = diag(λ1 , . . . , λn ). If x1 , . . . , xn ∈ Fn are linearly independent eigenvectors of A associated △ with λ1 , . . . , λn , respectively, then A = SBS −1, where S = [x1 · · · xn ]. Conversely, if S ∈ Fn×n is nonsingular and A = SBS −1, then, for all i ∈ {1, . . . , n}, coli (S ) is an associated eigenvector. Fact 7.15.8. Let A ∈ Cn×n, and, for all k ∈ {1, . . . , n}, define Ak ∈ Ck×k , where, for all i, j ∈ {1, . . . , k}, (Ak )(i, j) = tr Ai+ j−2 . Then, card[spec(A)] = max {k ∈ {1, . . . , n} : det Ak , 0}. In particular, A has n distinct eigenvalues if and only if An is nonsingular. Furthermore, the following statements are equivalent: i) A is diagonalizable over C. ii) card[spec(A)] = deg µA . iii) det Adeg µA , 0. Source: [995, 996]. Fact 7.15.9. Let A, S ∈ Fn×n, assume that S is nonsingular, let λ ∈ C, and assume that col1 (S −1AS ) = λe1 . Then, λ ∈ spec(A), and col1 (S ) is an associated eigenvector. Fact 7.15.10. Let A ∈ Fn×n. Then, A is cyclic if and only if there exists b ∈ Fn such that [b Ab · · · An−1b] is nonsingular. Source: Fact 16.21.14. Remark: (A, b) is controllable. See Corollary 16.6.3. △ Fact 7.15.11. Let A ∈ Fn×n, and define the positive integer m = maxλ∈spec(A) gmultA (λ). Then, m is the smallest integer such that there exists B ∈ Fn×m such that rank [B AB · · · An−1B] = n. Source: Fact 16.21.14. Remark: (A, B) is controllable. See Corollary 16.6.3. Fact 7.15.12. Let A ∈ Cn×n. Then, there exist v1 , . . . , vn ∈ Cn such that the following statements hold: i) v1 , . . . , vn ∈ Cn are linearly independent. ii) If λ ∈ spec(A) and A has a k × k Jordan block associated with λ, then there exist distinct integers i1 , . . . , ik such that Avi1 = λvi1 , Avi2 = λvi2 + vi1 , . . . , Avik = λvik + vik−1 . iii) Let λ and vi1 , . . . , vik be given by ii). Then, span {vi1 , . . . , vik } = N[(λI − A)k ]. Remark: v1, . . . , vn are generalized eigenvectors of A. Remark: (vi1 , . . . , vik ) is a Jordan chain of A associated with λ. See [1738, pp. 229–231]. Related: Fact 16.20.5. Fact 7.15.13. Let A ∈ Rn×n. Then, A is cyclic and semisimple if and only if A is simple. Fact 7.15.14. Let A = revdiag(a1 , . . . , an ) ∈ Rn×n. Then, A is semisimple if and only if, for all i ∈ {1, . . . , n}, ai and an+1−i are either both zero or both nonzero. Source: [1301, p. 116], [1604], and [2263, pp. 68, 86]. Fact 7.15.15. Let A ∈ Fn×n. Then, A has at least m real eigenvalues and m associated linearly independent eigenvectors if and only if there exists a positive-semidefinite matrix S ∈ Fn×n such that rank S = m and AS = SA∗. Source: [2263, pp. 68, 86]. Credit: M. A. Drazin and E. V. Haynsworth. Related: The case m = n is given by Proposition 7.7.13. Fact 7.15.16. Let A ∈ Fn×n, and let mspec(A) = {λ1 , . . . , λn }ms , where |λ1 | ≥ · · · ≥ |λn |. Then, the following statements are equivalent: i) A is normal. ii) There exists p ∈ F[s] such that A = p(A∗ ). iii) Every eigenvector of A is also an eigenvector of A∗. ∑ iv) There exist nonzero mutually orthogonal vectors x1 , . . . , xn ∈ Cn such that A = ni=1 λi xi xi∗.

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v) AA∗ − A∗A is either positive semidefinite or negative semidefinite. vi) For all x ∈ Fn, x∗A∗Ax = x∗AA∗ x. vii) For all x, y ∈ Fn, x∗A∗Ay = x∗AA∗ y. If these statements hold, then ρmax (A) = σmax (A). Source: [1242], [2263, p. 146], and [2991, p. 86]. Related: Fact 4.10.12, Corollary 7.5.8, Fact 7.12.30, Fact 10.13.2, and Fact 11.9.16 Fact 7.15.17. Let A ∈ Fn×n. Then, the following statements are equivalent: i) A is (simple, cyclic, derogatory, semisimple, defective, diagonalizable over F). ii) There exists α ∈ F such that A + αI is (simple, cyclic, derogatory, semisimple, defective, diagonalizable over F). iii) For all α ∈ F, A + αI is (simple, cyclic, derogatory, semisimple, defective, diagonalizable over F). △ Fact 7.15.18. Let x, y ∈ Fn, assume that xTy , 1, and define the elementary matrix A = I − xyT. Then, A is semisimple if and only if either xyT = 0 or xTy , 0. Source: Fact 7.15.5 and Fact 7.15.17. Fact 7.15.19. Let A ∈ Fn×n, and assume that A is nilpotent. Then, A is nonzero if and only if A is defective. Fact 7.15.20. Let A ∈ Fn×n, and assume that A is either involutory or skew involutory. Then, A is semisimple. Fact 7.15.21. Let A ∈ Rn×n, and assume that A is either involutory, idempotent, or tripotent. Then, A is diagonalizable over R. Fact 7.15.22. Let A ∈ Fn×n, let λ ∈ spec(A), assume that λ , 0, assume that λ is the unique eigenvalue of A on the circle {z ∈ C : |z| = |λ|}, and let k be a positive integer. Then, amultA(λ) = amultAk (λk ) and gmultA(λ) = gmultAk (λk ). Now, assume that A is nonsingular, and let k be a nonzero integer. Then, amultA(λ) = amultAk (λk ) and gmultA(λ) = gmultAk (λk ). In particular, amultA(λ) = amultA−1 (1/λ) and gmultA(λ) = gmultA−1 (1/λ). Related: Fact 8.10.7. Fact 7.15.23. Let A ∈ Fn×n. Then, the following statements are equivalent: i) A is semisimple. ii) A is group invertible, and, for all positive integers k, Ak is semisimple. iii) A is group invertible, and there exists a positive integer k such that Ak is semisimple. Now, assume that A is nonsingular. Then, the following statements are equivalent: iv) A is semisimple. v) For every integer k, Ak is semisimple. vi) There exists an integer k such that Ak is semisimple. Fact 7.15.24. Let A ∈ Fn×n. Then, the following statements hold: i) If spec(A) ⊆ {0, 1}, A is group invertible, and k is a positive integer, then Ak and A are similar. ii) If spec(A) = {1} and k is an integer, then Ak and A are similar. Fact 7.15.25. Let A ∈ Fn×n . Then, the following statements hold: i) If spec(A) ⊆ {−1, 0, 1}, A is group invertible, and k is an odd positive integer, then Ak and A are similar. ii) If spec(A) ⊆ {−1, 1} and k is an odd integer, then Ak and A are similar. Fact 7.15.26. Let A ∈ Fn×n . Then, the following statements hold: √ √ i) If spec(A) ⊆ {0, − 21 + 12 3 ȷ, − 21 − 12 3 ȷ, 1}, A is group invertible, and k is a positive integer, then A3k+1 and A are similar.

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√ √ ii) If spec(A) ⊆ {− 12 + 21 3 ȷ, − 12 − 12 3 ȷ, 1} and k is an integer, then A3k+1 and A are similar. Fact 7.15.27. Let A ∈ Fn×n and let m be a positive integer. Then, the following statements hold: i) If spec(A) ⊆ {e(2πi/m) ȷ : i = 0, . . . , m − 1} ∪ {0}, A is group invertible, and k is a nonnegative integer, then Akm+1 and A are similar. ii) If spec(A) ⊆ {e(2πi/m) ȷ : i = 0, . . . , m − 1} and k is an integer, then Akm+1 and A are similar. Fact 7.15.28. Let A ∈ Fn×n. Then, the following statements hold: i) If there exist distinct positive integers k and l such that Ak and Al are similar, then ind A ≤ min {k, l}. ii) If there exist distinct positive integers k and l such that Ak = Al , then ind A ≤ min {k, l} and (AD )D is semisimple. iii) If A is group invertible and there exist distinct positive integers k and l such that Ak = Al , then A is semisimple and A = A|l−k|+1 . iv) If A is nonsingular and there exist distinct nonzero integers k and l such that Ak = Al , then A is semisimple and Al−k = I. Source: Fact 7.15.23. Fact 7.15.29. Let A ∈ Fn×n, assume that A is group invertible, and assume that A3 = A2. Then, A is idempotent. Fact 7.15.30. Let A ∈ Cn×n, and let x ∈ Cn be an eigenvector of A associated with the eigenvalue λ. If A is nonsingular, then x is an eigenvector of AA associated with the eigenvalue (det A)/λ. If rank A = n − 1, then x is an eigenvector of AA associated with the eigenvalue tr AA or 0. Finally, if rank A ≤ n − 2, then x is an eigenvector of AA associated with the eigenvalue 0. Source: Use Fact 7.16.1 and AAA = AAA . See [791]. Related: Fact 3.19.3 and Fact 8.3.21. Fact 7.15.31. Let q(t) denote the displacement of a mass m > 0 connected to a wall by means of a dashpot c ≥ 0 and subject to a force f (t). Then, q(t) satisfies mq(t) ¨ + cq(t) ˙ = f (t), and thus q(t) ¨ + △

c 1 q(t) ˙ = f (t). m m

△

Next, x1 (t) = q(t) and x2 (t) = q(t) ˙ satisfy [ ] [ x˙1 (t) 0 = x˙2 (t) 0

] [ ] 0 x1 (t) + 1 f (t). x2 (t) m [0 1 ] △ The eigenvalues of the companion matrix Ac = 0 − mc are given by mspec(Ac ) = {0, − mc }ms . The matrix[ Ac] has a repeated eigenvalue if and only if c = 0, in which case mspec(Ac ) = {0, 0}ms , Ac = 00 10 is defective and nilpotent, and Ac is in Jordan form. Finally, if c > 0, then Ac = SAJ S −1 , ] [ ] [ ] [ where m 0 − mc △ 0 △ 1 −1 △ 1 c . AJ = , S = , S = 0 − mc 0 1 0 1 1 − mc

][

Remark: If c > 0, then this structure is a damped rigid body, whereas, if c = 0, then this structure is an undamped rigid body. Fact 7.15.32. Let q(t) denote the displacement of a mass m > 0 connected to a wall by means of a spring k > 0 and a dashpot c ≥ 0 and subject to a force f (t). Then, q(t) satisfies

mq(t) ¨ + cq(t) ˙ + kq(t) = f (t),

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and thus

c k 1 q(t) ˙ + q(t) = f (t). m m m √ △ △ √ Now, define the natural frequency ωn = k/m and damping ratio ζ = c/(2 km) to obtain q(t) ¨ +

q(t) ¨ + 2ζωn q(t) ˙ + ω2n q(t) = △

1 f (t). m

△

Next, x1 (t) = q(t) and x2 (t) = q(t) ˙ satisfy [ ] [ x˙1 (t) 0 = x˙2 (t) −ω2n

][ ] [ ] 0 1 x1 (t) + 1 f (t). −2ζωn x2 (t) m [ 0 ] △ 1 The eigenvalues of the companion matrix Ac = −ω2n −2ζωn are given by    {−ζωn − ωd ȷ, −ζωn + ωd ȷ}ms , 0 ≤ ζ ≤ 1,    mspec(Ac ) =  } { √ √     (−ζ − ζ 2 − 1)ωn , (−ζ + ζ 2 − 1)ωn , ζ > 1, √ △ where ωd = ωn 1 − ζ 2 is the damped natural frequency. The matrix Ac has a repeated eigenvalue if and only if ζ = 1, in which case mspec(Ac ) = {−ωn , −ωn }ms and [ Ac] is defective. In particular, △ 0 in the case where ζ = 1, it follows that Ac = SAJ S −1, where S = ω−1n −1 and AJ is the Jordan form [ ] △ −ωn 1 matrix AJ = 0 −ωn . If Ac is not defective, that is, if ζ , 1, then the Jordan form AJ of Ac is given by     −ζωn + ωd ȷ  0     0 ≤ ζ < 1,  ,      0 −ζωn − ωd ȷ    △  AJ =  )  √ (      −ζ − ζ 2 − 1 ωn  0    )  , ζ > 1. ( √    2 0 −ζ + ζ − 1 ω n

In the case where 0 ≤ ζ < 1, define the real normal form [ ] ωd △ −ζωn An = . −ωd −ζωn The matrices Ac , AJ , and An are related by the similarity transformations Ac = and AJ = S 3 An S 3−1 , where      1 1 ȷ  −ζωn − ωd ȷ △  −1   S 1 =   , S 1 =  2ωd  ζωn − ωd ȷ −ζωn + ωd ȷ −ζωn − ωd ȷ [ ] [ ] [ ] 1 1 −ȷ 1 0 ωd 0 △ 1 △ S2 = , S 2−1 = , S3 = , S 3−1 ζωn 1 ωd −ζωn ωd 2ωd 1 ȷ

S 1 AJ S 1−1 = S 2 An S 2−1  −1   , 1 [ ] 1 1 = ωd . ȷ −ȷ

In the case where ζ = 1, Ac and AJ are related by Ac = S 4 AJ S 4−1, where      −1 0   −1 0  △  −1  . S 4 =   , S 4 =  ωn −1 −ωn −1 In the case where ζ > 1, Ac and AJ are related by Ac = S 5 AJ S 5−1, where √     −ζωn − ωn ζ 2 − 1 1 1   −1 1 △    , S 5 =  S 5 =  √ √ √ √ −ζωn + ωn ζ 2 − 1 −ζωn − ωn ζ 2 − 1 −2ωn ζ 2 − 1 ζωn − ωn ζ 2 − 1

 −1   .  1

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Finally, define the energy-coordinates matrix [ 0 △ Ae = −ωn Then, Ae = S 6 Ac S 6−1, where

√ △

S6 =

] ωn . −2ζωn

[ m 1 2 0

0 1 ωn

] .

Remark: If c > 0, then this structure is a damped oscillator, whereas, if c = 0, then this structure is an undamped oscillator. Related: Fact 7.13.31.

7.16 Facts on Eigenstructure for Two or More Matrices Fact 7.16.1. Let A, B ∈ Cn×n, and assume that AB = BA. Furthermore, let x ∈ Cn be an

eigenvector of A associated with the eigenvalue λ ∈ C, and assume that Bx , 0. Then, Bx is an eigenvector of A associated with the eigenvalue λ ∈ C. Source: A(Bx) = BAx = B(λx) = λ(Bx). Fact 7.16.2. Let A, B ∈ Cn×n, and assume that rank [A, B] ≤ 1. Then, there exists a nonzero vector x ∈ Cn that is[ an eigenvector [ ] B. Source: [1448, p. 51] and [2991, p. 77]. ] [ ]of both A and 1 0 , and x = 1 . Then, rank [A, B] = 1, Ax = 0, and Bx = x. Example: Let A = 11 −1 , B = 10 1 −1 Related: Fact 7.16.15, Fact 7.19.2, Fact 7.19.6, and Fact 11.10.52. Fact 7.16.3. Let A, B ∈ Cn×n, and assume that A is simple. Then, AB = BA if and only if A and B have a common set of n linearly independent eigenvectors. Source: [2991, p. 76]. Fact 7.16.4. Let A, B ∈ Fn×n, and assume that [A, [A, B]] = 0. Then, [A, B] is nilpotent. Credit: N. Jacobson. See [1055], [1201], and [1448, p. 98]. Related: Fact 7.16.5. Fact 7.16.5. Let A, B ∈ Fn×n, assume that A is semisimple, and assume that [A, [A, B]] = 0. Then, [A, B] = 0. Source: [1201]. Credit: H. Shapiro. Related: Fact 7.16.4. Fact 7.16.6. Let A, B ∈ Fn×n, assume that A is cyclic, and assume that [A, [A, B]] = 0. Then, there exists nonsingular S ∈ Fn×n such that SAS −1 and SBS −1 are upper triangular. Source: [1201]. Credit: H. Shapiro. Fact 7.16.7. Let A, B ∈ Fn×n, assume that A2 is cyclic, and assume that A[A, B] = −[A, B]A. Then, [A, B] = 0. Source: [1201]. Remark: If A2 is cyclic, then A is cyclic, but the converse is false. Fact 7.16.8. Let A ∈ Fn×n. Then, the following statements are equivalent: i) A is cyclic. ii) Every matrix B ∈ Fn×n satisfying AB = BA is a polynomial in A. iii) For all C, D ∈ {B ∈ Fn×n : AB = BA}, CD = DC. Source: [1201] and [1450, p. 275]. The equivalence of ii) and iii) follows from Fact 7.16.9. Related: Fact 3.23.9, Fact 7.16.9, Fact 7.16.10, and Fact 9.5.3. Fact 7.16.9. Let A, B ∈ Fn×n. Then, B is a polynomial in A if and only if B commutes with every matrix that commutes with A. Source: [1450, pp. 276–278]. Related: Fact 3.23.9, Fact 6.8.15, Fact 7.16.8, Fact 7.16.10, Fact 9.5.3. Fact 7.16.10. Let A ∈ Fn×n, assume that A is simple, let B ∈ Fn×n, and assume that AB = BA. Then, B is a polynomial in A whose degree is less than or equal to n − 1. Source: [2983, p. 59] and [2991, pp. 73, 74]. Related: Fact 7.16.8. △ Fact 7.16.11. Let A, B ∈ Fn×n, and define D = AB − BAT . Then, the following statements hold: i) If AD = DAT , then D is singular. ii) If AD = DAT and A is semisimple, then D = 0. iii) If AD = DAT and DA = ATD, then D is nilpotent.

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iv) If AD = DAT and A is cyclic, then D is symmetric and card[spec(A)] ≤ gmultD (0). Source: [1201]. Credit: iv) is due to O. Taussky and H. Zassenhaus. Fact 7.16.12. Let A ∈ Cn×n, B ∈ Cm×m, and C ∈ Cn×m, assume that rank C = m, and assume that AC = CB. Then, mspec(B) ⊆ mspec(A). Source: [2991, p. 77]. Fact 7.16.13. Let A, B ∈ Cn×n, and assume that A and B are Hermitian. Then, spec([A, B]) ⊂ IA. Source: [2991, p. 305]. Fact 7.16.14. Let A, B ∈ Cn×n, assume that A and B are nonsingular, and assume that [A, B] is singular. Then, 1 ∈ spec(A−1B−1AB). Source: [2991, p. 77]. Fact 7.16.15. Let A, B ∈ Cn×n. Then, the following statements are equivalent: k l i) ∩n−1 k,l=1 N([A , B ]) , {0}. ∑n−1 k l ∗ k l ii) k,l=1 [A , B ] [A , B ] is singular. iii) A and B have a common eigenvector. Source: [1161]. Credit: D. Shemesh. Related: Fact 7.16.2, Fact 7.19.2, and Fact 11.10.52. Fact 7.16.16. Let A, B ∈ Fn×n. Then, the following statements hold: i) Assume that A and B are Hermitian. Then, AB is Hermitian if and only if AB = BA. ii) A is normal if and only if, for all C ∈ Fn×n such that AC = CA, it follows that A∗C = CA∗. iii) Assume that B is Hermitian and AB = BA. Then, A∗B = BA∗. iv) Assume that A and B are normal and AB = BA. Then, AB is normal. v) Assume that A and B are normal. Then, AB is normal if and only if BA is normal. vi) Assume that A is positive semidefinite and B is normal. Then, AB is normal if and only if AB = BA. vii) Assume that A and B are normal and either A or B has the property that distinct eigenvalues have unequal absolute values. Then, AB is normal if and only if AB = BA. Source: [798, 1168], [1305, p. 157], [2263, p. 102], [2875], and [2991, pp. 295, 305, 318]. Credit: v) is due to F. R. Gantmacher and M. G. Krein. See [1168]. Related: Fact 7.19.8. Fact 7.16.17. Let A, B, C ∈ Fn×n, [assume ] that [ A] and B are normal, and assume that AC = CB. Then, A∗C = CB∗. Source: Consider A0 B0 and 00 C0 in ii) of Fact 7.16.16. See [1302, p. 104] and [1305, p. 321]. Remark: This is the Putnam-Fuglede theorem. Fact 7.16.18. Let A, B ∈ Fn×n, and assume that A is dissipative and B is range Hermitian. Then, ind B = ind AB. Source: [433]. Fact 7.16.19. Let A ∈ Fn×n, B ∈ Fn×m, and C ∈ Fm×m. Then, [ ] A B max {ind A, ind C} ≤ ind ≤ ind A + ind C. 0 C [

If C is nonsingular, then ind whereas, if A is nonsingular, then

] A B = ind A, 0 C

[

] A B ind = ind C. 0 C

Source: [584, 2037]. Remark: The eigenstructure of a partitioned Hamiltonian matrix is considered in Fact 16.25.1. Related: Fact 8.12.3. Fact 7.16.20. Let A, B ∈ Rn×n, and assume that A and B are skew symmetric. Then, there exists

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an orthogonal matrix S ∈ Rn×n such that    0(n−l)×(n−l) A12   S T ,  A = S   T −A12 A22

  B11 B = S  − BT12

 B12  T  S , 0l×l

△

where l = ⌊n/2⌋. Consequently, T mspec(AB) = mspec(−A12 BT12 ) ∪ mspec(−A12 B12 ),

and thus every nonzero eigenvalue of AB has even algebraic multiplicity. Source: [72]. Fact 7.16.21. Let A, B ∈ Rn×n, and assume that A and B are skew symmetric. If n is even, then there exists a monic polynomial p of degree n/2 such that χAB (s) = p2 (s) and p(AB) = 0. If n is odd, then there exists a monic polynomial p(s) of degree (n − 1)/2 such that χAB (s) = sp2 (s) and ABp(AB) = 0. Consequently, if n is (even, odd), then χAB is (even, odd) and (every, every nonzero) eigenvalue of AB has even algebraic multiplicity and geometric multiplicity of at least 2. Source: [918, 1216].

7.17 Facts on Matrix Factorizations n 1×m Fact 7.17.1. Let A ∈ Fn×m and r ≥ rank A. Then, there exist x1 , . . . , xr[∈ F ] and y1 , . . . , yr ∈ F

such that A =

∑r

△

△

n×r and Y = i=1 xi yi = XY, where X = [x1 · · · xr ] ∈ F

y1

.. .

yr

∈ Fr×m . Furthermore,

rank X = rank Y = rank A if and only if r = rank A. Remark: The last statement gives a full-rank factorization of A. See Proposition 7.6.6. Fact 7.17.2. Let A ∈ Fn×n. Then, A is normal if and only if there exists a unitary matrix S ∈ Fn×n such that A∗ = AS. Now, assume that A is nonsingular. Then, A is normal if and only if A−1A∗ is unitary. Source: [2263, pp. 102, 113] and [2991, p. 304]. Fact 7.17.3. Let A ∈ Cn×n. Then, there exists a nonsingular matrix S ∈ Cn×n such that SAS −1 is symmetric. Source: [1448, p. 209]. Related: Corollary 7.4.9. Remark: The coefficient of the last matrix in [1448, p. 209] should be ȷ/2. Fact 7.17.4. Let A ∈ Cn×n, and assume that A2 is normal. Then, the following statements hold: i) There exists a unitary matrix S ∈ Cn×n such that SAS −1 is symmetric. ii) There exists a symmetric unitary matrix S ∈ Cn×n such that AT = SAS −1. Source: [2793]. Fact 7.17.5. Let A ∈ Fn×n, and assume that A is nonsingular. Then, A−1 and A∗ are similar if and only if there exists a nonsingular matrix B ∈ Fn×n such that A = B−1B∗. Furthermore, A is unitary if and only if there exists a normal, nonsingular matrix B ∈ Fn×n such that A = B−1B∗. Source: [876]. Sufficiency in the second statement follows from Fact 4.13.6. Fact 7.17.6. Let A, B ∈ Fn×n . Then, the following statements hold: i) AB and BA have the same nonsingular Jordan blocks. ii) If either A or B is nonsingular, then AB and BA are similar. iii) If rank AB = rank BA = rank A, then AB and BA are similar. iv) If A and B are normal, then rank AB = rank BA. v) If A and B are normal and rank A ∈ {0, 1, 2, n}, then AB and BA are similar. vi) A and B are Hermitian, then AB and BA are similar. vii) If A is normal and B is positive semidefinite, then AB and BA are similar. i ∞ viii) A and B are similar if and only if (rank (AB)i )∞ i=0 = (rank (BA) )i=0 . ix) If A + A∗ is positive semidefinite, rank(A + A∗ ) = rank A, and B is range Hermitian, then AB

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and BA are similar. x) If A and B are normal and either n ≤ 2 or rank A ≤ 1, then AB and BA are unitarily similar. Source: [1143]. Fact 7.17.7. Let A ∈ Fm×m and B ∈ Fn×n. Then, there exist C ∈ Fm×n and D ∈ Fn×m such that A = CD and B = DC if and only if both of the following statements hold: i) The Jordan blocks associated with nonzero eigenvalues are identical in A and B. ii) Let n1 ≥ n2 ≥ · · · ≥ nr denote the sizes of the Jordan blocks of A associated with 0 ∈ spec(A), and let m1 ≥ m2 ≥ · · · ≥ mr denote the sizes of the Jordan blocks of B associated with 0 ∈ spec(B), where ni = 0 or mi = 0 as needed. Then, |ni − mi | ≤ 1 for all i ∈ {1, . . . , r}. [ ] [ ] [ ] 001 001 000 △ △ Source: [1547, 1858]. Example: Let C = 0 0 0 and D = 1 0 0 . Then, A = CD = 0 0 0 and 010 000 100 [ ] 010 B = DC = 0 0 1 . Then, A2 = B3 = 0, whereas B2 , 0. Related: Fact 7.17.8. 000

Fact 7.17.8. Let A, B ∈ Fn×n, and assume that A and B are nonsingular. Then, A and B are

similar if and only if there exist nonsingular matrices C, D ∈ Fn×n such that A = CD and B = DC. Source: Sufficiency follows from Fact 7.11.14. Necessity is a special case of Fact 7.17.7. Fact 7.17.9. Let A, B ∈ Fn×n, and assume that A and B are nonsingular. Then, det A = det B if and only if there exist nonsingular matrices C, D, E ∈ Rn×n such that A = CDE and B = EDC. Credit: K. Shoda and O. Taussky-Todd. See [569]. Fact 7.17.10. Let A ∈ Fn×m, and assume that rank A = m. Then, there exist a unique matrix B ∈ Fn×m and a matrix C ∈ Fm×m such that B∗B = Im , C is upper triangular with positive diagonal entries, and A = BC. Source: [1448, p. 15] and [2314, p. 206]. Remark: C ∈ UT+ (n). See Fact 4.31.11. Remark: This factorization is a consequence of Gram-Schmidt orthonormalization. Fact 7.17.11. Let A ∈ Fn×n. Then, there exist B, C ∈ Fn×n such that B is unitary, C is upper triangular, and A = BC. If, in addition, A is nonsingular, then there exist unique matrices B, C ∈ Fn×n such that B is unitary, C is upper triangular with positive diagonal entries, and A = BC. Source: [1448, p. 112] and [2314, p. 362]. Remark: This is the QR decomposition. The orthogonal matrix B is constructed as a product of elementary reflectors. △ Fact 7.17.12. Let A ∈ Fn×n, let r = rank A, and assume that the first r leading principal subdeterminants of A are nonzero. Then, there exist B, C ∈ Fn×n such that B is lower triangular, C is upper triangular, and A = BC. Either B or C can be chosen to be nonsingular. Furthermore, both B and C are nonsingular if and only if A is nonsingular. Source: [1448, p. 160]. Remark: This is the LU decomposition. Remark: All LU factorizations of a singular matrix are characterized in [924]. Related: Fact 10.10.42. Fact 7.17.13. Let θ ∈ (−π, π). Then, ][ ][ ] [ ] [ 1 0 1 − tan 2θ cos θ − sin θ 1 − tan 2θ . = sin θ 1 0 sin θ cos θ 0 1 1 Remark: This is a ULU factorization involving three shear factors. The matrix −I2 requires four

shear factors. All shear factors may be different. See [2555, 2684]. Fact 7.17.14. Let A ∈ Fn×n. Then, A is nonsingular if and only if A is a product of elementary matrices. △ Fact 7.17.15. Let A ∈ Fn×n, assume that A is a projector, and let r = rank A. Then, there exist n ∗ nonzero vectors x1 , . . . , xn−r ∈ F such that xi x j = 0 for all i , j and such that A=

n−r [ ∏ i=1

] I − (xi∗xi )−1xi xi∗ .

607

MATRIX DECOMPOSITIONS

Source: A is unitarily similar to diag(1, . . . , 1, 0, . . . , 0), which is a product of elementary projectors. Remark: Every projector is the product of mutually orthogonal elementary projectors. Fact 7.17.16. Let A ∈ Fn×n, and assume that A , I. Then, A is a reflector if and only if there

exist a positive integer m ≤ n and nonzero vectors x1 , . . . , xm ∈ Fn such that xi∗x j = 0 for all i , j and such that m [ ∏ ] A= I − 2(xi∗xi )−1xi xi∗ . i=1

If these conditions hold, then m is the algebraic multiplicity of −1 ∈ spec(A). Source: A is unitarily similar to diag(±1, . . . , ±1), which can be written as the product of elementary reflectors. Remark: Every reflector is the product of mutually orthogonal elementary reflectors. Fact 7.17.17. Let A ∈ Rn×n. Then, A is orthogonal if and only if there exist a positive integer m and nonzero vectors x1 , . . . , xm ∈ Rn such that det A = (−1)m and m [ ∏ ] A= I − 2(xiTxi )−1xi xiT . i=1

If these conditions hold, then the minimal number of factors is rank(A−I). Source: [218, p. 24] and [2388, 2761]. The minimal number of factors is given in [2222, p. 143]. Remark: Every orthogonal matrix is the product of elementary reflectors. Credit: E. Cartan and J. Dieudonn´e. Related: Fact 4.11.5 and Fact 4.19.4. Problem: Extend this result to complex matrices. Fact 7.17.18. Let A ∈ Rn×n, where n ≥ 2. Then, A is orthogonal and det A = 1 if and only if there exist a positive integer m such that 1 ≤ m ≤ n(n − 1)/2, θ1 , . . . , θm ∈ R, and j1, . . . , jm , k1, . . . , km ∈ {1, . . . , n} such that m ∏ A= P(θi , ji , ki ), i=1

where

△

P(θ, j, k) = In + [(cos θ) − 1](E j, j + Ek,k ) + (sin θ)(E j,k − Ek, j ).

Source: [1018]. Remark: P(θ, j, k) is a plane or Givens rotation. See Fact 4.11.5. Remark:

Assume that det A = −1, and let B ∈ Rn×n be an elementary reflector. Then, AB ∈ SO(n). Therefore, the factorization given above holds with an additional elementary reflector. Remark: Related results are given in [1784]. Problem: Extend this result to complex matrices. Fact 7.17.19. Let A ∈ Fn×n. Then, A2∗A = A∗A2 if and only if there exist a projector B ∈ Fn×n and a Hermitian matrix C ∈ Fn×n such that A = BC. Source: [2291]. Fact 7.17.20. Let A ∈ Rn×n. Then, | det A| = 1 if and only if A is the product of n + 2 or fewer involutory matrices that have exactly one negative eigenvalue. In addition, the following statements hold: i) If n = 2, then 3 or fewer factors are needed. ii) If A , αI for all α ∈ R and det A = (−1)n, then n or fewer factors are needed. iii) If det A = (−1)n+1, then n + 1 or fewer factors are needed. Source: [656, 2289]. Remark: See [915] for the minimal number of factors for a unitary matrix. △ △ Fact 7.17.21. Let A ∈ Cn×n, and, for all k ∈ P, define r0 = n and rk = rank Ak . Then, the following statements are equivalent: i) There exists B ∈ Cn×n such that A = B2 . ii) The sequence (rk − rk+1 )∞ k=0 does not contain two components that are the same odd integer and, if r0 − r1 is odd, then 1 + 2r1 ≤ r0 + r2 . Now, assume that A ∈ Rn×n. Then, there exists B ∈ Rn×n such that A = B2 if and only if ii) holds

608

CHAPTER 7

and, for every negative eigenvalue λ of A and for every positive integer k, the Jordan form of A has an even number of k × k blocks associated with λ. Source: [1450, p. 472]. Remark: For all l ≥ 2, △ A = Nl does not have a square root. Remark: Uniqueness is discussed in [1545]. Square roots of A that are functions of A are defined in [1386]. Remark: The principal square root is considered in Theorem 12.9.1. Remark: mth roots are considered in [730, 1391, 2270, 2597]. Related: Fact 15.19.36. Fact 7.17.22. Let A ∈ Cn×n, and assume that A is group invertible. Then, there exists B ∈ Cn×n such that A = B2. Source: Fact 7.17.21 and [795, 2270]. Fact 7.17.23. Let A ∈ Fn×n, and assume that A is nonsingular and has no negative eigenvalues. △ △ n×n n×n Furthermore, define (Pk )∞ and (Qk )∞ by P0 = A, and Q0 = I, and, for all k ≥ 1, k=0 ⊂ F k=0 ⊂ F △

Pk+1 = 21 (Pk + Qk−1 ),

△

Qk+1 = 12 (Qk + P−1 k ).

△

Then, B = limk→∞ Pk exists, satisfies B2 = A, and is the unique square root of A satisfying spec(B) ⊂ ORHP. Furthermore, lim Qk = B−1. k→∞

Source: [875, 1385]. Remark: All indicated inverses exist. Remark: This sequence is related to Newton’s iteration for the matrix sign function. See Fact 12.15.2. Remark: An alternative algorithm for 3 × 3 matrices is given in [175]. Related: Fact 10.10.37. △ Fact 7.17.24. Let A ∈ Fn×n, assume that A is positive semidefinite, and let r = rank A. Then, n×r ∗ there exists B ∈ F such that A = BB . Fact 7.17.25. Let A ∈ Fn×n, and let k ≥ 1. Then, there exists a unique matrix B ∈ Fn×n such that A = B(B∗B)k. Source: [2250]. Fact 7.17.26. Let A ∈ Fn×n. Then, there exist symmetric matrices B, C ∈ Fn×n, at least one of which is nonsingular, such that A = BC. Source: [2263, p. 82]. Remark: Note that  −1      0 1 0   β1 β2 1   −β0 0 0        β2 1  0 1  =  β2 1 0   0  0      0 1 0 1 0 0 −β0 −β1 −β2

and use Theorem 7.3.3. Remark: B and C are symmetric for F = C. Remark: The equality is a Bezout matrix factorization; see Fact 6.8.8. See [527, 528, 1303]. Credit: F. G. Frobenius. Related: Corollary 6.3.12, Corollary 7.4.9, Proposition 7.7.13, and Fact 7.10.9. Fact 7.17.27. Let A ∈ Cn×n. Then, det A is real if and only if A is the product of four Hermitian matrices. Furthermore, four is the smallest number for which the previous statement is true. Source: [2919]. Fact 7.17.28. Let A ∈ Rn×n. Then, the following statements hold: i) A is the product of two positive-semidefinite matrices if and only if A is similar to a positivesemidefinite matrix. ii) If A is nilpotent, then A is the product of three positive-semidefinite matrices. iii) If A is singular, then A is the product of four positive-semidefinite matrices. iv) det A ≥ 0 if and only if A is the product of five positive-semidefinite matrices. v) det A > 0 and A , αI for all α ≤ 0 if and only if A is the product of four positive-definite matrices. vi) det A > 0 if and only if A is the product of five positive-definite matrices. Source: [263, 797, 1303, 2918, 2919]. Remark: See [797, 2919] for extensions to complex

609

MATRIX DECOMPOSITIONS

matrices. Example: [ ] [ ][ −1 0 2 0 5 = 0 −1 0 1/2 7

7 10

][

13/2 −5 −5 4

][

8 5

5 13/4

][

] 25/8 −11/2 . −11/2 10

Fact 7.17.29. Let A ∈ Rn×n. Then, the following statements hold:

i) A = BC, where B ∈ Rn×n is symmetric and C ∈ Rn×n is positive semidefinite, if and only if A2 is diagonalizable over R and spec(A) ⊂ [0, ∞). ii) A = BC, where B ∈ Rn×n is symmetric and C ∈ Rn×n is positive definite, if and only if A is diagonalizable over R. iii) A = BC, where B, C ∈ Rn×n are positive semidefinite, if and only if A = DE, where D ∈ Rn×n is positive semidefinite and E ∈ Rn×n is positive definite. iv) A = BC, where B ∈ Rn×n is positive semidefinite and C ∈ Rn×n is positive definite, if and only if A is diagonalizable over R and spec(A) ⊂ [0, ∞). v) A = BC, where B, C ∈ Rn×n are positive definite, if and only if A is diagonalizable over R and spec(A) ⊂ (0, ∞). Source: [797, 1441, 2910, 2918]. Fact 7.17.30. Let A ∈ Fn×n. Then, A is either singular or In if and only if A is the product of n or fewer idempotent matrices in Fn×n, each of whose rank is equal to rank A. Furthermore, rank(A −I) ≤ kdef A, where k ≥ 1, if and only if A is the product of k idempotent matrices. Source: [165, 271, 841, 999]. Example: [ ] [ ][ ] [ ] [ ][ ] 0 1 1 1/2 0 1/2 2 0 1 1 1 0 = , = . 0 0 0 0 0 1 0 0 0 0 1 0 Fact 7.17.31. Let A ∈ Rn×n, and assume that A is singular and is not a 2 × 2 nilpotent matrix.

Then, there exist nilpotent matrices B, C ∈ Rn×n such that A = BC and rank A = rank B = rank C. Source: [2499, 2917]. See also [2567]. △ Fact 7.17.32. Let A ∈ Fn×n, define r = rank A, let B ∈ Fn×r , let C ∈ Fr×n , and assume that A = BC. Then, A is idempotent if and only if CB = I. Source: [1275, p. 16]. Fact 7.17.33. Let A ∈ Fn×n, and assume that A is idempotent. Then, there exist B, C ∈ Fn×n such that B is positive definite, C is positive semidefinite, and A = BC. Source: iv) of Fact 7.17.29

and [2708]. Fact 7.17.34. Let A ∈ Rn×n, and assume that A is nonsingular. Then, A is similar to A−1 if

and only if A is the product of two involutory matrices. If, in addition, A is orthogonal, then A [is the product ] [ of two ] [ reflectors. ] Source: [269, 912, 2908, 2909] and [2263, p. 108]. Example: cos θ sin θ = 1 0 cos θ sin θ . 0 −1 sin θ − cos θ − sin θ cos θ Fact 7.17.35. Let A ∈ Rn×n. Then, | det A| = 1 if and only if A is the product of four or fewer involutory matrices. Source: [270, 1281, 2498]. Fact 7.17.36. Let A ∈ Rn×n, where n ≥ 2. Then, A is the product of two commutators. Source: [2919]. Fact 7.17.37. Let A ∈ Rn×n, and assume that det A = 1. Then, there exist nonsingular matrices B, C ∈ Rn×n such that A = BCB−1C −1. Source: [2430]. Remark: This is a multiplicative commutator. Remark: For nonsingular A, B, note that [A, B] = 0 if and only if ABA−1B−1 = I. Credit: K. Shoda. Related: Fact 7.17.38. Fact 7.17.38. Let A ∈ Rn×n, assume that A is orthogonal, and assume that det A = 1. Then, there exist reflectors B, C ∈ Rn×n such that A = BCB−1C −1. Source: [2610]. Related: Fact 7.17.37. Fact 7.17.39. Let A ∈ Fn×n, and assume that A is nonsingular. Then, there exist an involutory matrix B ∈ Fn×n and a symmetric matrix C ∈ Fn×n such that A = BC. Source: [1215].

610

CHAPTER 7

Fact 7.17.40. Let A ∈ Fn×n, and assume that n is even. Then, the following statements are

equivalent: i) A is the product of two skew-symmetric matrices. ii) Every elementary divisor of A has even algebraic multiplicity. [ ] iii) There exists B ∈ Fn/2×n/2 such that A is similar to B0 B0 . Remark: In i) the factors are skew symmetric in the case where A is complex. Source: [1216, 2919]. Fact 7.17.41. Let A ∈ Cn×n, and assume that n ≥ 4 and n is even. Then, A is the product of five skew-symmetric matrices in Cn×n. Source: [1720, 1721]. Fact 7.17.42. Let A ∈ Fn×n. Then, there exist a symmetric matrix B ∈ Fn×n and a skewsymmetric matrix C ∈ Fn×n such that A = BC if and only if A is similar to −A. Source: [2322]. Fact 7.17.43. Let A ∈ Fn×n. Then, A is diagonalizable over F with (nonnegative, positive) eigenvalues if and only if there exist (positive-semidefinite, positive-definite) matrices B, C ∈ Fn×n such that A = BC. Source: To prove sufficiency, use Theorem 10.3.6 and A = S −1 (SBS ∗ )(S −∗CS −1 )S. Fact 7.17.44. Let A ∈ Rn×n , and assume that A is a permutation matrix. Then, there exist symmetric permutation matrices A1 , . . . , An−1 ∈ Rn×n such that A = A1 · · · An−1 . Source: [2991, p. 163]. Fact 7.17.45. Let A ∈ Fn×n . Then, there exist Toeplitz matrices A1 , . . . , A2n+5 ∈ Fn×n and Hankel matrices B1 , . . . , B2n+5 ∈ Fn×n such that A = A1 · · · A2n+5 = B1 · · · B2n+5 . Source: [2956].

7.18 Facts on Companion, Vandermonde, Circulant, Permutation, and Hadamard Matrices Fact 7.18.1. Define A ∈ Fn×n by

        △  A =       

0

α1

0

0

0

0

0

0

··· .. . .. .

0

0

α2

0

0

0

0

.. .

.. .

.. .

..

.

..

.. .

0

0

0

···

0

αn−1

−β0

−β1

−β2

···

−βn−2

−βn−1

.

         ,     

and let χA (s) = sn + an−1 sn−1 + · · · + a1 s + a0 . Then, an−1 = βn−1 ,

an−2 = αn−1 βn−2 ,

an−3 = αn−1 αn−2 βn−3 ,

..., △

a0 = αn−1 αn−2 · · · α1 β0 .

Now, assume that α1 , . . . , αn−1 are nonzero, and define S = diag(1, α1 , α1 α2 . . . , α1 α2 · · · αn−1 ). Then, C(χA ) = SAS −1 . Source: [300, pp. 370, 371] and [2966]. Remark: A is a Leslie matrix, which has applications to population biology in the case where A is nonnegative. See [655, pp. 8–10]. In [2966], A is called a generalized Frobenius companion matrix. Fact 7.18.2. Let α1 , . . . , αn , β1 , . . . , βn , γ1 , . . . , γn , a0 , . . . , an−1 ∈ F, assume that α1 , . . . , αn are

611

MATRIX DECOMPOSITIONS

nonzero, and define A ∈ Fn×n by  β 1  − 1  α1 α1   γ β2  2 −  α α 2 2    γ3  0  α 3  △   .. A =  ..  . .   . ..  .. .     0 0    a  − 0 − a1 αn αn

0

0

···

0

1 α2

0

···

0

1 α3

..

.

0

−

β3 α3

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

−

0

···

a2 αn

···

γn−1 αn−1 − △

an−3 αn

−

βn−1 αn−1

γn − an−2 αn

       0     0     . ..  .   ..  .    1   αn−1   βn + an−1  − αn 0

△

Furthermore, define p0 , . . . , pn ∈ F[s] by p(s) = 1, p1 (s) = α1 s + β2 , and, for all i ∈ {2, . . . , n}, △ pi (s) = (αi s + βi )pi−1 (s) − γi pi−2 (s). Then, ( ) 1 χA (s) = ∏n [pn (s) + an−1 pn−1 (s) + · · · + a1 p1 (s) + a0 p0 (s)]. i=1 αi Source: [300, pp. 369–371]. Remark: A is a comrade matrix. Setting αi = 1, βi = 0, and γi = 0 for all i ∈ {1, . . . , n}, yields a companion matrix. Special cases include Leslie, Schwarz, and Routh matrices given, respectively, by Fact 7.18.1, Fact 15.19.25, and Fact 15.19.27. Remark: Choosing αi = 1, βi = 0, and γi = 1 for all i ∈ {1, . . . , n} yields a colleague matrix. Remark: Chebyshev, Legendre, Laguerre, and Hermite polynomials can be obtained with suitable choices of αi , βi , and γi . Remark: An alternative approach to forming the characteristic polynomial is given by the confederate matrix, which is a lower Hessenberg matrix. Companion, confederate, comrade, and colleague matrices are congenial matrices. See [295, 297, 300]. Fact 7.18.3. Let p ∈ F[s], where p(s) = sn + βn−1 sn−1 + · · · + β0 , and define Cb (p), Cr (p), Ct (p), Cl (p) ∈ Fn×n by   1 0 ··· 0 0   0    0 · · · 0 −β0    0 0 ..    0  .  1 0 0 1 0 0  0 · · · 0 −β1         0 1 .. 0 · · · 0 −β2   0 .   0 0 0 0   △ . △  .. ..  , ..  . . . . . , C (p) = Cb (p) =  .  r . . . . .. .. ..  .   . .. ..  . . .   . . .   . .   0 0 . . 0 −β    0 n−2     0 0 0 ··· 0 1     0 0 0 · · · 1 −βn−1 −β0 −β1 −β2 · · · −βn−2 −βn−1

612

CHAPTER 7

  −βn−1    1   .. △  Ct (p) =  .   0   0  0

−βn−2

···

−β2

−β1

0

···

0

0

..

..

.

.. .

.. .

.

0

0

.

0

..

0

..

.

1

0

0

···

0

1

Then,

 −β0    0   ..  .  ,   0    0   0

Cr (p) = CbT (p), ˆ b (p)I,ˆ Ct (p) = IC ˆ

Cl (p) = CbT (p),

  −βn−1   −βn−2  .  . △  Cl (p) =  .  −β 2   −β 1  −β0

0

0

0 .. .

··· .. . .. .

0 .. .

0 .. .

0 0 0

··· ··· ···

0 0 0

1 0 0

1

 0    0   ...   . 0   1   0

Cl (p) = CtT (p), ˆ r (p)I,ˆ Cl (p) = IC ˆ

Ct (p) = CrT (p),

χCb (p) = χCr (p) = χCt (p) = χCl (p) = p. Furthermore,

Cr (p) = SCb (p)S −1 ,

where S, Sˆ ∈ Fn×n are the Hankel matrices   β2 · · · βn−1 1   β1    ..  β . β 1 0  2 3  ..  △  . . . . . . . .  S =  . . . . .  ,   .  β . 0 0   n−1 1 .   1 0 ··· 0 0

ˆ t (p)Sˆ −1, Cl (p) = SC

 0  0   0 0   .. △ . ˆ ˆ ˆ S = IS I =  . ..   0 1   1 βn−1

··· . ..

0 1 . ..

. .. . ..

β3

···

β2

 1    βn−1   ..  .  .  β2   β1

△

Finally, defining βn = 1,

n−1 ∑ i=0

βi+1

i ∑

Cr (p) j E1,nCr (p)i− j = I.

j=0

Source: The last statement is given in [1815]. Remark: (Cb (p), Cr (p), Ct (p), Cl (p)) are the (bottom, right, top, left) companion matrices. Note that Cb (p) = C(p). See [300, p. 282] and [1573, p. 659]. Remark: S = B(p, 1), where B(p, 1) is a Bezout matrix. See Fact 6.8.8. Fact 7.18.4. Let p ∈ F[s], where p(s) = sn + βn−1 sn−1 + · · · + β0 , assume that β0 , 0, and define Cb (p) as in Fact 7.18.3. Then,    −β1/β0 · · · −βn−2/β0 −βn−1/β0 −1/β0     1 ··· 0 0 0     . .. .. ..  .. . Cb−1(p) = Ct ( p) ˆ =  .. . . .  ,    0 ··· 1 0 0     0 ··· 0 1 0 △

n where p(s) ˆ = β−1 0 s p(1/s). Related: Fact 6.9.5.

613

MATRIX DECOMPOSITIONS

Fact 7.18.5. Let n ≥ 2, let λ1 , . . . , λn ∈ F, and define the Vandermonde matrix V(λ1 , . . . , λn ) ∈

Fn×n by

  1  λ  1  2  λ1 △  V(λ1 , . . . , λn ) =  3  λ1  .  .  . λn−1 1

Then,

∏

det V(λ1 , . . . , λn ) =

1 λ2

··· ···

1 λn

λ22

···

λ2n

λ32 .. .

··· . · .· · ···

λ3n .. .

λn−1 2

(λ j − λi ) =

1≤i< j≤n

λn−1 n

n ∏ n ∏

       .     

(λ j − λi ).

i=1 j=i+1

Thus, V(λ1 , . . . , λn ) is nonsingular if and only if λ1 , . . . , λn are distinct. Remark: This result yields Proposition 6.5.4. Let x1 , . . . , xk be eigenvectors of V(λ1 , . . . , λn ) associated with distinct eigenvalues λ1 , . . . , λk of V(λ1 , . . . , λn ). Suppose that α1 x1 + · · · + αk xk = 0 so that V i(λ1 , . . . , λn )(α1 x1 + · · · + △ αk xk ) = α1λi1 xi + · · · + αk λik xk = 0 for all i ∈ {0, 1, . . . , k − 1}. Let X = [x1 · · · xk ] ∈ Fn×k and △ D = diag(α1 , . . . , αk ). Then, XDV T (λ1 , . . . , λk ) = 0, which implies that XD = 0. Hence, αi xi = 0 for all i ∈ {1, . . . , k}, and thus α1 = · · · = αk = 0. Remark: Connections between the Vandermonde matrix and the Pascal matrix, Stirling matrix, Bernoulli matrix, Bernstein matrix, and companion matrices are discussed in [9]. See also Fact 15.12.4. △ Fact 7.18.6. Let n ≥ 2, define V(1, . . . , n) by Fact 7.18.5, and define vn = det V(1, . . . , n). Then, vn =

n−1 ∏

i!.

i=1

In particular, (vi )8i=2 = (1, 2, 12, 288, 34560, 24883200, 125411328000). Furthermore, let k1 < · · · < kn be integers. Then, vn divides det V(k1 , . . . , kn ) Source: [2991, p. 145]. Remark: vn is a superfactorial. See [14]. Example: [det V(−1, 4, 7)]/v3 = 60 and [det V(−4, −1, 5, 8)]/v4 = 4374. Fact 7.18.7. Let n ≥ 2 and 1 ≤ k ≤ n, and define V(1, . . . , n) by Fact 7.18.5. Then, [ ] { } det V(1, . . . , n)[{k,...,n−1},{k+1,...,n}] det V(1, . . . , n)[k,n] n n = , = . k k det V(1, . . . , n − 1) det V(1, . . . , k) Source: [771, p. 227]. △ Fact 7.18.8. Let n and k be positive integers, let a, b ∈ Rn , and define A, B ∈ Rn×n by A = a11×n △

and B = 1n×1 bT . Then,

 [∏ (k)] ∏ k   (−1)⌊n/2⌋  1≤i< j≤n (a( j) − a(i) )(b( j) − b(i) ), k = n − 1, i=1 i   det (A + B)⊙k =     0, k ≤ n − 2. [ ] [ ] 456 111 Source: [2943]. Example: For A = 2 2 2 and B = 4 5 6 , det(A + B) = 0 and det (A + B)⊙2 = −8. 333

456

Fact 7.18.9. Let p ∈ F[s], where p(s) = sn + βn−1 sn−1 + · · · + β1 s + β0 , assume that p has distinct △

roots λ1 , . . . , λn ∈ C, and define V = V(λ1 , . . . , λn ). Then,

C(p) = V diag(λ1 , . . . , λn )V −1 . Consequently, for all i ∈ {1, . . . , n}, λi is an eigenvalue of C(p) with associated eigenvector coli (V). Finally, (VV T )−1CVV T = C T.

614

CHAPTER 7

Source: [295]. Remark: The case where C(p) has repeated eigenvalues is considered in [295]. Fact 7.18.10. Let a1 , . . . , an be distinct complex numbers, let x be a nonzero complex num△

ber, and define A = V(xa1 , . . . , xan )[V(a1 , . . . , an )]−1 . Then, mspec(A) = {xn−1 , xn−2 , . . . , x, 1}ms . Source: [801]. Fact 7.18.11. Let A ∈ Fn×n. Then, A is cyclic if and only if A is similar to a companion matrix. Source: This result follows from Corollary 7.4.3. Alternatively, let spec(A) = {λ1 , . . . , λr } and A = SBS −1, where S ∈ Cn×n is nonsingular and B = diag(B1 , . . . , Br ) is the Jordan form of A, where, for all i ∈ {1, . . . , r}, Bi ∈ Cni ×ni and λi , . . . , λi are the diagonal entries of Bi . Now, define R ∈ Cn×n △ by R = [R1 · · · Rr ] ∈ Cn×n, where, for all i ∈ {1, . . . , r}, Ri ∈ Cn×ni is the matrix   0 ··· 0   1   1 ··· 0   λi   .. .. . ..  · .· · △  . . .   Ri =  ( n−2 ) n−n −1  .  n−2 (n−2) n−3  i ···  λi 1 λi ni −1 λi   ( n−1 ) n−n   n−1 (n−1) n−2 i ··· λi ni −1 λi 1 λi Then, since λ1 , . . . , λr are distinct, it follows that R is nonsingular. Furthermore, C = RBR−1 is in companion form, and thus A = S R−1CRS. If ni = 1 for all i ∈ {1, . . . , r}, then R is a Vandermonde matrix. See Fact 7.18.5 and Fact 7.18.9. Fact 7.18.12. Let λ1 , . . . , λn ∈ F and, for all i ∈ {1, . . . , n}, define n ∏ △ pi (s) = (s − λ j ). j=1 j,i

Furthermore, define A ∈ Fn×n by   p1 (0)   . △  A =  ..   pn (0) Then,

1 ′ 1! p1 (0)

···

 (n−1) 1 (0)  (n−1)! p1

. · .· ·

. · .· ·

.. .

1 ′ 1! pn (0)

···

(n−1) 1 (0) (n−1)! pn

   .   

diag[p1 (λ1 ), . . . , pn (λn )] = AV(λ1 , . . . , λn ).

Source: [1040, p. 159]. Remark: p′ is the derivative of p. Fact 7.18.13. Let n ≥ 1, let a0 , . . . , an−1 ∈ F, and define circ(a0 , . . . , an−1 ) ∈ Fn×n by

       △  circ(a0 , . . . , an−1 ) =      

··· ··· .. .

a0 an−1

a1 a0

a2 a1

an−2 .. .

an−1 .. .

a0 .. .

a2

a3

a4

..

a1

a2

a3

···

..

an−2 an−3

an−1 an−2

.

an−4 .. .

an−3 .. .

.

a0

a1

an−1

a0

        .     

A matrix of this form is circulant. Furthermore, for all n ≥ 2, the n × n primary circulant matrix circ(0, 1, 0, . . . , 0) is the cyclic permutation matrix Pn defined by (4.1.1). Note that circ(1) = P1 = 1.

615

MATRIX DECOMPOSITIONS △

△

Finally, define p(s) = an−1 sn−1 + · · · + a1 s + a0 ∈ F[s], and let θ = e(2π/n) ȷ. Then, the following statements hold: i) p(Pn ) = circ(a0 , . . . , an−1 ). △ ii) Pn = C(q), where q ∈ F[s] is defined by q(s) = sn − 1 and C(q) is the companion matrix associated with q. ∑ n−1−i iii) q(s) = (s − 1)q0 (s), where q0 (s) = n−1 . i=0 s 2 n−1 iv) roots(q0 ) = {θ, θ , . . . , θ }. v) spec(Pn ) = roots(q) = {1, θ, θ2, . . . , θn−1 }. vi) det Pn = (−1)n−1. vii) mspec[circ(a0 , . . . , an−1 )] = {p(1), p(θ), p(θ2 ), . . . , p(θn−1 )}ms . viii) If A, B ∈ Fn×n are circulant and α, β ∈ F, then αA + βB is circulant. ix) If A, B ∈ Fn×n are circulant, then AB is circulant and AB = BA. x) If A is circulant, then A, AT , and A∗ are circulant. xi) If A is circulant and k is a nonnegative integer, then Ak is circulant. xii) If A is nonsingular and circulant, then A−1 is circulant. xiii) A ∈ Fn×n is circulant if and only if A = Pn APTn . xiv) Pn is an orthogonal matrix, and Pnn = In . xv) If A ∈ Fn×n is circulant, then A is reverse symmetric, Toeplitz, and normal. xvi) If A ∈ Fn×n is circulant and A < {αIn : α ∈ F}, then A is irreducible. xvii) A ∈ Fn×n is normal if and only if A is unitarily similar to a circulant matrix. Next, define the Fourier matrix S ∈ Cn×n by   1 1 ··· 1   1    1 θ θ2 · · · θn−1   1  △ S = n−1/2 V(1, θ, . . . , θn−1 ) = √  1 θ2 θ4 · · · θn−2  . n  . .. .. . ..   .. · .· · . . .    n−1 n−2 1 θ θ ··· θ Then, the following statements hold: xviii) S is symmetric and unitary. xix) If n ≥ 3, then S is not Hermitian. xx) S 4 = In and spec(S ) ⊆ {1, −1, ȷ, − ȷ}. xxi) Re S and Im S are symmetric, commute, and satisfy (Re S )2 + (Im S )2 = In . xxii) S −1Pn S = diag(1, θ, . . . , θn−1 ). xxiii) S −1 circ(a0 , . . . , an−1 )S = diag[p(1), p(θ), . . . , p(θn−1 )]. Source: Fact 3.16.2, [34, pp. 81–98], [839, p. 81], [2983, pp. 106–110], and [2991, pp. 138–142]. Remark: Circulant matrices play a role in digital signal processing, specifically, in the efficient implementation of the fast Fourier transform. See [2036, pp. 356–380], [2335], and [2776, pp. 206, 207]. Remark: S is a Fourier matrix and a Vandermonde matrix. Remark: If a real Toeplitz matrix is normal, then it must be either symmetric, skew symmetric, circulant, or skew circulant. See [167, 1019]. Remark: A unified treatment of the solutions of quadratic, cubic, and quartic equations using circulant matrices is given in [1577]. Remark: {I, Pk , P2k , . . . , Pk−1 k } is the multiplication group C(k). See Proposition 4.6.6 and Fact 4.31.17. Remark: Circulant matrices are generalized by cycle matrices, which correspond to visual geometric symmetries. See [1162].

616

CHAPTER 7

Fact 7.18.14. Let A ∈ Rn×n, and assume that A is a permutation matrix. Then, there exists a

permutation matrix S ∈ Rn×n such that

A = S diag(Pn1 , . . . , Pnr )S −1, where, for all i ∈ {1, . . . , r}, Pni is the ni × ni cyclic permutation matrix. Furthermore, the cyclic permutation matrices Pn1 , . . . , Pnr are unique up to a relabeling. Consequently, mspec(A) =

r ∪

{1, θi , . . . , θini −1 }ms ,

i=1 △

△

where θi = e(2π/ni ) ȷ. Hence, det A = (−1)n−r. Finally, An! = I, and m = lcm {n1, . . . , nr } is the smallest positive integer such that Am = I. Source: [839, p. 29]. The last statement follows from [966, pp. 32, 33] and [2991, p. 161]. Remark: This result provides a canonical matrix for permutation matrices under unitary similarity with a permutation matrix. A related result for nonnegative matrices is given by Fact 6.11.6. Remark: It follows that A can be written as the product   ] [ ]  I 0 0  [   I 0 P 0  S −1, A = S n1 · · ·  0 Pni 0  · · · 0 Pnr 0 I   0 0 I where the factors represent disjoint cycles. This factorization reveals the cycle decomposition for an element of the symmetric group Sn (see Fact 4.31.16) on a set having n elements, where Sn is represented by the multiplication group P(n) of n × n permutation matrices. See [966, pp. 29–32], [2345, p. 18], and Fact 4.31.14. Remark: The number of possible canonical matrices for n × n permutation matrices is pn , where pn is the number of integral partitions of n. For example, p1 = 1, p2 = 2, p3 = 3, p4 = 5, and p5 = 7. For all n, pn is given by the generating function 1+

∞ ∑ n=1

pn x n =

1 . (1 − x)(1 − x2 )(1 − x3 ) · · ·

See [118, p. 50] and [3024, p. 138]. Remark: The number of n × n permutation matrices with exactly k blocks is the number of permutations of an n-tuple that have exactly k cycles, which is [ ] given by the cycle number nk . See Fact 1.19.1. Fact 7.18.15. Let A ∈ Rn×n, and assume that A is a permutation matrix. Then, the following statements are equivalent: i) A is irreducible. ii) There exists a permutation matrix S ∈ Rn×n such that SAS −1 is the n × n cyclic permutation matrix Pn . iii) spec(A) = {1, e(2π/n) ȷ , e(4π/n) ȷ , . . . , e[(n−1)2π/n] ȷ }. iv) {k ≥ 1 : Ak = I} = {n, 2n, 3n, . . .}. Source: [2418, p. 177] and [2991, pp. 157, 161]. Fact 7.18.16. Let A ∈ Rn×n, assume that every entry of A is either 1 or −1, and assume that T AA = nI. (A is a Hadamard matrix.) Then, the following statements hold: i) Either n = 1, n = 2, or n is a multiple of 4. ii) n−1/2A is orthogonal. iii) | det A| = nn/2 . [ A] iv) AA −A is a Hadamard matrix. v) If B ∈ Rm×m is a Hadamard matrix, then A ⊗ B is a Hadamard matrix. vi) For every positive integer k, there exists a Hadamard matrix of size 2k.

617

MATRIX DECOMPOSITIONS

[

]

1 is a Source: [1445, p. 10], [2403, pp. 333–335], and [2991, pp. 150–154]. Remark: 11 −1 Hadamard matrix. Remark: It is not known whether there exists a Hadamard matrix for every

integer n that is divisible by 4. See [1445, p. 9]. Fact 7.18.17. Let Sn ⊂ Rn×n denote the set of matrices A such that every entry of A is either −1, 0, or 1, 11×n A = 11×n AT = 11×n , and, in every row and every column of A, every pair of nonzero entries that are either adjacent or separated by zeros have opposite signs. Then, card(Sn ) =

n−1 ∏ (3i + 1)! i=0

(i + n)!

.

Source: [570, 2971]. Remark: This is the alternating sign matrix conjecture. Remark: The set of n×n permutation matrices, whose cardinality is n!, is a proper subset of Sn . Remark: (card(Si ))10 i=1 =

(1, 2, 7, 42, 429, 7436, 218348, 10850216, 911835460, 129534272700).

7.19 Facts on Simultaneous Transformations Fact 7.19.1. Let S ⊂ Cn×n , and assume that there exists S ∈ Cn×n such that, for all A ∈ S,

S AS −1 is upper triangular. Then, there exists a unitary matrix U ∈ Cn×n such that, for all A ∈ S, UAU −1 is upper triangular. If, in addition, every matrix in S is normal, then there exists a unitary matrix U ∈ Cn×n such that, for all A ∈ S, UAU −1 is diagonal. Source: [2432, p. 153]. Fact 7.19.2. Let A, B ∈ Fn×n, and assume that there exists a nonsingular matrix S ∈ Fn×n such that SAS −1 and SBS −1 are upper triangular. Then, A and B have a common eigenvector with corresponding eigenvalues (SAS −1 )(1,1) and (SBS −1 )(1,1) . Source: [1161]. Related: Fact 7.16.2 and Fact 7.16.15. Fact 7.19.3. Let A, B ∈ Cn×n, and assume that PA,B is regular. Then, there exist unitary matrices S 1 , S 2 ∈ Cn×n such that S 1 AS 2 and S 1BS 2 are upper triangular. Source: [2539, p. 276]. Fact 7.19.4. Let A, B ∈ Rn×n, and assume that PA,B is regular. Then, there exist orthogonal matrices S 1 , S 2 ∈ Rn×n such that S 1 AS 2 is upper triangular and S 1BS 2 is upper Hessenberg with 2 × 2 diagonally located blocks. Source: [2539, p. 290]. Credit: C. Moler and G. W. Stewart. Fact 7.19.5. Let S ⊂ Fn×n, and assume that AB = BA for all A, B ∈ S. Then, there exists a unitary matrix S ∈ Fn×n such that, for all A ∈ S, SAS ∗ is upper triangular. Source: [1448, p. 81], [2290], [2432, pp. 153, 154], and [2991, pp. 75–77]. Related: Fact 7.19.10. Fact 7.19.6. Let A, B ∈ Cn×n, and assume that at least one of the following conditions holds: i) [A, [A, B]] = [B, [A, B]] = 0 ii) A[A, B] = [A, B]B = 0. iii) A[A, B] = B[A, B] = 0. iv) rank [A, B] ≤ 1. Then, there exists a nonsingular matrix S ∈ Cn×n such that SAS −1 and SBS −1 are upper triangular. Source: [1161, 1719, 2290, 2440]. Related: Fact 7.16.2. Fact 7.19.7. Let A, B ∈ Cn×n, and assume that A and B are idempotent. Then, [A, B] is nilpotent if and only if there exists a unitary matrix S ∈ Cn×n such that SAS ∗ and SBS ∗ are upper triangular. Source: [2581]. Remark: Necessity follows from Fact 4.22.13. Related: Fact 7.19.5. Fact 7.19.8. Let S ⊂ Fn×n, and assume that every matrix A ∈ S is normal. Then, AB = BA for all A, B ∈ S if and only if there exists a unitary matrix S ∈ Fn×n such that, for all A ∈ S, SAS ∗ is diagonal. Source: [1448, pp. 103, 172]. Related: Fact 7.16.16, Fact 7.19.9, and Fact 10.20.2. Fact 7.19.9. Let S ⊂ Fn×n, and assume that every matrix A ∈ S is diagonalizable over F. Then, AB = BA for all A, B ∈ S if and only if there exists a nonsingular matrix S ∈ Fn×n such that, for all A ∈ S, SAS −1 is diagonal. Source: Fact 7.19.8 and [1448, p. 52].

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CHAPTER 7

Fact 7.19.10. Let A, B ∈ Fn×n, and assume that {x ∈ Fn : x∗Ax = x∗Bx = 0} = {0}. Then, there exists a nonsingular matrix S ∈ Fn×n such that SAS ∗ and SBS ∗ are upper triangular. Source: [2263, p. 96]. Remark: A and B need not be Hermitian. Related: Fact 7.19.5 and Fact 10.20.8. Remark:

In the case where PA,B is a regular pencil, simultaneous triangularization of A and B by means of a biequivalence transformation is given in Proposition 7.8.3.

7.20 Facts on Additive Decompositions Fact 7.20.1. Let A ∈ Fn×n, assume that A is nonsingular, let B, C, D, E ∈ Hn , and assume that

A = B + ȷC and A−1 = D + ȷE. Then, In D = In B. In particular, A is dissipative if and only if A−1 is dissipative. Source: [1501]. Related: Fact 10.10.36, Fact 10.15.2, and Fact 10.21.18. Fact 7.20.2. Let A ∈ Fn×n. Then, there exist a unique Hermitian matrix B ∈ Fn×n and a unique △ △ skew-Hermitian matrix C ∈ Fn×n such that A = B + C. Now, define Bˆ = Re A and Cˆ = Im A so that ˆ ˆ A = B + ȷC. Then, B = 21 (A + A∗ ) = 12 ( Bˆ + Bˆ T ) + ȷ 12 (Cˆ − CˆT ), C = 21 (A − A∗ ) = 12 ( Bˆ − Bˆ T ) + ȷ 21 (Cˆ + CˆT ). Furthermore, A is normal if and only if BC = CB. Related: Fact 4.10.28, Fact 7.20.3, and Fact 15.14.4. Fact 7.20.3. Let A ∈ Fn×n. Then, there exist unique Hermitian matrices B, C ∈ Cn×n such that △ △ ˆ Then, A = B + ȷC. Now, define Bˆ = Re A and Cˆ = Im A so that A = Bˆ + ȷC. B = 12 (A + A∗ ) = 12 ( Bˆ + Bˆ T ) + ȷ 12 (Cˆ − CˆT ), C = 1 (A − A∗ ) = 1 (Cˆ + CˆT ) − ȷ 1 ( Bˆ − Bˆ T ). 2ȷ

2

2

Furthermore, A is normal if and only if BC = CB. Remark: B + ȷC is the Cartesian decomposition of A. Related: Fact 4.10.28, Fact 7.20.2, and Fact 15.14.4. Fact 7.20.4. Let A ∈ R(2n+1)×(2n+1) . Then, there exist a ∈ R, N ∈ R(2n+1)×(2n+1) , and S ∈ (2n+1)×(2n+1) R such that rank N ≤ n, S is skew symmetric, and A = aI + N + S. Source: [2838]. Fact 7.20.5. Let A ∈ Cn×n. Then, there exist unitary matrices B, C ∈ Cn×n such that A = 21 σmax (A)(B + C). Source: [1803, 2976]. Fact 7.20.6. Let A ∈ Rn×n. Then, there exist orthogonal matrices B, C, D, E ∈ Rn×n such that

A = 12 σmax (A)(B + C + D − E). Source: [1803]. See also [2976]. Remark: [1/σmax (A)]A is expressed as an affine combination of

B, C, D, E where the sum of the coefficients 12 , 12 , 12 , − 12 is 1.

△

Fact 7.20.7. Let A ∈ Rn×n, assume that σmax (A) ≤ 1, and define r = rank(I − A∗A). Then, A is a

convex combination of h(r) or fewer orthogonal matrices, where    r ≤ 4, △ 1 + r, h(r) =   3 + ⌊log2 r⌋, r > 4. Source: [1803]. Fact 7.20.8. Let n ≥ 2, and let A ∈ Rn×n. Then, A is the sum of a finite number of real, orthogonal matrices. Source: [2026]. Fact 7.20.9. Let A ∈ Fn×n. Then, the following statements are equivalent:

i) A is positive semidefinite, tr A is a positive integer, and rank A ≤ tr A. ∑ ii) There exist nonzero projectors B1 , . . . , Bl ∈ Fn×n, where l ≤ tr A, such that A = li=1 Bi .

MATRIX DECOMPOSITIONS

619

Source: [1049, 2920]. Remark: For some A, the smallest number of nonzero projectors is tr A. Related: Fact 7.20.12. Fact 7.20.10. Let A ∈ Fn×n, assume that A is Hermitian, 0 ≤ A ≤ I, and tr A is a rational number. Then, A is the average of a finite set of projectors in Fn×n. Source: [726]. Remark: The

required number of projectors can be arbitrarily large. Fact 7.20.11. Let A ∈ Fn×n, assume that A is Hermitian, and assume that 0 ≤ A ≤ I. Then, A is a convex combination of ⌊log2 n⌋ + 2 projectors in Fn×n. Source: [726]. Fact 7.20.12. Let A ∈ Fn×n. Then, the following statements are equivalent: i) tr A is an integer, and rank A ≤ tr A. ∑ ii) There exist idempotent matrices B1 , . . . , Bm ∈ Fn×n such that A = m i=1 Bi . iii) There exist a positive integer m and idempotent matrices B1 , . . . , Bm ∈ Fn×n such that, for ∑ all i ∈ {1, . . . , m}, rank Bi = 1 and R(Bi ) ⊆ R(A), and such that A = m i=1 Bi . ∑ △ n×n iv) There exist idempotent matrices B1 , . . . , Bl ∈ F , where l = tr A, such that A = li=1 Bi . Source: [1337, 2503, 2920]. Remark: The minimal number of idempotent matrices is discussed in [2824]. Related: Fact 7.20.13. Fact 7.20.13. Let A ∈ Fn×n, and assume that 2(rank A − 1) ≤ tr A ≤ 2n. Then, there exist idempotent matrices B, C, D, E ∈ Fn×n such that A = B + C + D + E. Source: [1764]. Related: Fact 7.20.15. Fact 7.20.14. Let A ∈ Fn×n. If n = 2 or n = 3, then there exist b, c ∈ F and idempotent matrices B, C ∈ Fn×n such that A = bB + cC. Furthermore, if n ≥ 4, then there exist b, c, d ∈ F and idempotent matrices B, C, D ∈ Fn×n such that A = bB + cC + dD. Source: [2287]. Fact 7.20.15. Let A ∈ Cn×n, and assume that A is Hermitian. If n = 2 or n = 3, then there exist b, c ∈ C and projectors B, C ∈ Cn×n such that A = bB + cC. Furthermore, if 4 ≤ n ≤ 7, then there exist b, c, d ∈ F and projectors B, C, D ∈ Fn×n such that A = bB + cC + dD. If n ≥ 8, then there exist b, c, d, e ∈ C and projectors B, C, D, E ∈ Cn×n such that A = bB + cC + dD + eE. Source: [2110]. Related: Fact 7.20.13.

7.21 Notes Much of the development in this chapter is based on [2221]. Additional discussions of the Smith and Smith-McMillan forms are given in [1573] and [2999]. The Smith form of a matrix whose entries are integers is considered in [1182, pp. 494–505]. The multicompanion form and the elementary multicompanion form are known as rational canonical forms [966, pp. 472–488], while the multicompanion form is traditionally called the Frobenius canonical form [302]. The derivation of the Jordan form by means of the elementary multicompanion form and the hypercompanion form follows [2221]. Corollary 7.4.9, Corollary 7.4.10, and Proposition 7.7.13 are given in [527, 528, 2591, 2592, 2595]. Corollary 7.4.10 is due to F. G. Frobenius. Canonical forms for congruence transformations are given in [1780, 2617]. The companion matrix is sometimes called a Frobenius matrix, see [9]. It is sometimes useful to define block-companion form matrices in which the scalars are replaced by matrix blocks [1186, 1187, 1189]. A variation of the Jordan form called the Weyr form is discussed in [144, 2166, 2431, 2432]. For Weyr form differs from the Jordan form in terms of the nilpotent component of the canonical form. Matrix pencils are discussed in [187, 341, 487, 1692, 2738, 2759]. Computational algorithms for the Kronecker canonical form are given in [1856, 2772]. Applications to linear system theory are discussed in [688, pp. 52–55] and [1583]. Since a pencil is a polynomial matrix, the terminology “regular pencil” conflicts with “regular polynomial matrix.” The polar decomposition is applied to the elastic deformation of solids in [2203, pp. 140–142].

Chapter Eight Generalized Inverses Generalized inverses provide a useful extension of the matrix inverse to singular matrices and to rectangular matrices that are neither left nor right invertible.

8.1 Moore-Penrose Generalized Inverse Let A ∈ Fn×m. If A is nonzero, then it follows from the singular value decomposition Theorem 7.6.3 that there exist orthogonal matrices S 1 ∈ Fn×n and S 2 ∈ Fm×m such that [ ] B 0r×(m−r) A = S1 S , (8.1.1) 0(n−r)×r 0(n−r)×(m−r) 2 △

△

where B = diag[σ1 (A), . . . , σr (A)], r = rank A, and σ1 (A) ≥ σ2 (A) ≥ · · · ≥ σr (A) > 0 are the positive singular values of A. In (8.1.1), some of the border zero matrices may be empty. Then, the (Moore-Penrose) generalized inverse A+ of A is the m × n matrix [ ] B−1 0r×(n−r) △ A+ = S 2∗ S∗. (8.1.2) 0(m−r)×r 0(m−r)×(n−r) 1 △

If A = 0n×m , then A+ = 0m×n , while, if m = n and det A , 0, then A+ = A−1. It is helpful to remember that A+ and A∗ are the same size. Note that A+ satisfies AA+A = A, A+AA+ = A+,

(8.1.3) (8.1.4)

(AA+ )∗ = AA+, (A+A)∗ = A+A.

(8.1.5) (8.1.6)

Hence, for each A ∈ Fn×m there exists X ∈ Fm×n satisfying the four statements AXA = A, XAX = X, (AX)∗ = AX,

(8.1.7) (8.1.8) (8.1.9)

(XA)∗ = XA.

(8.1.10)

We now show that X is uniquely defined by (8.1.7)–(8.1.10). Theorem 8.1.1. Let A ∈ Fn×m. Then, X = A+ is the unique matrix X ∈ Fm×n satisfying (8.1.7)– (8.1.10). Proof. Suppose there exists X ∈ Fm×n satisfying (8.1.7)–(8.1.10). Then, X = XAX = X(AX)∗ = XX ∗A∗ = XX ∗(AA+A)∗ = XX ∗A∗A+∗A∗ = X(AX)∗ (AA+ )∗ = XAXAA+ = XAA+ = (XA)∗A+ = A∗X ∗A+ = (AA+A)∗X ∗A+ = A∗A+∗A∗X ∗A+ = (A+A)∗ (XA)∗A+ = A+AXAA+ = A+AA+ = A+.



Given A ∈ Fn×m, X ∈ Fm×n is a (1)-inverse of A if (8.1.7) holds, a (1,2)-inverse of A if (8.1.7) and (8.1.8) hold, and so forth.

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Proposition 8.1.2. Let A ∈ Fn×m, assume that A is left invertible, and let X ∈ Fm×n. Then, the

following statements are equivalent: i) X is a left inverse of A. ii) X is a (1)-inverse of A. iii) X is a (1,2,4)-inverse of A. Proposition 8.1.3. Let A ∈ Fn×m, assume that A is right invertible, and let X ∈ Fm×n. Then, the following statements are equivalent: i) X is a right inverse of A. ii) X is a (1)-inverse of A. iii) X is a (1,2,3)-inverse of A. Proof. To prove i) =⇒ ii), note that AX = In, and thus AXA = A, which implies that X is a (1)-inverse of A. Conversely, since X is a (1)-inverse of A, it follows that AXA = A. Then, letting AR ∈ Fm×n denote a right inverse of A, it follows that In = AAR = AXAAR = AX. Hence, X is a right inverse of A. Finally, to prove ii) =⇒ iii), note that, since X is a right inverse of A, it is also a (2,3)-inverse of A.  It can now be seen that A+ is a particular (left, right) inverse if A is (left, right) invertible. Corollary 8.1.4. Let A ∈ Fn×m. If A is left invertible, then A+ is a left inverse of A. Furthermore, if A is right invertible, then A+ is a right inverse of A. The following result provides an explicit expression for A+ in the case where A is either left invertible or right invertible. It is helpful to note that A is (left, right) invertible if and only if (A∗A, AA∗ ) is positive definite. Proposition 8.1.5. Let A ∈ Fn×m. If A is left invertible, then A+ = (A∗A)−1A∗

(8.1.11)

+

and A is a left inverse of A. If A is right invertible, then A+ = A∗ (AA∗ )−1

(8.1.12)

+

and A is a right inverse of A. Proof. It suffices to verify (8.1.7)–(8.1.10) with X = A+. Corollary 8.1.6. Let x ∈ Fn , and assume that x is nonzero. Then, ∗ x+ = ∥x∥−2 2 x .

Proposition 8.1.7. Let A ∈ Fn×m. Then, the following statements hold:

i) ii) iii) iv) v) vi) vii) viii) ix) x) xi)

A = 0 if and only if A+ = 0. (A+ )+ = A. + A = A+ . △ A+T = (AT )+ = (A+ )T. △ A+∗ = (A∗ )+ = (A+ )∗. R(A) = R(AA∗ ) = R(AA+ ) = R(A+∗ ) = N[(AA+ )⊥ ] = N(A∗ )⊥. R(A∗ ) = R(A∗A) = R(A+A) = R(A+ ) = N[(A+A)⊥ ] = N(A)⊥. N(A) = N(A+A) = N(A∗A) = N(A+∗ ) = R[(A+A)⊥ ] = R(A∗ )⊥. N(A∗ ) = N(AA+ ) = N(AA∗ ) = N(A+ ) = R[(AA+ )⊥ ] = R(A)⊥. AA+ and A+A are positive semidefinite. spec(AA+ ) ⊆ {0, 1} and spec(A+A) ⊆ {0, 1}.

 (8.1.13)

GENERALIZED INVERSES

623

AA+ is the projector onto R(A). A+A is the projector onto R(A∗ ). (A+A)⊥ is the projector onto N(A). (AA+ )⊥ is the projector onto N(A∗ ). x ∈ R(A) if and only if x = AA+x. rank A = rank A+ = rank AA+ = rank A+A = tr AA+ = tr A+A. If A is idempotent, then rank A = tr AA+ = tr A2A+ = tr AA+A = tr A. rank (AA+ )⊥ = n − rank A. rank (A+A)⊥ = m − rank A. A = AA∗A∗+ = A∗+A∗A = A(A∗A)+A∗A = AA∗A(A∗A)+ = AA∗(AA∗ )+A = (AA∗ )+AA∗A. A∗ = A∗AA+ = A+AA∗. A+ = A∗(AA∗ )+ = (A∗A)+A∗ = A+A∗+A∗ = A∗A∗+A+ = A∗(A∗AA∗ )+A∗. A+∗ = (AA∗ )+A = A(A∗A)+ = AA+A+∗ = A+∗A+A = A(AA∗A)+A. (AA∗ )+ = A+∗A+. (A∗A)+ = A+A+∗. AA+ = (AA+ )∗ = A+∗A∗ = A(A∗A)+A∗ = AA∗ (AA∗ )+ = (AA∗ )+AA∗ = ([(AA+ )⊥ ]+ (AA+ )⊥ )⊥ . A+A = (A+A)∗ = A∗A+∗ = A∗(AA∗ )+A = A∗A(A∗A)+ = (A∗A)+A∗A = ([(A+A)⊥ ]+ (A+A)⊥ )⊥ . If S 1 ∈ Fn×n and S 2 ∈ Fm×m are unitary, then (S 1 AS 2 )+ = S 2∗ A+S 1∗. A is (range Hermitian, normal, Hermitian, skew Hermitian, positive semidefinite, positive definite) if and only if A+ is. xxxi) The following statements are equivalent: a) A is range Hermitian. b) R(A) = R(A+ ). c) AA+ = A+A. xxxii) If A is a projector, then A+ = A. xxxiii) If A is normal and k ≥ 1, then (Ak )+ = (A+ )k . xxxiv) If B ∈ Fn×m, then A = B if and only if A+ = B+ . xxxv) If B ∈ Fm×l and AB = 0, then B+A+ = 0. xxxvi) If B ∈ Fn×l , then A∗B = 0 if and only if A+B = 0. xxxvii) If B ∈ Fn×l , then R(AA+B) is the projection of R(B) into R(A). xxxviii) If B ∈ Fm×l , C ∈ Fn×l , and A∗AB = A∗C, then AB = AA+C. xxxix) If B ∈ Fl×m, then AB∗ = 0 if and only if A+AB+B = 0. xl) If S ∈ Fm×m is nonsingular, then AS (AS )+ = AA+ . Proof. The last three statements are given in [2238, pp. 186, 188, 312].  n×m n Theorem 3.7.5 shows that the equation Ax = b, where A ∈ F and b ∈ F , has a solution x ∈ Fm if and only if rank A = rank [A b]. In particular, Ax = b has a unique solution x ∈ Fm if and only if rank A = rank [A b] = m, while Ax = b has infinitely many solutions if and only if rank A = rank [A b] < m, and at least one solution if and only if b ∈ R(A). The following result expresses the last condition in terms of the (1)-inverse and the generalized inverse. Proposition 8.1.8. Let A ∈ Fn×m and b ∈ Fn. Then, the following statements are equivalent: i) b ∈ R(A). ii) If B ∈ Fm×n is a (1)-inverse of A, then ABb = b. xii) xiii) xiv) xv) xvi) xvii) xviii) xix) xx) xxi) xxii) xxiii) xxiv) xxv) xxvi) xxvii) xxviii) xxix) xxx)

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iii) There exists a (1)-inverse B ∈ Fm×n of A such that ABb = b. iv) AA+b = b. Proof. To show that i) =⇒ ii), let B ∈ Fm×n be a (1)-inverse of A, and let x ∈ Fm satisfy △ Ax = b. Then, ABb = ABAx = Ax = b. ii) =⇒ iii) is immediate. Next, iii) implies that x = Bb m satisfies Ax = b, and thus i) holds. Next, to show that i) =⇒ iv), let x ∈ F satisfy Ax = b. Then, △ AA+b = AA+Ax = Ax = b. Finally, iv) implies that x = A+b satisfies Ax = b, and thus i) holds.  Proposition 8.1.8 shows that, if B is a (1)-inverse of A, then Bb is a solution of Ax = b. However, it is not true that every solution x of Ax = b is of the form x = Bb for some (1)-inverse[ B] of A. For example, let A = [1 0] and b = 0 so that every (1)-inverse B of A is of the form B = β1 for some [ ] β ∈ F. Then, x = 01 satisfies Ax = b, but Bb = 0 , x for every (1)-inverse B of A. The following result assumes that Ax = b has at least one solution and characterizes all solutions in terms of the (1)-inverse and the generalized inverse. The case where A is right invertible is considered in Theorem 3.7.5. Proposition 8.1.9. Let A ∈ Fn×m and b ∈ Fn, assume that b ∈ R(A), let B ∈ Fm×n be a (1)inverse of A, and let x ∈ Fm . Then, the following statements are equivalent: i) Ax = b. ii) BAx = Bb. iii) x = Bb + (I − BA)x. iv) There exists y ∈ Fm such that x = Bb + (I − BA)y. v) A+Ax = A+b. vi) x = A+b + (I − A+A)x. vii) There exists y ∈ Fm such that x = A+b + (I − A+A)y. Now, assume that these statements hold and rank A = m. Then, the following statements hold: viii) If AL ∈ Fm×n is a left inverse of A, then x = AL b. ix) B is a (1)-inverse of A, and x = Bb. x) x = A+b. Proof. Immediately, i) =⇒ ii) =⇒ iii) =⇒ iv). To show that iv) =⇒ i), note that Ax = ABb + A(I − BA)y = ABb = b. Next, it follows from Proposition 8.1.8 that AA+b = b. Therefore, setting B = A+ yields i) =⇒ v) =⇒ vi) =⇒ vii) =⇒ i). viii) is immediate. To prove ix), note that Proposition 8.1.2 implies that B is a left inverse of A. Finally, x) follows from the fact that A+ is a left inverse of A.  Let y ∈ Fm and x = A+b + (I − A+A)y. Then, x∗x = b∗A+∗A+b + y∗(I − A+A)y. Therefore, x∗x is minimized by y = 0. Connections to least squares solutions are discussed in Fact 11.17.10. The following result extends Proposition 3.7.10. Proposition 8.1.10. Let A ∈ Fn×m and S ⊆ Fn. Then, Ainv (S) = A+ [S ∩ R(A)] + R(I − A+A).

(8.1.14)

AAinv (S) = S ∩ R(A).

(8.1.15)

Hence,

If A is right invertible, then AAinv (S) = S, +

+

+

(8.1.16)

A S ⊆ A S + R(I − A A) = A (S). inv

(8.1.17)

625

GENERALIZED INVERSES △

Proof. Let x ∈ Ainv (S). Then, y = Ax ∈ S. It thus follows from vii) of Proposition 8.1.9 that

there exists z ∈ F such that x = A y + (I − A+A)z ∈ A+ [S ∩ R(A)] + R(I − A+A). Conversely, let x ∈ A+ [S ∩ R(A)] + R(I − A+A). Then, there exist y ∈ S ∩ R(A) and z ∈ R(I − A+A) such that x = A+ y + (I − A+A)z. Hence, Ax = AA+ y + A(I − A+A)z = y.  The last statement of Proposition 8.1.10 is given by (3.7.9) of Proposition 3.7.10. The following corollary of Proposition 8.1.10 characterizes the solution set of Ax = b. Corollary 8.1.11. Let A ∈ Fn×m and b ∈ Fn. Then, the set of solutions x to the equation Ax = b is given by  + +    A b + R(I − A A), b ∈ R(A), inv A ({b}) =  (8.1.18)   ∅, b < R(A). m

+

The following result provides an expression for the generalized inverse in terms of the full-rank factorization given by Proposition 7.6.6. Proposition 8.1.12. Let A ∈ Fn×m , B ∈ Fn×r , and C ∈ Fr×m, and assume that rank A = rank B = r and A = BC. Then, A+ = C +B+ = C ∗ (CC ∗ )−1(B∗B)−1B∗ , AA+ = BB+ , A+A = C +C.

(8.1.19) (8.1.20)

A+ = C −1(B∗B)−1B∗.

(8.1.21)

A+ = C ∗ (CC ∗ )−1B−1.

(8.1.22)

If m = r, then If n = r, then If n = m = r, then A−1 = C −1B−1.

(8.1.23) ] △ [ Definition 8.1.13. Let A ∈ Fn×m, B ∈ Fn×l, C ∈ Fk×m, and D ∈ Fk×l, and define A = CA DB ∈ F(n+k)×(m+l). Then, the Schur complement A|A of A with respect to A is defined by △

A|A = D − CA+B.

(8.1.24)

Likewise, the Schur complement D|A of D with respect to A is defined by △

D|A = A − BD+C.

(8.1.25)

8.2 Drazin Generalized Inverse We now introduce a different type of generalized inverse, which applies only to square matrices yet is more useful in certain applications. Let A ∈ Fn×n. Then, A has a decomposition [ ] J1 0 −1 S , (8.2.1) A=S 0 J2 where S ∈ Fn×n is nonsingular, J1 ∈ Fm×m is nonsingular, and J2 ∈ F(n−m)×(n−m) is nilpotent. Then, the Drazin generalized inverse AD of A is the matrix ] [ −1 J1 0 −1 D △ S . (8.2.2) A =S 0 0 Let A ∈ Fn×n. Then, it follows from Definition 7.7.1 that ind A = indA (0). Furthermore, A is nonsingular if and only if ind A = 0, whereas ind A = 1 if and only if A is singular and the zero

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eigenvalue of A is semisimple. In particular, ind 0n×n = 1. Note that ind A is the largest size of all of the Jordan blocks of A associated with the zero eigenvalue of A. It can be seen that AD satisfies ADAAD = AD,

(8.2.3)

AA = A A,

(8.2.4)

A A =A,

(8.2.5)

D

D

k+1 D

k

where k = ind A. Hence, for all A ∈ Fn×n such that ind A = k there exists X ∈ Fn×n satisfying the three statements XAX = X,

(8.2.6)

AX = XA,

(8.2.7)

X=A.

k+1

A

k

(8.2.8)

We now show that X is uniquely defined by (8.2.6)–(8.2.8). △ Theorem 8.2.1. Let A ∈ Fn×n, and define k = ind A. Then, X = AD is the unique matrix n×n X ∈ F satisfying (8.2.6)–(8.2.8). Proof. Let X[ ∈ Fn×n satisfy (8.2.6)–(8.2.8). If k = 0, then it follows from (8.2.8) that X = A−1. ] Hence, let A = S J01 J02 S −1, where k = ind A ≥ 1, S ∈ Fn×n is nonsingular, J1 ∈ Fm×m is nonsingular, [ˆ ˆ ] △ and J2 ∈ F(n−m)×(n−m) is nilpotent. Now, let Xˆ = S −1XS = XXˆ 1 XXˆ12 be partitioned conformably with 21 2 [ ] △ ˆ it follows that J1 Xˆ1 = Xˆ1J1, J1 Xˆ 12 = Xˆ 12 J2 , Aˆ = S −1AS = J01 J02 . Since, by (8.2.7), Aˆ Xˆ = Xˆ A, J2 Xˆ 21 = Xˆ 21 J1, and J2 Xˆ 2 = Xˆ 2 J2 . Since J2k = 0, it follows that J1 Xˆ 12 J2k−1 = 0, and thus Xˆ 12 J2k−1 = 0. By repeating this argument, it follows that J1Xˆ 12 J2 = 0, and thus[ Xˆ 12 ]J2 = 0, which implies that ˆ J1 Xˆ 12 = 0, and thus Xˆ 12 = 0. Similarly, Xˆ 21 = 0, so that Xˆ = X01 Xˆ0 . Now, (8.2.8) implies that 2 J1k+1Xˆ 1 = J1k, and hence Xˆ 1 = J1−1. Next, (8.2.6) implies that Xˆ 2 J2 Xˆ 2 = Xˆ 2 , which, together with ˆ2 k ˆ k−1 J2 Xˆ 2 = Xˆ 2 J2 , yields Xˆ 22 J2 = Xˆ 2 . [ Consequently, ] [ ˆ0 = ] X2 J2 = X2 J2 , and thus, by repeating this −1 J 0 ˆ −1 = X. argument, Xˆ 2 = 0. Hence, AD = S 1 S −1 = S X1 0 S −1 = S XS  Proposition 8.2.2. Let A ∈ F

i) ii) iii) iv) v) vi) vii) viii) ix) x) xi) xii) xiii) xiv) xv)

D

0 0 n×n

0 0

△

, and define k = ind A. Then, the following statements hold:

A = AD . △ △ ADT = ATD = (AT )D = (AD )T. △ △ AD∗ = A∗D = (A∗ )D = (AD )∗. △ △ If r ∈ P, then ADr = ArD = (AD )r = (Ar )D. R(Ak ) = R(AD ) = R(AAD ) = N(I − AAD ). N(Ak ) = N(AD ) = N(AAD ) = R(I − AAD ). rank Ak = rank AD = rank AAD = def(I − AAD ). def Ak = def AD = def AAD = rank(I − AAD ). AAD is the idempotent matrix onto R(AD ) along N(AD ). AD = 0 if and only if A is nilpotent. AD is group invertible. ind AD = 0 if and only if A is nonsingular. ind AD = 1 if and only if A is singular. (AD )D = (AD )# = A2AD . The following statements are equivalent:

627

GENERALIZED INVERSES

a) A is group invertible. b) (AD )D = A. c) (AD )# = A. d) A2AD = A. xvi) [(AD )D ]D = [(AD )# ]D = (A2AD )D = AD . xvii) If A is idempotent, then k = 1 and AD = A. xviii) A = AD if and only if A is tripotent. xix) A − A2AD is nilpotent, and ind(A − A2AD ) = k. xx) (A − A2AD )A2AD = A2AD (A − A2AD ) = 0. xxi) (A − A2AD )AD = AD (A − A2AD ) = 0. xxii) rank A2AD = n − amultA (0), and def A2AD = amultA (0). xxiii) rank(A − A2AD ) = amultA (0) − gmultA (0), and def(A − A2AD ) = n − amultA (0) + gmultA (0). xxiv) Let X, Y ∈ Fn×n be such that XY = Y X = 0, ind X ≤ 1, Y is nilpotent, and ind Y = k. Then, X = A2AD and Y = A − A2AD . Let A ∈ Fn×n, and assume that ind A ≤ 1 so that, by Corollary 7.7.9, A is group invertible. In this case, the Drazin generalized inverse AD is denoted by A#, which is the group generalized inverse of A. Therefore, A# satisfies A#AA# = A#,

(8.2.9)

AA# = A#A,

(8.2.10)

AA A = A,

(8.2.11)

#

while A is the unique matrix X ∈ F #

n×n

satisfying XAX = X, AX = XA,

(8.2.12) (8.2.13)

AXA = A.

(8.2.14)

Proposition 8.2.3. Let A ∈ Fn×n, and assume that A is group invertible. Then, the following

statements hold: # i) A = A# . △ △ ii) A#T = AT# = (AT )# = (A# )T. △ △ iii) A#∗ = A∗# = (A∗ )# = (A# )∗. △ △ iv) If r ∈ P, then A#r = Ar# = (A# )r = (Ar )#. v) R(A) = R(AA# ) = N(I − AA# ) = R(AA+ ) = N(I − AA+ ). vi) N(A) = N(AA# ) = R(I − AA# ) = N(A+A) = R(I − A+A). vii) rank A = rank A# = rank AA# = rank A#A. viii) def A = def A# = def AA# = def A#A. ix) AA# is the idempotent matrix onto R(A) along N(A). x) A# = 0 if and only if A = 0. xi) A# is group invertible. xii) (A# )# = A. xiii) If X ∈ Fn×n is a (1)-inverse of A3 , then A# = AXA. xiv) If S ∈ Fn×n is nonsingular, then (SAS −1 )# = SA# S −1 . An alternative expression for the idempotent matrix onto R(A) along N(A) is given by Proposi-

628

CHAPTER 8

tion 4.8.11.

8.3 Facts on the Moore-Penrose Generalized Inverse for One Matrix Fact 8.3.1. Let A ∈ Fn×m . Then, A has a unique (1)-inverse if and only if n = m and A is nonsingular. Source: [281, p. 8]. Fact 8.3.2. Let A ∈ Fn×m , and let B ∈ Fm×n be a (1)-inverse of A. Then, AB and BA are

idempotent. Fact 8.3.3. Let A ∈ Rn×m , and let B ∈ Rm×n be a (1)-inverse of A. Then, the following statements are equivalent: i) AATBTB = AB. ii) tr AATBTB = tr AB. iii) ATBT = BA. Source: [2703]. Related: Fact 3.15.37. △ Fact 8.3.4. Let A ∈ Fn×m , let B ∈ Fm×n be a (1)-inverse of A, and define G ∈ Fm×n by G = BAB. Then, G is a (1,2)-inverse of A. Source: [281, p. 8]. △ Fact 8.3.5. Let A ∈ Fn×m , define r = rank A, let F ∈ Fn×r , let G ∈ Fr×m , assume that A = FG, let B ∈ Fr×n , and let C ∈ Fm×r . Then, the following statements are equivalent: i) BAC is nonsingular. ii) BF and GC are nonsingular. △ Now, assume that these statements hold, and define X = C(BAC)−1B. Then, the following statements hold: iii) X is a (1,2)-inverse of A. iv) If B = F ∗ , then X is a (1,2,3)-inverse of A. v) If C = G∗ , then X is a (1,2,4)-inverse of A. vi) If B = F ∗ and C = G∗ , then X = A+ . Source: [2238, p. 220]. Fact 8.3.6. Let A ∈ Fn×m and B ∈ Fm×n . Then, B is a (1,3)-inverse of A if and only if AB = AA+ . If these conditions hold, then the following statements hold: i) I − AB is a projector. ii) (I − AB)∗A = 0. iii) Let b ∈ Fn . Then, for all x ∈ Fm , ∥ABb − b∥2 ≤ ∥Ax − b∥2 . Source: [2238, p. 285]. Remark: Bb is a least-squares solution of Ax = b. See Fact 11.17.6. Fact 8.3.7. Let A ∈ Fn×m and B ∈ Fm×n . Then, B is a (1,4)-inverse of A if and only if BA = A+A. If these conditions hold, then the following statements hold: i) I − BA is a projector. ii) (I − BA)A∗ = 0. iii) Let b ∈ R(A). Then, Bb = A+ b, and, for all x ∈ Fm such that Ax = b and x , Bb, ∥Bb∥2 < ∥x∥2 . Source: [2238, p. 284]. Remark: Bb is a minimum-norm solution of Ax = b. See Fact 11.17.7. Fact 8.3.8. Let A ∈ Fn×m and B, C ∈ Fm×n , assume that B is a (1,3)-inverse of A, and assume that C is a (1,4)-inverse of A. Then, A+ = CAB. Source: [360, p. 48] and [2238, p. 285]. Credit: N. S. Urquhart. Fact 8.3.9. Let A ∈ Fn×m and D ∈ Fm×n . Then, the following statements are equivalent: i) D is a (2)-inverse of A. ii) There exist projectors C ∈ Fn×n and B ∈ Fm×m such that D = (CAB)+ .

629

GENERALIZED INVERSES

Source: [2238, p. 297]. Fact 8.3.10. Let A ∈ Fn×m . Then, the following statements hold:

i) If B ∈ Fn×n is a (1,2)-inverse of AA∗ , then A∗B is a (1,2)-inverse of A. ii) If B ∈ Fm×m is a (1,2)-inverse of A∗A, then BA∗ is a (1,2)-inverse of A. Source: [2238, p. 289]. Fact 8.3.11. Let A ∈ Fn×m, B ∈ Fk×n, and C ∈ Fm×l, and assume that B is left inner and C is right inner. Then, (BAC)+ = C ∗A+B∗. Source: [1343, p. 506]. Fact 8.3.12. Let A ∈ Fn×m, assume that A is right invertible, let AR ∈ Fm×n , and assume that AR is a right inverse of A. Then, AR = A+ if and only if ARA is Hermitian. Fact 8.3.13. Let A ∈ Fn×m, assume that A is left invertible, let AL ∈ Fm×n , and assume that AL is a left inverse of A. Then, AL = A+ if and only if AAL is Hermitian. Fact 8.3.14. Let A ∈ Fn×m, assume that A is left invertible, and let AL ∈ Fm×n be a left inverse of A. Then, B ∈ Fm×n is a left inverse of A if and only if there exists S ∈ Fm×n such that B = AL + S (In − AA+ ). Related: Fact 3.18.3, Fact 11.17.4, and [2238, p. 150]. Fact 8.3.15. Let A ∈ Fn×m, assume that A is right invertible, and let AR ∈ Fm×n be a right inverse of A. Then, B ∈ Fm×n is a right inverse of A if and only if there exists S ∈ Fm×n such that B = AR + (Im − A+A)S. Related: Fact 3.18.4 and Fact 11.17.5. Fact 8.3.16. Let A ∈ Fn×m, b ∈ Fn, and y ∈ Fm, assume that A is right invertible, and define △ x = A+b + (I − A+A)y, which satisfies Ax = b. Furthermore, let z = (I − A+A)y, and assume that there m×n T exists S ∈ F such that z Sb , 0, and define AR = A+ +

1 zzTS. zTSb

Then, AR ∈ Fm×n is a right inverse of A, and x = AR b. Related: See the comments following Theorem 3.7.5. Problem: Assuming that A is right invertible, find necessary and sufficient conditions under which every solution x of Ax = b is given by x = AR b for some right inverse of A. Fact 8.3.17. Let A ∈ Fn×m, and assume that rank A = 1. Then, A+ = (tr AA∗ )−1A∗. Consequently, if x ∈ Fn and y ∈ Fn are nonzero, then (xy∗ )+ = (x∗xy∗y)−1 yx∗ =

1 ∥x∥22 ∥y∥22

yx∗.

In particular,

1 1m×n . nm Fact 8.3.18. Let x ∈ Fn, and assume that x is nonzero. Then, the projector A ∈ Fn×n onto span {x} is given by A = (x∗x)−1xx∗. 1+n×m =

Fact 8.3.19. Let x, y ∈ Fn, assume that x, y are nonzero, and assume that x∗y = 0. Then, the

projector A ∈ Fn×n onto span {x, y} is given by

A = (x∗x)−1xx∗ + (y∗y)−1yy∗. Fact 8.3.20. Let x, y ∈ Fn, and assume that x, y are linearly independent. Then, the projector

A ∈ Fn×n onto span {x, y} is given by

A = (x∗xy∗ y − |x∗y|2 )−1 (y∗ yxx∗ − y∗ xyx∗ − x∗ yxy∗ + x∗xyy∗ ).

630

CHAPTER 8 △

Furthermore, define z = [I − (x∗x)−1 xx∗ ]y. Then, A = (x∗x)−1 xx∗ + (z∗z)−1zz∗. Remark: For F = R, this result is given in [2474, p. 178]. Fact 8.3.21. Let n ≥ 2, let A ∈ Fn×n, assume that rank A = n − 1, let x ∈ N(A) be nonzero, let

y ∈ N(A∗ ) be nonzero, let α = 1 if spec(A) = {0} and the product of the nonzero eigenvalues of A △ otherwise, and define k = amultA (0). Then, AA = In particular,

(−1)k+1α ∗ xy . y∗(Ak−1 )+ x

NnA = (−1)n+1E1,n .

If, in addition, k = 1, then

AA =

α ∗ xy . y∗ x

Source: [1924, pp. 40, 41] and Fact 4.22.4. Remark: This result provides an expression for ii) of Fact 3.19.3. Remark: If A is range Hermitian, then N(A) = N(A∗ ) and y∗x , 0, and thus Fact 7.15.5 implies that AA is semisimple. Related: Fact 7.15.30. Fact 8.3.22. Let A ∈ Fn×m, and assume that rank A = n − 1. Then,

A+ =

1 A∗ [AA∗ + (AA∗ )A ]A. det[AA∗ + (AA∗ )A ]

Source: [772]. Remark: Extensions to matrices of arbitrary rank are given in [772]. Expressions

for the generalized inverse in terms of submatrices are given in [2822]. △ △ Fact 8.3.23. Let A ∈ Fn×n, assume that A is nonzero, and define r = rank A, and B = diag[σ1 (A), . . . , σr (A)]. Then, there exist S ∈ Fn×n , K ∈ Fr×r , and L ∈ Fr×(n−r) such that S is unitary, KK ∗ + LL∗ = Ir , and    ∗  BK BL   S . A = S  0(n−r)×r 0(n−r)×(n−r) Furthermore,

 ∗ −1  K B A = S  ∗ −1 LB +

 0r×(n−r)  ∗  S .

0(n−r)×(n−r)

Source: Fact 7.10.27 and [240, 1338]. Related: Fact 8.5.13. Fact 8.3.24. Let A ∈ Fn×n . Then,

rank [A, A+ ] = 2 rank [A A∗ ] − 2 rank A, rank [AA+ , A+A] = 2 rank [A A∗ ] + 2 rank A2 − 4 rank A. If, in addition, k ≥ 1, then rank [Ak , A+ ] = rank

[

] Ak + rank [Ak A∗ ] − 2 rank A. A∗

Source: [2657, 2672]. Fact 8.3.25. Let A ∈ Fn×m and B ∈ Fn×n . Then, the following statements are equivalent:

i) B = AA+. ii) R(B) = R(A) and B = BB∗. iii) R(B) ⊆ R(A) and A∗B = A∗.

631

GENERALIZED INVERSES

iv) R(B∗ ) ⊆ R(A) and BA = A. v) For all X ∈ Fn×n such that XA = A, tr BB∗ ≤ tr XX ∗. vi) A∗B = A∗ , BA = A, and rank B = rank A. Source: [2657]. Fact 8.3.26. Let A ∈ Fn×m, and assume that rank A = m. Then, (AA∗ )+ = A(A∗A)−2A∗. Related: Fact 8.4.18. Fact 8.3.27. Let A ∈ Fn×m. Then, A+ = lim A∗(AA∗ + αI)−1 = lim (A∗A + αI)−1A∗, α↓0

+

∗

α↓0

∗

−1

A A = lim A (AA + αI) A, α↓0

+

AA = lim A(A∗A + αI)−1A∗. α↓0

Fact 8.3.28. Let A ∈ Fn×m, let χAA∗ (s) = sn + βn−1 sn−1 + · · · + β1 s + β0 , and let k denote the largest

integer in {0, . . . , n − 1} such that βn−k , 0. Then,

∗ ∗ k−1 A+ = −β−1 + βn−1 (AA∗ )k−2 + · · · + βn−k+1 I]. n−k A [(AA )

Source: [863]. Fact 8.3.29. Let A ∈ Fn×n, and assume that A is Hermitian. Then, In A = In A+ = In AD. If, in

addition, A is nonsingular, then In A = In A−1. Fact 8.3.30. Let A ∈ Fn×n. Then, the following statements are equivalent: i) A is Hermitian. ii) A∗ = A2A+ . Source: [257]. △ Fact 8.3.31. Let A ∈ Fn×m, and define r = rank A. Then, the following statements are equivalent: i) A is a partial isometry. ii) A∗A is a projector. iii) AA∗A = A. iv) A∗AA∗ = A∗. v) A+ = A∗. vi) σ1 (A) = σr (A) = 1. In particular, Nn+ = NnT. Source: [360, pp. 219–220]. Remark: The partial isometry A preserves lengths and distances with respect to the Euclidean norm on R(A∗ ). See [360, p. 219]. Related: Fact 7.12.34 and Fact 8.6.1. Fact 8.3.32. Let A ∈ Fn×n. Then, the following statements are equivalent: i) A+A∗ = A∗A+. ii) AA+A∗A = AA∗A+A. iii) AA∗A2 = A2A∗A. If these statements hold, then A is star-dagger. If A is star-dagger, then A2(A+ )2 and (A+ )2A2 are positive semidefinite. Source: [1338, 2628]. Related: Fact 8.6.1. △ Fact 8.3.33. Let A ∈ Fn×m, assume that A is nonzero, and define r = rank A. Then, for all + i ∈ {1, . . . , r}, the singular values of A are given by −1 σi (A+ ) = σr+1−i (A).

In particular,

σr (A) = 1/σmax (A+ ).

632

CHAPTER 8

If, in addition, A ∈ Fn×n and A is nonsingular, then σmin (A) = 1/σmax (A−1 ). Fact 8.3.34. Let A ∈ Fn×m. Then, X = A+ is the unique matrix satisfying

[

rank

A A+A

] AA+ = rank A. X

Source: [1043]. Related: Fact 3.22.11 and Fact 8.10.11. Fact 8.3.35. Let A ∈ Fn×n, and assume that A is centrohermitian. Then, A+ is centrohermitian. Source: [1776]. Fact 8.3.36. Let A ∈ Fn×m. Then, I + A+A and I + AA+ are positive definite, and A+ = 4(I +

A+A)−1A+(I + AA+ )−1. Fact 8.3.37. Let A ∈ Fn×n, and assume that A is unitary. Then,

1∑ i A = I − (A − I)(A − I)# = I − (A − I)(A − I)+. k→∞ k i=0 k−1

lim

Source: Fact 15.22.14 and Fact 15.22.17. Since A − I is normal, it is range Hermitian. Fact 8.10.16 implies that (A − I)# = (A − I)+. See [1301, p. 185]. Remark: I − (A − I)(A − I)+ is the projector onto N[(A − I)∗ ] = N(A − I) = {x: Ax = x}. See Fact 4.9.6. Remark: This is the ergodic theorem. △ Fact 8.3.38. Let A ∈ Fn×m, and define the sequence (Bi )∞ i=1 by Bi+1 = 2Bi − Bi ABi , where △ ∗ 2 + B0 = αA and α ∈ (0, 2/σmax (A)). Then, limi→∞ Bi = A . Source: [300, p. 259] and [624, p. 250]. Remark: This is a Newton-Raphson algorithm. Remark: B0 satisfies ρmax (I − B0 A) < 1. Remark: For the case where A is square and nonsingular, see Fact 3.20.22. Credit: A. Ben-Israel. Problem:

Determine whether convergence holds for all B0 ∈ Fn×n satisfying ρmax (I − B0 A) < 1. n×m Fact 8.3.39. Let A ∈ Fn×m, let (Ai )∞ , and assume that limi→∞ Ai = A. Then, the i=1 ⊂ F following statements are equivalent: i) limi→∞ A+i = A+ . ii) There exists a positive integer k such that, for all i > k, rank Ai = rank A. iii) (σmax (A+i ))∞ i=1 is bounded. iv) limi→∞ Ai A+i = AA+ . Source: [403], [624, pp. 218, 219], [1275, p. 211], and [2403, pp. 199, 200].

8.4 Facts on the Moore-Penrose Generalized Inverse for Two or More Matrices Fact 8.4.1. Let A, B ∈ Fn×m . Then, N(I −B+A) = N(A−B)∩R(B+ ). Source: Let x ∈ N(I −B+A).

Then, x = B+Ax ∈ R(B+ ) ⊆ N(A − B) ∩ R(B+ ). Now, let x ∈ N(A − B) ∩ R(B+ ). Then, Ax = Bx, and there exists y ∈ Fn such that x = B+ y. Therefore, x − B+Ax = x − B+Bx = x − B+BB+ y = x − B+ y = 0. Remark: This result generalizes (34) of [620]. △ △ Fact 8.4.2. Let A ∈ Fn×m and B ∈ Fn×l , and define P = AA+ and Q = BB+ . Then, rank A∗B = rank A∗Q = rank PB = rank PQ = rank B − dim[N(P) ∩ R(Q)] =

1 2

rank [P, Q] + dim[R(A) ∩ R(B)]

= rank A + rank B − dim(R(B) + [R(A) ∩ N(B∗ )]) = rank(I − P − Q) − rank P⊥ Q⊥ = rank(PQ + QP) − 12 rank [P, Q]

633

GENERALIZED INVERSES

= n + rank [P, Q] − rank PQ⊥ − rank P⊥ Q − rank P⊥ Q⊥ = rank A + rank B + rank P⊥ Q⊥ − n = rank A + rank B − rank [A B] + 12 rank [P, Q] = 12 [rank A + rank B + rank(I − P − Q) − n] = n + 12 rank [P, Q] − rank(I − PQ) = 12 [rank A + rank B + rank [P, Q] − rank(P − Q)], rank [A B] = rank A + rank P⊥ B = rank B + rank Q⊥ A = rank(P + Q − PQ) = rank(P − Q) + dim[R(A) ∩ R(B)] = 12 [rank A + rank B + rank(P − Q)] = n − rank P⊥ Q⊥ + 21 rank [P, Q] = rank(PQ + QP) + dim([N(A∗ ) ∩ R(B)] + [R(A) ∩ N(B∗ )]) = n + rank [P, Q] − dim([R(A) + N(B∗ )] ∩ [N(A∗ ) + R(B)] ∩ [N(A∗ ) + N(B∗ )]), rank A + 21 rank [P, Q] ≤ rank [A B], 1 2

rank [P, Q] = rank A∗Q⊥ P⊥ B ≤ rank A∗B ≤ rank(PQ + QP).

Furthermore, the following statements are equivalent: i) rank A∗B = rank A. ii) R(A) ∩ N(B∗ ) = {0}. iii) R(A) ⊆ N(A∗ ) + R(B). iv) P = PQ(PQ)+ . v) P is the projector onto R(P) ∩ [N(P) + R(Q)]. Finally, the following statements are equivalent: vi) rank A∗B = rank B. vii) N(A∗ ) ∩ R(B) = {0}. viii) N(A∗ ) ⊆ R(A) + N(B∗ ). ix) P⊥ = P⊥ Q⊥ (P⊥ Q⊥ )+ . x) P⊥ is the projector onto N(P) ∩ [R(P) + N(Q)]. Proof. See [252], [2238, p. 312], and [3014]. Fact 8.4.3. Let A ∈ Fn×m and B ∈ Fn×l. Then, R(A) ⊆ R(B) if and only if BB+A = A. Source: [31, p. 35]. Fact 8.4.4. Let A ∈ Fn×m and B ∈ Fm×n. Then, the following statements are equivalent: i) AB = 0. ii) B = (A+A)⊥B. Source: [2238, p. 311]. Fact 8.4.5. Let A, B ∈ Fn×n , assume that B is Hermitian, and assume that AB = BA. Then, the following statements hold: i) A(B+B)⊥ = (B+B)⊥ A. ii) (A+A)⊥ B = B(A+A)⊥ . Source: [2238, p. 311]. Fact 8.4.6. Let A ∈ Fn×m and B ∈ Fm×n. Then, the following statements are equivalent:

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CHAPTER 8

i) B = A+. ii) A∗AB = A∗ and B∗BA = B∗. iii) BAA∗ = A∗ and ABB∗ = B∗. Remark: See [1343, pp. 503, 513]. Fact 8.4.7. Let A, B ∈ Fn×m . Then, the following statements are equivalent: [ ] A i) rank [A B] = rank . B ii) rank(AA∗ + BB∗ ) = rank(A∗A + B∗B). iii) rank(AA+ + BB+ ) = rank(A+A + B+B). iv) rank(AA+ − BB+ ) = rank(A+A − B+B). [ ∗ ] [ ∗ ] AA B A A B∗ v) = rank . B∗ 0 B 0 [ ∗ ] [ ∗ ] BB A B B A∗ vi) = rank . A∗ 0 A 0 [ + ] [ + ] AA BB+ A A B+B vii) rank = rank . BB+ AA+ B+B A+A viii) rank [I − AA+ I − BB+ ] + m = rank [I − A+A I − B+B] + n. ix) dim[R(A) ∩ R(B)] = dim[R(A∗ ) ∩ R(B∗ )]. x) dim[N(A) ∩ N(B)] + m = dim[N(A∗ ) ∩ N(B∗ )] + n. Related: Fact 3.14.11. Credit: Y. Tian. Fact 8.4.8. Let A ∈ Fn×m and B ∈ Fn×l. Then, R(A) ∩ R(B) = R[AA+ (AA+ + BB+ )+BB+ ], dim[R(A) ∩ R(B)] = rank AA+ (AA+ + BB+ )+BB+ = rank A + rank B − rank [A B]. Now, assume that A∗B = 0. Then, R(A) ∩ R(B) = {0}, AA+ (AA+ + BB+ )+BB+ = 0, and rank [A B] = rank A + rank B. Source: Fact 8.8.19. For the second statement, Proposition 8.1.7 implies that A+B = 0, and thus Fact 8.8.19 implies that AA+ (AA+ + BB+ )+BB+ = AA+BB+ (AA+ + BB+ )+ AA+BB+ = 0. Alternatively, Fact 3.13.37 implies that R(A) and R(B) are mutually orthogonal subspaces, and thus R(A) ∩ R(B) = {0}. For the last statement, use Fact 3.14.8, Fact 3.14.18, and Fact 8.4.8. See [2238, p. 306]. Remark: See Theorem 3.1.3, Fact 3.13.37, Fact 3.14.15, Fact 8.8.19, Fact 8.9.5, Fact 8.9.26, and Fact 10.24.20. Fact 8.4.9. Let A ∈ Fn×m and B ∈ Fn×l. Then, R(A) + R(B) = R([A B][A B]+ ). Source: [2238, p. 306]. Related: Fact 8.9.20 and Fact 8.9.23. Fact 8.4.10. Let A ∈ Fn×m, and let x ∈ Fn and y ∈ Fm be nonzero. Furthermore, define △

△

d = A+ x, △

δ = d∗d, △

if and only if

△

f = (I − AA+ )x, △

η = e∗e,

λ = 1 + y∗A+ x, Then,

△

e = A+∗ y,

△

ϕ = f ∗ f,

△

µ = |λ|2 + δψ,

g = (I − A+A)y, △

ψ = g∗g, △

ν = |λ|2 + ηϕ.

rank(A + xy∗ ) = rank A − 1 x ∈ R(A),

y ∈ R(A∗ ),

λ = 0.

635

GENERALIZED INVERSES

If these conditions hold, then (A + xy∗ )+ = A+ − δ−1dd∗A+ − η−1A+ee∗ + (δη)−1d∗A+ede∗. Furthermore,

rank(A + xy∗ ) = rank A

if and only if exactly one of the following statements holds:    x ∈ R(A), y ∈ R(A∗ ), λ , 0,     x ∈ R(A), y < R(A∗ ),       x < R(A), y ∈ R(A∗ ). If these conditions hold, then, respectively,    (A + xy∗ )+ = A+ − λ−1 de∗,     (A + xy∗ )+ = A+ − µ−1 (ψdd∗A+ + δge∗ ) + µ−1 (λgd∗A+ − λde∗ ),      (A + xy∗ )+ = A+ − ν−1 (ϕA+ ee∗ + ηd f ∗ ) + ν−1 (λA+ef ∗ − λde∗ ). Finally, assume that A does not have full rank. Then, rank(A + xy∗ ) = rank A + 1 if and only if

x < R(A),

y < R(A∗ ).

If these conditions hold, then (A + xy∗ )+ = A+ − ϕ−1d f ∗ − ψ−1ge∗ + λ(ϕψ)−1gf ∗. Source: [227]. To prove sufficiency in the first alternative of the third statement, let xˆ ∈ Fm and

yˆ ∈ Fn satisfy x = A xˆ and y = A∗ yˆ . Then, A + xy∗ = A(I + xˆy∗ ). Since α , 0 it follows that −1 , y∗A+ x = yˆ ∗AA+A xˆ = yˆ ∗A xˆ = y∗ xˆ. It now follows that I + xˆy∗ is an elementary matrix and thus, by Fact 4.10.19, is nonsingular. Remark: An equivalent version of the first statement is given in [734] and [1467, p. 33]. A detailed treatment of the generalized inverse of an outer-product perturbation is given in [2821, pp. 152–157]. Remark: The equality rank(A + xy∗ ) = rank A − 1 is the Wedderburn rank-one reduction formula. See [734] and Fact 8.9.13. Related: Fact 3.13.32. Fact 8.4.11. Let A ∈ Fn×n, assume that A is Hermitian and nonsingular, and let x ∈ Fn and y ∈ Fn be nonzero. Then, A + xy∗ is singular if and only if y∗A−1 x + 1 = 0. If these conditions hold, then (A + xy∗ )+ = (I − aa+ )A−1(I − bb+ ), △

△

where a = A−1 x and b = A−1y. Source: [2403, pp. 197, 198]. △ Fact 8.4.12. Let A ∈ Fn×n, assume that A is Hermitian, let b ∈ Fn, and define S = I − A+A. Then,   [I − (b∗(A+ )2b)−1A+bb∗A+ ]A+ [I − (b∗A+2b)−1A+bb∗A+ ], 1+ b∗A+b = 0,       + (A + bb∗ )+ =  A − (1 + b∗A+b)−1A+bb∗A+, 1+ b∗A+b , 0,      [I − (b∗Sb)−1Sbb∗ ]A+ [I − (b∗Sb)−1bb∗S ] + (b∗Sb)−2Sbb∗S, b∗Sb , 0. Source: [2048]. Fact 8.4.13. Let A ∈ Fn×n, assume that A is positive semidefinite, let C ∈ Fm×m, assume that C

is positive definite, and let B ∈ Fn×m. Then,

(A + BCB∗ )+ = A+ − A+B(C −1 + B∗A+B)−1B∗A+

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if and only if AA+B = B. Source: [2146]. Remark: AA+B = B is equivalent to R(B) ⊆ R(A). See Fact 8.4.3. Remark: Extensions of the matrix inversion lemma are considered in [849, 1047, 2048, 2310] and [1343, pp. 426–428, 447, 448]. Fact 8.4.14. Let A ∈ Fn×m and B ∈ Fm×l. Then, AB = 0 if and only if B+A+ = 0. Source: ix) =⇒ i) of Fact 8.4.26. Fact 8.4.15. Let A ∈ Fn×m and B ∈ Fn×l. Then, A+B = 0 if and only if A∗B = 0. Source: Proposition 8.1.7. Fact 8.4.16. Let A ∈ Fn×m, let B ∈ Fm×p, and assume that rank B = m. Then, AB(AB)+ = AA+. Source: [2403, p. 215]. Fact 8.4.17. Let A ∈ Fn×m, let B ∈ Fm×m, and assume that B is positive definite. Then,

ABA∗ (ABA∗ )+A = A. Source: [2403, p. 215]. Fact 8.4.18. Let A ∈ Fn×m, assume that rank A = m, let B ∈ Fm×m, and assume that B is positive

definite. Then,

(ABA∗ )+ = A(A∗A)−1B−1 (A∗A)−1A∗.

Source: Fact 8.3.26. △ Fact 8.4.19. Let A ∈ Fn×m, let S ∈ Fm×m, assume that S is nonsingular, and define B = AS . Then, BB+ = AA+. Source: [2418, p. 144]. Fact 8.4.20. Let A ∈ Fn×r and B ∈ Fr×m, and assume that rank A = r and rank B = m. Then,

B+A+ is a left inverse of AB, and

B+A+ = (B∗B)−1B∗ (A∗A)−1A∗. If, in addition, m = r, then

(AB)+ = B+A+ = B−1 (A∗A)−1A∗.

Remark: A and B are left invertible. Related: Fact 3.18.9. Fact 8.4.21. Let A ∈ Fn×r and B ∈ Fr×m, and assume that rank A = n and rank B = r. Then,

B+A+ is a right inverse of AB, and

B+A+ = B∗(BB∗ )−1A∗(AA∗ )−1 . If, in addition, n = r, then

(AB)+ = B+A+ = B∗(BB∗ )−1A−1.

Remark: A and B are right invertible. Related: Fact 3.18.10. △ △ Fact 8.4.22. Let A ∈ Fn×m and B ∈ Fm×l, and define B1 = A+AB and A1 = AB1 B+1 . Then,

(AB)+ = (A1B1 )+ = B+1 A1+. Source: [2238, p. 188] and [2403, pp. 191, 192]. Remark: Products of generalized inverses are

considered in [2663]. Fact 8.4.23. Let A ∈ Fn×m and B ∈ Fm×l. Then, (AB)+ = (A+AB)+ (ABB+ )+ = (A+∗B)+ (B+A+ )∗ (AB+∗ )+ ,

(AB)+AB = (A+AB)+A+AB.

Furthermore, if R(B) = R(A∗ ), then A+AB = B,

ABB+ = A,

(AB)+ = B+A+.

Source: [2238, p. 312], [2403, p. 192], [2652, 2665]. Credit: R. E. Cline and T. N. E. Greville.

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Fact 8.4.24. Let A ∈ Fn×m and B ∈ Fm×l, and assume that at least one of the following statements

holds: i) (AB)+ = B+ (A+ABB+ )+A+ . ii) (AB)+ = B∗ (A∗ABB∗ )+A∗ . iii) (AB)+ = B+A+ − B+ [(I − BB+ )(I − A+A)]+A+ . iv) R(AA∗AB) = R(AB) and R[(ABB∗B)∗ ] = R[(AB)∗ ]. Then, all four statements hold. Source: [2637]. Fact 8.4.25. Let A ∈ Fn×m and B ∈ Fm×l. rank[(ABB+ )+ − BB+A+ ] = rank [A∗AB B] − rank B, [ ] A rank[(A+AB)+ − B+A+A] = rank − rank A, ABB∗ rank[(A+ABB+ )+ − A+ABB+ ] = rank [A∗ B] + rank AB − rank A − rank B. Source: [2657]. Fact 8.4.26. Let A ∈ Fn×m and B ∈ Fm×l. Then, the following statements are equivalent:

(AB)+ = B+A+. R(A∗AB) ⊆ R(B) and R(BB∗A∗ ) ⊆ R(A∗ ). R(AA∗AB) = R(AB) and R(BB∗A∗ ) ⊆ R(A∗ ). R(A∗AB) ⊆ R(B) and R[(ABB∗B)∗ ] = R[(AB)∗ )]. (AB)+ = B+A+ABB+A∗ . (AB)+ = B+ (ABB+ )+ and (ABB+ )+ = BB+A+ . (AB)+ = (A+AB)+A+ and (A+AB)+ = B+A+A. (AB)+ = B+ (A+ABB+ )+A+ and (A+ABB+ )+ = BB+A+A. (AB)+ = (A∗AB)+A∗ and (A∗AB)+ = B+ (A∗A)+ . (AB)+ = B∗ (ABB∗ )+ and (ABB∗ )+ = (BB∗ )+ A+ . (AB)+ = B∗ (A∗ABB∗ )+A∗ and (A∗ABB∗ )+ = (BB∗ )+ (A∗A)+ . AB(AB)+ = ABB+A+ and (AB)+AB = B+A+AB. AB(AB)+A = ABB+ and A+AB = B(AB)+AB. (ABB+ )+ = BB+A+ and (A+AB)+ = B+A+A. B+(ABB+ )+ = B+A+ and (A+AB)+A = B+A+. A+ABB+ = BB+A+A. A∗ABB+ and A+ABB∗ are Hermitian. A∗AB = BB+A∗AB and ABB∗ = ABB∗A+A. A+AB = B(AB)+AB and BB+A∗ = A∗AB(AB)+ . A∗ABB∗ = BB+A∗ABB∗A+A. (A+∗B)+ = B+A∗ . (AB+∗ )+ = B∗A+ . Source: [31, p. 53], [1231], [2238, pp. 187, 188], [2403, pp. 190, 191], [2639, 2652], and [2853]. Remark: Conditions under which B+A+ is a (1)-inverse of AB are given in [2639]. Credit: The equivalence of i) and ii) is due to T. N. E. Greville. Fact 8.4.27. Let A ∈ Fn×n and B ∈ Fm×n, and assume that A is idempotent. Then, i) ii) iii) iv) v) vi) vii) viii) ix) x) xi) xii) xiii) xiv) xv) xvi) xvii) xviii) xix) xx) xxi) xxii)

A∗ (BA)+ = (BA)+.

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Source: [1343, p. 514]. Fact 8.4.28. Let A, B ∈ Fn×n , assume that A and B are idempotent, and assume that AB = BA. Then, A + B − 32 AB is a 1-inverse of A + B, and A − B − AB is a 1-inverse of A − B. Source: [235]. Related: Fact 8.8.5. Fact 8.4.29. Let A ∈ Fn×m and B ∈ Fm×l, and assume that rank B = m. Then, AB(AB)+ = AA+. Source: [2238, p. 188]. Fact 8.4.30. Let A ∈ Fn×m, B ∈ Fm×n, and C ∈ Fm×n, and assume that BAA∗ = A∗ and A∗AC = A∗. Then, A+ = BAC. Source: [31, p. 36]. Credit: H. P. Decell. Fact 8.4.31. Let A, B ∈ Fn×n, and assume that A + B is nonsingular. Then, the following

statements are equivalent: i) rank A + rank B = n. ii) A(A + B)−1B = 0. iii) B(A + B)−1A = 0. iv) A(A + B)−1A = A. v) B(A + B)−1B = B. vi) A(A + B)−1B + B(A + B)−1A = 0. vii) A(A + B)−1A + B(A + B)−1B = A + B. viii) (A + B)−1 = [(I − BB+ )A(I − B+B)]+ + [(I − AA+ )B(I − A+A)]+. Source: [2666]. Related: Fact 3.14.11 and Fact 10.24.24. Fact 8.4.32. Let A ∈ Fn×m and B ∈ Fn×p. Then, the following statements are equivalent: i) [AA+ , BB+ ] = 0. ii) rank [A B] = rank A + rank B − rank A∗B. Source: [2657]. Related: Fact 4.18.7. Fact 8.4.33. Let A ∈ Fn×m and B ∈ Fn×p. Then, (AA∗ + BB∗ )+ = (I − C +∗B∗ )A+∗EA+ (I − BC + ) + (CC ∗ )+, where △

C = (I − AA+ )B,

△

E = I − A∗B(I − C +C)[I + (I − C +C)B∗(AA∗ )+B(I − C +C)]−1 (A+B)∗.

Source: [536, p. 16] and [2403, p. 196]. Related: Fact 8.9.20. Fact 8.4.34. Let A, B ∈ Fn×m, and assume that BA∗ = 0. Then,

(A + B)+ = A+ + (I − A+B)(C + + D), where △

C = (I − AA+ )B,

△

D = (I − C +C)[I + (I − C +C)B∗(AA∗ )+B(I − C +C)]−1B∗(AA∗ )+(I − BC + ).

Source: [536, p. 17] and [2403, p. 196]. Remark: As noted in Fact 8.9.25, if A∗B = 0, then C = B

and D = 0. This observation yields Fact 8.4.35. Fact 8.4.35. Let A, B ∈ Fn×m, and assume that A∗B = 0 and BA∗ = 0. Then, (A + B)+ = A+ + B+.

Source: Fact 3.13.38, Fact 8.4.34, Fact 8.4.37, [762], [1343, p. 513], and [2403, p. 197]. Credit:

R. Penrose. Fact 8.4.36. Let A, B ∈ Fn×m, assume that rank(A + B) = rank A, and assume that I + A+B is

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nonsingular. Then, (A + B)+ = (A+A + X ∗ )[A+A − X(I + X ∗X)−1X ∗ ](I + A+B)−1 [AA+ − Y ∗ (I + YY ∗ )−1Y](AA+ + Y ∗ ), where

△

X = (I + A+B)−1A+B(I − A+A),

△

Y = (I − AA+ )BA+ (I + BA+ )−1 .

Source: [1275, p. 220]. Fact 8.4.37. Let A, B ∈ Fn×m, and assume that rank(A + B) = rank A + rank B. Then,

(A + B)+ = (I − C +B)A+ (I − BC + ) + C +, where

△

C = (I − AA+ )B(I − A+A).

Furthermore, where

(A + B)+ = (I − S )A+ (I − T ) + SB+ T, △

S = [B+B(I − A+A)]+ ,

△

T = [(I − AA+ )BB+ ]+ .

Source: [762] and [2238, p. 197]. Fact 8.4.38. Let A, B ∈ Fn×m. Then,

(A + B)+ = (I + A+B)+ (A+ + A+BA+ )(I + BA+ )+ if and only if AA+B = B = BA+A. Furthermore, if n = m and A is nonsingular, then (A + B)+ = (I + A−1B)+ (A−1 + A−1BA−1 )(I + BA−1 )+. Source: [762]. Fact 8.4.39. Let A, B ∈ Fn×m. Then,

A+ − B+ = B+ (B − A)A+ + (I − B+B)(A∗ − B∗ )A+∗A+ + B+B+∗ (A∗ − B∗ )(I − AA+ ) = A+ (B − A)B+ + (I − A+A)(A∗ − B∗ )B+∗B+ + A+A+∗ (A∗ − B∗ )(I − BB+ ). Furthermore, if B is left invertible, then A+ − B+ = B+ (B − A)A+ + B+B+∗ (A∗ − B∗ )(I − AA+ ), while, if B is right invertible, then A+ − B+ = A+ (B − A)B+ + (I − A+A)(A∗ − B∗ )B+∗B+. Source: [624, p. 224]. Fact 8.4.40. Let A ∈ Fn×m, B ∈ Fl×k, and C ∈ Fn×k. Then, there exists X ∈ Fm×l satisfying

AXB = C if and only if AA+CB+B = C. Furthermore, X satisfies AXB = C if and only if there exists Y ∈ Fm×l such that X = A+CB+ + Y − A+AYBB+. Finally, if Y = 0, then tr X ∗X is minimized. Source: Proposition 8.1.8. See [1924, p. 37] and, for Hermitian X, see [1608]. Fact 8.4.41. Let A ∈ Fn×m, and assume that rank A = m. Then, AL ∈ Fm×n is a left inverse of A if and only if there exists B ∈ Fm×n such that AL = A+ + B(I − AA+ ). Source: Use Fact 8.4.40 with A = C = In .

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Fact 8.4.42. Let A ∈ Fn×m, and assume that rank A = n. Then, AR ∈ Fm×n is a right inverse of A

if and only if there exists B ∈ Fm×n such that

AR = A+ + (I − A+A)B. Source: Use Fact 8.4.40 with B = C = In . Fact 8.4.43. Let A, B ∈ Fn×m. Then, the following statements are equivalent: ∗

i) A ≤ B. ii) A+A = A+B and AA+ = BA+. ∗

iii) B − A ≤ B. If these statements hold, then the following statements hold: iv) AB+ , B+A, A+B, and BA+ are Hermitian. v) BA+B = A and B+AB+ = A+ . Source: [1339, 2058] and [2059, pp. 127–131]. Related: Fact 4.30.8 and Fact 8.5.17. Fact 8.4.44. Let A, B ∈ Fn×m. Then, the following statements are equivalent: rs

i) A ≤ B. rs

ii) B+A ≤ B+B and R(A) ⊆ R(B). rs

iii) AB+ ≤ BB+ and R(A∗ ) ⊆ R(B∗ ). rs

iv) B+AB+ ≤ B+ , R(A) ⊆ R(B), and R(A∗ ) ⊆ R(B∗ ). Furthermore, the following statements are equivalent: rs

v) B+A ≤ B+B and R(A∗ ) ⊆ R(B∗ ). rs

vi) B+AB+ ≤ B+ and R(A∗ ) ⊆ R(B∗ ). Furthermore, the following statements are equivalent: rs

vii) AB+ ≤ BB+ and R(A) ⊆ R(B). rs

viii) B+AB+ ≤ B+ and R(A) ⊆ R(B). Source: [1880]. Related: Fact 4.30.4. Fact 8.4.45. Let A, C ∈ Fn×m, and let B, D ∈ Fn×l. Then, the following statements hold: rs

rs

rs

i) If A ≤ C, B ≤ D, and R(C) ∩ R(D) = {0}, then [A B] ≤ [C D]. rs

rs

ii) Assume that A ≤ C and R(A) ⊆ R(C). Then, [A B] ≤ [C D] if and only if B = AC +D. Source: [1880]. Fact 8.4.46. Let A, C ∈ Fn×m, and let B, D ∈ Fl×m. Then, the following statements hold: rs rs [ ] rs [ ] i) Assume that A ≤ C, B ≤ D, and R(C ∗ ) ∩ R(D∗ ) = {0}. Then, AB ≤ CD . rs [ ] rs [ ] ii) Assume that A ≤ C and R(D∗ ) ⊆ R(C ∗ ). Then, AB ≤ CD if and only if B = DC +A. Source: [1880]. Fact 8.4.47. Let A1 , . . . , Ak ∈ Fn×m, and assume that, for all distinct i, j ∈ {1, . . . , k}, A∗i A j = 0. Then,  k + k ∑ ∑    = A A+i . i   i=1

Source: [2238, p. 186].

i=1

GENERALIZED INVERSES

641

8.5 Facts on the Moore-Penrose Generalized Inverse for Range-Hermitian, Range-Disjoint, and Range-Spanning Matrices Fact 8.5.1. Let A ∈ Fn×n and k ≥ 1. Then, the following statements are equivalent:

i) Ak = A+ . ii) Ak+2 = A, and A is range Hermitian. Furthermore, if k ≥ 2 and Ak = A∗ , then i) and ii) hold. Source: [233]. Fact 8.5.2. Let A ∈ Fn×n. Then, rank [A, A+ ] = rank [AA+ A+A] = 2 rank [A A∗ ] − 2 rank A = rank(A − A2A+ ) = rank(A − A+A2 ). Furthermore, the following statements are equivalent: i) A is range Hermitian. ii) AA+ = A+A. iii) rank [A A∗ ] = rank A. iv) A = A2A+. v) A = A+A2. Source: [2673]. Related: Fact 4.9.6, Fact 8.5.3, and Fact 8.5.8. Fact 8.5.3. Let A ∈ Fn×n. Then, the following statements are equivalent: i) A is range Hermitian. ii) R(A) = R(A+ ). iii) A+A = AA+. iv) (I − A+A)⊥ = AA+. v) A = A2A+. vi) A = A+A2. vii) AA+ = A2(A+ )2. viii) (AA+ )2 = A2(A+ )2. ix) (A+A)2 = (A+ )2A2. x) ind A ≤ 1, and (A+ )2 = (A2 )+. xi) ind A ≤ 1, and AA+A∗A = A∗A2A+. xii) A2A+ + A∗A+∗A = 2A. xiii) A2A+ + (A2A+ )∗ = A + A∗. xiv) R(A − A+ ) = R(A − A3 ). xv) R(A + A+ ) = R(A + A3 ). Source: [719, 2628, 2647, 2718] and Fact 8.10.16. Related: Fact 4.9.6, Fact 8.5.2, and Fact 8.5.8. Fact 8.5.4. Let A ∈ Fn×n. Then, the following statements are equivalent: i) A = A+ . ii) A2 = AA+ . iii) A+ = A2A+ . iv) AA2∗ = A. v) A2∗A = A. vi) A2 = AA+ = A+A. vii) (A+ )2 = AA+ = A+A. viii) A is range Hermitian, and A2 is idempotent.

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A is range Hermitian, and (A+ )2 is idempotent. A is range Hermitian, and A∗A = A∗A+ . A is range Hermitian, and AA∗ = A+A∗ . A is range Hermitian, and A∗AA∗ = A∗A+A∗ . A is tripotent and range Hermitian. A∗ is tripotent and range Hermitian. A+ is tripotent and range Hermitian. A is tripotent, and A2 is Hermitian. A is group invertible, and A2 is a projector. A is group invertible, and (A+ )2 is a projector. A is group invertible, and Ac = A. A is group invertible, and A# = A2A+ . [ ] AA∗A A xxi) rank = rank A. A A∗ ([ ]) ([ ]) A AA∗A xxii) R =R . A∗ A

ix) x) xi) xii) xiii) xiv) xv) xvi) xvii) xviii) xix) xx)

△

n×n r×r xxiii) There exists a unitary [ ] matrix S ∈ F and an involutory matrix B ∈ F , where r = rank A, ∗ B 0 such that A = S 0 0 S . Source: [360, p. 49] and [233, 257]. Fact 8.5.5. Let A ∈ Fn×n. Then, the following statements are equivalent: i) A is Hermitian and tripotent. ii) A is a partial isometry and tripotent. iii) A is Hermitian, and A = A+ . iv) A is a partial isometry, tripotent, and either 21 (AA+ − A) or 12 (AA+ + A) is idempotent. v) A is tripotent, range Hermitian, and semicontractive. vi) A is group invertible, and A = A+ = A∗ = A# = Ac . Source: [257]. Related: Fact 7.10.27. Fact 8.5.6. Let A ∈ Fn×n. Then, the following statements are equivalent: i) A + A+ = 2AA+. ii) A + A+ = 2A+A. iii) A + A+ = AA+ + A+A. iv) A is range Hermitian, and A2 + AA+ = 2A. v) A is range Hermitian, and (I − A)2A = 0. Source: [2707, 2717]. Fact 8.5.7. Let A ∈ Fn×n and k ≥ 3. Then, the following statements are equivalent: i) A is normal, and Ak+1 = A. ii) A is a partial isometry, and Ak+1 = A. iii) A is a partial isometry, and Ak = AA∗ . iv) A is group invertible and a partial isometry, and Ak−1 = A# . v) A is group invertible, and Ak−1 = A∗ = A# . vi) Ak−1 = A∗ .

GENERALIZED INVERSES

643

The following statements are equivalent: vii) A is range Hermitian, and Ak+1 = A. viii) Ak = AA+ . The following statements are equivalent: ix) A is range Hermitian and a partial isometry. x) A = A∗A2 . xi) A = A2A∗ . Now, let k ≥ 1. Then, following statements are equivalent: xii) Ak = A. xiii) A is group invertible, and Ak+1 = A2 . Source: [257]. △ Fact 8.5.8. Let A ∈ Fn×n, let r = rank A, let B ∈ Fn×r and C ∈ Fr×n, and assume that A = BC and rank B = rank C = r. Then, the following statements are equivalent: i) A is range Hermitian. ii) BB+ = C +C. iii) N(B∗ ) = N(C). iv) B = C +CB and C = CBB+. v) B+ = B+C +C and C = CBB+. vi) B = C +CB and C + = BB+C +. vii) B+ = B+C +C and C + = BB+C +. Source: [950]. Remark: A = BC is a full-rank factorization. Related: Fact 4.9.6, Fact 8.5.2, and Fact 8.5.3. Fact 8.5.9. Let A, B ∈ Fn×n, and assume that A is range Hermitian. Then, AB = BA if and only + if A B = BA+. Source: [2627]. Fact 8.5.10. Let A, B ∈ Fn×n, and assume that A and B are range Hermitian. Then, the following statements are equivalent: i) AB = BA. ii) A+B = BA+. iii) AB+ = B+A. iv) A+B+ = B+A+. Source: [2627]. Fact 8.5.11. Let A, B ∈ Fn×n, assume that A and B are range Hermitian, and assume that (AB)+ = + + A B . Then, AB is range Hermitian. Source: [1335]. Related: Fact 10.24.23. Fact 8.5.12. Let A, B ∈ Fn×n, and assume that A and B are range Hermitian. Then, the following statements are equivalent: i) AB is range Hermitian. ii) AB(I − A+A) = 0 and (I − B+B)AB = 0. iii) N(A) ⊆ N(AB) and R(AB) ⊆ R(B). iv) N(AB) = N(A) + N(B) and R(AB) = R(A) ∩ R(B). Source: [1335, 1658]. △ △ Fact 8.5.13. Let A ∈ Fn×n be nonzero, and define r = rank A and B = diag[σ1 (A), . . . , σr (A)].

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Then, there exist S ∈ Fn×n , K ∈ Fr×r , and L ∈ Fr×(n−r) such that S is unitary, KK ∗ + LL∗ = Ir , and    ∗  BK BL  S .  A = S  0(n−r)×r 0(n−r)×(n−r) △

△

Now, define the projectors P = AA+ and Q = A+A. Then, the following statements hold: i) PQ = QP if and only if K is a partial isometry. ii) rank PQ = rank K. iii) rank(P + Q) = r + rank L. iv) rank(P − Q) = 2 rank L. v) rank(PQ + QP) = 2 rank K + rank L − r. vi) rank(PQ − QP) = 2(rank K + rank L − r). vii) rank(I − PQ) = n − r + rank L. viii) rank(P + Q − PQ) = r + rank L. ix) rank(I − P − Q) = n − 2(r − rank K). Furthermore, the following statements are equivalent: x) A is group invertible. xi) K is nonsingular. xii) rank P = rank PQ. xiii) rank(P + Q) = rank(PQ + QP). xiv) rank(P − Q) = rank(PQ − QP). xv) σmax (P − Q) < 1. xvi) R(A) ∩ N(A) = {0}. xvii) R(A∗ ) ∩ N(A∗ ) = {0}. xviii) R(A) = R(PQ). xix) R(A∗ ) = R(QP). xx) 1 < spec(P + Q). xxi) 1 < spec(P − Q). xxii) −1 < spec(P − Q). If x)–xxii) hold, then    (BK)−1 (KBK)−1L  ∗ # A = S   S . 0(n−r)×r 0(n−r)×(n−r) Finally, let α, β ∈ R be nonzero. Then, αP + βQ is a projector if and only if at least one of the following statements holds: xxiii) α = β = 1 and A2 = 0. xxiv) α = −β = 1 and A is range Hermitian. xxv) −α = β = 1 and A is range Hermitian. Source: [240, 247, 250, 251, 1338]. Related: Fact 4.18.20, Fact 7.10.27, and Fact 8.3.23. △ Fact 8.5.14. Let A ∈ Fn×n , assume that A is nonzero, and define the projectors P = AA+ and △ + Q = A A. Then, the following statements are equivalent: i) A is range disjoint. ii) A+ is range disjoint.

GENERALIZED INVERSES

iii) R(P) ∩ R(Q) = {0}. iv) PQ is range disjoint. v) rank(P − Q) = rank P + rank Q. vi) R(P − Q) = R(P) + R(Q). vii) rank PQ⊥ + rank P⊥ Q = rank P + rank Q. viii) R(PQ⊥ ) + R(P⊥ Q) = R(P) + R(Q), and R(PQ⊥ ) and R(P⊥ Q) are mutually orthogonal. ix) rank(P − Q) = rank(P + Q). x) R(P − Q) = R(P + Q). xi) rank(PQ + QP) = rank PQ + rank QP. xii) I − PQ is nonsingular. xiii) rank P⊥ Q = rank A. xiv) rank Q⊥ P = rank A. xv) rank(P + Q) = 2 rank A. xvi) rank(P + Q − PQ) = 2 rank A. xvii) (Q⊥ P)+ = P(P + Q − QP)+ . xviii) (Q⊥ P)+ A = A. xix) A(QP⊥ )+ = A. xx) (QP⊥ )+ = [I − (Q⊥ P)+∗ ]Q. xxi) (Q⊥ P)+ = P[I − (QP⊥ )+∗ ]. xxii) (QP⊥ )+A+ (Q⊥ P)+ is a 1-inverse of A. xxiii) (QP⊥ )+ + (Q⊥ P)+∗ is the projector onto R(A) + R(A∗ ). xxiv) (QP⊥ )+∗ + (Q⊥ P)+ is the projector onto R(A) + R(A∗ ). xxv) [R(P) + N(Q)] ∩ [N(P) + N(Q)] = R(P) + N(Q). xxvi) [R(P) + R(Q)] ∩ [N(P) + N(Q)] = R(P + Q). xxvii) rank [A A∗ ] = 2 rank A. xxviii) rank(AA∗ + A∗A) = 2 rank A. xxix) A∗ (AA∗ + A∗A)+A is idempotent. xxx) A∗ (AA∗ + A∗A)+A = A+A. xxxi) A∗ (AA∗ + A∗A)+A∗ = 0. xxxii) 2 < spec(P + Q). Furthermore, the following statements hold: xxxiii) If A , 0, then P + Q , 0. xxxiv) P − Q , I, PQ + QP , I, and P + Q , PQ. xxxv) P + Q = I if and only if A2 = 0 and A is range spanning. xxxvi) The following statements are equivalent: a) A2 = 0. b) PQ = 0. c) P⊥ − Q is a projector. d) PQP = 0. e) PQ + QP is idempotent. f ) PQ + QP = 0.

645

646

xxxvii)

xxxviii)

xxxix)

xl)

xli)

xlii)

xliii)

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g) spec(PQ + QP) ∩ (0, 2] = ∅. The following statements are equivalent: a) A is range Hermitian. b) P − Q = 0. c) P⊥ + Q is a projector. d) P + Q⊥ is a projector. e) P⊥ − Q⊥ is a projector. f ) rank A = tr PQ. The following statements are equivalent: a) A is nonsingular. b) PQ = I. c) PQ is nonsingular. d PQP = I. e) PQP is nonsingular. The following statements are equivalent: a) A is range spanning. b) P + Q is nonsingular. c) P + Q − PQ is nonsingular. d) (Q⊥ P)+ = (I − QP)+ Q⊥ . The following statements are equivalent: a) A is group invertible and range spanning. b) PQ is range spanning. c) PQ + QP is nonsingular. The following statements are equivalent: a) A is range disjoint and range spanning. b) P − Q is nonsingular. c) (I − QP)+ Q⊥ = P(P + Q − QP)+ . P − Q is nonsingular if and only if A is range spanning and at least one of the following statements holds: a) spec(P + Q) ⊂ [0, 2). b) rank(P − Q) = rank P + rank Q. c) R(P − Q) = R(P) + R(Q). d) rank PQ⊥ + rank P⊥ Q = rank P + rank Q. e) R(PQ⊥ ) + R(P⊥ Q) = R(P) + R(Q) and R(PQ⊥ ) ∩ R(P⊥ Q) = {0}. f ) (Q⊥ P)+ = P(P + Q − QP)+ . The following statements are equivalent: a) PQ = QP. b) P⊥ Q is a projector. c) PQ⊥ is a projector. d) P⊥ Q⊥ is a projector. e) PQP is idempotent. f ) I − PQ is idempotent.

GENERALIZED INVERSES

647

g) PQ − QP is idempotent. h) P + Q − PQ is idempotent. xliv) The following statements are equivalent: a) A is range spanning, and PQ = QP. b) P⊥ + Q⊥ is a projector. c) P − Q⊥ is a projector. xlv) A is nilpotent if and only if tr PQ = 0. xlvi) Assume that A is range disjoint. Then, (A + A∗ )+ = A+ + A+∗ if and only if A2 = 0. xlvii) A is range spanning and PQ = QP if and only if P + Q = I + PQ. xlviii) A is group invertible, range disjoint, and range spanning if and only if [P, Q] is nonsingular. xlix) Let B ∈ Fn×n , and assume that AB = BA and either A or B is range disjoint. Then, AB is range disjoint. l) spec(PQ) ⊂ [0, 1], spec(P − Q) ⊂ [−1, 1], spec(P + Q) ⊂ [0, 2], spec(I − PQ) ⊂ [0, 1], spec(PQ + QP) ⊂ [− 14 , 2], spec([P, Q]) ⊂ ȷ[−1, 1], spec(P + Q − PQ) ⊂ [0, 1]. Source: [234, 250, 251]. Related: Fact 7.13.4. Fact 8.5.15. Let A ∈ Cn×n . Then, the following statements are equivalent: i) A is group invertible. ii) There exists Ac ∈ Fn×n such that AAc = AA+ and R(Ac ) ⊆ R(A). iii) There exists a unique matrix Ac ∈ Fn×n such that AAc = AA+ and R(Ac ) ⊆ R(A). △ Now, assume that A is group invertible, define r = rank A, and define B ∈ Rr×r , K ∈ Fr×r , L ∈ Fr×(n−r) , and S ∈ Fn×n as in Fact 8.5.13. Then, the following statements hold: iv) If A is nonzero, then K is nonsingular, and    (BK)−1 0r×(n−r)  ∗ c  S .  A = S  0(n−r)×r 0(n−r)×(n−r) v) Ac = 0 if and only if A = 0. vi) The following statements are equivalent: a) A is range Hermitian. b) A# = Ac . c) A+ = Ac . d) (Ac )c = A. e) AcA = AAc . f ) (A+ )c = A. g) (A+ )c = (Ac )+ . vii) Ac = AA+ if and only if A is idempotent. viii) Ac = A if and only if A is tripotent and range Hermitian. ix) Ac = A∗ if and only if A is a partial isometry and is range Hermitian. x) Ac = A# AA+ . xi) Ac is range Hermitian. xii) Ac is a (1,2)-inverse of A. xiii) (Ac )+ = (Ac )c = A2A+ . xiv) (Ac )2A = A# .

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If k ≥ 1, then (Ac )k = (Ak )c . AcA = A#A. If A is a projector and B ∈ Fn×n is a projector, then (AB)c = (ABA)+ . If B ∈ Fn×n is a (1)-inverse of A, then ABAc = Ac . Ac is a (2)-inverse of B if and only if A+ BA# = Ac . Source: [247, 1934]. Remark: Ac is the core inverse of A. xv) xvi) xvii) xviii) xix)

#

Fact 8.5.16. Let A, B ∈ Fn×n, assume that A and B are group invertible, and let A ≤ B denote #

A#A = A#B = BA# . Then, “≤” is a partial ordering on {A ∈ Fn×n : A is group invertible}. Furthermore, the following statements are equivalent: #

i) A ≤ B. ii) A2 = AB = BA. #

iii) B − A ≤ B. rs

iv) A ≤ B and AB = BA. If these statements hold, then the following statements hold: v) BA#B = A and B#AB# = A# . vi) AB = BA. vii) AB# = B#A. #

viii) For all k ≥ 1, Ak ≤ Bk . #

Source: [2024, 2058] and [2059, Chapter 4]. Remark: “≤” is the sharp partial ordering. c

Fact 8.5.17. Let A, B ∈ Fn×n, assume that A and B are group invertible, and let A ≤ B denote c

AcA = AcB and AAc = BAc . Then, “≤” is a partial ordering on {A ∈ Fn×n : A is group invertible}. Furthermore, the following statements are equivalent: c

i) A ≤ B. ii) A2 = BA and A+A = A+B. iii) There exist an idempotent matrix P ∈ Fn×n and a projector Q ∈ Fn×n such that A = BP = QB and PA# = A# . If i)–iii) hold, then the following statements hold: iv) BAcB = A, BcABc = Ac , and BcBAc = AcBBc = Ac . v) The following statements are equivalent: a) A is range Hermitian. b) AcB = BAc . c) ABc = BcA. If A is range Hermitian, then the following statements are equivalent: ∗

vi) A ≤ B. vii) AB = BA = A2 . viii) (AB)+ = B+A+ = A+B+ and A = AA+B. ix) (AB)+ = B+A+ = A+B+ and A = BAA+ . If either A is range Hermitian, AcB = BAc , or ABc = BcA, then the following statements are equivalent:

649

GENERALIZED INVERSES c

x) A ≤ B. ∗

xi) A ≤ B. #

xii) A ≤ B. If A is range Hermitian, then the following statements are equivalent: c

xiii) A ≤ B. ∗

xiv) A2 ≤ B2 and AB = BA = A2 . c

xv) A2 ≤ B2 and AB = BA = A2 . If B is range Hermitian, then the following statements hold: ∗

xvi) If A ≤ B, then ABB+ = A = BB+A. xvii) If ABB+ = BB+A, then BB+AA+ = AA+BB+ , BB+A+A = A+ABB+ , B+A+ is a (1,2,3)-inverse of AB, and A+B+ is a (1,2,4)-inverse of BA. ∗

If B is range Hermitian and A ≤ B, then the following statements hold: xviii) (AB)+ = B+A+ if and only if BB∗A+A = A+ABB∗ . xix) (BA)+ = A+B+ if and only if B∗BAA+ = AA+B∗B. xx) If B is a partial isometry, then (AB)+ = B+A+ and (BA)+ = A+B+ . c

If A ≤ B and B is range Hermitian, then the following statements hold: xxi) Bc Ac is a (1,2,3)-inverse of AB. xxii) (AB)c = (AB)+ = B+A+ = BcAc . xxiii) If A is range Hermitian, then (AB)+ = BcAc . #

If A is group invertible, B is range Hermitian, and A ≤ B, then the following statements hold: xxiv) B+A+ is a (1,2,3)-inverse of AB, and A+B+ is a (1,2,4)-inverse of BA. xxv) (AB)+ = B+A+ if and only if BB∗A+A = A+ABB∗ . xxvi) (BA)+ = A+B+ if and only if B∗BAA+ = AA+B∗B. The following statements hold: rs

xxvii) If A is range Hermitian and A ≤ B, then B+A+ is a (1,2,3)-inverse of AB and A+B+ is a (1,2,4)-inverse of BA. ∗

#

xxviii) If A is range Hermitian and B is group invertible, then A ≤ B if and only if A ≤ B. ∗

rs

xxix) If A and B are partial isometries, then A ≤ B if and only if A ≤ B. #

xxx) If A and B are group invertible and A ≤ B, then AB is group invertible and (AB)# = B#A# = A#B# . c ∗ Source: [405, 1934]. Remark: “≤” is the core partial ordering. Remark: “≤” is the star partial ordering. See Fact 4.30.8 and Fact 8.4.43.

8.6 Facts on the Moore-Penrose Generalized Inverse for Normal Matrices, Hermitian Matrices, and Partial Isometries Fact 8.6.1. Let A ∈ Fn×n. Then, the following statements are equivalent:

i) A is normal. ii) A+ is normal. iii) AA∗A+ = A+AA∗.

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A is range Hermitian, and A+A∗ = A∗A+. A(AA∗A)+ = (AA∗A)+A. AA+A∗A2A+ = AA∗. A(A∗ + A+ ) = (A∗ + A+ )A. A∗A(AA∗ )+A∗A = AA∗. 2AA∗(AA∗ + A∗A)+AA∗ = AA∗. There exists X ∈ Fn×n such that AA∗X = A∗A and A∗AX = AA∗. There exists X ∈ Fn×n such that AX = A∗ and A+∗X = A+. A+A∗ is a partial isometry. Source: [719] and [1242]. Related: Fact 4.10.12, Fact 4.13.6, Fact 7.17.5, Fact 8.3.32, and Fact 8.10.17. Fact 8.6.2. Let A ∈ Fn×n. Then, the following statements are equivalent: i) A is Hermitian. ii) AA+ = A∗A+. iii) A2A+ = A∗. iv) AA∗A+ = A. Source: [240]. Fact 8.6.3. Let A ∈ Fn×n. Then, 2 rank A ≤ tr A∗A + tr (A∗A)+ . Furthermore, equality holds if and only if A is a partial isometry. Source: [234]. iv) v) vi) vii) viii) ix) x) xi) xii)

8.7 Facts on the Moore-Penrose Generalized Inverse for Idempotent Matrices Fact 8.7.1. Let A ∈ Fn×n , and assume that A is idempotent. Then, A+ = A+A2A+ = A∗A∗+AA+ . Source: [778]. Remark: A+ is the product of the projector A+A onto N(A)⊥ = R(A∗ ) and the projector AA+ onto R(A). Credit: T. N. E. Greville. Related: Fact 8.7.2. Fact 8.7.2. Let A ∈ Fn×n, and consider the following statements:

i) A is idempotent. ii) rank A = tr A. iii) rank A ≤ tr A2A+A∗. Then, i) =⇒ ii) =⇒ iii). Furthermore, the following statements are equivalent: iv) A is idempotent. v) rank A + rank(I − A) = n. vi) rank A = tr A and rank(I − A) = tr(I − A). vii) rank A ≤ tr A and rank(I − A) ≤ tr(I − A). viii) rank A = tr A and there exist distinct k, l ≥ 1 such that Ak = Al . ix) rank A = tr A = tr A2A+A∗. x) rank A + tr A2A+A∗ = tr(A + A∗ ). xi) There exist projectors B, C ∈ Fn×n such that A = (BC)+ . xii) A∗A+ = A+. xiii) A+A∗ = A+. xiv) A∗A+A = A+A. xv) AA+A∗ = AA+. xvi) A+A + (I − A)(I − A)+ = I.

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GENERALIZED INVERSES

xvii) AA+ + (I − A)+(I − A) = I. Finally, let B and C satisfy x). Then, R(B) = R(A∗ ) and R(C) = R(A). Source: [234, 718, 1134, 1247, 1607] and [2418, p. 166]. The last statement is given in [2238, p. 298]. Remark: Note that A∗A+A is a projector, and R(A∗A+A) = R(A∗ ) = R(A+A). Remark: N(A) = R(I − A+A) = R(I − A) = R[(I − A)(I − A+ )]. Remark: xvi) states that the projector onto the null space of A is the projector onto the range of I − A, while xvii) states that the projector onto the range of A is the projector onto the null space of I − A. Remark: xi), which is due to R. Penrose, follows from Fact 8.8.9 by setting A = I. Credit: Sufficiency of xiv)–xvii) is due to G. Trenkler. Related: Fact 4.18.19 and Fact 7.13.28. Fact 8.7.3. Let A ∈ Fn×n, and assume that A is idempotent. Then, A + A∗ − I is nonsingular, and (A + A∗ − I)−1 = AA+ + A+A − I. Source: Fact 8.7.2. Related: Fact 4.18.19, Fact 7.13.28, and [2036, p. 457] for a geometric interpretation of this equality. Fact 8.7.4. Let A ∈ Fn×n, and assume that A is idempotent. Then, 2A(A + A∗ )+A∗ is the projector onto R(A) ∩ R(A∗ ). Source: [2702]. Fact 8.7.5. Let A ∈ Fn×n, and assume that A is idempotent. Then, I − A+ is range Hermitian. Source: [258]. Fact 8.7.6. Let A ∈ Fn×n. Then, the following statements are equivalent: i) A+ is idempotent. ii) AA∗A = A2. If A is range Hermitian, then the following statements are equivalent: iii) A+ is idempotent. iv) AA∗ = A∗A = A. The following statements are equivalent: v) A+ is a projector. vi) A is a projector. vii) A is idempotent, and A and A+ are similar. viii) A is idempotent, and A = A+. ix) A is idempotent, and AA+ = AA∗. x) A+ = A, and A2 = A∗. xi) A and A+ are idempotent. xii) A = AA+. Source: [2418, pp. 167, 168] and [2628, 2710, 2869]. Related: Fact 4.17.4. Fact 8.7.7. Let A ∈ Fn×n, and assume that A and B are idempotent. Then, σmax (AA+ − BB+ ) ≤ σmax (A − B),

σmax (A+ − B+ ) ≤ 2σmax (A − B).

Source: [778] and [1590, p. 58]. Fact 8.7.8. Let A ∈ Fn×n . Then, A is idempotent if and only if there exist B ∈ Fn×n and C ∈ Fl×n

such that A = B(CB)+C. If these conditions hold, then

R(A) = R(AA∗B∗ ) = R(AA∗B∗B) = R(A) ∩ [(AA∗ )+ (R(A) ∩ N(B))]⊥ , N(A) = N(A∗B∗B) = N(AA∗B∗B) = N(B) + (B∗B)+ [R(A) + N(B)]⊥ . Source: [659]. Related: Fact 8.7.9. Fact 8.7.9. Let A ∈ Fn×m and B ∈ Fl×n , assume that R(A) and N(B) are complementary sub-

spaces, and let C ∈ Fn×n be the idempotent matrix onto R(A) along N(B). Then, C = A(BA)+B. If, in addition, A and B are projectors, then C = (BA)+ . Source: [659]. Related: Fact 4.15.3, Fact 8.7.8,

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and Fact 8.8.14.

8.8 Facts on the Moore-Penrose Generalized Inverse for Projectors Fact 8.8.1. Let A ∈ Fn×n. Then, the following statements are equivalent:

i) ii) iii) iv)

A is a projector. A is idempotent, and rank A = tr A+ . A is idempotent, and tr A = tr AA∗ . A is idempotent, and A+ = A. Source: [234]. Fact 8.8.2. Let A ∈ Fn×n. Then, the following statements are equivalent: i) A is a projector. ii) A = A+A. iii) A = AA+ . iv) A = A2 = A+ . Source: [247]. Fact 8.8.3. Let A, B ∈ Fn×n, and assume that A and B are projectors. Then, the following statements hold: i) (AB)+ = B(AB)+. ii) (AB)+ = (AB)+A. iii) (AB)+ = B(AB)+A. iv) (AB)+ = BA − B(B⊥A⊥ )+A. v) (AB)+, B(AB)+, (AB)+A, B(AB)+A, and BA − B(B⊥A⊥ )+A are idempotent. vi) AB = A(AB)+B. vii) (AB)2 = AB + AB(B⊥A⊥ )+AB. viii) AB = AB(AB)+BA(BA)+ = (BA)+AB. ix) AB(AB + BA)+BA = A(A + B)+B. x) (ABA)+ = (BA)+ (AB)+ = A[I − (A⊥ B⊥ )+ ]B[I − (B⊥ A⊥ )+ ]A. xi) (AB)+ = (AB)+A = B(AB)+ . xii) (AB)+AB = BA(BA)+ = (AB)+B = B(BA)+ . xiii) [(AB)2 ]+ = B[I − (B⊥ A⊥ )+ ]A[I − (A⊥ B⊥ )+ ]B[I − (B⊥ A⊥ )+ ]A. xiv) (AB)# = (BA)+ (AB)+ (BA)+ = A[I − (A⊥ B⊥ )+ ]B[I − (B⊥ A⊥ )+ ]A[I − (A⊥ B⊥ )+ ]B. Source: To prove i) note that R[(AB)+ ] = R[(AB)∗ ] = R(BA), and thus R[B(AB)+ ] = R[B(AB)∗ ] = R(BA). Hence, R[(AB)+ ] = R[B(AB)+ ]. It now follows from Fact 4.18.3 that (AB)+ = B(AB)+. iv) follows from Fact 8.4.24; v) and vi) follow from iii); vii) follows from iv) and vi); viii)–xiv) are given in [245, 2657]. Remark: The fact that the first expression in v) is idempotent is given by Fact 8.7.2. Remark: See [2637, 2869]. Related: Fact 7.13.2, Fact 8.4.23, Fact 8.7.6, and Fact 8.8.23. Fact 8.8.4. Let A, B ∈ Fn×n, and assume that A and B are projectors. Then, the following statements hold: i) A(A − B)+B = B(A − B)+A = 0. ii) A − B = A(A − B)+A − B(B − A)+B. iii) (A − B)+ = (A − AB)+ + (AB − B)+. iv) (A − B)+ = (A − BA)+ + (BA − B)+. v) (A − B)+ = A − B + B(A − BA)+ − (B − BA)+A.

GENERALIZED INVERSES

653

(A − B)+ = A − B + (A − AB)+B − A(B − AB)+. (I − A − B)+ = (A⊥B⊥ )+ − (AB)+. (I − A − B)+ = (B⊥A⊥ )+ − (BA)+. R([A, B]) = R[(A − B)+ − (A − B)]. [ ]+ [ ] A A x) [A B][A B]+ = (A + B)(A + B)+ = . B B

vi) vii) viii) ix)

xi) rank[AB − (AB)2 ] = rank B(B⊥ A⊥ )+A = rank B⊥ (BA)+A⊥ . xii) rank [AB, (AB)+ ] = 2 rank [AB BA] − 2 rank AB. xiii) rank(A − B) = tr (A − AB)+ + tr (B − BA)+ . Now, let C, D ∈ Fn×n denote the projectors onto R(A) ∩ [R(A) ∩ R(B)]⊥ and R(B) ∩ [R(B) ∩ R(A)]⊥ , respectively. Then, the following statements hold: xiv) σmax (A − B) = σmax (C − D). xv) tr (A − B)2 = tr (C − D)2 . xvi) rank(A − B) = dim[R(A) + R(B)] − dim[R(A) ∩ R(B)]. xvii) rank(A − B) = rank[A + A⊥ (A⊥ B)+ ] − rank[A − A(AB⊥ )+ ]. Finally, let C, D ∈ Fn×n denote the projectors onto R(A) ∩ R(I − B) and R(I − A) ∩ R(B), respectively. Then, the following statement holds: xviii) C = A − A(AB)⊥ and D = A⊥ − A⊥ (A⊥ B⊥ )+ . xix) rank(C − D) = dim[R(C) + R(D)] = rank[C + D⊥ (C⊥ D)+ ]. Source: [249, 718, 2636]. Related: Fact 4.16.5. Fact 8.8.5. Let A, B ∈ Fn×n, and assume that A and B are projectors. Then, the following statements are equivalent: i) AB is a projector. ii) (AB)+ = AB. iii) (AB)+ = BA. iv) (A − B)+ = A − B. v) B(A − BA)+ = (B − BA)+A. vi) A − B is tripotent. vii) [AB, (AB)+ ] = 0. viii) [AB(AB)+ , (AB)+AB] = 0. ix) [AB(AB)∗ , (AB)+AB] = 0. x) [AB(AB)+ , (AB)∗AB] = 0. xi) [(AB)+ ]2 = [(AB)2 ]+. xii) I − AB is normal. xiii) I − AB is a partial isometry. xiv) I − AB is star-dagger. xv) (I − AB)+ is idempotent. xvi) [(AB)2 ]+ = [(AB)+ ]2 . xvii) (I − AB)+ = I − AB. xviii) (ABA)+ = ABA. xix) AB(AB)+ = ABA. xx) (AB)+ = A(AB)+ .

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(AB)+ = B(BA)+ . (AB − BA)+ = (AB)+ − (BA)+ . (ABA)+ is an orthogonal projector. tr (ABA)+ = tr (AB)+ . rank[I − (AB)+ (BA)+ ] = def AB. (AB)+ = (AB)# . A + B − 32 AB is a 1-inverse of A + B. A − B − AB is a 1-inverse of A − B. B = AB(AB)+ + A⊥ − A⊥ (A⊥ B⊥ )+ . AB(AB)+ = A − A(AB⊥ )+ . I − A − B = (I − A − B)+ . A + B − AB = (A + B − AB)+ . rank ABA = tr (ABA)+ . tr ABA = tr (ABA)+ . rank(AB − BA) = tr (AB − BA)+ . rank(A + B − AB) = tr (A + B − AB)+ . Source: [235, 245, 252, 261, 718, 2636, 2657, 2705, 2706]. Remark: Star-dagger is defined in Fact 8.3.32. Related: Fact 4.18.7, Fact 7.13.5, Fact 7.13.6, Fact 8.4.28, and Fact 8.11.13. Fact 8.8.6. Let A, B ∈ Fn×n, assume that A and B are projectors. Then, the following statements are equivalent: i) A + B is a projector. ii) (A + B)+ = A + B. iii) AB + BA = (AB + BA)+ . iv) rank(I − A − B) = tr (I − A − B)+ . v) AB is a projector, and rank(A + B) = tr (A + B)+ . Source: [261, 2657]. Related: Fact 4.18.9. Fact 8.8.7. Let A, B ∈ Fn×n, assume that A and B are projectors. Then, the following statements are equivalent: i) A − B is a projector. ii) rank(A − B) = tr (A − B)+ . Source: [261]. Related: Fact 4.18.10. Fact 8.8.8. Let A, B ∈ Fn×n, assume that A and B are projectors, let P ∈ Fn×n be the projector onto N([A, B]), and let α, β ∈ F, where α and β are nonzero. Then, the following statements hold: i) rank [A, B] ≤ rank(αA + βB). ii) If [A, B] is nonsingular, then αA + βB. iii) rank [A, B] = rank (αA + βB)P⊥ . iv) αA + βB is nonsingular if and only if rank (αA + βB)P + rank(αA + βB) = n. v) A − B is nonsingular if and only if (A + B)P = P. Now, assume that AB = BA. Then, the following statements hold: vi) spec(αA + βB) ⊂ {0, α, β, α + β}. vii) If α + β , 0, then rank(αA + βB) = rank(A + B). viii) (αA + βB)+ = [(α + β)+ − α1 − β1 ]AB + α1 A + β1 B.

xxi) xxii) xxiii) xxiv) xxv) xxvi) xxvii) xxviii) xxix) xxx) xxxi) xxxii) xxxiii) xxxiv) xxxv) xxxvi)

GENERALIZED INVERSES

655

Source: [406]. Related: Fact 4.18.20. Fact 8.8.9. Let A ∈ Fn×m and B ∈ Fm×n . Then, BAB = B if and only if there exist projectors C ∈ Fn×n and D ∈ Fm×m such that B = (CAD)+. Source: [1232]. Fact 8.8.10. Let A ∈ Fn×n. Then, the following statements are equivalent:

i) A2 = AA∗A. ii) A is the product of two projectors. iii) A = A(A+ )2A. Credit: T. Crimmins. See [2291]. Fact 8.8.11. Let A ∈ Fn×n, and assume that A is idempotent. Then, A is the idempotent matrix onto R(AA+ ) along R(I − A+A). Source: Use Fact 8.10.2 with A# = A, or note that R(A) = R(AA+ ) and N(A) = R(I − A+A). Fact 8.8.12. Let A, B ∈ Fn×n, and assume that A and B are projectors. Then, the following statements hold: i) (A⊥B)+ = (I − BA)+B(I − BA) = (I − BA)+ BA⊥ . ii) (A⊥B)+ is the idempotent matrix onto R(B) ∩ [N(A) + N(B)] along R(A) + [N(A) ∩ N(B)]. iii) If R(A) + R(B) = Fn , then (A⊥B)+ is the idempotent matrix onto R(B) ∩ [N(A) + N(B)] along R(A). iv) If R(A) ∩ R(B) = {0}, then I − BA is nonsingular and (A⊥B)+ = (I − BA)−1B(I − BA) = (I − BA)−1 BA⊥ = (I − AB)−1 A⊥ is the idempotent matrix onto R(B) along R(A)+[N(A)∩N(B)]. v) (I − BA)+ B⊥ is the idempotent matrix onto (R(B) ∩ [N(A) + N(B)]) + [N(A) ∩ N(B)] along R(B). vi) A(A + B− BA)+ is the idempotent matrix onto R(A) along (R(A) ∩ [N(A) + N(B)]) + [N(A) ∩ N(B)]. vii) R[(I − BA)+ B⊥ ] = N[B(A + B − AB)]+ and N[(I − BA)+ B⊥ ] = R[B(A + B − AB)]+ . viii) R(A) + R(B) = Fn if and only if (A⊥B)+ = (I − AB)+ A⊥ . ix) R(A) ∩ R(B) = {0} if and only if (A⊥B)+ = B(A + B − BA)+ . x) R(A) and R(B) are complementary subspaces if and only if (I − AB)+ A⊥ = B(A + B − BA)+ . xi) (A⊥B)+ = (I − BAB)+ BA⊥ . xii) (A⊥B)+ = B(B − A)+ . xiii) (B − A)+ = (A⊥B)+ − (BA⊥ )+ . xiv) Let C denote the projector onto [R(A) ∩ R(B)] + [N(A) ∩ N(B)]. Then, (B − A)+ = (A⊥B)+ + (BA⊥ )+ − I + C. xv) Let C denote the projector onto [R(A) ∩ R(B)] + [N(A) ∩ N(B)], and let D denote the projector onto [R(A) + R(B)] ∩ [N(A) + N(B)]. Then, C + D = I. xvi) (B − A)+ = (A + B)+ (B − A)(A + B)+ . xvii) (A + B)+ = (B − A)+ (A + B)(B − A)+ + A(A + B)+B. xviii) (AB)+ = (A + B − I)+ A = B(A + B − I)+ . xix) [(A + B − I)+ ]2 A = (ABA)+ = A[(A + B − I)+ ]2 . xx) (BA)+ = (BA)+B = A(A + B − I)+B = A[(A + B − I)+ ]2B. xxi) For all k ≥ 0, A[(A + B − I)+ ]2k+1B = A[(A + B − I)+ ]2k+2B. xxii) (BA)+ = AB(AB)# . xxiii) (AB)# = A[(A + B − I)+ ]3B = A[(A + B − I)+ ]4B. xxiv) A[(A + B − I)+ ]3B = (BA)+ (A + B − I)+ (BA)+ .

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CHAPTER 8

A[(A + B − I)+ ]2B + B[(A + B − I)+ ]2A = (A + B − I)+ + (A + B − I)(A + B − I)+ . (A⊥ B)+ A = A(BA⊥ )+ = 0. (A⊥ B)+ = B(A⊥ B)+ = (A⊥ B)+B. (A + B)(A − B)+ + (A − B)+ (A + B) = 2(A − B)+ . (A − B)(A − B)+ = [(A + B)(A − B)+ ]2 = (AB⊥ )+ + (BA⊥ )+ = (B⊥ A)+ + (A⊥ B)+ = AB⊥ (AB⊥ )+ + A⊥ B(A⊥ B)+ . Source: Fact 4.18.19, Fact 8.8.14, and [248]. Fact 8.8.13. Let A, B ∈ Fn×n, assume that A and B are projectors. Then, (AB)+ is the idempotent matrix onto R(BA) along N(BA) = R(AB)⊥ . Source: [778]. Remark: Corollary 10.3.7 implies that BA is diagonalizable over F and thus group invertible. Corollary 4.8.10 thus implies that R(BA) and N(BA) are complementary subspaces. Credit: T. N. E. Greville. Related: Fact 8.8.14. Fact 8.8.14. Let A, B ∈ Fn×n, assume that A and B are projectors, and assume that R(A) and R(B) are complementary subspaces. Then, (B⊥A)+ is the idempotent matrix onto R(A) along R(B). Source: Fact 8.8.15, [1246], and [1504]. Remark: Fact 8.8.3 implies that (B⊥A)+ is idempotent. Fact 4.18.19 implies that I − BA is nonsingular and (B⊥A)+ = (I − AB)−1A(I − AB) = A(I − BA)−1 (I − AB) = (I − BA)−1 (I − B) = B(A + B − BA)−1 . Related: Proposition 4.8.6, Fact 4.16.4, Fact 8.8.13, and Fact 8.8.15. Fact 8.8.15. Let A, B ∈ Fn×n, and assume that R(A) and R(B) are complementary subspaces. △ △ Furthermore, define P = AA+ and Q = BB+. Then, (P⊥Q)+ is the idempotent matrix onto R(A) along R(B). Source: [1232]. Related: Fact 4.16.4, Fact 4.18.19, Fact 8.8.14, and Fact 8.10.2. Fact 8.8.16. Let A, B ∈ Fn×n , and assume that A and B are projectors. Then, the following statements hold: i) A + A⊥ (A⊥ B)+ is the projector onto R(A) + R(B). ii) A + A⊥ (A⊥ B)+ = (A + B)(A + B)+ = A + (AB⊥ )+AB⊥ . iii) A − A(AB⊥ )+ is the projector onto R(A) ∩ R(B). iv) A − A(AB⊥ )+ = A − AB⊥ (AB⊥ )+ = A − (B⊥ A)+B⊥ A = 2A(A + B)+B = 2[A − A(A + B)+B]. Source: [2239]. Related: Fact 4.18.11 and Fact 8.8.17. Fact 8.8.17. Let A, B ∈ Fn×n , and assume that A and B are projectors. Then, the following statements hold: ([ ]) A⊥ + i) R[A − A(AB⊥ ) ] = N . B⊥ ii) σmax (AB) = σmax [B − A⊥ (A⊥ B)+ ]. iii) If R(A) ∩ R(B) = {0}, then

xxv) xxvi) xxvii) xxviii) xxix)

σ2max [(A⊥ B)+ ] =

1 1−

σ2max (AB)

.

iv) If A , 0 and R(A) ∩ R(B) = {0}, then σ2max (AB) < v) vi) vii) viii) ix)

σ2max [(A⊥ B)+ ] < 1. 1 + σ2max [(A⊥ B)+ ]

(A − B)(A − B)+ + [(A − B)(A − B)+ ]⊥ [[(A − B)(A − B)+ ]⊥ (I − A − B)(I − A − B)+ ]+ = I. (A − B)(A − B)+ (I − A − B)(I − A − B)+ = [A, B][A, B]+ . (A − B)(A − B)+ + (I − A − B)(I − A − B)+ − [A, B][A, B]+ = I. (A + B)(A + B)+ + [(A − B)(A − B)+ ]⊥ [[(A + B)(A + B)+ ]⊥ (I − A − B)(I − A − B)+ ]+ = I. (A + B)(A + B)+ (I − A − B)(I − A − B)+ = (AB + BA)(AB + BA)+ .

657

GENERALIZED INVERSES

x) (A − B)(A − B)+ + (I − A − B)(I − A − B)+ − (AB + BA)(AB + BA)+ = I. In addition, the following statements are equivalent: xi) R([A, B]) = R(A − B). xii) R(AB + BA) = R(A + B). xiii) I − A − B is nonsingular. xiv) A = AB(AB)+ and B = BA(BA)+ . Finally, the following statements are equivalent: xv) R([A, B]) is nonsingular. xvi) A − B and I − A − B are nonsingular. xvii) [R(A) + R(B)] ∩ [N(A) + N(B)] = Fn and [R(A) + N(B)] ∩ [N(A) + R(B)] = Fn . xviii) R(A) + R(B) = Fn , R(A) + N(B) = Fn , N(A) + R(B) = Fn , and N(A) + N(B) = Fn . Source: [242, 244]. Related: Fact 4.18.11, Fact 4.18.12, Fact 7.13.27, and Fact 7.13.28. Fact 8.8.18. Let A, B ∈ Fn×n, and assume that A and B are complementary projectors. Then, the following statements hold: i) (B⊥A)+ = AB⊥ − A(A⊥B)+ B⊥ . ii) rank[(B⊥A)+ − AB⊥ ] = rank AB. iii) The following statements are equivalent: a) (B⊥A)+ = A. b) AB = 0. c) R(A)⊥ = R(B). Source: [2657]. Fact 8.8.19. Let A, B ∈ Fn×n, and assume that A and B are projectors. Then, glb({A, B}) = 2A(A + B)+B = 2B(A + B)+A. Furthermore, glb({A, B}) is the projector onto R(A) ∩ R(B); that is, R(A) ∩ R(B) = R[2A(A + B)+ B]. In addition, glb({A, B}) = lim A(BA)k = A#B k→∞

= 2AB(A + B)+A = 2BA(A + B)+B = 2A(A + B)+BA = 2B(A + B)+AB = 2AB(A + B)+AB = 2BA(A + B)+BA = 2[A − A(A + B)+A] = 2[B − B(A + B)+B] = A − A(AB⊥ )+ = B − B(BA⊥ )+ = A − (B⊥A)+ B⊥A = B − (A⊥B)+ A⊥B = 2AA+ (AA+ + BB+ )+BB+ = 2BB+ (AA+ + BB+ )+AA+ ([ ]+ [ ]) A⊥ A⊥ = . B⊥ B⊥ ⊥ Source: [84], [1302, pp. 64, 65, 121, 122], [1381, p. 191], [2238, pp. 304–306], and Fact 4.18.12. Related: Fact 8.4.8, Fact 10.11.68, and Fact 10.24.20. Fact 8.8.20. Let A, B ∈ Fn×n, and assume that A and B are projectors. Then,

lub({A, B}) = (A + B)(A + B)+ = (A + B)+ (A + B). Furthermore, lub({A, B}) is the projector onto R(A + B); that is, R(A + B) = R[(A + B)(A + B)+ ]. In

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CHAPTER 8

addition, lub({A, B}) = I − lim A⊥(B⊥A⊥ )k k→∞

= I − 2A⊥(A⊥ + B⊥ )+B⊥ = I − 2B⊥(B⊥ + A⊥ )+A⊥ = A + (AB⊥ )+AB⊥ = B + (BA⊥ )+BA⊥ = A + (BA⊥ )+BA⊥ = B + (AB⊥ )+AB⊥ = A + A⊥ (A⊥B)+ = B + B⊥ (B⊥A)+ = [A B][A B]+ . Source: For the first equality, use Fact 3.12.16 and Fact 10.7.3. For the second equality, see [84], [1302, pp. 64, 65, 121, 122], and [2238, pp. 303–306]. Remark: Fact 4.10.31 implies that R(A) + R(B) = R([A B]) = R(A + B) = span[R(A) ∪ R(B)]. Related: Fact 4.18.11, Fact 8.4.9, and

Fact 10.24.20. Fact 8.8.21. Let A, B ∈ Fn×n, and assume that A and B are projectors. Then, the following statements hold: i) glb({A, B}) ≤ lub({A, B}). ii) glb({A, B}) + lub({A⊥ , B⊥ }) = I. iii) lub({A⊥ , B⊥ }) = glb({A, B})⊥ and glb({A⊥ , B⊥ }) = lub({A, B})⊥ . iv) rank glb({A, B}) = rank AA+ (AA+ + BB+ )+BB+ = rank A + rank B − rank [A B]. v) rank lub({A, B}) = rank [A B]. vi) rank glb({A, B}) + rank lub({A, B}) = rank A + rank B. vii) tr glb({A, B}) + tr lub({A, B}) = tr A + tr B. If A ≤ B, then the following statement holds: viii) B = lub[{A, glb({A⊥ , B})}]. If AB = BA. Then, the following statements hold: ix) 2AB ≤ A + B. x) glb({A, B}) = AB = BA. xi) lub({A, B}) = A + B − AB = B + A − BA. If AB = BA = 0, then the following statements hold: xii) glb({A, B}) = 0. xiii) lub({A, B}) = A + B. Source: [2238, pp. 306, 311], Fact 8.4.8, and Fact 8.4.9. vi) is equivalent to Theorem 3.1.3. [ ] △ Fact 8.8.22. Let A ∈ Fn×n , B ∈ Fn×m , and C ∈ Fm×m , define A = BA∗ CB , and assume that A is a projector. Then, rank A = rank A + rank C − rank B,

rank(A − B∗C +B) = rank C − rank B∗C +B. Source: [2657]. Fact 8.8.23. Let k ≥ 1, let A1 , . . . , Ak ∈ Fn×n, assume that A1 , . . . , Ak are projectors, and define

B1 , . . . , Bk−1 ∈ Fn×n by △

Bi = (A1 · · · Ak−i+1 )+A1 · · · Ak−i ,

i = 1, . . . , k − 2;

△

Bk−1 = A2 · · · Ak (A1 · · · Ak )+.

Then, B1 , . . . , Bk−1 are idempotent, and (A1 · · · Ak )+ = B1 · · · Bk−1 .

659

GENERALIZED INVERSES

Source: [2651]. Remark: For k = 2, the result B1 is idempotent is given by vi) of Fact 8.8.3.

8.9 Facts on the Moore-Penrose Generalized Inverse for Partitioned Matrices Fact 8.9.1. Let A ∈ Fn×m and B ∈ Fn×l. Then,

R(A) ∩ R(B) = [A 0n×l ]N([A B]) ∩ [0n×m B]N([A B]) = [A 0n×l ]R(Im+l − [A B]+ [A B]) ∩ [0n×m B]R(Im+l − [A B]+ [A B]). Source: Fact 3.14.14. Fact 8.9.2. Let A ∈ Fn×m and B ∈ Fn×l. Then,

([

R(A) ∩ R(B) = N

I − AA+ I − BB+

])

,

[

] I − AA+ dim[R(A) ∩ R(B)] = def . I − BB+

Related: Fact 4.18.12. Fact 8.9.3. Let A ∈ Fn×m and B ∈ Fl×m. Then,

N(A) + N(B) = R(I − A+A) + R(I − B+B) = R([I − A+A I − B+B]), dim[N(A) + N(B)] = rank [I − A+A I − B+B], N

([ ]) A = N(A) ∩ N(B) = R(I − A+A) ∩ R(I − B+B), B

[ ] A def = def [I − A+A I − B+B] − def A − def B. B Source: The last equality follows from Fact 3.14.14. Fact 8.9.4. Let A ∈ Fn×m and B ∈ Fm×l. Then,

N(A) + R(B) = R(I − A+A) + R(B) = R([I − A+A B]), [ ] A dim[N(A) + R(B)] = rank [I − A+A B] = rank + rank B − rank A, I − BB+ ([ ]) A N(A) ∩ R(B) = N(A) ∩ N(I − BB+ ) = N , I − BB+ [ ] [ ] A A dim[N(A) ∩ R(B)] = def = m − rank , I − BB+ I − BB+ [ ] A rank A + rank B ≤ rank + rank B = m − dim[N(A) ∩ R(B)] + rank B = m + rank AB. I − BB+ Source: The fourth equality follows from Fact 8.9.15. The last equality follows from (3.6.19). Remark: This result provides an alternative proof of Sylvester’s inequality given by Proposition 3.6.11. Related: Fact 3.13.21. Fact 8.9.5. Let A ∈ Fn×m and B ∈ Fn×l. Then, the following statements are equivalent:

i) ii) iii) iv) v)

rank [A B] = rank A + rank B. R(A) ∩ R(B) = {0}. rank(AA∗ + BB∗ ) = rank A + rank B. A∗(AA∗ + BB∗ )+A is idempotent. A∗(AA∗ + BB∗ )+A = A+A.

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CHAPTER 8

vi) A∗(AA∗ + BB∗ )+B = 0. Source: [1924, pp. 56, 57]. Remark: Additional equivalent statements are given in Fact 8.9.26. Related: Fact 3.14.15, Fact 8.4.8, and Fact 8.9.21. △ △ Fact 8.9.6. Let A ∈ Fn×m and B ∈ Fn×l, and define the projectors P = AA+ and Q = BB+. Then, the following statements are equivalent: i) rank [A B] = rank A + rank B = n. ii) P − Q is nonsingular. If these statements hold, then (P − Q)−1 = (P − PQ)+ + (PQ − Q)+ = (P − QP)+ + (QP − Q)+ = P − Q + Q(P − QP)+ − (Q − QP)+P. Source: [718]. △ Fact 8.9.7. Let A ∈ Fn×m, B ∈ Fn×l, C ∈ Fk×m, and D ∈ Fk×l, and define X = B − AA+B, △

△

Y = C − CA+A, and Z = (Ik − YY + )(D − CA+B)(Il − X +X). Then,

rank [A B] = rank A + rank(B − AA+B) = rank B + rank(A − BB+A) = rank A + rank B − dim[R(A) ∩ R(B)], [ ] A rank = rank A + rank(C − CA+A) = rank C + rank(A − AC +C) C = rank A + rank C − dim[R(A∗ ) ∩ R(C ∗ )], [

0 C

] B = rank B + rank C + rank (Ik − CC + )D(Il − B+B), D

[

A rank C

] 0 = rank A + rank D + rank (Ik − DD+ )C(Im − A+A), D

[

] B = rank A + rank D + rank (In − AA+ )B(Il − D+D), D

rank

A rank 0 [ rank [

] B = rank B + rank C + rank (In − BB+ )A(Im − C +C), 0

A C

] [ B I = D CA+

A C [

A rank C

0 I

]  

A C − CA+A

[ B − AA+B  I  D − CA+B 0

] A+B , I

  ]  0 B − AA+B  B  = rank A + rank  D C − CA+A D − CA+B [ ] A = rank + rank [A B] − rank A + rank Z C = rank A + rank X + rank Y + rank Z ≤ rank A + rank B + rank C + rank(D − CA+B),

[

AA∗ rank B∗

] [ B −AA∗ = rank 0 B∗

] B = rank [A B] + rank B, 0

661

GENERALIZED INVERSES

[ rank

A∗AA∗ CA∗

] A∗B = rank A + rank(D − CA+B). D

Furthermore, the following statements hold: i)

[

] B . D

A rank A + rank(D − CA B) ≤ rank C +

ii) If AA+B = B and CA+A = C, then rank A + rank(D − CA+B) = rank

[

] B . D

A C

[ ] iii) rank A + rank(D − CA+B) = rank CA DB if and only if the following statements hold: a) N(D − CA+B) ⊆ N[(I − AA+ )B]. b) N[(D − CA+ B)∗ ] ⊆ N[(I − A+A)C ∗ ]. c) (I − AA+ )B(D − CA+ B)+C(I − A+A) = 0. iv) If n = m and A is nonsingular, then [ ] A B −1 n + rank(D − CA B) = rank . C D v) If k = l and D is nonsingular, then k + rank(A − BD−1C) = rank

[

A C

] B . D

vi) Let A = AL AR , X = XL XR , Y = YL YR , and Z = ZL ZR be full rank factorizations. Then,   A+L B  [ ] [ ]  AR   0 ZR A B AL 0 0 XL   = C D CA+R ZL YL (Ik − YY + )(D − CA+B)XR+  YR YL+ (D − CA+B)  0 XR is a full-rank factorization. Source: The first expression for

[A B]

C D follows from Fact 3.14.11. The inequality follows from the previous equality and Fact 3.14.20. See [212, 639, 1966, 2035], [2238, [ p. ]194], [2634, 2638, 2656, 2941], Fact 3.14.15, and Fact 3.14.17. Remark: Since rank C ≤ rank CA D0 , it follows that

rank C ≤ rank A + rank D + rank (In − DD+ )C(Im − A+A). Using Fact 3.14.20, this inequality can be interpolated by noting that rank C ≤ rank CA+A + rank C(Im − A+A) + rank DD+C + rank (Ik − DD+ )C − rank C ≤ rank A+A + rank C(Im − A+A) + rank DD+ + rank (Ik − DD+ )C − rank C = rank A + rank C(Im − A+A) + rank D + rank (Ik − DD+ )C − rank C ≤ rank A + rank D + rank (Ik − DD+ )C(Im − A+A). Related: Proposition 3.9.3, Fact 8.9.34, and Proposition 10.2.4. △ Fact 8.9.8. Let A ∈ Fn×n, B ∈ Fn×m, and C ∈ Fm×m, and define X = B − AA+B. Then,

([

ν−

A B∗

B 0

])

= rank B + ν− [(I − BB+ )A(I − BB+ )] [ ] A B = rank ∗ − rank B − ν+ [(I − BB+ )A(I − BB+ )], B 0

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CHAPTER 8

([ ν+

([ ν−

([ ν+

A B∗

B C

A B∗

B C

])

])

A B∗

B 0

])

= rank B + ν+ [(I − BB+ )A(I − BB+ )] [ ] A B = rank ∗ − rank B − ν− [(I − BB+ )A(I − BB+ )], B 0

= rank [A B] − ν+ (A) + ν+ [(I − X +X)(C − B∗A+B)(I − X +X)] [ ] A B = rank ∗ − rank [A B] + ν− (A) − ν− [(I − X +X)(C − B∗A+B)(I − X +X)], B C = rank [A B] − ν− (A) + ν− [(I − X +X)(C − B∗A+B)(I − X +X)] [ ] A B = rank ∗ − rank [A B] + ν+ (A) − ν+ [(I − X +X)(C − B∗A+B)(I − X +X)]. B C

Source: [2656]. [ ] △ Fact 8.9.9. Let A ∈ Fn×n, B ∈ Fn×m, and C ∈ Fm×m, define A = BA∗ CB , and assume that A is Hermitian and B = AA+B. Then, In A = In A + In(A|A). Remark: This is the Haynsworth inertia additivity formula. See [2273]. Remark: If A is positive semidefinite, then B = AA+B. See

Proposition 10.2.5 and [2991, p. 257]. Fact 8.9.10. Let A ∈ Fn×m, B ∈ Fk×l, and C ∈ Fn×l. Then, [ ] A C min rank(AX + YB + C) = rank − rank A − rank B. 0 B X∈Fm×l,Y∈Fn×k Furthermore, X, Y is a minimizing solution if and only if there exist U ∈ Fm×k, U1 ∈ Fm×l, and U2 ∈ Fn×k such that X = −A+C + UB + (Im − A+A)U1,

Y = (AA+ − I)CB+ − AU + U2 (Ik − BB+ ).

Finally, all such matrices X ∈ Fm×l and Y ∈ Fn×k satisfy AX + YB + C = 0 [

if and only if rank

] A C = rank A + rank B. 0 B

Source: [2633, 2668]. Related: Fact 7.11.25. Note that A and B are square in Fact 7.11.25. Fact 8.9.11. Let A ∈ Fn×n, B ∈ Fn×m, and C ∈ Fm×m, and assume that

[

A B B∗ C

]

is a projector. Then,

rank(C − B∗A+B) = rank C − rank B∗A+B. Source: [226, 2646]. Fact 8.9.12. Let A ∈ Fn×m, B ∈ Fn×l, C ∈ Fl×m, and D ∈ Fl×l, and assume that D is nonsingular.

Then,

rank A = rank(A − BD−1C) + rank BD−1C

if and only if there exist X ∈ Fm×l and Y ∈ Fl×n such that B = AX, C = YA, and D = YAX. Source: [734]. Related: Fact 8.4.10 for the case l = 1. Fact 8.9.13. Let A ∈ Fn×m, B ∈ Fn×l, and C ∈ Fl×m, and assume that YAX ∈ Fl×l is nonsingular. Then, rank[A − B(YAX)−1C] = rank A − rank B(YAX)−1C.

663

GENERALIZED INVERSES

If, in addition, l = 1, then

) 1 1 BC = rank A − rank BC. rank A − YAX YAX (

Source: Fact 8.4.10 and Fact 8.9.12. Remark: The second equality is the Wedderburn rank-one

reduction formula. See [734] and [1451, p. 14]. [ △ Fact 8.9.14. Let A11 ∈ Fn×m, A12 ∈ Fn×l, A21 ∈ Fk×m, and A22 ∈ Fk×l, and define A = AA11 21 [B B ] △ F(n+k)×(m+l) and B = AA+ = BT11 B1222 , where B11 ∈ Fn×n, B12 ∈ Fn×k, and B22 ∈ Fk×k. Then,

A12 A22

]

∈

12

rank B12 = rank [A11 A12 ] + rank [A21 A22 ] − rank A. Source: [2675]. Related: Fact 4.15.22 and Fact 4.17.13. Fact 8.9.15. Let A ∈ Fn×m and B ∈ Fm×l. Then,

[

0 rank n×l B

] [ ] A A + = rank A + rank [I − A A B] = rank + rank B Im I − BB+

= m + rank AB = rank A + rank B + rank (I − BB+ )(I − A+A), max {0, rank A + rank B − m} ≤ rank A + rank B − rank [A∗ B] ≤ rank AB ≤ min {rank A, rank B} ≤ min {n, m, l}. The following statements are equivalent: i) rank AB = rank A. ii) R(AB) = R(A). iii) [B I − A+A] is right invertible. The following statements are equivalent: iv) rank AB = rank B. v) N(AB) = N(B). [ ] A vi) I−BB + is left invertible. The following statements are equivalent: [ ] A vii) rank 0n×l B Im = rank A + rank B. viii) rank AB = rank A + rank B − m. ix) rank AB = rank A + rank B − rank [A∗ B], and rank [A∗ B]. x) There exist X ∈ Fl×m and Y ∈ Fm×n such that BX + Y A = I. xi) (I − BB+ )(I − A+A) = 0. xii) N(A) ⊆ R(B). xiii) N(B∗ ) ⊆ R(A∗ ). Source: [1966]. Note that [ ] [ ][ ] [ ][ ] [ I 0 0 A 0 A I B+ −I 0n×l A = + = = 0 B Im A I B I − A+A B I − BB+ 0 I

A I

][

AB 0 B I

]

and that the rows of [0 A] are orthogonal to the rows of [B I − A+A]. Remark: The generalized inverses can be replaced by arbitrary (1)-inverses. Credit: Y. Tian. Fact 8.9.16. Let A ∈ Fn×m, B ∈ Fm×l, and C ∈ Fl×k. Then, [ ] 0 AB rank = rank B + rank ABC BC B = rank AB + rank BC + rank [(I − BC)(BC)+ ]B[(I − (AB)+ (AB)],

664

CHAPTER 8

max {0, rank A + rank B + rank C − m − l} ≤ max {0, rank A + rank B + rank C − rank [A∗ B] − rank [B∗ C]} ≤ max {0, rank AB + rank BC − rank B} ≤ rank ABC ≤ min {rank AB, rank BC} ≤ min {rank A, rank B, rank C} ≤ min {n, m, l, k}. Furthermore, the following statements are equivalent: i) rank ABC = rank B. ii) rank AB = rank BC = rank B. iii) R[(AB)∗ ] = R(BC) = R(B). The following statements are equivalent: [ ] 0 AB = rank AB + rank BC. iv) rank BC B v) rank ABC = rank AB + rank BC − rank B. vi) There exist X ∈ Fk×l and Y ∈ Fm×n such that BCX + YAB = B. vii) [(I − BC)(BC)+ ]B[(I − (AB)+ (AB)] = 0. The following statements are equivalent: viii) rank ABC = rank A + rank B + rank C − rank [A∗ B] − rank [B∗ C]. ix) rank ABC = rank AB + rank BC − rank B, rank [A∗ B] = rank A + rank B + rank AB, and rank [B∗ C] = rank B + rank C − rank BC. The following statements are equivalent: x) rank ABC = rank A + rank B + rank C − m − l. xi) rank ABC = rank AB + rank BC − rank B, rank AB = rank A + rank B − m, rank BC = rank B + rank C − l. xii) rank ABC = rank A + rank B + rank C − rank [A∗ B] − rank [B∗ C], rank [A∗ B] = m, and rank [B∗ C] = l. Source: [1966, 2675] and Fact 7.11.25. Credit: Y. Tian. Related: Fact 3.14.20. Fact 8.9.17. Let x, y ∈ R3, and assume that x and y are linearly independent. Then,   +  x (I3 − yϕT )  + [x y] =   , ϕT △

△

where x+ = (xTx)−1xT , α = yT(I − xx+ )y, and ϕ = α−1 (I − xx+ )y. Now, let x, y, z ∈ R3, and assume that x and y are linearly independent. Then,    (I2 − βwwT )[x y]+  +   , [x y z] =   T + βw [x y] △

△

where w = [x y]+z and β = 1/(1 + wTw). Source: [2701]. Fact 8.9.18. Let x, y ∈ R3 . Then, the following statements hold: i) If x , 0, then K +(x) = −(xTx)−1K(x). ii) K(x)K(x)+ = I − xx+ . iii) K(x)K(x)+ y = y if and only if xTy = 0. iv) (K(x) + ȷI)+ = − 14 [K(x) + 3xx+ ȷ + ȷI]. v) If xTy , 0, then [K(x)K(y)]+ =

1 (xx+ xT y

+ yy+ − I) −

1 yxT . xT xyTy

665

GENERALIZED INVERSES

vi) If xTy = 0 and xTx + yTy , 0, then +     K(x) y  −1  K(x) y   =   . xT x + yTy  −yT 0  −yT 0 Source: [2699, 2723]. Related: Fact 4.12.1. Fact 8.9.19. Let A ∈ Fn×m and b ∈ Fn. Then,

  +  A (In − bϕ∗ )   , [A b] =  ϕ∗ +

where

  [b A] =  +

ϕ∗ A+(In − bϕ∗ )

   ,

 + +∗ +    (b − AA b) , b , AA b, ϕ=   γ−1 (AA∗ )+ b, b = AA+b. △

△

and γ = 1 + b∗(AA∗ )+b. Source: [31, p. 44], [1040, p. 270], and [2423, p. 148]. Credit: T. N. E. Greville. Fact 8.9.20. Let A ∈ Fn×m and B ∈ Fn×l. Then, [ ∗ ] A (AA∗ + BB∗ )+ [A B]+ = ∗ , [A B][A B]+ = (AA∗ + BB∗ )(AA∗ + BB∗ )+ . B (AA∗ + BB∗ )+ Source: [2645]. Related: Fact 8.4.33. Fact 8.9.21. Let A ∈ Fn×m and B ∈ Fn×l. Then,

  +  A − A+B(C + + D)   ,  [A B] =  C+ + D +

where △

C = (I − AA+ )B, Furthermore,

△

△

D = (I − C +C)[I + (I − C +C)B∗(AA∗ )+B(I − C +C)]−1B∗(AA∗ )+(I − BC + ).   ∗  A (AA∗ + BB∗ )−1           B∗(AA∗ + BB∗ )−1  ,          −1        A∗A A∗B   A∗  + [A B] =     ,  ∗    B A B∗B   B∗              A∗(AA∗ )−1(I − BE)       ,    E

rank [A B] = n,

rank [A B] = m + l, rank A = n, △

where E = [I + B∗(AA∗ )−1B]−1B∗(AA∗ )−1. Finally, define M = [I + B∗ (AA∗ )+B]−1 . Then, the following statements hold: [ ] A+ − A+ BMB∗ (AA∗ )+ i) [A B]+ = if and only if C +CB∗ (AA∗ )+ B = 0. C + + MB∗ (AA∗ )+ [ + ] A − A+ BMB∗ (AA∗ )+ if and only if C = 0. ii) [A B]+ = MB∗ (AA∗ )+ ] [ + A − A+ BC + iii) [A B]+ = if and only if C +CB∗ (AA∗ )+B = B∗ (AA∗ )+B. C+

666

CHAPTER 8

[

] A+ if and only if C = B. B+ Source: [761], [536, pp. 14–18] and [2403, pp. 193–195]. Remark: If [A B] is square and nonsingular and A∗B = 0, then the second expression yields Fact 3.22.9. See Fact 8.4.34. Related: Fact 8.9.5 and Fact 8.9.26. Fact 8.9.22. Let A ∈ Fn×m and B ∈ Fn×l, and assume that R(B) ⊆ R(A). Then, [ + ] A − A+BM −1B∗ (AA∗ )+ [A B]+ = , M −1B∗ (AA∗ )+ iv) [A B]+ =

△

where M = I + B∗ (AA∗ )+B. Source: [1880]. Fact 8.9.23. Let A ∈ Fn×m and B ∈ Fn×l. Then, [A B][A B]+ = AA+ + B1 B∗1 − B1 A+1 (I − AA+ ), △

△

where A1 = (I − BB+ )AA+ and B1 = (I − AA+ )BB+ . Source: [2679]. Fact 8.9.24. Let A ∈ Fn×m and B ∈ Fn×l. Then, [ ]+ A = [A+ − (E + + T )BA+ E + + T ], B where △

E = B(I − A+A),

△

T = (I − E +B)(A∗A)+ B∗ [I + (I − EE + )B(A∗A)+B∗ (I − EE + )]−1 (I − EE + ).

Source: [2238, p. 188]. Fact 8.9.25. Let A ∈ Fn×m and B ∈ Fn×l. Then,

( [ + ]) A + rank [A B] − + = rank [AA∗B BB∗A]. B

Hence, A∗B = 0 if and only if

] A+ [A B] = + . B +

[

Source: [2637]. Remark: If A∗B = 0, then C = B and D = 0, where C and D are defined in Fact 8.9.21. Related: Fact 8.9.26. Fact 8.9.26. Let A ∈ Fn×m and B ∈ Fn×l. Then, the following statements are equivalent:

i) [A B][A B]+ = 12 (AA+ + BB+ ). ii) R(A) = R(B). Furthermore, the following statements are equivalent: [ ] 1 A+ + iii) [A B] = . 2 B+ iv) AA∗ = BB∗. Furthermore, the following statements are equivalent: [ +] A v) [A B]+ = + . B vi) R(A) ⊆ N(B∗ ). vii) B∗A = 0. Finally, the following statements are equivalent: ([ + ]) ([ ∗ ]) A A = R . viii) R B+ B∗

667

GENERALIZED INVERSES

ix) R(A) ∩ R(B) = {0}. x) AA+ (AA+ + BB+ )+BB+ = 0. [ ] [(I − BB+ )A]+ + xi) [A B] = . [(I − AA+ )B]+ [ + ] A − A+B[(I − AA+ )B]+ + xii) [A B] = + . B − B+A[(I − BB+ )A]+ xiii) [(I − BB+ )A]+ = A+ − A+B[(I − AA+ )B]+ . xiv) [(I − AA+ )B]+ = B+ − B+A[(I − BB+ )A]+ . If these conditions hold, then the following equalities hold: xv) [(I − BB+ )A]+B = 0 and [(I − AA+ )B]+A = 0. xvi) [(I − BB+ )A]+A = A+A and [(I − AA+ )B]+B = B+B. [ + ] xvii) [A B]+ [A B] = A0A B0+B . Source: [222, 2631, 2654] and Fact 8.4.8. Related: Fact 8.9.25. Additional conditions that are equivalent to R(A) ∩ R(B) = {0} are given by Fact 8.9.5. Fact 8.9.27. Let A, B ∈ Fn×m. Then, the following statements are equivalent: ([ ]) ([ ]) i) R AA∗A = R BB∗B . ([ ]) ([ ]) ii) R AA+A = R BB+B . iii) A = B. Source: [2669]. Fact 8.9.28. Let A ∈ Fn×m and B ∈ Fk×l. Then, [ ]+ [ + ] A 0 A 0 = . 0 B 0 B+ Fact 8.9.29. Let A ∈ Fn×m. Then,

[

In

A

0m×n

0m×m

]+

  (In + AA∗ )−1 =  ∗ A (In + AA∗ )−1

 0n×m   . 0m×m

Source: [35, 2710]. Fact 8.9.30. Let A ∈ Fn×n, let B ∈ Fn×m, and assume that BB∗ = I. Then,

[

A B∗

B 0

]+

[

=

0 B∗

] B . −B∗AB

Source: [970, p. 237]. Fact 8.9.31. Let A ∈ Fn×n, assume that A is positive semidefinite, and let B ∈ Fn×m. Then,

[

△

A B∗

B 0 △

]+

 +  C − C +BD+B∗C + =  (C +BD+ )∗

 C +BD+   , DD+ − D+

where C = A + BB∗ and D = B∗C +B. Source: [1924, p. 58]. Remark: Representations of the generalized inverse of a partitioned matrix are given in [232, 284, 358], [360, Chapter 5], [608, 624, 652, 653, 654, 1248], [1275, pp. 161–165], [1329, 1331, 1489, 1814, 2034, 2035, 2037, 2038, 2041, 2140, 2301, 2327, 2624, 2655, 2677, 2678, 2855]. Problem: Show that the generalized inverses in this result and in Fact 8.9.30 are identical in the case where A is positive semidefinite and BB∗ = I.

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CHAPTER 8

Fact 8.9.32. Let A, B ∈ Fn×m. Then,

(A + B)+ = 21 [Im Im ]

[

A B

B A

]+ [ ] In . In

Source: [2624, 2629, 2666]. Related: Fact 3.22.5 and Fact 3.24.7. Fact 8.9.33. Let A1 , . . . , Ak ∈ Fn×m. Then,

(A1 + · · · + Ak )+ = 1k [Im

  A1   Ak · · · Im ]  .  ..  A2

A2

···

A1 .. .

··· .. .

A3

···

+ Ak      In    Ak−1   .    .  . ..   .  .     In A1

Source: [2629]. Remark: The partitioned matrix is block circulant. See Fact 3.22.7 and Fact

8.12.5. Fact 8.9.34. Let A ∈ Fn×n, x, y ∈ Fn, and a ∈ F, and assume that x ∈ R(A). Then,

[

A yT

] [ x I = T a y

0 1

]  

[  I  T +  0 a−yA x

A

0

yT − yTA

] A+x . 1

Remark: This factorization holds in the case where A is singular and a = 0. See Fact 3.17.11, Fact 3.21.4, and Fact 8.9.7, and note that x = AA+x. Problem: Obtain a factorization in the case where

x < R(A) (and thus x is nonzero and A is singular) and a = 0.  A1    △ n×m Fact 8.9.35. Let A ∈ F , assume that A =  ...  , and define B = [A1+ · · · A+k ]. Then, the Ak

following statements hold: i) det AB = 0 if and only if rank A < n. ii) 0 < det AB ≤ 1 if and only if rank A = n. iii) If rank A = n, then det AA∗ det AB = ∏k , ∗ i=1 det Ai Ai

det AA∗ ≤

k ∏

det Ai A∗i .

i=1

iv) det AB = 1 if and only if AB = I. v) AB is group invertible. vi) Every eigenvalue of AB is nonnegative. vii) rank A = rank B = rank AB = rank BA. ∑ Now, assume that rank A = ki=1 rank Ai , and let β denote the product of the positive eigenvalues of AB. Then, the following statements hold: viii) 0 < β ≤ 1. ix) β = 1 if and only if B = A+. Source: [1765, 2563]. Remark: iii) yields Hadamard’s inequality given by Fact 10.15.10 in the case where A is square and each Ai has a single row. Fact 8.9.36. Let A ∈ Fn×m and B ∈ Fn×l. Then, [ ∗ ] A A A∗B det ∗ = (det A∗A) det B∗(I − AA+ )B = (det B∗B) det A∗(I − BB+ )A. B A B∗B Related: Fact 3.17.28.

669

GENERALIZED INVERSES

Fact 8.9.37. Let A ∈ Fn×n, B ∈ Fn×m, C ∈ Fm×n, and D ∈ Fm×m, assume that either rank [A B] =

rank A or rank

[A] C

= rank A, and let A− ∈ Fn×n be a (1)-inverse of A. Then, [ ] A B det = (det A)det(D − CA−B). C D

Source: [300, p. 266]. △

[A

]

△

∈ F(n+m)×(n+m), B ∈ F(n+m)×l, C ∈ Fl×(n+m), D ∈ Fl×l, and A = and assume that A, A11 , and A11 |A are nonsingular. Then, Fact 8.9.38. Let A =

11 A12 A21 A22

A|A = (A11 |A)|(A11 |A). Source: [2238, pp. 195, 196] and [2263, pp. 18, 19]. Remark: Partitioning B =

[A B] C D

[B ]

C = [C1 C2 ] conformably with A, the equality can be written as

1

B2

,

and

−1 −1 −1 −1 D − CA−1B = D − C1 A−1 11 B1 − (C 2 − C 1 A11 A12 )(A22 − A21 A11 A12 ) (B2 − A21 A11 B1 ).

Remark: This is the Crabtree-Haynsworth quotient formula. See [1458]. Remark: Extensions are given in [2238, pp. 195, 196] and [2988]. Fact 8.9.39. Let A, B ∈ Fn×m. Then, the following statements are equivalent: rs

A ≤ B. BB+A = AB+B = AB+A = A. [ ] rank A = rank [A B] = rank AB and AB+A = A. R(A) ⊆ R(B), N(B) ⊆ N(A), and AB+A = A. Source: [402] and [2418, p. 45]. Related: Fact 10.24.8. i) ii) iii) iv)

8.10 Facts on the Drazin and Group Generalized Inverses for One Matrix Fact 8.10.1. Let A ∈ Fn×n. Then, the following statements hold:

i) ind A = ind(A − A2AD ). ii) For all k ≥ 1, Ak − Ak+1AD = (A − A2AD )k . iii) Let X ∈ Fn×n. Then, X = AD if and only if the following statements hold: a) XAX = X. b) AX = XA. c) A − A2 X is nilpotent. Source: [1657]. △ △ Fact 8.10.2. Let A ∈ Fn×n, and define the projectors P = AA+ and Q = I − A+A. Then, P and Q are complementary projectors if and only if A is group invertible. If these conditions hold, then AA# is the idempotent matrix onto R(P) along R(Q). Source: Corollary 4.8.10. Remark: P and Q are the projectors onto R(A) and N(A), respectively. Related: Fact 8.10.3. Fact 8.10.3. Let A ∈ Fn×n . Then, the following statements are equivalent: i) A is group invertible. ii) rank [AA+ I − A+A] = rank AA+ + rank(I − A+A) = n. In this case, AA# = (A+A2A+ )+ . Source: Proposition 4.8.6 and Corollary 4.8.10. The expression for the idempotent matrix AA# follows from Fact 8.8.14. Remark: This result shows that the range and null space of a group-invertible matrix are complementary subspaces. n×m , for all i ≥ 1, assume that Ai is group invertible, Fact 8.10.4. Let A ∈ Fn×m, let (Ai )∞ i=1 ⊂ F and assume that limi→∞ Ai = A. Then, the following statements are equivalent: i) A is group invertible, and limi→∞ A#i = A# .

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CHAPTER 8

ii) (σmax (A#i ))∞ i=1 is bounded. iii) limi→∞ Ai A#i = AA# . Source: [403]. Related: Fact 8.10.6. Fact 8.10.5. Let A ∈ Fn×n, and assume that A is group invertible and rank A = 1. Then, A# =

1 A. tr A2

Consequently, if x, y ∈ Fn satisfy x∗y , 0, then (xy∗ )# = (x∗y)−2 xy∗ . In particular, 1#n×n = n−2 1n×n . n×m Fact 8.10.6. Let A ∈ Fn×m, let (Ai )∞ , and assume that limi→∞ Ai = A. Then, the i=1 ⊂ F following statements are equivalent: i) limi→∞ ADi = AD . ii) (σmax (ADi ))∞ i=1 is bounded. Source: [403]. Related: Fact 8.10.4. Fact 8.10.7. Let A ∈ Fn×n, let λ ∈ spec(A), assume that λ , 0, and let k ≥ 1. Then, amultA(λ) = amultAkD (1/λk ) and gmultA(λ) = gmultAkD (1/λk ). Related: Fact 7.15.23. Fact 8.10.8. Let A ∈ Fn×n . Then, there exists p ∈ F[s] such that p(A) = AD . Source: [2238, p. 227]. Fact 8.10.9. Let A ∈ Fn×n . Then, AD is group invertible. Furthermore, (AD )# = A2AD . Source: [1275, p. 63]. △ Fact 8.10.10. Let A ∈ Fn×n , define k = ind A, and let b ∈ Fn . Then, the following statements are equivalent: i) x = AD b satisfies Ax = b. ii) b ∈ R(Ak ). iii) There exists x ∈ R([b Ab · · · An−1b]) such that Ax = b. △ Now, assume that these statements hold, and define m = deg µA . Then, there exists a unique vector x ∈ R([b Ab · · · Am−k−1b]) such that Ax = b. Source: [2238, pp. 228, 229]. Fact 8.10.11. Let A ∈ Fn×n. Then, X = AD is the unique matrix satisfying [ ] A AAD rank D = rank A. A A X Source: [2854, 2993]. Related: Fact 3.22.11 and Fact 8.3.34. △ Fact 8.10.12. Let A ∈ Fn×n, and let k = ind A. Then,

AD = Ak (A2k+1 )+Ak . If, in particular, ind A ≤ 1, then A# = A(A3 )+A,

(A# )+ = A+A3A+ .

Source: [360, pp. 165, 174] and [403]. Fact 8.10.13. Let A ∈ Fn×n, assume that A is group invertible, and let S, B ∈ Fn×n, where S is nonsingular, B is a Jordan form of A, and A = SBS −1. Then, A# = SB#S −1 = SB+S −1. Source: Since

B is range Hermitian, it follows from Fact 8.10.16 that B# = B+. See [360, p. 158]. Fact 8.10.14. Let A ∈ Fn×n and k ≥ 1. Then, the following statements are equivalent: i) k ≥ ind A.

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GENERALIZED INVERSES

ii) limα→0 αk(A + αI)−1 exists. iii) limα→0 (Ak+1 + αI)−1Ak exists. If these statements hold, then

AD = lim (Ak+1 + αI)−1Ak , α→0

lim αk(A + αI)−1

α→0

   (−1)k−1 (I − AAD )Ak−1, k = ind A > 0,     −1 = A , k = ind A = 0,     0, k > ind A.

Source: [2037]. Fact 8.10.15. Let A ∈ Fn×n. Then, A is group invertible if and only if limα→0 (A + αI)−1A exists.

If these conditions hold, then

lim (A + αI)−1A = AA#.

α→0

Source: [624, p. 138]. Fact 8.10.16. Let A ∈ Fn×n. Then, the following statements are equivalent:

i) ii) iii) iv) v) vi) vii) viii) ix) x) xi) xii) xiii) xiv) xv) xvi) xvii) xviii) xix) xx) xxi) xxii) xxiii) xxiv) xxv) xxvi) xxvii)

A is range Hermitian. A+ = AD. A is group invertible, and A+ = A# . A is group invertible, and A(A+ )2 = A#. A is group invertible, and AA#A+ = A#. A is group invertible, and A∗AA# = A∗. A is group invertible, and A+AA# = A+. A is group invertible, and A#A+A = A+. A is group invertible, and AA# = A+A. A is group invertible, and A∗A+ = A∗A#. A is group invertible, and A+A∗ = A#A∗. A is group invertible, and (A+ )2 = A+A#. A is group invertible, and (A+ )2 = A#A+. A is group invertible, and (A+ )2 = A2#. A is group invertible, and A+A# = A2#. A is group invertible, and A#A+ = A2#. A is group invertible, and A+A# = A#A+. A is group invertible, and AA+A∗ = A∗AA+. A is group invertible, and AA+A# = A+A#A. A is group invertible, and AA+A# = A#AA+. A is group invertible, and AA#A∗ = A∗AA#. A is group invertible, and AA#A+ = A+AA#. A is group invertible, and AA#A+ = A#A+A. A is group invertible, and A∗A+A = A+AA∗. A is group invertible, and A+AA# = A#A+A. A is group invertible, and (A+ )2A# = A+A#A+. A is group invertible, and (A+ )2A# = A#(A+ )2.

672

A is group invertible, and A+A#A+ = A# (A+ )2. A is group invertible, and A+A2# = A#A+A#. A is group invertible, and A+A2# = A2#A+. A is group invertible, and A2#A+ = A#A+A#. A is group invertible, and A∗A#A + AA#A∗ = 2A∗. A is group invertible, and A+A#A + AA#A+ = 2A+. Source: [240, 719]. Related: Fact 8.5.3. Fact 8.10.17. Let A ∈ Fn×n. Then, the following statements are equivalent: i) A is normal. ii) A is group invertible, and A∗A+ = A#A∗. iii) A is group invertible, and A∗A# = A+A∗. iv) A is group invertible, and A∗A# = A#A∗. v) A is group invertible, and AA∗A# = A∗A#A. vi) A is group invertible, and AA∗A# = A#AA∗. vii) A is group invertible, and AA#A∗ = A#A∗A. viii) A is group invertible, and A∗AA# = A#A∗A. ix) A is group invertible, and A2∗A# = A∗A#A∗. x) A is group invertible, and A∗A+A# = A#A∗A+. xi) A is group invertible, and A∗A#A∗ = A#A2∗. xii) A is group invertible, and A∗A#A+ = A+A∗A#. xiii) A is group invertible, and A∗A2# = A#A∗A#. xiv) A is group invertible, and A+A∗A# = A#A+A∗. xv) A is group invertible, and A+A#A∗ = A#A∗A+. xvi) A is group invertible, and A#A∗A# = A2#A∗. Source: [240, 719]. Related: Fact 4.10.12, Fact 4.13.6, Fact 7.17.5, and Fact 8.6.1. Fact 8.10.18. Let A ∈ Fn×n. Then, the following statements are equivalent: i) A is Hermitian. ii) A is group invertible, and AA# = A∗A+. iii) A is group invertible, and AA# = A∗A#. iv) A is group invertible, and AA# = A+A∗. v) A is group invertible, and A+A = A#A∗. vi) A is group invertible, and A∗AA# = A. vii) A is group invertible, and A2∗A# = A∗. viii) A is group invertible, and A∗(A+ )2 = A#. ix) A is group invertible, and A∗A+A# = A+. x) A is group invertible, and A∗A+A# = A#. xi) A is group invertible, and A∗A2# = A#. xii) A is group invertible, and A#A∗A# = A+. Source: [240]. Fact 8.10.19. Let A ∈ Fn×n. Then, the following statements are equivalent: i) A is a projector.

xxviii) xxix) xxx) xxxi) xxxii) xxxiii)

CHAPTER 8

GENERALIZED INVERSES

673

ii) A is idempotent, and A∗ = A# . Source: [1249]. Fact 8.10.20. Let A ∈ Rn×n. Then, the following statements are equivalent: i) A is idempotent. ii) A is group invertible, and A2# = A. Remark: The result is false if A is complex. Fact 8.10.21. Let A ∈ Fn×n. Then, the following statements are equivalent: i) A is tripotent. ii) A is group invertible, and A# = A. iii) A is group invertible, and A3# = A. iv) A is group invertible, and AA# = A2 . v) A is group invertible, and 12 (AA# − A) is idempotent. Source: [257]. Fact 8.10.22. Let A ∈ Fn×n. Then, the following statements are equivalent: i) A is a generalized projector. ii) A∗ = A2 . iii) A = A2∗ . iv) A = A∗A+ . v) A = A+A∗ . vi) A∗ = A2 = A+ . vii) A is group invertible, and A = A∗A# . viii) A is group invertible, and A = A#A∗ . Now, assume that these statements hold. Then, the following statements hold: ix) A∗A is a projector. x) For all k ≥ 0, A = A3k+1 and A∗ = A2 = A# = AD = A+ = A2 = A3k+2 . Source: [233, 237, 1249]. Related: Fact 8.10.23. Fact 8.10.23. Let A ∈ Fn×n. Then, the following statements are equivalent: i) A+ = A2 . ii) A = (A2 )+ . iii) A = (A+ )2 . iv) A2∗ = A∗+ . v) A is group invertible, and A = A+A# . vi) A is group invertible, and A = A#A+ . vii) A is group invertible, and A2# = (A# )+ . viii) A3 is a projector, and R(A3 ) = R(A). Source: [225, 233, 237, 1249]. Remark: If i)–viii) hold, then A is a hypergeneralized projector. Related: Fact 8.10.22. Fact 8.10.24. Let A ∈ Fn×n. Then, the following statements are equivalent: i) A is a projector. ii) A is an idempotent generalized projector. iii) A is an idempotent hypergeneralized projector. Furthermore, the following statements hold:

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CHAPTER 8

iv) A is a projector if and only if there exists a generalized projector B ∈ Fn×n such that A = B∗B. v) If A is a generalized projector, then A and I − A are normal. vi) Assume that A is a generalized projector. Then, I − A is a generalized projector if and only if A is a projector. Source: [224, 233, 225, 1249]. v) follows from Fact 4.9.6. Fact 8.10.25. Let A ∈ Fn×n, assume that A is a hypergeneralized projector, and define B ∈ Fn×n △ by B = 31 (A + A2 + A3 ). Then, the following statements hold: i) B is idempotent. ii) B is a projector if and only if I − A is range Hermitian. If these conditions hold, then B = I − (I − A)(I − A)+ . iii) AB = BA = B. iv) R(B) = N(I − A) and N(B) = R(I − A). Source: [1249]. Fact 8.10.26. Let A, B ∈ Fn×n, and assume that A and B are (idempotent, projectors, generalized projectors). Then, so is A + B if and only if AB = BA = 0. Source: [1249]. Fact 8.10.27. Let A, B ∈ Fn×n, assume that A and B are hypergeneralized projectors, and assume that AB = BA = 0. Then, A + B is a hypergeneralized projector. Source: [1249].

8.11 Facts on the Drazin and Group Generalized Inverses for Two or More Matrices Fact 8.11.1. Let A ∈ Fn×r and B ∈ Fr×n, and assume that rank AB = r. Then, AB is group invertible if and only if BA is nonsingular. If these conditions hold, then (AB)# = A(BA)−2 B. Source: [360, p. 157] and [2238, p. 231]. Credit: R. E. Cline. Related: Fact 8.11.6. Fact 8.11.2. Let A, B ∈ Fn×n, and assume that AB and BA are group invertible. Then,

(AB)# = A(BA)2#B. Source: [873]. Fact 8.11.3. Let A, B ∈ Fn×n, assume that A and B are group invertible, and consider the fol-

lowing statements: i) ABA = B. ii) BAB = A. iii) A2 = B2. Then, if two of the above statements hold, then the third statement holds. Furthermore, if i)–iii) hold, then the following statements hold: iv) A and B are group invertible. v) A# = A3 and B# = B3. vi) A5 = A and B5 = B. vii) A4 = B4 = (AB)4. viii) If A and B are nonsingular, then A4 = B4 = (AB)4 = I. Source: [1015]. △ Fact 8.11.4. Let A ∈ Rn×n, where n ≥ 2, assume that A is positive, define B = ρmax (A)I − A, n T let x, y ∈ R be positive, and assume that Ax = ρmax (A)x and A y = ρmax (A)y. Then, the following statements hold: i) B + x1Ty xyT is nonsingular.

675

GENERALIZED INVERSES

ii) B# = (B + iii) I − BB# =

1 xyT )−1 (I xTy 1 xyT. xTy

iv) B# = limk→∞

(∑ k−1 i=0

−

1

1 xyT ). xTy

ρimax (A)

Ai −

k xyT xTy

)

.

Source: [2344, p. 9-4]. Related: Fact 6.11.5. Fact 8.11.5. Let A, B ∈ Fn×n, and assume that A and B are similar. Then, AD and BD are similar. Fact 8.11.6. Let A ∈ Fn×m and B ∈ Fm×n. Then, (AB)D = A(BA)2D B. Source: [624, pp. 149, 150] and [657]. Remark: This is Cline’s formula. Related: Fact 8.11.1. Fact 8.11.7. Let A, B ∈ Fn×n, and assume that AB = BA. Then,

(AB)D = BDAD = AD BD ,

ADB = BAD,

ABD = BDA.

Furthermore, ind AB ≤ max {ind A, ind B}. Source: [624, pp. 149, 150] and [2238, pp. 227, 228]. Fact 8.11.8. Let A, B ∈ Fn×n, and assume that AB = BA = 0. Then, (A + B)D = AD + BD. Source: [1340]. Credit: M. P. Drazin. Fact 8.11.9. Let A, B ∈ Fn×n, and assume that A and B are idempotent. Then, the following

statements hold: i) If AB = 0, then (A + B)D = A + B − 2BA and (A − B)D = A − B. ∑ △ i (i+1)D . ii) If AB = 0, B is nilpotent, and k = ind B, then (A + B)D = k−1 i=0 B A D D iii) If BA = 0, then (A + B) = A + B − 2AB and (A − B) = A − B. ∑ △ (i+1)D i B. iv) If BA = 0, B is nilpotent, and k = ind B, then (A + B)D = k−1 i=0 A 1 3 D D v) If AB = A, then (A + B) = 4 A + B − 4 BA and (A − B) = BA − B. vi) If AB = B, then (A + B)D = A + 14 B − 43 BA and (A − B)D = A − BA. vii) If BA = A, then (A + B)D = 14 A + B − 34 AB and (A − B)D = AB − B. viii) If BA = B, then (A + B)D = A + 41 B − 34 AB and (A − B)D = A − AB. ix) x) xi) xii)

If AB = BA, then (A + B)D = A + B − 23 AB and (A − B)D = A − B. If ABA = 0, then (A + B)D = A + B − 2AB − 2BA + 3BAB and (A − B)D = A − B − BAB. If BAB = 0, then (A + B)D = A + B − 2AB − 2BA + 3ABA and (A − B)D = A − B + ABA. If ABA = A, then (A + B)D = 81 (A + B)2 + 78 BA⊥B and (A − B)D = −BA⊥B.

xiii) If BAB = B, then (A + B)D = 81 (A + B)2 + 78 AB⊥A and (A − B)D = AB⊥A. xiv) If ABA = B, then (A + B)D = A − 12 B and (A − B)D = A − B. xv) If BAB = A, then (A + B)D = − 21 A + B and (A − B)D = A − B. xvi) If ABA = AB, then (A + B)D = A + B− 2BA − 43 AB+ 54 BAB and (A − B)D = A − B− AB+ BAB. xvii) If ABA = BA, then (A+ B)D = A+ B−2AB− 43 BA+ 54 BAB and (A− B)D = A− B− BA+ BAB. Source: [653, 872, 870]. Related: Fact 8.11.12. Fact 8.11.10. Let A, B ∈ Fn×n, and assume that A and B are idempotent. Then, the following statements hold: i) (A − B)D = (I − AB)D (A − AB) + (A + B − AB)D (AB − B). ii) [A, B]D = (ABA)D (A − B)D − (A − B)D (ABA)D . iii) (AB + BA)D = (A + B)D (A + B − I)D . iv) (ABA)D = (I − A − B)2D A.

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CHAPTER 8

(AB)D = (ABA)2D B = (I − A − B)4D AB. (A − AB)D = (I − ABA)2D (A − AB) = (A − ABA)2D (I − AB). (A − BA)D = (A − BA)(I − ABA)2D = (I − BA)(A − ABA)2D . (I − AB)D A = A(A − BA)D = (A − AB)D A = A(I − ABA)D = (I − ABA)D A = A(I − BA)D = (A − ABA)D = A(A − B)2D = (A − B)2D A = A(I − ABA)D = (I − ABA)D A. ix) (I − ABA)D = I − A + A(A − B)2D . Source: [872]. Fact 8.11.11. Let A, B ∈ Fn×n, and assume that A and B are projectors. Then, the following statements hold: i) If AB = A, then (A + B)D = − 21 A + B and (A − B)D = A − B. v) vi) vii) viii)

ii) If AB = B, then (A + B)D = A − 12 B and (A − B)D = A − B. iii) If AB = 0, then (A − B)D = A − B. iv) The following statements are equivalent: a) AB = BA. b) (AB)D = AB. c) (A − B)D = A − B. v) AB = BA = 0 if and only if (A + B)D = A + B. vi) (A − B)D = (A − B)2 [(A − BA)D − (B − BA)D ]. vii) A[(A + B)D − (A − B)D ](A − B)2 = 0. viii) (AB)D = (ABA)D − A(B⊥ B⊥ )D . ix) (AB)D AB = (ABA)D AB. Source: [653, 871, 872]. Related: Fact 8.11.12. Fact 8.11.12. Let A, B ∈ Fn×n, assume that A and B are group invertible, let α, β ∈ F, and assume that α and β are nonzero. Then, the following statements hold: i) If ABB# = BAA# , then αA + βB is group invertible. If, in addition, α + β , 0, then ( ) ( ) 1 1 1 1 1 (αA+βB)# = (A# + B# − A#BB# )+ − (I − BB# )A# + − (I − AA# )B# . α+β α α+β β α+β ii) If ABB# = BAA# , then A − B is group invertible and (A − B)# = (A − B)(A# − B# )2 . iii) If BB#A = AA#B, then αA + βB is group invertible. If, in addition, α + β , 0, then ( ) ( ) 1 1 1 1 1 (A# + B# − B#BA# )+ − A# (I − BB# )+ − B# (I −AA# ). (αA+βB)# = α+β α α+β β α+β iv) If BB#A = AA#B, then A − B is group invertible and (A − B)# = (A# − B# )2 (A − B). v) If ABB# = BAA# and BB#A = AA#B, then αA + βB is group invertible. If, in addition, α + β , 0, then (αA + βB)# =

1 1 1 A#BB# + (I − BB# )A + (I − AA# )B# . α+β α β

vi) If ABB# = BAA# and BB#A = AA#B, then A − B and AB are group invertible, and (A − B)# = A# − B# and (AB)# = (BA)# = (A#BB# )2 . vii) If BA#A = A and α + β , 0, then αA + βB is group invertible and (αA+βB)# =

α β α2 + 2αβ α A#+ B#+ (I−AA# )B# (I−AA# )− A# (B−A)B# . 2 2 2 (α + β) (α + β) β(α + β) (α + β)2

677

GENERALIZED INVERSES

viii) Assume that AB = BA. Then, αA + βB is group invertible if and only if αABB# + βBAA# is group invertible. If these conditions hold, then (αA + βB)# = (αABB# + βBAA# )# +

1 1 # A (I − BB# ) + B# (I − AA# ). α β

ix) If AB = BA, then AB is group invertible, (AB)# = B#A# = A#B# , A#B = BA# , and B#A = AB# . x) If A and B are idempotent and AB = BA, then A + B is group invertible, (A + B)# = A + B − 23 AB, and (A − B)# = A − B. xi) If A and B are tripotent and AB2 = BA2 , then A + B is group invertible, (A + B)# = A + B − 1 2 1 2 1 2 # 3 2 2 2 AB − 2 A B − 2 B A, and (A − B) = (A − B) = A − B + B A − A B + BAB − ABA. xii) If A and B are tripotent and A2B = B2A, then A + B is group invertible, (A + B)# = A + B − 1 1 2 1 2 2 # 3 2 2 2 AB − 2 B A − 2 BA , and (A − B) = (A − B) = A − B + BA − AB + BAB − ABA. Source: [1879]. Related: Fact 8.11.9. Fact 8.11.13. Let A, B ∈ Fn×n, and assume that A and B are projectors. Then, (AB)# = (BA)+ (AB)+ (BA)+ = [(BA)2 ]+ . Furthermore, the following statements are equivalent: i) AB is a projector. ii) (AB)# is Hermitian. iii) (AB)# is idempotent. iv) rank AB = tr (AB)# . v) AB(AB)# = BA(AB)# . vi) (I − AB)# = I − (AB)# . vii) tr[I − (AB)# ] = rank[I − (AB)# ]. Source: [245]. Related: Fact 8.8.5. Fact 8.11.14. Let A, B ∈ Fn×n, assume that A and B are idempotent, assume that A − B is group △ △ △ invertible, and define F = A(A − B)# , G = (A − B)#A, and H = (A − B)# (A − B). Then, the following statements hold: i) F, G, and H are idempotent. ii) F = (A − B)# B⊥ and R(F) = R(AH). iii) G = B⊥ (A − B)# and N(G) = N(AH). iv) B(A − B)# = (A − B)#A⊥ and A⊥ (A − B)# = (A − B)#B. v) (A − B)# = F + G − H. vi) AH = H if and only if A + B is group invertible and (A + B)# = (A − B)# (A + B)(A − B)# = (2G − H)(F + G − H). vii) FA = AG = AH = HA. viii) BHB = BH = HB = HBH. ix) B⊥ F⊥ = G⊥ B⊥ = B⊥ F⊥ B⊥ = A⊥ H⊥ . x) (I − AB)# = FG + F⊥ A⊥ . xi) (I − ABA)# = FG + A⊥ . xii) (A − ABA)# = FG. xiii) (A − AB)# = (FG)2 B⊥ . xiv) (A − BA)# = B⊥ (FG)2 . Source: [873]. Related: Fact 4.16.10.

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CHAPTER 8

8.12 Facts on the Drazin and Group Generalized Inverses for Partitioned Matrices △

Fact 8.12.1. Let A ∈ Fn×m, B ∈ Fm×n, and define A =

[

AD =

0 B(AB)D

[

]

. Then, ind A ≤ 2 ind AB + 1, and ] (AB)D A 0 A(BA)D = . 0 (BA)D B 0 ]

[

0 A B 0

Source: [657, 873]. [ ] △ Fact 8.12.2. Let A ∈ Fn×m, B ∈ Fm×n, and define A = B0 A0 . Then, the following statements are

equivalent: i) A is group invertible. ii) AB and BA are group invertible, and [I − AB(AB)# ]A = B[I − AB(AB)# ] = 0. If these statements hold, then [ ] [ ] 0 (AB)#A 0 A(BA)# # A = = . B(AB)# 0 (BA)#B 0 Source: [873]. [ ] △ Fact 8.12.3. Let A ∈ Fn×n , B ∈ Fn×m , and C ∈ Fm×m , and define A = A0 CB . Then, A is group

invertible if and only if A and C are group invertible and (I − AA# )B(I −CC # ) = 0. If these conditions hold, then [ # ] A A2#B(I − CC # ) + (I − AA# )BC 2# − A#BC # A# = . 0 C# Source: [402, 2037]. Credit: C. G. Cao. Related: Fact 7.16.19. ] △ [ Fact 8.12.4. Let A ∈ Fn×n , B ∈ Fn×m , C ∈ Fm×n , and D ∈ Fm×m , and define A = CA DB and △

S = D − CAD B. Then, the following statements hold: i) If S is nonsingular, (I − AAD )B = 0, and C(I − AAD ) = 0, then [ D ] A + AD BS −1CAD −AD BS −1 AD = . −S −1CAD S −1 ii) If S = 0, (I − AAD )B = 0, and C(I − AAD ) = 0, then [ ] I AD = A(WA)2D [I AD B], CAD △

where W = AAD + AD BCAD . iii) Assume that A is nonsingular. Then, rank A = n if and only if S = 0. △ iv) Assume that A is nonsingular and S = 0. Then, A is group invertible if and only if W = −1 −1 I + A BCA is nonsingular. If these conditions hold, then [ ] I # A = (WAW)−1 [I A−1B]. CA−1 v) If ind A = 1, ind S ≤ 1, and (I − AA# )B = 0, then [ # ][ A + A#BS #CA# −A#BS # I − A#BS #C(I − AA# ) A= −S #CA# S# S #C(I − AA# ) Source: [653].

] A#B(I − SS # ) . I

679

GENERALIZED INVERSES

Fact 8.12.5. Let A1 , . . . , Ak ∈ Fn×n. Then,

(A1 + · · · + Ak )D =

1 [In k

  A1   Ak · · · In ]  .  ..  A2

A2

···

A1 .. .

··· .. .

A3

···

D Ak      In    Ak−1   .    .  . ..   .  .     In A1

Source: [2629]. Related: Fact 8.9.33.

8.13 Notes A brief history of the generalized inverse is given in [359] and [360, p. 4]. The proof of the uniqueness of A+ is given in [1924, p. 32]. Additional books on generalized inverses include [360, 536, 2299, 2821]. The terminology “range Hermitian” is used in [360]; the terminology “EP” is more common. Generalized inverses are widely used in least squares methods; see [489, 624, 1766]. Applications to singular differential equations are considered in [623]. Applications to Markov chains are discussed in [1490].

Chapter Nine Kronecker and Schur Algebra In this chapter we introduce Kronecker matrix algebra, which is useful for solving linear matrix equations.

9.1 Kronecker Product For A ∈ Fn×m define the vec operator as

   col1 (A)    △  ..  ∈ Fnm , vec A =  .   colm (A)

(9.1.1)

which is the column vector of size nm × 1 obtained by stacking the columns of A. We recover A from vec A by writing A = vec−1(vec A).

(9.1.2)

Note that, if x ∈ F , then vec x = vec x = x. Proposition 9.1.1. Let A ∈ Fn×m and B ∈ Fm×n. Then, T

n

tr AB = (vec AT )T vec B = (vec BT )T vec A.

(9.1.3)

Proof. Note that

tr AB =

n ∑

n ∑ [coli (AT )]T coli (B) i=1    col (B)  ]  1.  colTn (AT )  ..  = (vec AT )T vec B.   coln (B)

rowi (A)coli (B) =

i=1

[

= colT1 (AT ) · · ·



Next, we introduce the Kronecker product. Definition 9.1.2. Let A ∈ Fn×m and B ∈ Fl×k. Then, the Kronecker product A ⊗ B ∈ Fnl×mk of A and B is the partitioned matrix    A(1,1) B A(1,2) B · · · A(1,m) B   △  .. . ..  . (9.1.4) A ⊗ B =  ... · · · . . .    A(n,1) B A(n,2) B · · · A(n,m) B Unlike matrix multiplication, the Kronecker product A ⊗ B does not entail a restriction on either the size of A or the size of B. The following results are immediate consequences of the definition of the Kronecker product. Proposition 9.1.3. Let α ∈ F, A ∈ Fn×m, and B ∈ Fl×k. Then, α ⊗ A = A ⊗ α = αA, A ⊗ B = A ⊗ B,

A ⊗ (αB) = (αA) ⊗ B = α(A ⊗ B),

(A ⊗ B)T = AT ⊗ BT,

(A ⊗ B)∗ = A∗ ⊗ B∗.

(9.1.5) (9.1.6)

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Proposition 9.1.4. Let A, B ∈ Fn×m and C ∈ Fl×k. Then,

(A + B) ⊗ C = A ⊗ C + B ⊗ C, C ⊗ (A + B) = C ⊗ A + C ⊗ B.

(9.1.7) (9.1.8)

The next result shows that the Kronecker product is associative. Proposition 9.1.5. Let A ∈ Fn×m, B ∈ Fl×k, and C ∈ F p×q. Then, A ⊗ (B ⊗ C) = (A ⊗ B) ⊗ C.

(9.1.9)

We thus write A ⊗ B ⊗ C for A ⊗ (B ⊗ C) and (A ⊗ B) ⊗ C. The next result shows how matrix multiplication interacts with the Kronecker product. Proposition 9.1.6. Let A ∈ Fn×m, B ∈ Fl×k, C ∈ Fm×q, and D ∈ Fk×p. Then, (A ⊗ B)(C ⊗ D) = AC ⊗ BD.

(9.1.10)

Proof. Note that the i j block of (A ⊗ B)(C ⊗ D) is given by

[

[(A ⊗ B)(C ⊗ D)]i j = A(i,1) B · · ·

=

m ∑

  C D ]  (1,.j)  A(i,m) B  ..    C(m, j) D

A(i,k)C(k, j) BD = (AC)(i, j) BD = (AC ⊗ BD)i j .



k=1

Next, we consider the inverse of a Kronecker product. Proposition 9.1.7. Assume that A ∈ Fn×n and B ∈ Fm×m are nonsingular. Then, A ⊗ B is nonsingular, and (A ⊗ B)−1 = A−1 ⊗ B−1. (9.1.11) Proof. Note that (A ⊗ B)(A−1 ⊗ B−1 ) = AA−1 ⊗ BB−1 = In ⊗ Im = Inm . Proposition 9.1.8. Let x ∈ Fn and y ∈ Fm. Then,

xyT = x ⊗ yT = yT ⊗ x,

vec xyT = y ⊗ x.

 (9.1.12)

The following result concerns the vec of the product of three matrices. Proposition 9.1.9. Let A ∈ Fn×m, B ∈ Fm×l, and C ∈ Fl×k. Then, vec(ABC) = (CT ⊗ A)vec B.

(9.1.13)

Proof. Using (9.1.10) and (9.1.12), it follows that

vec ABC = vec

l ∑

Acoli (B)eTi C =

i=1

l ∑

vec[Acoli (B)(CTei )T ]

i=1

l l ∑ ∑ = (CTei ) ⊗ [Acoli (B)] = (CT ⊗ A) ei ⊗ coli (B) i=1

= (CT ⊗ A)

i=1 l ∑

vec[coli (B)eTi ] = (CT ⊗ A)vec B.



i=1

The following result concerns the eigenvalues and eigenvectors of the Kronecker product of two matrices.

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Proposition 9.1.10. Let A ∈ Fn×n and B ∈ Fm×m. Then,

mspec(A ⊗ B) = {λµ : λ ∈ mspec(A), µ ∈ mspec(B)}ms .

(9.1.14)

If, in addition, x ∈ C is an eigenvector of A associated with λ ∈ spec(A) and y ∈ C is an eigenvector of B associated with µ ∈ spec(B), then x ⊗ y is an eigenvector of A ⊗ B associated with λµ. Proof. (9.1.10) implies (A ⊗ B)(x ⊗ y) = (Ax) ⊗ (By) = (λx) ⊗ (µy) = λµ(x ⊗ y).  Using the Minkowski product, (9.2.5) can be written as n

m

mspec(A ⊗ B) = mspec(A) mspec(B).

(9.1.15)

Proposition 9.1.10 shows that mspec(A ⊗ B) = mspec(B ⊗ A). Hence, det(A ⊗ B) = det(B ⊗ A) and tr(A ⊗ B) = tr(B ⊗ A). Proposition 9.1.11. Let A ∈ Fn×n and B ∈ Fm×m. Then, det(A ⊗ B) = det(B ⊗ A) = (det A)m (det B)n.

(9.1.16)

Proof. Let mspec(A) = {λ1 , . . . , λn }ms and mspec(B) = {µ1 , . . . , µm }ms . Then, Proposition 9.1.10 implies that     n,m m m ∏    ∏  ∏ m m    µ j  det(A ⊗ B) = µ j  · · · λn λi µ j = λ1 j=1

j=1

i, j=1

= (λ1 · · · λn )m (µ1 · · · µm )n = (det A)m (det B)n.



Proposition 9.1.12. Let A ∈ Fn×n and B ∈ Fm×m. Then,

tr(A ⊗ B) = tr(B ⊗ A) = (tr A)(tr B).

(9.1.17)

Proof. Note that

tr(A ⊗ B) = tr(A(1,1) B) + · · · + tr(A(n,n) B) = [A(1,1) + · · · + A(n,n) ] tr B = (tr A)(tr B).



Next, define the Kronecker permutation matrix Pn,m ∈ Fnm×nm by △

Pn,m =

n,m ∑

Ei, j,n×m ⊗ E j,i,m×n .

(9.1.18)

i, j=1

Proposition 9.1.13. Let A ∈ Fn×m. Then,

vec AT = Pn,m vec A.

(9.1.19)

9.2 Kronecker Sum and Linear Matrix Equations Next, we define the Kronecker sum of two square matrices. Definition 9.2.1. Let A ∈ Fn×n and B ∈ Fm×m. Then, the Kronecker sum A ⊕ B ∈ Fnm×nm of A and B is △ A ⊕ B = A ⊗ Im + In ⊗ B. (9.2.1) Proposition 9.2.2. Let α ∈ F, A ∈ Fn×n, and B ∈ Fm×m. Then,

(αA) ⊕ (αB) = α(A ⊕ B), A ⊕ B = A ⊕ B, Proposition 9.2.3. Let A ∈ F

(A ⊕ B)T = AT ⊕ BT, ,B∈F

n×n

(A ⊕ B)∗ = A∗ ⊕ B∗.

(9.2.2) (9.2.3)

, and C ∈ F . Then,

m×m

l×l

A ⊕ (B ⊕ C) = (A ⊕ B) ⊕ C.

(9.2.4)

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Hence, we write A ⊕ B ⊕ C for A ⊕ (B ⊕ C) and (A ⊕ B) ⊕ C. Proposition 9.1.10 shows that, if λ ∈ spec(A) and µ ∈ spec(B), then λµ ∈ spec(A ⊗ B). The following result involving the Kronecker sum is analogous. Proposition 9.2.4. Let A ∈ Fn×n and B ∈ Fm×m. Then, mspec(A ⊕ B) = {λ + µ : λ ∈ mspec(A), µ ∈ mspec(B)}ms .

(9.2.5)

Now, let x ∈ C be an eigenvector of A associated with λ ∈ spec(A), and let y ∈ C be an eigenvector of B associated with µ ∈ spec(B). Then, x ⊗ y is an eigenvector of A ⊕ B associated with λ + µ. Proof. Using (9.1.10), we have n

m

(A ⊕ B)(x ⊗ y) = (A ⊗ Im )(x ⊗ y) + (In ⊗ B)(x ⊗ y) = (Ax ⊗ y) + (x ⊗ By) = (λx ⊗ y) + (x ⊗ µy) = λ(x ⊗ y) + µ(x ⊗ y) = (λ + µ)(x ⊗ y).



Using the Minkowski sum, (9.2.5) can be written as mspec(A ⊕ B) = mspec(A) + mspec(B).

(9.2.6)

The next result concerns the existence and uniqueness of solutions to Sylvester’s equation. See Fact 7.11.26 and Proposition 15.10.3. Proposition 9.2.5. Let A ∈ Fn×n, B ∈ Fm×m, and C ∈ Fn×m. Then, X ∈ Fn×m satisfies AX + XB + C = 0

(9.2.7)

if and only if X satisfies (BT ⊕ A) vec X + vec C = 0. Consequently, B ⊕ A is nonsingular if and only if there exists a unique matrix X ∈ F (9.2.7). If these conditions hold, then X is given by T

X = − vec−1 [(BT ⊕ A)−1 vec C].

(9.2.8) n×m

satisfying (9.2.9)

Furthermore, B ⊕ A is singular and rank B ⊕ A = rank[B ⊕ A vec C] if and only if there exist infinitely many matrices X ∈ Fn×m satisfying (9.2.7). In this case, for each X ∈ Fn×m satisfying (9.2.7), the set of solutions of (9.2.7) is given by vec−1 [vec X + N(BT ⊕ A)]. Proof. Note that (9.2.7) is equivalent to T

T

T

0 = vec(AXI + IXB) + vec C = (I ⊗ A) vec X + (BT ⊗ I) vec X + vec C = (BT ⊗ I + I ⊗ A) vec X + vec C = (BT ⊕ A) vec X + vec C, which yields (9.2.8). The remaining results follow from Corollary 3.7.8.  For the following corollary, note Fact 7.11.26. Corollary 9.2.6. Let A ∈ Fn×n, B ∈ Fm×m, and C ∈ Fn×m, and assume that spec(A) ∩ spec(−B) [ =] 0 ∅. Then, there exists a unique matrix X ∈ Fn×m satisfying (9.2.7). Furthermore, the matrices A0 −B [ ] C are similar and satisfy and A0 −B [ ] [ ][ ][ ] A C I X A 0 I −X = . (9.2.10) 0 −B 0 I 0 −B 0 I

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KRONECKER AND SCHUR ALGEBRA

9.3 Schur Product An alternative form of vector and matrix multiplication is given by the Schur product. If A ∈ Fn×m and B ∈ Fn×m, then A ⊙ B ∈ Fn×m is defined by △

(A ⊙ B)(i, j) = A(i, j) B(i, j) .

(9.3.1)

Hence, A ⊙ B is formed by means of entry-by-entry multiplication. For matrices A, B, C ∈ Fn×m, the commutative, associative, and distributive equalities

hold. For all A ∈ F

A ⊙ B = B ⊙ A, A ⊙ (B ⊙ C) = (A ⊙ B) ⊙ C,

(9.3.2) (9.3.3)

A ⊙ (B + C) = A ⊙ B + A ⊙ C

(9.3.4)

A ⊙ 1n×m = 1n×m ⊙ A = A.

(9.3.5)

,

n×m

Furthermore, if A is square, then I ⊙ A is the diagonal part of A. Next, let A ∈ Fn×m and α ∈ [0, ∞). Then, the Schur power A⊙α ∈ Fn×m is defined by ( ) △ = (A(i, j) )α . A⊙α (i, j)

(9.3.6)

Thus, A⊙2 = A ⊙ A and A⊙0 = 1n×m. Furthermore, α < 0 is allowed in the case where A has no zero entries. In particular, A⊙−1 is the matrix whose entries are the reciprocals of the entries of A. Proposition 9.3.1. Let A, B ∈ Fn×m. Then, the following statements hold: i) (A ⊙ B)T = AT ⊙ BT , A ⊙ B = A ⊙ B, (A ⊙ B)∗ = A∗ ⊙ B∗ . ii) If either A or B is (diagonal, lower bidiagonal, upper bidiagonal, tridiagonal, lower Hessenberg, upper Hessenberg), then so is A ⊙ B. iii) If A and B are (Toeplitz, Hankel), then so is A ⊙ B. iv) Let α ∈ [0, ∞). Then, (A ⊙ B)⊙α = A⊙α ⊙ B⊙α . v) Assume that A and B have no zero entries, and let α ∈ R. Then, (A ⊙ B)⊙α = A⊙α ⊙ B⊙α . Now, assume that n = m. Then, the following statements hold: vi) If A and B are (symmetric, Hermitian), then so is A ⊙ B. vii) If A and B are (skew symmetric, skew Hermitian), then A ⊙ B is (symmetric, Hermitian). viii) If A is symmetric and B is skew symmetric, then A ⊙ B is skew symmetric. ix) If A is Hermitian and B is skew Hermitian, then A ⊙ B is skew Hermitian. The following result shows that A ⊙ B is a submatrix of A ⊗ B. Proposition 9.3.2. Let A, B ∈ Fn×m. Then, A ⊙ B = (A ⊗ B)({1,n+2,2n+3,...,n2 },{1,m+2,2m+3,...,m2 }) .

(9.3.7)

A ⊙ B = (A ⊗ B)({1,n+2,2n+3,...,n2 }) ,

(9.3.8)

If, in addition, n = m, then and thus A ⊙ B is a principal submatrix of A ⊗ B. Proof. See [1450, p. 304] and [1951].

9.4 Facts on the Kronecker Product Fact 9.4.1. Let x, y ∈ Fn. Then, x ⊗ y = (x ⊗ In )y = (In ⊗ y)x. Fact 9.4.2. Let x, y, w, z ∈ Fn. Then, xTwyTz = (xT ⊗ yT )(w ⊗ z) = (x ⊗ y)T(w ⊗ z).



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Fact 9.4.3. Let A ∈ Fn×m and B ∈ F1×m. Then, A(B ⊗ Im ) = B ⊗ A. Fact 9.4.4. Let A ∈ Fn×n and B ∈ Fm×m, and assume that A and B are (diagonal, upper triangular,

lower triangular). Then, so is A ⊗ B. Fact 9.4.5. Let A ∈ Fn×n, B ∈ Fm×m, and l ≥ 0. Then, (A ⊗ B)l = Al ⊗ Bl. Fact 9.4.6. Let A ∈ Fn×m. Then, vec A = (Im ⊗ A) vec Im = (AT ⊗ In ) vec In . Fact 9.4.7. Let A ∈ Fn×m and B ∈ Fm×l. Then, m ∑ vec AB = (Il ⊗ A)vec B = (BT ⊗ A) vec Im = coli (BT ) ⊗ coli (A). i=1

Fact 9.4.8. Let A ∈ F

,B∈F

n×m

, and C ∈ F . Then,

m×l

l×n

tr ABC = (vec A)T(B ⊗ In )vec CT. Fact 9.4.9. Let A, B, C ∈ Fn×n, and assume that C is symmetric. Then,

(vec C)T(A ⊗ B) vec C = (vec C)T(B ⊗ A)vec C. Fact 9.4.10. Let A ∈ Fn×m, B ∈ Fm×l, C ∈ Fl×k, and D ∈ Fk×n. Then,

tr ABCD = (vec A)T (B ⊗ DT ) vec CT. Fact 9.4.11. Let A ∈ Fn×m, B ∈ Fm×l, and k ≥ 1. Then,

(AB)⊗k = A⊗kB⊗k, △

where A⊗k = A ⊗ A ⊗ · · · ⊗ A, with A appearing k times. Fact 9.4.12. Let A, C ∈ Fn×m and B, D ∈ Fl×k, assume that A is (left equivalent, right equivalent, biequivalent) to C, and assume that B is (left equivalent, right equivalent, biequivalent) to D. Then, A ⊗ B is (left equivalent, right equivalent, biequivalent) to C ⊗ D. Fact 9.4.13. Let A, B, C, D ∈ Fn×n, assume that A is (similar, congruent, unitarily similar) to C, and assume that B is (similar, congruent, unitarily similar) to D. Then, A ⊗ B is (similar, congruent, unitarily similar) to C ⊗ D. Fact 9.4.14. Let A, C ∈ Fn×m and B, D ∈ Fk×l. Then, the following statements are equivalent: i) A ⊗ B = C ⊗ D , 0. ii) A and B are nonzero, and there exist a, b ∈ F such that A = aC, B = bD, and ab = 1. Source: [2991, p. 120]. Fact 9.4.15. Let A ∈ Fn×m and B ∈ Fl×k. Then, R(A ⊗ B) = R(A ⊗ Il ) ∩ R(In ⊗ B). Source: [2641]. Fact 9.4.16. For all i ∈ {1, . . . , k}, let Ai ∈ Fni ×mi and Bi ∈ Fni ×pi be nonzero matrices. Then, the following statements are equivalent: i) For all i ∈ {1, . . . , k}, R(Ai ) ⊆ R(Bi ). ii) R(A1 ⊗ · · · ⊗ Ak ) ⊆ R(B1 ⊗ · · · ⊗ Bk ). Source: [2649]. Fact 9.4.17. Let A ∈ Fn×n and B ∈ Fm×m, and let γ ∈ spec(A ⊗ B). Then, ∑ ∑ gmultA (λ)gmultB (µ) ≤ gmultA⊗B (γ) ≤ amultA⊗B (γ) = amultA(λ)amultB (µ), where both sums are taken over all λ ∈ spec(A) and µ ∈ spec(B) such that λµ = γ. Fact 9.4.18. Let A ∈ Fn×n and B ∈ Fm×m. Then, ρmax (A ⊗ B) = ρmax (A)ρmax (B). Furthermore, ρmax (A ⊗ A) = ρ2max (A).

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Fact 9.4.19. Let A ∈ Fn×m and B ∈ Fl×k. Then, msval(A ⊗ B) = msval(A)msval(B). Source:

[2979, pp. 36, 37]. Fact 9.4.20. Let A ∈ Fn×m and B ∈ Fl×k. Then, rank(A ⊗ B) = (rank A)(rank B) = rank(B ⊗ A). Consequently, A ⊗ B = 0 if and only if either A = 0 or B = 0. Source: Fact 9.4.19. Related: Fact 10.25.15. Fact 9.4.21. Let A ∈ Fn×n and B ∈ Fm×m, and let γ ∈ spec(A ⊗ B). Then, indA⊗B (γ) = 1 if and only if indA(λ) = 1 and indB (µ) = 1 for all λ ∈ spec(A) and µ ∈ spec(B) such that λµ = γ. Fact 9.4.22. Let A ∈ Fn×n and B ∈ Fn×n, and assume that A and B are (group invertible, range Hermitian, range symmetric, Hermitian, symmetric, normal, positive semidefinite, positive definite, unitary, orthogonal, projectors, reflectors, involutory, idempotent, tripotent, nilpotent, semisimple). Then, so is A ⊗ B. Related: Fact 9.4.37. Fact 9.4.23. Let A1 , . . . , Al ∈ Fn×n, and assume that A1 , . . . , Al are skew Hermitian. If l is (even, odd), then A1 ⊗ · · · ⊗ Al is (Hermitian, skew Hermitian). Fact 9.4.24. Let Ai, j ∈ Fni ×n j for all i ∈ {1, . . . , k} and j ∈ {1, . . . , l}. Then,      A11 ⊗ B A22 ⊗ B · · · A1l ⊗ B   A11 A22 · · · A1l      . .  A21 ⊗ B A22 ⊗ B · .· · A2l ⊗ B   A21 A22 · .· · A2l      .. . . ..  . . . ..  ⊗ B =   ..  · .· · · .· · · .· · · .· · . .  .    .    Ak1 ⊗ B Ak2 ⊗ B · · · Akl ⊗ B Ak1 Ak2 · · · Akl Fact 9.4.25. Let x ∈ Fk, and, for all i ∈ {1, . . . , l}, let Ai ∈ Fn×ni . Then,

x ⊗ [A1 · · · Al ] = [x ⊗ A1 · · · x ⊗ Al ]. Fact 9.4.26. Let x ∈ Fm, A ∈ Fn×m, and B ∈ Fm×l. Then,

(A ⊗ x)B = (A ⊗ x)(B ⊗ 1) = (AB) ⊗ x. Fact 9.4.27. Let A ∈ Fn×n and B ∈ Fm×m. Then, the eigenvalues of

∑k,l

∑k,l i, j=1,1

γi j Ai ⊗ B j are of

the form i, j=1,1 γi j λ µ , where λ ∈ spec(A) and µ ∈ spec(B), and an associated eigenvector is given by x ⊗ y, where x ∈ Fn is an eigenvector of A associated with λ ∈ spec(A) and y ∈ Fn is an eigenvector of B associated with µ ∈ spec(B). Source: Let Ax = λx and By = µy. Then, γi j (Ai ⊗ B j )(x ⊗ y) = γi j λi µ j (x ⊗ y). See [1097], [1738, p. 411], and [1914, p. 83]. Fact 9.4.28. Let A, B ∈ Fn×m . Then,  ( [ ]) (rank A) rank [A B] + rank AB − 2(rank A)2    [A] 2 2  (rank A) rank [A B] + (rank B) rank B − (rank A) − (rank B)   2    2(rank A) rank [A B] − (rank A) ≤ rank(A ⊗ B + B ⊗ A) ≤    2(rank A) rank [ A ] − (rank A)2 . B i j

Now, assume that n = m and B is nonsingular. Then, 2n rank A − 2(rank A)2 ≤ rank(A ⊗ B + B ⊗ A) ≤ 2n rank A − (rank A)2 . Credit: Y. Tian. Fact 9.4.29. Let A ∈ Fn×m, B ∈ Fl×k, C ∈ Fn×p, D ∈ Fl×q. Then,

(rank A) rank [B D] + (rank D)(rank C − rank A) ≤ rank [A ⊗ B C ⊗ D]

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    (rank A) rank [B D] + (rank D) rank [A C] − (rank A) rank D ≤   (rank B) rank [A C] + (rank C) rank [B D] − (rank B) rank C. If, in addition, either rank [A C] = rank A + rank C or rank [B D] = rank B + rank D, then rank [A ⊗ B C ⊗ D] = (rank A) rank B + (rank C) rank D. Source: [2644, 2650]. Fact 9.4.30. Let A ∈ Fn×n and B ∈ Fm×m. Then,

rank(I − A ⊗ B) ≤ nm − [n − rank(I − A)][m − rank(I − B)]. Source: [743]. Fact 9.4.31. Let A ∈ Fn×n and B ∈ Fm×m. Then, ind A ⊗ B = max {ind A, ind B}. Fact 9.4.32. Let A ∈ Fn×m and B ∈ Fm×n. Then, |n − m|min {n, m} ≤ amultA⊗B (0). Source:

[1450, p. 249]. Fact 9.4.33. Let A ∈ Fn×m and B ∈ Fl×k, and assume that nl = mk and n , m. Then, A ⊗ B and B ⊗ A are singular. Source: [1450, p. 250]. Fact 9.4.34. Let A ∈ Fn×m and B ∈ Fl×k. Then, A and B are left invertible if and only if A ⊗ B is

left invertible. If these conditions hold, AL is a left inverse of A, and BL is a left inverse of B, then BL ⊗ AL is a left inverse of A ⊗ B. Remark: If A and B have full column rank, then so does A ⊗ B. Related: Fact 3.18.9. Fact 9.4.35. Let A ∈ Fn×m and B ∈ Fl×k. Then, A and B are right invertible if and only if A ⊗ B is right invertible. If these conditions hold, AR is a right inverse of A, and BR is a right inverse of B, then BR ⊗ AR is a right inverse of A ⊗ B. Remark: If A and B have full row rank, then so does A ⊗ B. Related: Fact 3.18.10. Fact 9.4.36. Let A ∈ Fn×m and B ∈ Fl×k. Then, (A ⊗ B)+ = A+ ⊗ B+. Fact 9.4.37. Let A ∈ Fn×n and B ∈ Fm×m. Then, (A ⊗ B)D = AD ⊗ BD. Now, assume that A and B are group invertible. Then, A ⊗ B is group invertible, and (A ⊗ B)# = A# ⊗ B#. Related: Fact 9.4.22. Fact 9.4.38. The Kronecker permutation matrix Pn,m ∈ Rnm×nm has the following properties: i) Pn,m is a permutation matrix. ii) PTn,m = P−1 n,m = Pm,n . iii) Pn,m is orthogonal. iv) Pn,m Pm,n = Inm . v) Pn,n is orthogonal, symmetric, and involutory. vi) Pn,n is a reflector. ∑ vii) Pn,m = ni=1 ei,n ⊗ Im ⊗ eTi,n. viii) Pnp,m = (In ⊗ P p,m )(Pn,m ⊗ Ip ) = (Ip ⊗ Pn,m )(P p,m ⊗ In ). ix) sig Pn,n = tr Pn,n = n. x) The inertia of Pn,n is given by In Pn,n = [ 12 (n2 − n) 0 21 (n2 + n)]T . xi) xii) xiii) xiv) xv)

det Pn,n = (−1)(n −n)/2. P1,m = Im and Pn,1 = In . If x ∈ Fn and y ∈ Fm, then Pn,m (y ⊗ x) = x ⊗ y. If A ∈ Fn×m and b ∈ Fk, then Pk,n (A ⊗ b) = b ⊗ A and Pn,k (b ⊗ A) = A ⊗ b. If A ∈ Fn×m and B ∈ Fl×k, then 2

Pl,n (A ⊗ B)Pm,k = B ⊗ A,

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vec(A ⊗ B) = (Im ⊗ Pk,n ⊗ Il )[(vec A) ⊗ (vec B)], vec(AT ⊗ B) = (Pnk,m ⊗ Il )[(vec A) ⊗ (vec B)], vec(A ⊗ BT ) = (Im ⊗ Pl,nk )[(vec A) ⊗ (vec B)]. xvi) If A ∈ Fn×m, B ∈ Fl×k, and nl = mk, then tr(A ⊗ B) = [vec(Im ) ⊗ (Ik )]T [(vec A) ⊗ (vec BT )]. xvii) If A ∈ Fn×n and B ∈ Fl×l, then Pl,n (A ⊗ B)Pn,l = Pl,n (A ⊗ B)P−1 l,n = B ⊗ A. Hence, A ⊗ B and B ⊗ A are similar. xviii) If A ∈ Fn×m and B ∈ Fm×n, then tr AB = tr[Pm,n (A ⊗ B)]. xix) Pnp,m = Pn,pm P p,nm = P p,nm Pn,pm . xx) Pnp,m P pm,n Pmn,p = I. △

△

Now, let A ∈ Fn×m , define r = rank A, and define K = Pn,m (A∗ ⊗ A). Then, the following statements hold: xxi) K is Hermitian, rank K = r2, tr K = tr A∗A, K 2 = (AA∗ ) ⊗ (A∗A). xxii) mspec(K) = {σ21 (A), . . . , σ2r (A)} ∪ {±σi (A)σ j (A): i < j, i, j ∈ {1, . . . , r}}. Source: [2403, pp. 308–311, 342, 343]. 2 2 △ Fact 9.4.39. Define Ψn ∈ Rn ×n by Ψn = 12 (In2 + Pn,n ), let x, y ∈ Fn, and let A, B ∈ Fn×n. Then, the following statements hold: i) Ψn is a projector. ii) Ψn = Ψn Pn,n = Pn,n Ψn . iii) Ψn (x ⊗ y) = 21 (x ⊗ y + y ⊗ x). Ψn vec(A) = 12 vec(A + AT ). Ψn (A ⊗ B)Ψn = Ψn (B ⊗ A)Ψn . Ψn (A ⊗ A)Ψn = Ψn (A ⊗ A) = (A ⊗ A)Ψn . Ψn (A ⊗ B + B ⊗ A)Ψn = Ψn (A ⊗ B + B ⊗ A) = (A ⊗ B + B ⊗ A)Ψn = 2Ψn (B ⊗ A)Ψn . (A ⊗ A)Ψn (AT ⊗ AT ) = Ψn (AAT ⊗ AAT ). Source: [2403, p. 312]. Fact 9.4.40. For all i ∈ {1, . . . , p}, let Ai ∈ Fni ×ni . Then, iv) v) vi) vii) viii)

mspec(A1 ⊗ · · · ⊗ A p ) = {λ1 · · · λ p : λi ∈ mspec(Ai ) for all i ∈ {1, . . . , p}}ms . Finally, for all i ∈ {1, . . . , p}, let xi ∈ Cni be an eigenvector of Ai associated with λi ∈ spec(Ai ). Then, x1 ⊗ · · · ⊗ x p is an eigenvector of A1 ⊗ · · · ⊗ A p associated with the eigenvalue λ1 · · · λ p .  Fact 9.4.41. Let A1 , . . . , Ak ∈ Fn×m , and define the k-tensor f : ki=1 Fm×n 7→ F on Fm×n by △

f (X1 , . . . , Xk ) = tr[(A1 ⊗ · · · ⊗ Ak )(X1 ⊗ · · · ⊗ Xk )]. Then, for all X1 , . . . , Xk ∈ Fm×n , f (X1 , . . . , Xk ) = tr[(A1 X1 ) ⊗ · · · ⊗ (Ak Xk )] =

k ∏ i=1

tr Ai Xi .

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Now, let σ be a permutation of (1, . . . , k), and define fσ : △

k

i=1 F

fσ (X1 , . . . , Xk ) = f (Xσ(1) , . . . , Xσ(k) ) =

m×n

k ∏

7→ F by

tr Ai Xσ(i) .

i=1

Furthermore, let g be the l-tensor on Fm×n defined by △

g(X1 , . . . , Xl ) = tr[(B1 ⊗ · · · ⊗ Bl )(X1 ⊗ · · · ⊗ Xl )], where B1 , . . . , Bl ∈ Fn×m . Then, f ⊗ g defined by △

( f ⊗ g)(X1 , . . . , Xk+l ) = tr[(A1 ⊗ · · · ⊗ Ak ⊗ B1 ⊗ · · · ⊗ Bl )(X1 ⊗ · · · ⊗ Xk+l )] = tr[(A1 X1 ) ⊗ · · · ⊗ (Ak Xk ) ⊗ (B1 Xk+1 ) ⊗ · · · ⊗ (Bl Xk+l )]  k  l ∏  ∏  =  tr Ai Xi  tr Bi Xk+i i=1

i=1

is a (k + l)-tensor on F . Next, f is an alternating k-tensor on Fm×n if, for every permutation σ of (1, . . . , k), fσ = sign(σ) f. If k = 1, then f is alternating. Now, define the k-tensor A( f ) on Fm×n by ∑ △ A( f ) = sign(σ) fσ , m×n

where the sum is taken over all k! permutations σ of (1, . . . , k). Then, A( f ) is alternating. Finally, define the alternating (k + l)-tensor f ∧ g on Fm×n by 1 A( f ⊗ g). k!l! Now, let f, g, and h be k-, l-, and j-tensors, respectively, on Fm×n . Then, the following statements hold: [ A1 ] . △ i) Let n = 1 and k = m, and define A = .. . Then, for all X1 , . . . , Xm ∈ Fm , A( f )(X1 , . . . , Xm ) △

f ∧g=

Am

△

ii) iii)

iv) v) vi) vii) viii) ix)

= det AX, where X = [X1 · · · Xm ]. If f is alternating, then A( f ) = k! f. The following statements are equivalent: a) f is alternating. b) For all X1 , . . . , Xk ∈ Fm×n that are not distinct, f (X1 , . . . , Xk ) = 0. A( f ⊗ g) = k!1 A[A( f ) ⊗ g] = l!1 A[ f ⊗ A(g)]. A(g ⊗ f ) = (−1)kl A( f ⊗ g) and g ∧ f = (−1)kl f ∧ g. If α, β ∈ F, then (α f ) ∧ βg = αβ( f ∧ g). If k = l = 1, then f ∧ g = f ⊗ g − g ⊗ f. △ f ⊗ g ⊗ h = f ⊗ (g ⊗ h) = ( f ⊗ g) ⊗ h is a (k + l + j)-tensor on Fm×n . △ f ∧ g ∧ h = f ∧ (g ∧ h) = ( f ∧ g) ∧ h is an alternating (k + l + j)-tensor on Fm×n . Furthermore, 1 1 f ∧ A(g ⊗ h) = A[ f ⊗ (g ∧ h)] l! j! k!(l + j)! 1 1 = A( f ⊗ g) ∧ h = A[( f ∧ g) ⊗ h] k!l! (k + l)! j! 1 = A( f ⊗ g ⊗ h). k!l! j!

f ∧g∧h=

KRONECKER AND SCHUR ALGEBRA

691

x) If k = l = j = 1, then f ∧ g ∧ h = f ⊗ g ⊗ h + g ⊗ h ⊗ f + h ⊗ f ⊗ g − h ⊗ g ⊗ f − f ⊗ h ⊗ g − g ⊗ f ⊗ h. Source: [2742, pp. 18–33]. Remark: f ∧ g is the wedge product of f and g. Alternating tensors

play a central role in geometric algebra, see [784, 908, 925, 1178, 1268, 1375, 1376, 1759, 1892, 1977, 2265], as well as in integration on manifolds, including Green’s theorem and Stokes’s theorem, see [553, 1265, 2101, 2505, 2742, 2837, 2856]. Remark: The determinant function is an alternating m-tensor on the columns of m × m matrices. Remark: f is a covariant tensor on Fm×n and a contravariant tensor on Fn×m . See [553, pp. 218–262]. △ Fact 9.4.42. Let A, B, C ∈ Fn×m , and define A ∧ B = A ⊗ B − B ⊗ A and △

A ∧ B ∧ C = A ⊗ B ⊗ C + B ⊗ C ⊗ A + C ⊗ A ⊗ B − C ⊗ B ⊗ A − A ⊗ C ⊗ B − B ⊗ A ⊗ C. Then, the following statements hold: i) A ∧ B = 0 if and only if A and B are linearly dependent. ii) A ∧ B ∧ C = 0 if and only if A, B, C are linearly dependent. iii) For all α ∈ F, (αA) ∧ B = A ∧ (αB) = α(A ∧ B). iv) For all α ∈ F, (αA) ∧ B ∧ C = A ∧ (αB) ∧ C = A ∧ B ∧ (αC) = α(A ∧ B ∧ C). v) A ∧ B = −(B ∧ A). vi) A ∧ B ∧ C = −B ∧ A ∧ C = −A ∧ C ∧ B = −C ∧ B ∧ A = B ∧ C ∧ A = C ∧ A ∧ B. ∑ vii) A ∧ B ∧ C = 12 sign(σ)[A ⊗ (B ∧ C)]σ , where the sum is taken over all permutations σ of A, B, C, and, writing A1 , A2 , A3 for A, B, C, respectively, [A1 ⊗ (A2 ∧ A3 )]σ denotes [Aσ(1) ⊗ (Aσ(2) ∧ Aσ(3) )]σ . Remark: A, B, C represent 1-tensors on Fm×n . See Fact 9.4.41.

9.5 Facts on the Kronecker Sum Fact 9.5.1. Let A ∈ Fn×n. Then, (A ⊕ A)2 = A2 ⊕ A2 + 2A ⊗ A. Fact 9.5.2. Let A ∈ Fn×n. Then, det(A ⊕ A) = (−1)n det χA (−A). Source: [2991, p. 121]. Fact 9.5.3. Let A ∈ Fn×n. Then,

n ≤ def(AT ⊕ −A) = dim {X ∈ Fn×n : AX = XA}, rank(AT ⊕ −A) = dim {[A, X]: X ∈ Fn×n } ≤ n2 − n. Source: Fact 3.23.9. Remark: def(AT ⊕ −A) is the dimension of the centralizer (also called the commutant) of A. See Fact 3.23.9. Related: Fact 7.16.8 and Fact 7.16.9. Problem: Characterize

rank(AT ⊕ −A) in terms of the eigenstructure of A. Fact 9.5.4. Let A ∈ Fn×n, assume that A is nilpotent, and assume that AT ⊕ −A = 0. Then, A = 0. Source: Note that AT ⊗ Ak = I ⊗ Ak+1, and use Fact 9.4.20. Fact 9.5.5. Let A ∈ Fn×n, and assume that, for all X ∈ Fn×n, AX = XA. Then, there exists α ∈ F such that A = αI. Source: It follows from Proposition 9.2.4 that all of the eigenvalues of A are equal. Hence, there exists α ∈ F such that A = αI + B, where B is nilpotent. Now, Fact 9.5.4 implies that B = 0. Fact 9.5.6. Let A ∈ Fn×n and B ∈ Fm×m, and let γ ∈ spec(A ⊕ B). Then, ∑ ∑ gmultA(λ)gmultB (µ) ≤ gmultA⊕B (γ) ≤ amultA⊕B (γ) = amultA(λ)amultB (µ), where both sums are taken over all λ ∈ spec(A) and µ ∈ spec(B) such that λ + µ = γ. Fact 9.5.7. Let A ∈ Fn×n and B ∈ Fm×m. Then, αmax (A ⊕ B) = αmax (A) + αmax (B). Furthermore, αmax (A ⊕ A) = 2αmax (A).

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Fact 9.5.8. Let A ∈ Fn×n and B ∈ Fm×m, and let γ ∈ spec(A ⊕ B). Then, indA⊕B (γ) = 1 if and

only if indA(λ) = 1 and indB (µ) = 1 for all λ ∈ spec(A) and µ ∈ spec(B) such that λ + µ = γ. Fact 9.5.9. Let A ∈ Fn×n and B ∈ Fm×m, and assume that A and B are (group invertible, range Hermitian, Hermitian, symmetric, skew Hermitian, skew symmetric, normal, positive semidefinite, positive definite, semidissipative, dissipative, nilpotent, semisimple). Then, so is A ⊕ B. Fact 9.5.10. Let A ∈ Fn×n and B ∈ Fm×m. Then, Pm,n (A ⊕ B)Pn,m = Pm,n (A ⊕ B)P−1 m,n = B ⊕ A. Hence, A ⊕ B and B ⊕ A are similar, and thus rank(A ⊕ B) = rank(B ⊕ A). Source: xiii) of Fact 9.4.38. Fact 9.5.11. Let A ∈ Fn×n and B ∈ Fm×m. Then,

n rank B + m rank A − 2(rank A)(rank B) ≤ rank(A ⊕ B) { nm − [n − rank(I + A)][m − rank(I − B)] ≤ nm − [n − rank(I − A)][m − rank(I + B)]. If, in addition, −A and B are idempotent, then rank(A ⊕ B) = n rank B + m rank A − 2(rank A)(rank B). Equivalently,

rank(A ⊕ B) = (rank (−A)⊥ ) rank B + (rank B⊥ ) rank A.

Source: [743]. Remark: The second inequality may be strict if −A and B are idempotent. △ Fact 9.5.12. Let A ∈ Fn×n and B ∈ Fm×m, assume that A is positive definite, define p(s) =

det(I − sA), and let mroots(p) = {λ1 , . . . , λn }ms . Then, det(A ⊕ B) = (det A)m

n ∏

det(λi B + I).

i=1

Source: Specialize Fact 9.5.13. Remark: In the case where rank C = 1, an expression for

det(vec−1 [(A ⊕ B)−1 vec C]) is given in [2390]. Fact 9.5.13. Let A, C ∈ Fn×n and B, D ∈ Fm×m, assume that A is positive definite, assume that △ C is positive semidefinite, define p(s) = det(C − sA), and let mroots(p) = {λ1 , . . . , λn }ms . Then, det(A ⊗ B + C ⊗ D) = (det A)m

n ∏

det(λi D + B).

i=1

Source: [2043, pp. 40, 41]. Remark: The Kronecker product definition in [2043] follows the convention of [1914], where “A ⊗ B” denotes B ⊗ A. Fact 9.5.14. Let A, C ∈ Fn×n and B, D ∈ Fm×m, assume that rank C = 1, and assume that A is nonsingular. Then,

det(A ⊗ B + C ⊗ D) = (det A)m (det B)n−1 det[B + (tr CA−1 )D]. Source: [2043, p. 41]. Fact 9.5.15. Let A ∈ Fn×n and B ∈ Fm×m. Then, the following statements are equivalent:

i) spec(A) ∩ spec(−B) = ∅. ii) AT ⊕ B is nonsingular. iii) A ⊕ B is nonsingular.

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iv) For all C ∈ Fn×m, there exists a unique solution X ∈ Fn×m of AX + XB + C = 0. v) For all C ∈ Fn×m, there exists a solution X ∈ Fn×m of AX + XB + C = 0. vi) There exists C ∈ Fn×m such that AX + XB + C = 0 has a unique solution X ∈ Fn×m . [ ] [ ] 0 and A C are similar. vii) For all C ∈ Fn×m, A0 −B 0 −B Source: The equivalence of i)–vi) follows from Corollary 3.7.9 and Proposition 9.2.5. The equivalence of v) and vii) follows from Fact 7.11.26. Fact 9.5.16. Let A ∈ Fn×n, B ∈ Fm×m, and C ∈ Fn×m, and assume that det(BT ⊕ A) , 0. Then, X ∈ Fn×m satisfies A2X + 2AXB + XB2 + C = 0 if and only if

X = −vec−1 [(BT ⊕ A)−2 vec C].

Fact 9.5.17. For all i ∈ {1, . . . , p}, let Ai ∈ Fni ×ni . Then,

mspec(A1 ⊕ · · · ⊕ A p ) = {λ1 + · · · + λ p : λi ∈ mspec(Ai ) for all i ∈ {1, . . . , p}}ms . If, in addition, for all i ∈ {1, . . . , p}, xi ∈ Cni is an eigenvector of Ai associated with λi ∈ spec(Ai ), then x1 ⊕ · · · ⊕ x p is an eigenvector of A1 ⊕ · · · ⊕ A p associated with λ1 + · · · + λ p . Fact Let A ∈ Fn×m , and, for 1 ≤ k ≤ min {n, m}, define the kth compound A(k) to be (n) (9.5.18. ) m the k × k matrix whose entries are k × k subdeterminants of A, where the k-element subsets of rows and columns of A are ordered lexicographically. (Example: For k = 3, n = 4, and m = 5, the four 3-element subsets of the rows of A are ordered as {1, 2, 3}, {1, 2, 4}, {1, 3, 4}, {2, 3, 4}, and the ten 3-element subsets of the columns of A are ordered as {1, 2, 3}, {1, 2, 4}, {1, 2, 5}, {1, 3, 4}, {1, 3, 5}, {1, 4, 5}, {2, 3, 4}, {2, 3, 5}, {2, 4, 5}, {3, 4, 5}.) Specifically, (A(k) )(i, j) is the determinant of the k × k submatrix of A obtained from the ith subset of k rows of A and the jth subset of k columns of A. n Furthermore, for all x, y ∈ Fn , define x ∧ y ∈ F(2) by [ ] x y(i p ) △ (x ∧ y)(p) = det (i p ) , x( j p ) x( j p ) () where, for all p ∈ {1, . . . , n2 }, the values of 1 ≤ i p < j p ≤ n are given by the pth component of the (n) 2 -tuple whose components (i, j) are ordered lexicographically. (Example: For n = 4, this 6-tuple is ((1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)).) Then, the following statements hold: i) A(0) = 1, A(1) = A. ii) A(2) (x ∧ y) = (Ax) ∧ (Ay). iii) If α ∈ F, then (αA)(k) = αkA(k). (k)

iv) (AT )(k) = (A(k) )T , A = A(k) , (A∗ )(k) = (A(k) )∗, (A+ )(k) = (A(k) )+ . ( A) v) If k ≤ rank A, then rank A(k) = rank . In particular, if k = rank A, then rank A(k) = 1. k vi) If k > rank A, then A(k) = 0. vii) If B ∈ Fm×l and 1 ≤ k ≤ min {n, m, l}, then (AB)(k) = A(k)B(k). viii) If B ∈ Fm×n, then det AB = A(n)B(n). ix) msval(A(k) ) = {σi1 (A) · · · σik (A) : 1 ≤ i1 < · · · < ik ≤ min {n, m}}ms . Now, assume that m = n, let 1 ≤ k ≤ n, and let mspec(A) = {λ1 , . . . , λn }ms . Then, the following statements hold: x) If A is (diagonal, lower triangular, upper triangular, symmetric, Hermitian, positive semidefinite, positive definite, normal, unitary), then so is A(k).

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xi) Assume that A is skew Hermitian. If k is odd, then A(k) is skew Hermitian. If k is even, then A(k) is Hermitian. xii) Assume that A is diagonal, upper triangular, or lower triangular, and let 1 ≤ i1 < · · · < ik ≤ n. Then, the (i1 + · · · + ik , i1 + · · · + ik ) entry of A(k) is A(i1,i1 ) · · · A(ik ,ik ) . In particular, In(k) = I(nk) . xiii) xiv) xv) xvi) xvii)

n−1

det A(k) = (det A)(k−1), A(n) = det A. △ SA(n−1)TS = AA, where S = diag(1, −1, 1, . . .). det A(n−1) = det AA = (det A)n−1, tr A(n−1) = tr AA. If A is nonsingular, then (A(k) )−1 = (A−1 )(k). mspec(A(k) ) = {λi1 · · · λik : 1 ≤ i1 < · · · < ik ≤ n}ms . In particular, mspec(A(2) ) = {λi λ j : i, j = 1, . . . , n, i < j}ms .

∑ xviii) tr A(k) = 1≤i1 0. Let Hn, Nn, and Pn denote, respectively, the Hermitian, positive-semidefinite, and positivedefinite matrices in Fn×n. If F is not specified, then F = C. Hence, Pn ⊂ Nn ⊂ Hn. If A ∈ Nn, then we write A ≥ 0, whereas, if A ∈ Pn, then we write A > 0. If A, B ∈ Hn, then A − B ∈ Nn is possible in the case where neither A nor B is positive semidefinite. In this case, we write A ≥ B or B ≤ A. Similarly, A − B ∈ Pn is denoted by A > B or B < A. This notation is consistent with the case n = 1, where H1 = R, N1 = [0, ∞), and P1 = (0, ∞). Since 0 ∈ Nn, it follows that Nn is a pointed cone. Furthermore, if A, −A ∈ Nn, then x∗Ax = 0 for all x ∈ Fn, which implies that A = 0. Hence, Nn is a one-sided cone. Finally, Nn is a convex cone since, if A, B ∈ Nn and α, β > 0, then αA + βB ∈ Nn . Likewise, Pn is a convex cone. The following result shows that the relation “≤” is a partial ordering on Hn. Proposition 10.1.1. The relation “≤” is reflexive, antisymmetric, and transitive on Hn; that is, if A, B, C ∈ Hn, then the following statements hold: i) A ≤ A. ii) If A ≤ B and B ≤ A, then A = B. iii) If A ≤ B and B ≤ C, then A ≤ C. Proof. Since Nn is a pointed, one-sided, convex cone, it follows from Proposition 3.1.7 that the relation “≤” is reflexive, antisymmetric, and transitive.  Additional properties of “≤” and “ 0, then αA > 0 for all α > 0, and αA < 0 for all α < 0. iii) αA + βB ∈ Hn for all α, β ∈ R.

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iv) If A ≥ 0 and B ≥ 0, then αA + βB ≥ 0 for all α, β ≥ 0. v) If A ≥ 0 and B > 0, then A + B > 0. vi) A2 ≥ 0. vii) A2 > 0 if and only if det A , 0. viii) If A ≤ B and B < C, then A < C. ix) If A < B and B ≤ C, then A < C. x) If A ≤ B and C ≤ D, then A + C ≤ B + D. xi) If A ≤ B and C < D, then A + C < B + D. Furthermore, let S ∈ Fm×n. Then, the following statements hold: xii) If A ≤ B, then SAS ∗ ≤ SBS ∗. xiii) If A < B and rank S = m, then SAS ∗ < SBS ∗. xiv) If SAS ∗ ≤ SBS ∗ and rank S = n, then A ≤ B. xv) If SAS ∗ < SBS ∗ and rank S = n, then m = n and A < B. xvi) Assume that A ≤ B. Then, SAS ∗ < SBS ∗ if and only if rank S = m and R(S ∗ ) ∩ N(B − A) = {0}. Proof. i)–xi) are immediate. To prove xiii), note that A < B implies that (B − A)1/2 is positive definite. Thus, rank S (A − B)1/2 = m, which implies that S (A − B)S ∗ is positive definite. To prove xiv), note that, since rank S = n, it follows that S has a left inverse S L ∈ Fn×m. Thus, xii) implies that A = S LSAS ∗S L∗ ≤ S LSBS ∗S L∗ = B. To prove xv), note that, since S (B − A)S ∗ is positive definite, it follows that rank S = m. Hence, m = n and S is nonsingular. Thus, xiii) implies that A = S −1SAS ∗S −∗ < S −1SBS ∗S −∗ = B. xvi) is proved in [627].  The following result is an immediate consequence of Theorem 7.5.7. Corollary 10.1.3. Let A, B ∈ Hn, and assume that A and B are congruent. Then, A is positive semidefinite if and only if B is positive semidefinite. Furthermore, A is positive definite if and only if B is positive definite.

10.2 Submatrices and Schur Complements We first consider some factorizations of a partitioned Hermitian matrix. Note that A11|A and A22 |A are Schur complements defined by Definition 8.1.13. [A A ] 12 ∈ Hn+m. If Proposition 10.2.1. Let A = A∗11 12 A22 + A12 , A12 = A11 A11

then

[

A11 A∗12

] [ A12 I = ∗ + A22 A12 A11

0 I

][

A11 0

(10.2.1) 0

][

A11 |A

I 0

] A+11 A12 , I

(10.2.2)

where A11|A = A22 − A∗12 A+11 A12 .

(10.2.3)

A12 = A12 A+22 A22 ,

(10.2.4)

Alternatively, if

then

[

A11 A∗12

] [ A12 I = A22 0

A12 A+22 I

][

A22 |A 0

0 A22

][

I A+22 A∗12

] 0 , I

(10.2.5)

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POSITIVE-SEMIDEFINITE MATRICES

where A22 |A = A11 − A12 A+22 A∗12 .

(10.2.6)

The following result shows that, if A is positive semidefinite, then (10.2.1) and (10.2.4) hold. [A A ] 11 12 n+m . Then, (10.2.1) and (10.2.4) are satisfied. A∗12 A22 ∈ N [ ] Proof. Since A ≥ 0, it follows from Corollary 7.5.5 that A = BB∗, where B = BB12 ∈ F(n+m)×r

Lemma 10.2.2. Let A = △

and r = rank A. Thus, A11 = B1B∗1 , A12 = B1B∗2 , and A22 = B2 B∗2 . Since A11 is Hermitian, it follows △ from xxvii) of Proposition 8.1.7 that A+11 is also Hermitian. Next, defining S = B1 − B1B∗1 (B1 B∗1 )+B1 , it follows that SS ∗ = 0, and thus tr SS ∗ = 0. Hence, Lemma 3.3.3 implies that S = 0, and thus B1 = B1B∗1 (B1B∗1 )+B1. Consequently, B1B∗2 = B1B∗1 (B1B∗1 )+B1B∗2 ; that is, (10.2.1) holds. The proof of (10.2.4) is analogous.  [A A ] 12 n+m ∈ N . Then, the following statements hold: Corollary 10.2.3. Let A = A∗11 12 A22 i) R(A12 ) ⊆ R(A11 ). ii) R(A∗12 ) ⊆ R(A22 ). iii) rank [A11 A12 ] = rank A11 . iv) rank [A∗12 A22 ] = rank A22 . v) A11|A and A22 |A are positive semidefinite. Proof. i) and ii) follow from (10.2.1) and (10.2.4); iii) and iv) are consequences of i) and ii); and v) follows from (10.2.2), (10.2.5), and xiv) of Proposition 10.1.2.  [ ] △ A A12 n+m Proposition 10.2.4. Let A = A∗11 ∈ N . Then, 12 A22 rank A = rank A11 + rank A11 |A = rank A22 |A + rank A22

(10.2.7) (10.2.8)

≤ rank A11 + rank A22 .

(10.2.9)

det A = (det A11 ) det(A11 |A),

(10.2.10)

Furthermore, det A = (det A22 ) det(A22 |A). (10.2.11) [A A ] 12 ∈ Hn+m. Then, the following statements are equivalent: Proposition 10.2.5. Let A = A∗11 12 A22 i) A ≥ 0. + + ii) A11 ≥ 0, A12 = A11 A11 A12 , and A∗12 A11 A12 ≤ A22 . + + ∗ iii) A22 ≥ 0, A12 = A12 A22 A22 , and A12 A22 A12 ≤ A11 . The following statements are also equivalent: iv) A > 0. v) A11 > 0 and A∗12 A−1 11 A12 < A22 . −1 ∗ vi) A22 > 0 and A12 A22 A12 < A11 . The following result follows from (3.9.17) and (3.9.18) or from (10.2.2) and (10.2.5). [ ] △ A A12 ∗ Proposition 10.2.6. Let A = A11 ∈ Hn+m. Then, the following statements hold: 12 A22 i) Assume that A11 is nonsingular. Then, A is nonsingular if and only if A11 |A is nonsingular. If these conditions hold, then  −1  −1 ∗ −1 −1   A11 + A−1 −A−1  11 A12 (A11 |A) A12 A11 11 A12 (A11 |A) −1  A =  (10.2.12)  , −1 ∗ −1 −1 −(A11 |A) A12 A11 (A11 |A) △

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where ∗ −1 A11 |A = A22 − A12 A11 A12 .

(10.2.13)

ii) Assume that A22 is nonsingular. Then, A is nonsingular if and only if A22 |A is nonsingular. If these conditions hold, then     (A22 |A)−1 −(A22 |A)−1 A12 A−1 22 −1  , (10.2.14) A =  ∗ −1 ∗ −1 −1 −1  −A−1 A−1 22 A12 (A22 |A) 22 A12 (A22 |A) A12 A22 + A22 where ∗ A22 |A = A11 − A12 A−1 22 A12 . [B B ] iii) Assume that A is nonsingular, and let A−1 = B1112∗ B1222 ∈ Hn+m . Then,

B11 |A−1 = A−1 22 ,

B22 |A−1 = A−1 11 .

(10.2.15)

(10.2.16)

iv) If A is positive definite, then A is nonsingular and A−1 , A11 , A22 , A11 |A, and A22 |A are positive definite. [ ] △ Lemma 10.2.7. Let A ∈ Fn×n, b ∈ Fn, and a ∈ R, and define A = bA∗ ba . Then, the following statements are equivalent: i) A is positive semidefinite. ii) A is positive semidefinite, b = AA+b, and b∗A+b ≤ a. iii) Either A is positive semidefinite, a = 0, and b = 0 or a > 0 and bb∗ ≤ aA. Furthermore, the following statements are equivalent: iv) A is positive definite. v) A is positive definite, and b∗A−1b < a. vi) a > 0 and bb∗ < aA. If iv)–vi) hold, then det A = (det A)(a − b∗A−1b). (10.2.17) For the following result note that every matrix is a principal submatrix of itself, while the determinant of a matrix is also a principal subdeterminant of the matrix. Proposition 10.2.8. Let A ∈ Hn. Then, χA ∈ R[s]. Furthermore, the following statements are equivalent: i) A is positive semidefinite. ii) Every principal submatrix of A is positive semidefinite. iii) Every principal subdeterminant of A is nonnegative. iv) For all i ∈ {1, . . . , n}, the sum of all i × i principal subdeterminants of A is nonnegative. v) For all i ∈ {1, . . . , n − 1}, (−1)n−i βi ≥ 0, where χA (s) = sn + βn−1 sn−1 + · · · + β1 s + β0 . Proof. To prove i) =⇒ ii), let Aˆ ∈ Fm×m be the principal submatrix of A obtained from A by △ retaining rows and columns i1, . . . , im . Then, Aˆ = S TAS, where S = [ei1 · · · eim ] ∈ Rn×m. Now, let m ∗ˆ ∗ T xˆ ∈ F . Since A is positive semidefinite, it follows that xˆ A xˆ = xˆ S AS xˆ ≥ 0, and thus Aˆ is positive semidefinite. Next, ii) =⇒ iii) =⇒ iv) are immediate. To prove iv) =⇒ i), Proposition 6.4.6 implies χA (s) =

n ∑ i=0

βi si =

n n ∑ ∑ (−1)n−i γn−i si = (−1)n γn−i (−s)i , i=0

(10.2.18)

i=0

where, for all i ∈ {1, . . . , n}, γi is the sum of all i × i principal subdeterminants of A, and βn = γ0 = 1. By assumption, γi ≥ 0 for all i ∈ {1, . . . , n}. Now, suppose there exists λ ∈ spec(A) such that λ < 0.

707

POSITIVE-SEMIDEFINITE MATRICES

∑ Then, 0 = (−1)n χA (λ) = ni=0 γn−i (−λ)i > 0, which is a contradiction. Hence, spec(A) ⊂ [0, ∞). The equivalence of iv) and v) follows from Proposition 6.4.6.  n Proposition 10.2.9. Let A ∈ H . Then, the following statements are equivalent: i) A is positive definite. ii) Every principal submatrix of A is positive definite. iii) Every principal subdeterminant of A is positive. iv) Every leading principal submatrix of A is positive definite. v) Every leading principal subdeterminant of A is positive. vi) For all i ∈ {1, . . . , n}, the sum of all i × i principal subdeterminants of A is positive. vii) For all i ∈ {1, . . . , n − 1}, (−1)n−i βi > 0, where χA (s) = sn + βn−1 sn−1 + · · · + β1 s + β0 . Proof. To prove i) =⇒ ii), let Aˆ ∈ Fm×m and S be as in the proof of Proposition 10.2.8, and let xˆ be nonzero so that S xˆ is nonzero. Since A is positive definite, it follows that xˆ∗Aˆ xˆ = xˆ∗S TAS xˆ > 0, and hence Aˆ is positive definite. Next, ii) =⇒ iii) =⇒ v) and ii) =⇒ iv) =⇒ v) are immediate. To prove v) =⇒ i), suppose that the leading principal submatrix Ai ∈ Fi×i has positive determinant for all i ∈ {1, . . . , n}. The result is true for [ A bn] = 1. For n ≥ 2, we show that, if Ai is positive definite, then so is Ai+1 . Writing Ai+1 = b∗ii aii , it follows from Lemma 10.2.7 that det Ai+1 = (det Ai )(ai − b∗i A−1 i bi ) > 0, and hence ai − b∗i A−1 b = det A /det A > 0. Lemma 10.2.7 now implies that A is positive definite. Using i+1 i i+1 i i this argument for all i ∈ {2, . . . , n} implies that A is positive definite.  [ ] 0 0 The example A = 0 −1 shows that every principal subdeterminant of A, rather than just the leading principal subdeterminants of[A, must A is positive semidefinite. ] be checked to determine whether √ √ 111 A less obvious example is A = 1 1 1 , whose eigenvalues are 0, 1 + 3, and 1 − 3. In this case, 110 [ ] the principal subdeterminant det A[1,1] = det 11 10 < 0. Note that iii) of Proposition 10.2.9 includes det A > 0 since the determinant of A is also a subde[ ] 3/2 −1 1

terminant of A. The matrix A = −1 2 1 has the property that every 1 × 1 and 2 × 2 subdeterminant 1 1 2 is positive but is not positive definite. This example shows, if the requirement that the determinant of A be positive is omitted, then iii) =⇒ ii) of Proposition 10.2.9 is false.

10.3 Simultaneous Diagonalization This section considers the simultaneous diagonalization of a pair of matrices A, B ∈ Hn. There are two types of simultaneous diagonalization. Cogredient diagonalization involves a nonsingular matrix S ∈ Fn×n such that SAS ∗ and SBS ∗ are both diagonal, whereas contragredient diagonalization involves a nonsingular matrix S ∈ Fn×n such that SAS ∗ and S −∗BS −1 are both diagonal. Both types of simultaneous transformation involve congruence transformations. We begin by assuming that one of the matrices is positive definite. The first result is cogredient diagonalization. Theorem 10.3.1. Let A, B ∈ Hn, and assume that A is positive definite. Then, there exists a nonsingular matrix S ∈ Fn×n such that SAS ∗ = I and SBS ∗ is diagonal. Proof. Setting S 1 = A−1/2 , it follows that S 1 AS 1∗ = I. Now, since S 1 BS 1∗ is Hermitian, Corollary 7.5.5 implies that there exists a unitary matrix S 2 ∈ Fn×n such that SBS ∗ = S 2 S 1BS 1∗ S 2∗ is diagonal, where S = S 2 S 1. Finally, SAS ∗ = S 2 S 1 AS 1∗S 2∗ = S 2 IS 2∗ = I.  An analogous result holds for contragredient diagonalization. Theorem 10.3.2. Let A, B ∈ Hn, and assume that A is positive definite. Then, there exists a nonsingular matrix S ∈ Fn×n such that SAS ∗ = I and S −∗BS −1 is diagonal. Proof. Setting S 1 = A−1/2, it follows that S 1 AS 1∗ = I. Since S 1−∗BS 1−1 is Hermitian, it follows that there exists a unitary matrix S 2 ∈ Fn×n such that S −∗BS −1 = S 2−∗S 1−∗BS 1−1S 2−1 = S 2 (S 1−∗BS 1−1 )S 2∗ is

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diagonal, where S = S 2 S 1. Finally, SAS ∗ = S 2 S 1 AS 1∗S 2∗ = S 2 IS 2∗ = I.  n Corollary 10.3.3. Let A, B ∈ H , and assume that A is positive definite. Then, AB is diagonalizable over F, every eigenvalue of AB is real, and In(AB) = In(B). Corollary 10.3.4. Let A, B ∈ Pn. Then, there exists a nonsingular matrix S ∈ Fn×n such that SAS ∗ and S −∗BS −1 are equal and diagonal. Proof. By Theorem 10.3.2 there exists a nonsingular matrix S 1 ∈ Fn×n such that S 1 AS 1∗ = I and △ ∗ −∗ −1 B1 = S 1−∗BS 1−1 is diagonal. Defining S = B1/4 = B1/2  1 S 1 yields SAS = S BS 1 . The transformation S of Corollary 10.3.4 is a balancing transformation. See Definition 16.9.28. Next, we weaken the requirement in Theorem 10.3.1 and Theorem 10.3.2 that A be positive definite by assuming only that A is positive semidefinite. In this case, however, we assume that B is also positive semidefinite. n n×n Theorem such that [ ] 10.3.5. Let A, B ∈ N . Then, there exists a nonsingular matrix S ∈ F ∗ ∗ SAS = 0I 00 and SBS is diagonal. [ ] Proof. Let the nonsingular matrix S 1 ∈ Fn×n satisfy S 1 AS 1∗ = 0I 00 , and similarly partition [B B ] [ + ] △ S 1BS 1∗ = B11∗12 B1222 , which is positive semidefinite. Letting S 2 = 0I −B12I B22 , it follows from Lemma ] [ 10.2.2 that B11 − B12 B+22 B∗12 0 ∗ ∗ . S 2 S 1BS 1 S 2 = 0 B22 Next, let U1 and U2 be unitary such that U1 (B11 − B12 B+22 B∗12 )U1∗ and[U2]B22 U2∗ are di[ Umatrices ] △ △ 0 agonal. Then, defining S 3 = 01 U2 and S = S 3 S 2 S 1 , it follows that SAS ∗ = 0I 00 and SBS ∗ = ∗ ∗ ∗ S 3 S 2 S 1BS 1 S 2S 3 is diagonal.  n n×n Theorem such that [ ] 10.3.6. Let A, B ∈ N . Then, there exists a nonsingular matrix S ∈ F SAS ∗ = 0I 00 and S −∗BS −1 is diagonal. [ ] Proof. Let S 1 ∈ Fn×n be a nonsingular matrix such that S 1AS 1∗ = 0I 00 , and similarly partition [B B ] [ + ] △ S 1−∗BS 1−1 = B11∗12 B1222 , which is positive semidefinite. Letting S 2 = 0I B11IB12 , it follows that [ ] B11 0 −∗ −∗ −1 −1 S 2 S 1 BS 1 S 2 = . 0 B22 − B∗12 B+11B12 ∗ ∗ + ∗ Now, let U1 and U2 be unitary matrices such [ ] that△the matrices U1B11 U1 and U2 (B22 [ − ]B12 B11B12 )U2 △ U1 0 ∗ −∗ −1 I 0 are diagonal. Then, defining S 3 = 0 U2 and S = S 3 S 2 S 1 , it follows that SAS = 0 0 and S BS = S 3−∗S 2−∗S 1−∗BS 1−1S 2−1S 3−1 is diagonal.  Corollary 10.3.7. Let A, B ∈ Nn. Then, AB is diagonalizable over F, and every eigenvalue of AB is nonnegative. If, in addition, A and B are positive definite, then every eigenvalue of AB is positive. Proof. It follows from Theorem 10.3.6 that there exists a nonsingular matrix S ∈ Fn×n such that A1 = SAS ∗ and B1 = S −∗BS −1 are diagonal with nonnegative diagonal entries. Hence, AB = S −1A1B1 S is semisimple and all of its eigenvalues are nonnegative.  A more direct approach to showing that AB has nonnegative eigenvalues is to use Corollary 6.4.11 and note that λi (AB) = λi (B1/2 AB1/2 ) ≥ 0. Corollary 10.3.8. Let A, B ∈ Nn, and assume that rank A = rank B = rank AB. Then, there exists a nonsingular matrix S ∈ Fn×n such that SAS ∗ = S −∗BS −1 and SAS ∗ is diagonal. [ ] Proof. By Theorem 10.3.6 there exists a nonsingular matrix S 1 ∈ Fn×n such that S 1 AS 1∗ = I0r 00 , [ ] △ where r = rank A, and such that B1 = S 1−∗BS 1−1 is diagonal. Hence, AB = S 1−1 I0r 00 B1S 1. Since rank A [ ] = rank B = rank AB = r, it follows that B1 = Bˆ 1 0 , where Bˆ 1 ∈ Fr×r is a diagonal matrix all of 0 0

709

POSITIVE-SEMIDEFINITE MATRICES

[ ] [ ˆ 1/4 ] △ whose diagonally located entries are positive. Hence, S 1−∗BS 1−1 = Bˆ01 00 . Now, define S 2 = B1 0 0 I [ ˆ 1/2 ] △ and S = S 2 S 1 . Then, SAS ∗ = S 2 S 1AS 1∗ S 2∗ = B1 0 = S 2−∗S 1−∗BS 1−1S 2−1 = S −∗BS −1.  0 0

10.4 Eigenvalue Inequalities Next, we turn our attention to inequalities for eigenvalues. We begin with several lemmas. Lemma 10.4.1. Let A ∈ Hn and β ∈ R. Then, the following statements hold: i) βI ≤ A if and only if β ≤ λmin (A). ii) βI < A if and only if β < λmin (A). iii) A ≤ βI if and only if λmax (A) ≤ β. iv) A < βI if and only if λmax (A) < β. Proof. To prove i), assume that βI ≤ A, and let S ∈ Fn×n be a unitary matrix such that B = SAS ∗ is diagonal. Then, βI ≤ B, which yields β ≤ λmin (B) = λmin (A). Conversely, let S ∈ Fn×n be a unitary matrix such that B = SAS ∗ is diagonal. Since the diagonal entries of B are the eigenvalues of A, it follows that λmin (A)I ≤ B, which implies that βI ≤ λmin (A)I ≤ S ∗BS = A. ii), iii), and iv) are proved in a similar manner.  Corollary 10.4.2. Let A ∈ Hn. Then, λmin (A)I ≤ A ≤ λmax (A)I.

(10.4.1)

Proof. Use i) and iii) of Lemma 10.4.1 with β = λmin (A) and β = λmax (A), respectively.



The following result concerns the maximum and minimum values of the Rayleigh quotient. Lemma 10.4.3. Let A ∈ Hn. Then, x∗Ax , x∈F \{0} x∗x

λmin (A) = min n

x∗Ax . \{0} x∗x

λmax (A) = max n x∈F

(10.4.2)

Proof. It follows from (10.4.1) that λmin (A) ≤ x∗Ax/x∗x for all nonzero x ∈ Fn. Letting x ∈ Fn

be an eigenvector of A associated with λmin (A), it follows that this lower bound is attained. An analogous argument yields the second equality.  The following result is the Cauchy interlacing theorem. Lemma 10.4.4. Let A ∈ Hn, and let A0 be an (n − 1) × (n − 1) principal submatrix of A. Then, for all i ∈ {1, . . . , n − 1}, λi+1 (A) ≤ λi (A0 ) ≤ λi (A).

(10.4.3)

Proof. Note that (10.4.3) is the chain of inequalities

λn (A) ≤ λn−1 (A0 ) ≤ λn−1 (A) ≤ · · · ≤ λ2 (A) ≤ λ1 (A0 ) ≤ λ1 (A). Suppose that this chain of inequalities does not hold. In particular, first suppose that the rightmost inequality that is not true is λ j (A0 ) ≤ λ j (A), so that λ j (A) < λ j (A0 ). Choose δ such that λ j (A) < δ < λ j (A0 ) and such that δ is not an eigenvalue of A0 . If j = 1, then A − δI is negative definite, while, if j ≥ 2, then λ j (A) < δ < λ j (A0 ) ≤ λ j−1 (A0 ) ≤ λ j−1 (A), so that A − δI has j − 1 positive eigenvalues. Thus, ν+ (A − δI) = j − 1. Furthermore, since δ < λ j (A0 ), it follows that ν+ (A0 − δI) ≥ j. Now, assume for convenience that the rows and columns [ of] A are ordered so that A0 is the (n −1) × (n −1) leading principal submatrix of A. Thus, A = Aβ∗0 γβ , where β ∈ Fn−1 and γ ∈ F. Next, note the equality [ ][ ][ ] I 0 A0 − δI 0 I (A0 − δI)−1β A − δI = ∗ , β (A0 − δI)−1 1 0 γ − δ − β∗ (A0 − δI)−1β 0 1

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where A0 − δI is nonsingular since δ is chosen to not be an eigenvalue of A0 . Since the right-hand side of this equality involves a congruence transformation, and since ν+ (A0 − δI) ≥ j, it follows from Theorem 7.5.7 that ν+ (A−δI) ≥ j. However, this inequality contradicts the fact that ν+ (A−δI) = j−1. Finally, suppose that the right-most inequality in (10.4.3) that is not true is λ j+1 (A) ≤ λ j (A0 ), so that λ j (A0 ) < λ j+1 (A). Choose δ such that λ j (A0 ) < δ < λ j+1 (A) and such that δ is not an eigenvalue of A0 . Then, it follows that ν+ (A − δI) ≥ j + 1 and ν+ (A0 − δI) = j − 1. Using the congruence transformation as in the previous case, it follows that ν+ (A − δI) ≤ j, which contradicts the fact that  ν+ (A − δI) ≥ j + 1. The following result is the eigenvalue interlacing theorem. Theorem 10.4.5. Let A ∈ Hn, and let A0 ∈ Hk be a k × k principal submatrix of A. Then, for all i ∈ {1, . . . , k}, λi+n−k (A) ≤ λi (A0 ) ≤ λi(A). (10.4.4) Proof. For k = n−1, the result is given by Lemma 10.4.4. Hence, let k = n−2, and let A1 denote an (n −1) × (n −1) principal submatrix of A such that the (n − 2) × (n − 2) principal submatrix A0 of A is also a principal submatrix of A1 . Therefore, Lemma 10.4.4 implies that λn (A) ≤ λn−1(A1 ) ≤ · · · ≤ λ2 (A1 ) ≤ λ2 (A) ≤ λ1 (A1 ) ≤ λ1 (A) and λn−1(A1 ) ≤ λn−2 (A0 ) ≤ · · · ≤ λ2 (A0 ) ≤ λ2 (A1 ) ≤ λ1 (A0 ) ≤ λ1 (A1 ). Combining these inequalities yields λi+2 (A) ≤ λi (A0 ) ≤ λi (A) for all i = 1, . . . , n − 2, while proceeding in a similar manner with k < n − 2 yields (10.4.4).  Corollary 10.4.6. Let A ∈ Hn, and let A0 ∈ Hk be a k × k principal submatrix of A. Then,

λmin (A) ≤ λmin (A0 ) ≤ λmax (A0 ) ≤ λmax (A),

(10.4.5)

λmin (A0 ) ≤ λk (A).

(10.4.6)

The following corollary of both Lemma 10.4.3 and Theorem 10.4.5 compares the maximum and minimum eigenvalues with the maximum and minimum diagonal entries. Corollary 10.4.7. Let A ∈ Hn. Then, λmin (A) ≤ dmin (A) ≤ dmax (A) ≤ λmax (A).

(10.4.7)

Lemma 10.4.8. Let A, B ∈ Hn, and assume that A ≤ B and mspec(A) = mspec(B). Then,

A = B.

△

△

ˆ where Aˆ = A + αI and Bˆ = B + αI. Note that mspec(A) ˆ = Proof. Let α ≥ 0 satisfy 0 < Aˆ ≤ B, ˆ−1/2 ˆ ˆ−1/2

ˆ and thus det Aˆ = det B. ˆ Next, it follows that I ≤ A BA . Hence, i) of Lemma 10.4.1 mspec( B), ˆ Aˆ = 1, which implies implies that λmin (Aˆ−1/2Bˆ Aˆ−1/2 ) ≥ 1. Furthermore, det(Aˆ−1/2Bˆ Aˆ−1/2 ) = det B/det ˆ Hence, that, for all i ∈ {1, . . . , n}, λi (Aˆ−1/2Bˆ Aˆ−1/2 ) = 1. Hence, Aˆ−1/2Bˆ Aˆ−1/2 = I, and thus Aˆ = B. A = B.  The following result is the monotonicity theorem or Weyl’s inequality. Theorem 10.4.9. Let A, B ∈ Hn, and assume that A ≤ B. Then, for all i ∈ {1, . . . , n}, λi (A) ≤ λi (B).

(10.4.8)

If A , B, then there exists i ∈ {1, . . . , n} such that λi (A) < λi (B).

(10.4.9)

If A < B, then (10.4.9) holds for all i ∈ {1, . . . , n}. Proof. Since A ≤ B, Corollary 10.4.2 implies that λmin (A)I ≤ A ≤ B ≤ λmax (B)I. Hence, it follows from iii) and i) of Lemma 10.4.1 that λmin (A) ≤ λmin (B) and λmax (A) ≤ λmax (B). Next, let S ∈ Fn×n be a unitary matrix such that SAS ∗ = diag[λ1 (A), . . . , λn (A)]. Furthermore, for 2 ≤ i ≤ n−1, let A0 = diag[λ1 (A), . . . , λi (A)], and let B0 denote the i × i leading principal submatrices of SAS ∗ and

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POSITIVE-SEMIDEFINITE MATRICES

SBS ∗, respectively. Since A ≤ B, it follows that A0 ≤ B0 , which implies that λmin (A0 ) ≤ λmin (B0 ). It now follows from (10.4.6) that λi (A) = λmin (A0 ) ≤ λmin (B0 ) ≤ λi (SBS ∗ ) = λi (B), which proves (10.4.8). If A , B, then Lemma 10.4.8 implies that mspec(A) , mspec(B), and thus there exists i ∈ {1, . . . , n} such that (10.4.9) holds. In the case where A < B, it follows that λmin (A0 ) < λmin (B0 ), which implies (10.4.9) for all i ∈ {1, . . . , n}.  n Corollary 10.4.10. Let A, B ∈ H . Then, the following statements hold: i) If A ≤ B, then tr A ≤ tr B. ii) If A ≤ B and tr A = tr B, then A = B. iii) If A < B, then tr A < tr B. iv) If 0 ≤ A ≤ B, then 0 ≤ det A ≤ det B. v) If 0 ≤ A < B, then 0 ≤ det A < det B. vi) If 0 < A ≤ B and det A = det B, then A = B. Proof. i), iii), iv), and v) follow from Theorem 10.4.9. To prove ii), note that, since A ≤ B and tr A = tr B, Theorem 10.4.9 implies that mspec(A) = mspec(B). Now, Lemma 10.4.8 implies that A = B. A similar argument yields vi).  The following result, which is a generalization of Theorem 10.4.9, is due to H. Weyl. Theorem 10.4.11. Let A, B ∈ Hn. If i + j ≥ n + 1, then λi (A) + λ j (B) ≤ λi+ j−n (A + B).

(10.4.10)

λi+ j−1 (A + B) ≤ λi (A) + λ j (B).

(10.4.11)

λi (A) + λmin (B) ≤ λi (A + B) ≤ λi (A) + λmax (B), λmin (A) + λmin (B) ≤ λmin (A + B) ≤ λmin (A) + λmax (B),

(10.4.12) (10.4.13)

λmax (A) + λmin (B) ≤ λmax (A + B) ≤ λmax (A) + λmax (B).

(10.4.14)

If i + j ≤ n + 1, then In particular, for all i ∈ {1, . . . , n},

Furthermore, if rank B ≤ r, then, for all i ∈ {1, . . . , n − r}, λi+r (A) ≤ λi (A + B),

(10.4.15)

λi+r (A + B) ≤ λi (A).

(10.4.16)

If, in particular, x ∈ Fn , then, for all i ∈ {1, . . . , n − 1}, λi+1 (A) ≤ λi+1 (A + xx∗ ) ≤ λi (A) ≤ λi (A + xx∗ ).

(10.4.17)

ν− (A + B) ≤ ν− (A) + ν− (B), ν+ (A + B) ≤ ν+ (A) + ν+ (B).

(10.4.18) (10.4.19)

Finally,

Proof. See [862], [1451, pp. 239–242], [2403, pp. 114, 115], and [2979, pp. 53, 54]. Lemma 10.4.12. Let A, B, C ∈ Hn. If A ≤ B and C is positive semidefinite, then



tr AC ≤ tr BC.

(10.4.20)

tr AC < tr BC.

(10.4.21)

If A < B and C is positive definite, then

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Proof. Since C 1/2AC 1/2 ≤ C 1/2 BC 1/2 , i) of Corollary 10.4.10 implies that

tr AC = tr C 1/2AC 1/2 ≤ tr C 1/2BC 1/2 = tr BC. (10.4.21) follows from ii) of Corollary 10.4.10 in a similar fashion. Proposition 10.4.13. Let A, B ∈ Fn×n, and assume that B is positive semidefinite. Then, 1 2 λmin (A

+ A∗ ) tr B ≤ Re tr AB ≤ 12 λmax (A + A∗ ) tr B.



(10.4.22)

If, in addition, A is Hermitian, then λmin (A) tr B ≤ tr AB ≤ λmax (A) tr B.

(10.4.23)

Proof. It follows from Corollary 10.4.2 that 12 λmin (A + A∗ )I ≤ 12 (A + A∗ ), while Lemma 10.4.12

implies that 21 λmin (A + A∗ ) tr B = tr 21 λmin (A + A∗ )IB ≤ tr 21 (A + A∗ )B = Re tr AB, which proves the left-hand inequality of (10.4.22). Similarly, the right-hand inequality holds.  For results relating to Proposition 10.4.13, see Fact 7.13.12, Fact 7.13.13, Fact 7.13.16, and Fact 10.22.25. Proposition 10.4.14. Let A ∈ Nn, let B ∈ Pn, and assume that det B = 1. Then, (det A)1/n ≤ n1 tr AB.

(10.4.24)

1/n −1

Furthermore, equality holds if and only if A = (det A) B . Proof. The arithmetic-mean–geometric-mean inequality given by Fact 2.11.81 implies that 1/n  n n  ∏ 1∑ 1 1/2 1/2  1/n 1/2 1/2 1/n   λi (B AB ) ≤ (det A) = (det B AB ) =  λi (B1/2AB1/2 ) = tr AB. n n i=1 i=1 Equality holds if and only if there exists β > 0 such that B1/2AB1/2 = βI. If these conditions hold, then β = (det A)1/n and A = (det A)1/nB−1 .  n For A, B ∈ N , the following corollary of Proposition 10.4.14 includes Minkowski’s determinant theorem, which is the second inequality in the string (det A + det B)1/n ≤ (det A)1/n + (det B)1/n ≤ [det(A + B)]1/n .

(10.4.25)

Corollary 10.4.15. Let A, B ∈ Nn, and let p and q be real numbers such that 1 ≤ p ≤ n ≤ q.

Then, det A + det B ≤ [(det A)1/p + (det B)1/p ] p ≤ [(det A) + (det B) ] { [(det A)1/q + (det B)1/q ]q ≤ det(A + B). 1/n

1/n n

(10.4.26) (10.4.27) (10.4.28)

Furthermore, the following statements hold: i) If p = 1, then (10.4.26) is an equality. ii) If n = 1, then (10.4.26), (10.4.27), and lower (10.4.28) are equalities. iii) If either A or B is singular, then (10.4.26), (10.4.27), and upper (10.4.28) are equalities. iv) If either q = 1, A = 0, B = 0, or A + B is singular, then (10.4.26)–(10.4.28) are equalities. Now, assume that n ≥ 2. Then, the following statements hold: v) If A is positive definite and (det A + det B)1/n = (det A)1/n + (det B)1/n , then det B = 0. vi) The following statements are equivalent: a) Either A = 0, B = 0, A + B is singular, or there exists α ≥ 0 such that either B = αA or

713

POSITIVE-SEMIDEFINITE MATRICES

A = αB. b) (det A)1/n + (det B)1/n = [det(A + B)]1/n . vii) If A + B is positive definite and (det A)1/n + (det B)1/n = [det(A + B)]1/n , then A and B are positive definite and there exists α > 0 such that B = αA. viii) If A is positive definite and det A + det B = det(A + B), then B = 0. ix) Either A = 0, B = 0, or A + B is singular if and only if det A + det B = det(A + B). x) Either B = 0 or A + B is singular if and only if det A = det(A + B). Proof. Inequalities (10.4.26), (10.4.27), and upper (10.4.28) are consequences of the reverse power-sum inequality given by Fact 2.11.91 and Fact 11.8.21. To prove lower (10.4.28), note that, in the case where A + B is singular, lower (10.4.28) is immediate. In the case where A + B is positive definite, it follows from Proposition 10.4.14 that (det A)1/n + (det B)1/n ≤

1 n

tr[A[det(A + B)]1/n (A + B)−1 ] + n1 tr[B[det(A + B)]1/n (A + B)−1 ]

= [det(A + B)]1/n. Next, v) follows from Fact 2.11.91; vi) is given in [2991, p. 215]; vii) follows from vi). To prove viii), note that (10.4.26) and (10.4.27) hold as equalities. Hence, v) implies that det B = 0. Consequently, det A = det(A + B). Since 0 < A ≤ A + B, vi) of Corollary 10.4.10 implies that B = 0. ix) and x) are given in [2991, p. 215]. 

10.5 Exponential, Square Root, and Logarithm of Hermitian Matrices Let B ∈ Rn×n be diagonal, let D ⊆ R, let f : D 7→ R, and assume that, for all i ∈ {1, . . . , n}, B(i,i) ∈ D. Then, we define △

f (B) = diag[ f (B(1,1) ), . . . , f (B(n,n) )]. ∗

Furthermore, let A = SBS ∈ F be Hermitian, where S ∈ F and assume that spec(A) ⊂ D. Then, we define f (A) ∈ Hn by n×n

n×n

(10.5.1)

is unitary, B ∈ R

△

f (A) = Sf (B)S ∗.

n×n

is diagonal, (10.5.2)

Hence, with an obvious extension of notation, f : {X ∈ H : spec(X) ⊂ D} 7→ H . If f : D 7→ R is one-to-one, then its inverse f −1: {X ∈ Hn : spec(X) ⊂ f (D)} 7→ Hn exists. It remains to be shown, however, that the definition of f (A) given by (10.5.2) is independent of the matrices S and B in the decomposition A = SBS ∗. The following lemma is needed. ˆ ∈ Rn×n denote the diagonal matrices D = Lemma 10.5.1. Let S ∈ Fn×n be unitary, let D, D diag(λ1In1 , . . . , λr Inr ) and Dˆ = diag(µ1In1 , . . . , µr Inr ), where λ1 , . . . , λr , µ1 , . . . , µr ∈ R, and assume ˆ that SD = DS . Then, S Dˆ = DS. [ S ] 12 Proof. Let r = 2, and partition S = SS 11 . Then, it follows from SD = DS that λ2 S 12 = λ1 S 12 21 S 22 [ ] and λ1S 21 = λ2 S 21 . Since λ1 , λ2 , it follows that S 12 = 0 and S 21 = 0. Therefore, S = S011 S022 , and ˆ A similar argument holds for r ≥ 3. thus S Dˆ = DS.  ∗ ∗ n n×n Proposition 10.5.2. Let A = RBR = SCS ∈ H , where R, S ∈ F are unitary and B, C ∈ Rn×n are diagonal. Furthermore, let D ⊆ R, let f : D 7→ R, and assume that all diagonal entries of B are contained in D. Then, Rf (B)R∗ = Sf (C)S ∗. Proof. Let spec(A) = {λ1 , . . . , λr }. Then, the columns of R and S can be rearranged to obtain △ ˜ S˜ ∈ Fn×n such that A = RD ˜ R˜ ∗ = SD ˜ S˜ ∗ , where D = unitary matrices R, diag(λ1In1 , . . . , λnr Inr ). △ ∗ ˜ It thus follows from Lemma 10.5.1 that UDˆ = DU, ˆ where Hence, UD = DU, where U = S˜ R. △ Dˆ = f (D) = diag[ f (λ1 )In1 , . . . , f (λnr )Inr ]. Hence, R˜ Dˆ R˜ ∗ = S˜ Dˆ S˜ ∗, while rearranging the columns of R˜ and S˜ as well as the diagonal entries of Dˆ yields R f (B)R∗ = S f (C)S ∗.  n

n

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CHAPTER 10

Let A = SBS ∗ ∈ Fn×n be Hermitian, where S ∈ Fn×n is unitary and B ∈ Rn×n is diagonal. Then, the matrix exponential is defined by △

eA = SeBS ∗ ∈ Hn ,

(10.5.3)

△

where, for all i ∈ {1, . . . , n}, (eB )(i,i) = eB(i,i) . Let A = SBS ∗ ∈ Fn×n be positive semidefinite, where S ∈ Fn×n is unitary and B ∈ Rn×n is diagonal with nonnegative entries. Then, for all r ≥ 0 (not necessarily an integer), Ar = SBrS ∗ is △ positive semidefinite, where, for all i ∈ {1, . . . , n}, (Br )(i,i) = [B(i,i) ]r . Note that A0 = I. In particular, the positive-semidefinite matrix A1/2 = SB1/2S ∗

(10.5.4)

A1/2A1/2 = SB1/2S ∗SB1/2S ∗ = SBS ∗ = A.

(10.5.5)

is a square root of A since

The uniqueness of the positive-semidefinite square root of A given by (10.5.4) follows from Theorem 12.9.1; see also [1450, p. 410] and [1767]. Uniqueness is shown in [970, pp. 265, 266], [1448, p. 405], [1655], and [2991, p. 81]. Hence, if C ∈ Fn×m, then C ∗C is positive semidefinite, and we define △

⟨C ⟩ = (C ∗C)1/2 .

(10.5.6)

If A is positive definite, then Ar is positive definite for all r ∈ R, and, if r , 0, then (Ar )1/r= A. Now, assume that A ∈ Fn×n is positive definite. Then, the matrix logarithm is defined by △

log A = S (log B)S ∗ ∈ Hn ,

(10.5.7)

△

where, for all i ∈ {1, . . . , n}, (log B)(i,i) = log[B(i,i) ]. In chapters 10 and 11, the matrix exponential, square root, and logarithm are extended to matrices that are not necessarily Hermitian.

10.6 Matrix Inequalities Lemma 10.6.1. Let A, B ∈ Hn, and assume that 0 ≤ A ≤ B. Then, R(A) ⊆ R(B). Proof. Let x ∈ N(B). Then, x∗Bx = 0, and thus x∗Ax = 0, which implies that Ax = 0. Hence,

N(B) ⊆ N(A), and thus N(A)⊥ ⊆ N(B)⊥. Since A and B are Hermitian, it follows from Theorem 3.5.3 that R(A) = N(A)⊥ and R(B) = N(B)⊥. Hence, R(A) ⊆ R(B).  The following result is the Douglas-Fillmore-Williams lemma [930, 1050]. Theorem 10.6.2. Let A ∈ Fn×m and B ∈ Fn×l. Then, the following statements are equivalent: i) There exists C ∈ Fl×m such that A = BC. ii) There exists α > 0 such that AA∗ ≤ αBB∗. iii) R(A) ⊆ R(B). Proof. First we prove i) =⇒ ii). Since A = BC, it follows that AA∗ = BCC ∗B∗. Since CC ∗ ≤ △ λmax (CC ∗ )I, it follows that AA∗ ≤ αBB∗, where α = λmax (CC ∗ ). To prove ii) =⇒ iii), let x ∈ N(B∗ ). ∗ ∗ ∗ ∗ ⊥ Then, 0 ≤ x AA x = αx BB x = 0, and thus x ∈ N(A∗ ). Hence, R(B)⊥ = N(B∗ ) ⊆ N(A∗ ) =[R(A) ] , D 0 which is equivalent to iii). Finally, to prove iii) =⇒ i), use Theorem 7.6.3 to write B = S 1 0 0 S 2 , where S 1 ∈ Fn×n and S 2 ∈ Fl×l are unitary and D ∈ Rr×r [is diagonal with positive diagonal ] [ ] entries, △ ∗ ∗ ∗ ∗ D 0 where r = rank B. Since R(S 1 A) ⊆ R(S 1 B) and S 1B = 0 0 S 2 , it follows that S 1 A = A01 , where A1 ∈ Fr×m. Consequently, [ ] [ ] [ −1 ][ ] A1 D 0 0 A1 ∗ D A = S1 = S1 S S = BC, 0 0 0 2 2 0 0 0

715

POSITIVE-SEMIDEFINITE MATRICES △

where C = S 2∗

[

D−1 0 0 0

][ ]

∈ Fl×m.  ∞ n Proposition 10.6.3. Let (Ai )i=1 ⊂ N satisfy 0 ≤ Ai ≤ Aj for all i ≤ j, and assume that there △ exists B ∈ Nn satisfying Ai ≤ B for all i ≥ 1. Then, A = limi→∞ Ai exists and satisfies 0 ≤ A ≤ B. Proof. Let k ∈ {1, . . . , n}, and let i < j. Since Ai ≤ Aj ≤ B, it follows that the sequence (Ar(k,k) )∞ r=1 △ is nondecreasing and bounded from above by B(k,k) . Hence, A(k,k) = limr→∞ Ar(k,k) exists. Now, let l ∈ {1, . . . , n}, where l , k. Since Ai ≤ Aj , it follows that (ek + el )TAi (ek + el ) ≤ (ek + el )TAj (ek + el ), which implies that Ai(k,l) − A j(k,l) ≤ 21 [A j(k,k) − Ai(k,k) + A j(l,l) − Ai(l,l) ]. Likewise, (ek − el )TAi (ek − el ) ≤ (ek − el )TAj (ek − el ) implies that A j(k,l) − Ai(k,l) ≤ 12 [A j(k,k) − Ai(k,k) + A j(l,l) − Ai(l,l) ]. Hence, ∞ |A j(k,l) − Ai(k,l) | ≤ 21 [A j(k,k) − Ai(k,k) ] + 12 [A j(l,l) − Ai(l,l) ]. Next, since (Ar(k,k) )∞ r=1 and (Ar(l,l) )r=1 are ∞ convergent sequences and thus Cauchy sequences, it follows that (Ar(k,l) )r=1 is a Cauchy sequence. △ ∞ Consequently, (Ar(k,l) )∞ r=1 is convergent, and thus A(k,l) = limi→∞ Ai(k,l) exists. Therefore, (Ai )i=1 is △ convergent, and thus A = limi→∞ Ai exists. Since Ai ≤ B for all i ≥ 1, it follows that A ≤ B.  n Proposition 10.6.4. Let A ∈ P and p ∈ (0, ∞). Then, A1 0

A−1 (A − I) ≤ log A ≤ p−1 (Ap − I), −1

log A = lim p (A − I). p

p↓0

(10.6.1) (10.6.2)

Proof. Use Fact 2.15.3.  n −1 −1 Lemma 10.6.5. Let A ∈ P . If A ≤ I, then I ≤ A . Furthermore, if A < I, then I < A . Proof. Since A ≤ I, it follows from xii) of Proposition 10.1.2 that I = A−1/2AA−1/2 ≤ A−1/2IA−1/2

= A−1. Similarly, A < I implies that I = A−1/2AA−1/2 < A−1/2IA−1/2 = A−1.  Proposition 10.6.6. Let A, B ∈ Hn, and assume that either A and B are positive definite or A and B are negative definite. If A ≤ B, then B−1 ≤ A−1. If, in addition, A < B, then B−1 < A−1. Proof. Suppose that A and B are positive definite. Since A ≤ B, it follows that B−1/2AB−1/2 ≤ I. Now, Lemma 10.6.5 implies that I ≤ B1/2A−1B1/2 , which implies that B−1 ≤ A−1. If A and B are negative definite, then A ≤ B is equivalent to −B ≤ −A. The case where A < B is proved in a similar manner.  The following result is the Furuta inequality. Proposition 10.6.7. Let A, B ∈ Nn, and assume that 0 ≤ A ≤ B. Furthermore, let p, q, r ∈ R satisfy p ≥ 0, q ≥ 1, r ≥ 0, and p + 2r ≤ (1 + 2r)q. Then, A(p+2r)/q ≤ (ArB pAr )1/q , r p r 1/q

(B A B )

≤B

.

(p+2r)/q

Proof. See [1116] and [1124, pp. 129, 130]. Corollary 10.6.8. Let A, B ∈ Nn, and assume that 0 ≤ A ≤ B. Then,

(10.6.3) (10.6.4) 

A2 ≤ (AB2A)1/2 ,

(10.6.5)

≤B.

(10.6.6)

2

(BA B)

1/2

2

Proof. In Proposition 10.6.7 set r = 1, p = 2, and q = 2. Corollary 10.6.9. Let A, B, C ∈ Nn, and assume that 0 ≤ A ≤ C ≤ B. Then,

(CA2C)1/2 ≤ C 2 ≤ (CB2C)1/2 . Proof. Use Corollary 10.6.8. See also [2820].

The following result provides representations for Ar, where r ∈ (0, 1).

 (10.6.7) 

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CHAPTER 10

Proposition 10.6.10. Let A ∈ Nn and r ∈ (0, 1). Then,

) ∫ ( ( rπ ) sin rπ ∞ xr+1 −1 r dx, Ar = cos I − (A + xI) x I+ 2 π 1 + x2 0 ∫ sin rπ ∞ (A + xI)−1Axr−1 dx. Ar = π 0

(10.6.8) (10.6.9)

Proof. Let t ≥ 0. As shown in [443], [449, p. 143],

∫

0

∞

(

) xr+1 xr π (r rπ ) − dx = t − cos . sin rπ 2 1 + x2 t + x

Solving for tr and replacing t by A yields (10.6.8). Likewise, replacing t by A in Fact 14.4.6 yields (10.6.9).  The following result is the L¨owner-Heinz inequality. Corollary 10.6.11. Let A, B ∈ Nn, assume that 0 ≤ A ≤ B, and let r ∈ [0, 1]. Then, Ar ≤ Br. If, in addition, A < B and r ∈ (0, 1], then Ar < Br. Proof. Let A ≤ B and r ∈ [0, 1]. In Proposition 10.6.7, replace p, q, r with r, 1, 0. The first result follows from (10.6.3). Now, let A < B and r ∈ (0, 1). Then, it follows from (10.6.8) of Proposition 10.6.10 as well as Proposition 10.6.6 that ∫ sin rπ ∞ r r B −A = [(A + xI)−1 − (B + xI)−1 ]xr dx > 0. π 0 Hence, Ar < Br. Alternatively, let A < B and r ∈ (0, 1). Then, Proposition 10.6.6 implies that, for all x ≥ 0, (A + xI)−1A = I − x(A + xI)−1 < I − x(B + xI)−1 = (B + xI)−1B. It thus follows from (10.6.9) of Proposition 10.6.10 that ∫ sin rπ ∞ [(B + xI)−1B − (A + xI)−1A]xr−1 dx > 0. Br − Ar = π 0 Hence, Ar < Br. Additional proofs are given in [1124, p. 127] and [2977, p. 2]. In the case where A ≤ B and r = 1/2, let λ ∈ R be an eigenvalue of B1/2 − A1/2, and let x ∈ Fn be an associated eigenvector. Then, λx∗ (B1/2 + A1/2 )x = x∗ (B1/2 + A1/2 )(B1/2 − A1/2 )x = x∗ (B − B1/2A1/2 + A1/2B1/2 − A)x = x∗ (B − A)x ≥ 0. Since B1/2 + A1/2 is positive semidefinite, it follows that either λ ≥ 0 or x∗ (B1/2 + A1/2 )x = 0. In the latter case, B1/2 x = A1/2 x = 0, which implies that λ = 0.  [ ] [ ] △ △ The L¨owner-Heinz inequality does not extend to r > 1. In fact, A = 21 11 and B = 10 00 satisfy A ≥ B ≥ 0, whereas, for all r > 1, Ar ̸≥ Br. For details, see [1124, pp. 127, 128]. Many of the results given so far involve functions that are nondecreasing or increasing on suitable sets of matrices. Definition 10.6.12. Let D ⊆ Hn and ϕ: D 7→ Hm. Then, the following terminology is defined: i) ϕ is nondecreasing if, for all A, B ∈ D such that A ≤ B, it follows that ϕ(A) ≤ ϕ(B). ii) ϕ is increasing if ϕ is nondecreasing and, for all A, B ∈ D such that A < B, it follows that ϕ(A) < ϕ(B).

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POSITIVE-SEMIDEFINITE MATRICES

iii) ϕ is strongly increasing if ϕ is nondecreasing and, for all A, B ∈ D such that A ≤ B and A , B, it follows that ϕ(A) < ϕ(B). iv) ϕ is (nonincreasing, decreasing, strongly decreasing) if −ϕ is (nondecreasing, increasing, strongly increasing). Proposition 10.6.13. The following functions are nondecreasing: △ i) ϕ: Hn 7→ Hm defined by ϕ(A) = BAB∗, where B ∈ Fm×n. △ ii) ϕ: Hn 7→ R defined by ϕ(A) = tr AB, where B ∈ Nn. ] [ △ △ A A iii) ϕ: Nn+m 7→ Nn defined by ϕ(A) = A22 |A, where A = A∗1112 A1222 . △

iv) ϕ: Nn × Nm 7→ Nnm defined by ϕ(A, B) = Ar1 ⊗ Br2 , where r1, r2 ∈ [0, 1] satisfy r1 + r2 ≤ 1. △ v) ϕ: Nn × Nn 7→ Nn defined by ϕ(A, B) = Ar1 ⊙ Br2, where r1, r2 ∈ [0, 1] satisfy r1 + r2 ≤ 1. The following functions are increasing: △ vi) ϕ: Hn 7→ R defined by ϕ(A) = λi (A), where i ∈ {1, . . . , n}. △ vii) ϕ: Nn 7→ Nn defined by ϕ(A) = Ar , where r ∈ [0, 1]. △ viii) ϕ: Pn 7→ −Pn defined by ϕ(A) = −A−r , where r ∈ [0, 1]. △ ix) ϕ: −Pn 7→ Pn defined by ϕ(A) = (−A)−r, where r ∈ [0, 1]. △ x) ϕ: Hn 7→ Hm defined by ϕ(A) = BAB∗, where B ∈ Fm×n and rank B = m. [ ] △ △ A A xi) ϕ: Pn+m 7→ Pn defined by ϕ(A) = A22 |A, where A = A∗1112 A1222 . [ ] △ △ A A xii) ϕ: Pn+m 7→ Pn defined by ϕ(A) = −(A22 |A)−1, where A = A∗1112 A1222 . △

xiii) ϕ: Pn 7→ Hn defined by ϕ(A) = log A. The following functions are strongly increasing: △ xiv) ϕ: Hn 7→ [0, ∞) defined by ϕ(A) = tr BAB∗, where B ∈ Fm×n and rank B = m. △ xv) ϕ: Hn 7→ R defined by ϕ(A) = tr AB, where B ∈ Pn. △ xvi) ϕ: Nn 7→ [0, ∞) defined by ϕ(A) = tr Ar, where r > 0. △ xvii) ϕ: Nn 7→ [0, ∞) defined by ϕ(A) = det A. Proof. For the proof of iii), see [1800]. To prove xviii), let A, B ∈ Pn, and assume that A ≤ B. Then, for all r ∈ [0, 1], it follows from vii) that r−1 (Ar − I) ≤ r−1 (Br − I). Letting r ↓ 0 and using Proposition 10.6.4 yields log A ≤ log B, which proves that log is nondecreasing. See [1124, p. 139] and [2586, p. 256]. To prove that log is increasing, assume that A < B, and let ε > 0 satisfy A + εI < B. Then, log A < log(A + εI) ≤ log B.  Finally, we consider convex functions defined with respect to matrix inequalities. The following definition generalizes Definition 1.6.6 in the case where n = m = p = 1. Definition 10.6.14. Let D ⊆ Fn×m be a convex set, and let ϕ: D 7→ H p. Then, the following terminology is defined: i) ϕ is convex if, for all α ∈ [0, 1] and A1 , A2 ∈ D, ϕ[αA1 + (1 − α)A2 ] ≤ αϕ(A1 ) + (1 − α)ϕ(A2 ).

(10.6.10)

ii) ϕ is concave if −ϕ is convex. iii) ϕ is strictly convex if, for all α ∈ (0, 1) and distinct A1 , A2 ∈ D, ϕ[αA1 + (1 − α)A2 ] < αϕ(A1 ) + (1 − α)ϕ(A2 ).

(10.6.11)

iv) ϕ is strictly concave if −ϕ is strictly convex. Theorem 10.6.15. Let S1 , S2 ⊆ R, let ϕ : S1 7→ S2 , and assume that ϕ is continuous. Then, the following statements hold:

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i) Let S1 = S2 = (0, ∞), and assume that, for all n ≥ 1, ϕ: Pn 7→ Pn is increasing. Then, ψ : Pn 7→ Pn defined by ψ(x) = 1/ϕ(x) is convex. ii) Let S1 = S2 = [0, ∞). Then, for all n ≥ 1, ϕ: Nn 7→ Nn is increasing if and only if, for all n ≥ 1, ϕ: Nn 7→ Nn is concave. iii) Let S1 = [0, ∞) and S2 = R. Then, for all n ≥ 1, ϕ: Nn 7→ Hn is convex and ϕ(0) ≤ 0 if and only if, for all n ≥ 1, ψ : Pn 7→ Hn defined by ψ(x) = ϕ(x)/x is increasing. Proof. See [449, pp. 120–122].  Lemma 10.6.16. Let D ⊆ Fn×m and S ⊆ H p be convex sets, and let ϕ1 : D 7→ S and ϕ2 : S 7→ Hq. Then, the following statements hold: i) If ϕ1 is convex and ϕ2 is nondecreasing and convex, then ϕ2 ◦ ϕ1: D 7→ Hq is convex. ii) If ϕ1 is concave and ϕ2 is nonincreasing and convex, then ϕ2 ◦ ϕ1: D 7→ Hq is convex. iii) If S is symmetric, ϕ2 (−A) = −ϕ2 (A) for all A ∈ S, ϕ1 is concave, and ϕ2 is nonincreasing and concave, then ϕ2 ◦ ϕ1: D 7→ Hq is convex. iv) If S is symmetric, ϕ2 (−A) = −ϕ2 (A) for all A ∈ S, ϕ1 is convex, and ϕ2 is nondecreasing and concave, then ϕ2 ◦ ϕ1: D 7→ Hq is convex. Proof. To prove i) and ii), let α ∈ [0, 1] and A1 , A2 ∈ D. In both cases it follows that ϕ2 (ϕ1 [αA1 + (1 − α)A2 ]) ≤ ϕ2 [αϕ1 (A1 ) + (1 − α)ϕ1 (A2 )] ≤ αϕ2 [ϕ1 (A1 )] + (1 − α)ϕ2 [ϕ1 (A2 )]. iii) and iv) follow from i) and ii), respectively. Proposition 10.6.17. The following functions are convex: △ i) ϕ: Nn 7→ Nn defined by ϕ(A) = Ar , where r ∈ [1, 2]. △ ii) ϕ: Nn 7→ Nn defined by ϕ(A) = A2. △ iii) ϕ: Pn 7→ Pn defined by ϕ(A) = A−r, where r ∈ [0, 1]. △ iv) ϕ: Pn 7→ Pn defined by ϕ(A) = A−1. △ v) ϕ: Pn 7→ Pn defined by ϕ(A) = A−1/2. △ vi) ϕ: Nn 7→ −Nn defined by ϕ(A) = −Ar, where r ∈ [0, 1]. △ vii) ϕ: Nn 7→ −Nn defined by ϕ(A) = −A1/2. △ viii) ϕ: Nn 7→ Hm defined by ϕ(A) = γBAB∗, where γ ∈ R and B ∈ Fm×n. △ ix) ϕ: Nn 7→ Nm defined by ϕ(A) = BArB∗, where B ∈ Fm×n and r ∈ [1, 2]. △ x) ϕ: Pn 7→ Nm defined by ϕ(A) = BA−rB∗, where B ∈ Fm×n and r ∈ [0, 1]. △ xi) ϕ: Nn 7→ −Nm defined by ϕ(A) = −BArB∗, where B ∈ Fm×n and r ∈ [0, 1]. △ xii) ϕ: Pn 7→ −Pm defined by ϕ(A) = −(BA−rB∗ )−p, where B ∈ Fm×n , rank B = m, and r, p ∈ [0, 1]. △ xiii) ϕ: Fn×m 7→ Nn defined by ϕ(A) = ABA∗, where B ∈ Nm. △ xiv) ϕ: Pn × Fm×n 7→ Nm defined by ϕ(A, B) = BA−1B∗. △ xv) ϕ: {A ∈ Fn×n : A + A∗ > 0} 7→ Pn defined by ϕ(A) = (A−1 + A−∗ )−1 . △ xvi) ϕ: Nn × Nn 7→ Nn defined by ϕ(A, B) = −A(A + B)+B. [ ] △ △ A A xvii) ϕ: Nn+m 7→ Nn defined by ϕ(A) = −A22 |A, where A = A∗1112 A1222 . [ ] △ △ A A xviii) ϕ: Pn+m 7→ Pn defined by ϕ(A) = (A22 |A)−1, where A = A∗1112 A1222 . △

xix) ϕ: Hn 7→ [0, ∞) defined by ϕ(A) = tr Ak, where k is a nonnegative even integer. △ xx) ϕ: Pn 7→ (0, ∞) defined by ϕ(A) = tr A−r, where r > 0. △ xxi) ϕ: Pn → 7 (−∞, 0) defined by ϕ(A) = −(tr A−r )−p, where r, p ∈ [0, 1].



719

POSITIVE-SEMIDEFINITE MATRICES △

xxii) ϕ: Nn × Nn 7→ (−∞, 0] defined by ϕ(A, B) = − tr (Ar + Br )1/r, where r ∈ [0, 1]. △ xxiii) ϕ: Nn × Nn 7→ [0, ∞) defined by ϕ(A, B) = tr (A2 + B2 )1/2. △ xxiv) ϕ: Nn × Nm 7→ R defined by ϕ(A, B) = − tr Ar XB pX ∗, where X ∈ Fn×m, r, p ≥ 0, and r + p ≤ 1. △ xxv) ϕ: Nn 7→ (−∞, 0) defined by ϕ(A) = − tr Ar XApX ∗, where X ∈ Fn×n, r, p ≥ 0, and r + p ≤ 1. △ xxvi) ϕ: Pn × Pm × Fm×n 7→ R defined by ϕ(A, B, X) = (tr A−pXB−r X ∗ )q , where r, p ≥ 0, r + p ≤ 1, and q ≥ (2 − r − p)−1. △ xxvii) ϕ: Pn × Fn×n 7→ [0, ∞) defined by ϕ(A, X) = tr A−pXA−r X ∗, where r, p ≥ 0 and r + p ≤ 1. △ xxviii) ϕ: Pn × Fn×n 7→ [0, ∞) defined by ϕ(A) = tr A−pXA−r X ∗, where r, p ∈ [0, 1] and X ∈ Fn×n. △ xxix) ϕ: Pn 7→ R defined by ϕ(A) = − tr([Ar, X][A1−r, X]), where r ∈ (0, 1) and X ∈ Hn. △ xxx) ϕ: Pn 7→ Hn defined by ϕ(A) = −log A. △ xxxi) ϕ: Pn 7→ Hm defined by ϕ(A) = Alog A. △ xxxii) ϕ: Nn \{0} 7→ R defined by ϕ(A) = − log tr Ar, where r ∈ [0, 1]. △ xxxiii) ϕ: Pn 7→ R defined by ϕ(A) = log tr A−1. △ xxxiv) ϕ: Pn × Pn 7→ (0, ∞) defined by ϕ(A, B) = tr[A(log A − log B)]. △ xxxv) ϕ: Pn × Pn → [0, ∞) defined by ϕ(A, B) = −e[1/(2n)]tr(log A+log B). △ xxxvi) ϕ: Hn 7→ R defined by ϕ(A) = log tr eA . △ xxxvii) ϕ: Nn 7→ (−∞, 0] defined by ϕ(A) = −(det A)1/n. △ xxxviii) ϕ: Pn 7→ (0, ∞) defined by ϕ(A) = log det BA−1B∗, where B ∈ Fm×n and rank B = m. △ xxxix) ϕ: Pn 7→ R defined by ϕ(A) = −log det A. △ xl) ϕ: Pn 7→ (0, ∞) defined by ϕ(A) = det A−1. △ xli) ϕ: Pn 7→ R defined by ϕ(A) = log(det Ak /det A), where k ∈ {1, . . . , n − 1} and Ak is the leading k × k principal submatrix of A. △ xlii) ϕ: Pn 7→ R defined by ϕ(A) = − det A/det A(S) , where S ⊆ {1, . . . , n}. △

Nn ×Nm 7→ −Nnm defined by ϕ(A, B) = −Ar1 ⊗Br2 , where r1, r2 ∈ [0, 1] satisfy r1 +r2 ≤ 1. △ Pn × Nm 7→ Nnm defined by ϕ(A, B) = A−r ⊗ B1+r, where r ∈ [0, 1]. △ Nn × Nn 7→ −Nn defined by ϕ(A, B) = −Ar1 ⊙ Br2, where r1, r2 ∈ [0, 1] satisfy r1 + r2 ≤ 1. △ ∑ Hn 7→ R defined by ϕ(A) = ki=1 λi (A), where k ∈ {1, . . . , n}. ∑ △ xlvii) ϕ: Hn 7→ R defined by ϕ(A) = − ni=k λi (A), where k ∈ {1, . . . , n}. Proof. i) and iii) are given in [88] and [449, p. 123]. Let α ∈ [0, 1] for the remainder of the proof. To prove ii), let A1 , A2 ∈ Hn. Since xliii) xliv) xlv) xlvi)

ϕ: ϕ: ϕ: ϕ:

α(1 − α) = (α − α2 )1/2 [(1 − α) − (1 − α)2 ]1/2, it follows that 0 ≤ [(α − α2 )1/2 A1 − [(1 − α) − (1 − α)2 ]1/2A2 ]2 = (α − α2 )A21 + [(1 − α) − (1 − α)2 ]A22 − α(1 − α)(A1 A2 + A2 A1 ). Hence,

[αA1 + (1 − α)A2 ]2 ≤ αA21 + (1 − α)A22 ,

which shows that ϕ(A) = A2 is convex.

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To prove iv), let A1 , A2 ∈ Pn. Then, [ −1 ] [ −1 A I A α 1 + (1 − α) 2 I A1 I

[ −1 ] and AI2 AI are positive semidefinite, and thus 2 ] [ −1 ] I αA1 + (1 − α)A−1 I 2 = A2 I αA1 + (1 − α)A2

[ A−1

I 1 I A1

]

is positive semidefinite. It now follows from Proposition 10.2.5 that [αA1 + (1 − α)A2 ]−1 ≤ αA−1 1 + −1 (1 − α)A−1 2 , which shows that ϕ(A) = A is convex. △

△

To prove v), note that ϕ(A) = A−1/2 = ϕ2 [ϕ1 (A)], where ϕ1 (A) = A1/2 and ϕ2 (B) = B−1. It follows from vii) that ϕ1 is concave, while it follows from iv) that ϕ2 is convex. Furthermore, x) of Proposition 10.6.13 implies that ϕ2 is nonincreasing. It thus follows from ii) of Lemma 10.6.16 that ϕ(A) = A−1/2 is convex. △ To prove vi), let A ∈ Pn, and note that ϕ(A) = −Ar = ϕ2[ϕ1 (A)], where ϕ1 (A) = A−r and △ −1 ϕ2 (B) = −B . It follows from iii) that ϕ1 is convex, while it follows from iv) that ϕ2 is concave. Furthermore, x) of Proposition 10.6.13 implies that ϕ2 is nondecreasing. It thus follows from iv) of Lemma 10.6.16 that ϕ(A) = Ar is convex on Pn. Continuity implies that ϕ(A) = Ar is convex on Nn. To prove vii), let A1 , A2 ∈ Nn. Then, 2 0 ≤ α(1 − α)(A11/2 − A1/2 2 ) ,

which is equivalent to

1/2 2 [αA1/2 1 + (1 − α)A2 ] ≤ αA1 + (1 − α)A2 .

Using viii) of Proposition 10.6.13 yields 1/2 1/2 αA1/2 1 + (1 − α)A2 ≤ [αA1 + (1 − α)A2 ] .

Finally, multiplying by −1 shows that ϕ(A) = −A1/2 is convex. The proof of viii) is immediate. ix), x), and xi) follow from i), iii), and vi), respectively. To prove xii), note that ϕ(A) = −(BA−rB∗ )−p = ϕ2 [ϕ1 (A)], where ϕ1 (A) = −BA−rB∗ and ϕ2 (C) = −p C . x) implies that ϕ1 is concave, while iii) implies that ϕ2 is convex. Furthermore, ix) of Proposition 10.6.13 implies that ϕ2 is nonincreasing. It thus follows from ii) of Lemma 10.6.16 that ϕ(A) = −(BA−rB∗ )−p is convex. To prove xiii), let A1 , A2 ∈ Fn×m, and let B ∈ Nm. Then, 0 ≤ α(1 − α)(A1 − A2 )B(A1 − A2 )∗ = αA1 BA1∗ + (1 − α)A2 BA∗2 − [αA1 + (1 − α)A2 ]B[αA1 + (1 − α)A2 ]∗. Thus,

[αA1 + (1 − α)A2 ]B[αA1 + (1 − α)A2 ]∗ ≤ αA1BA1∗ + (1 − α)A2 BA∗2 ,

which shows that ϕ(A) = ABA∗ is convex. n m×n ] xiv), [ let A1 , ]A2 ∈ P and B1 , B2 ∈ F . Then, it follows from Proposition 10.2.5 that [ To prove ∗ B1 A−1 1 B1 B1 B∗1 A1

and

 ∗  B1 A−1 1 B1 α B∗1

∗ B2 A−1 2 B2 B2 B∗2 A2

are positive semidefinite, and thus

   B A−1B∗ B1   + (1 − α) 2 2 2 A1 B∗2

  ∗ −1 ∗ B2   αB1A−1 1 B1 + (1 − α)B2 A2 B2  =  A2 αB∗1 + (1 − α)B∗2

 αB1 + (1 − α)B2   αA1 + (1 − α)A2

is positive semidefinite. It thus follows from Proposition 10.2.5 that ∗ −1 ∗ [αB1 + (1 − α)B2 ][αA1 + (1 − α)A2 ]−1 [αB1 + (1 − α)B2 ]∗ ≤ αB1 A−1 1 B1 + (1 − α)B2 A2 B2 ,

which shows that ϕ(A, B) = BA−1B∗ is convex. xv) is given in [1987]. xvi) follows from Fact 10.24.20.

POSITIVE-SEMIDEFINITE MATRICES

721

] ] [ [ △ △ A A To prove xvii), let A = A1112∗ A1222 ∈ Pn+m and B = BB1112 BB1222 ∈ Pn+m. Then, it follows from xiv) with A1 , B1 , A2 , B2 replaced by A22 , A12 , B22 , B12 , respectively, that ∗ −1 ∗ [αA12 + (1 − α)B12 ][αA22 + (1 − α)B22 ]−1[αA12 + (1 − α)B12 ]∗ ≤ αA12 A−1 22 A12 + (1 − α)B12 B22 B12 .

Hence, −[αA22 + (1 − α)B22 ]|[αA + (1 − α)B] = [αA12 + (1 − α)B12 ][αA22 + (1 − α)B22 ]−1 [αA12 + (1 − α)B12 ]∗ − [αA11 + (1 − α)B11 ] ∗ −1 ∗ ≤ α(A12 A−1 22 A12 − A11 ) + (1 − α)(B12 B22 B12 − B11 )

= α(−A22 |A) + (1 − α)(−B22 |B), △

which shows that ϕ(A) = −A22 |A is convex. By continuity, the result holds for A ∈ Nn+m. To prove xviii), note that ϕ(A) = (A22 |A)−1 = ϕ2 [ϕ1 (A)], where ϕ1 (A) = A22 |A and ϕ2 (B) = B−1. It follows from xv) that ϕ1 is concave, while it follows from iv) that ϕ2 is convex. Furthermore, x) of Proposition 10.6.13 implies that ϕ2 is nonincreasing. It thus follows from Lemma 10.6.16 that △ ϕ(A) = (A22 |A)−1 is convex. xix) is given in [523, p. 106]. xx) is given in by Theorem 9 of [1817]. To prove xxi), note that ϕ(A) = −(tr A−r )−p = ϕ2 [ϕ1 (A)], where ϕ1 (A) = tr A−r and ϕ2 (B) = −B−p. iii) implies that ϕ1 is convex and that ϕ2 is concave. Furthermore, ix) of Proposition 10.6.13 implies that ϕ2 is nondecreasing. It thus follows from iv) of Lemma 10.6.16 that ϕ(A) = −(tr A−r )−p is convex. xxii) and xxiii) are proved in [633]. xxiv)–xxviii) are given by Corollary 1.1, Theorem 1, Corollary 2.1, Theorem 2, and Theorem 8, respectively, of [633]. A proof of xxiv) in the case where p = 1 − r is given in [449, p. 273]. xxix) is proved in [449, p. 274] and [633]. xxx) is given in [454, p. 113]. xxxi) is given in [449, p. 123], [454, p. 113], and [1123]. To prove xxxii), note that ϕ(A) = − log tr Ar = ϕ2 [ϕ1 (A)], where ϕ1 (A) = tr Ar and ϕ2 (x) = − log x. vi) implies that ϕ1 is concave. Furthermore, ϕ2 is convex and nonincreasing. It thus follows from ii) of Lemma 10.6.16 that ϕ(A) = − log tr Ar is convex. xxxiii) is given in [2099]. xxxiv) is given in [449, p. 275]. xxxv) is given in [102]. xxxvi) is given in [1381, p. 144]. To prove xxxvii), let A1 , A2 ∈ Nn. From Corollary 10.4.15 it follows that (det A1 )1/n +(det A2 )1/n ≤ [det(A1 + A2 )]1/n. Replacing A1 and A2 by αA1 and (1 − α)A2 , respectively, and multiplying by −1 shows that ϕ(A) = −(det A)1/n is convex. xxxviii) is proved in [2099]. xxxix) is a special case of xxxvii), which is due to K. Fan. See [788] and [789, p. 679]. To prove xxxviii), note that ϕ(A) = −nlog[(det A)1/n ] = ϕ2[ϕ1 (A)], where ϕ1 (A) = (det A)1/n and ϕ2 (x) = −nlog x. It follows from xix) that ϕ1 is concave. Since ϕ2 is nonincreasing and convex, it follows from ii) of Lemma 10.6.16 that ϕ(A) = − log det A is convex. To prove xl), note that ϕ(A) = det A−1 = ϕ2 [ϕ1 (A)], where ϕ1 (A) = log det A−1 and ϕ2 (x) = e x. It follows from xx) that ϕ1 is convex. Since ϕ2 is nondecreasing and convex, it follows from i) of Lemma 10.6.16 that ϕ(A) = det A−1 is convex. xli) and xlii) are given in [788] and [789, pp. 684, 685]. Next, xliii) is given in [449, p. 273], [454, p. 114], and [2977, p. 9]. xliv) is given in [454, p. 114]. xlv) is given in [2977, p. 9]. Finally, xlvi) is given in [1969, p. 478] and [1971, p. 688]. xlvii) follows from xlvi).  The following result follows from xvii) of Proposition 10.6.17 by setting α = 1/2. Versions of this result appear in [639, 1355, 1800, 1869] and [2263, p. 152]. [ ] [ ] △ A A12 △ B B ∗ Corollary 10.6.18. Let A = A11 ∈ Fn+m and B = B1112∗ B1222 ∈ Fn+m, and assume that A and B 12 A22

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are positive semidefinite. Then, A11 |A + B11|B ≤ (A11 + B11 )|(A + B).

(10.6.12)

The following result, which follows from xlv) and xlvi) of Proposition 10.6.17, yields inequalities for the eigenvalues of a pair of Hermitian matrices. These are the Lidskii-Wielandt inequalities. Corollary 10.6.19. Let A, B ∈ Hn. Then, for all k ∈ {1, . . . , n} and 1 ≤ i1 < · · · < ik ≤ n, k ∑ j=1

λi j (A) +

n ∑

λi j (B) ≤

j=n−k+1

k ∑

λi (A + B) ≤

i=1

k ∑ [λi (A) + λi (B)]

(10.6.13)

i=1

with equality in both inequalities for k = n. Furthermore, for all k ∈ {1, . . . , n}, n n ∑ ∑ λi (A + B) [λi (A) + λi (B)] ≤ i=k

(10.6.14)

i=k

with equality for k = 1. Proof. See [449, p. 71], [1450, p. 201], [1969, p. 688], [1971, p. 478], [2403, p. 116], and [2991, p. 356]. 

10.7 Facts on Range and Rank Fact 10.7.1. Let A, B ∈ Fn×n, and assume that A and B are positive semidefinite. Then, there

exists α > 0 such that A ≤ αB if and only if R(A) ⊆ R(B). If these conditions hold, then rank A ≤ rank B. Source: Theorem 10.6.2. Fact 10.7.2. Let A ∈ Nn . Then, R(A) = R(A1/2 ) = R(A2 ). Fact 10.7.3. Let A, B ∈ Fn×n, assume that A is positive semidefinite, and assume that B is either positive semidefinite or skew Hermitian. Then, the following statements hold: i) N(A + B) = N(A) ∩ N(B). ii) R(A + B) = R(A) + R(B) = R([A B]) = span[R(A) ∪ R(B)]. [ ] iii) rank(A + B) = dim[R(A) + R(B)] = rank [A B] = rank AB . Source: In i), “⊇” is given by Fact 3.13.15. To prove “⊆,” let x ∈ N(A + B), which implies that Ax + Bx = 0, and thus x∗Ax + x∗Bx = 0. In the case where B is positive semidefinite, it follows that x∗Ax = x∗Bx = 0, and thus Ax = Bx = 0. In the case where B is skew Hermitian, it follows that x∗Ax = −x∗Bx = 0, and thus Ax = 0 and therefore Bx = 0. To prove ii) in the case where B is positive semidefinite, it follows from i) and Fact 3.12.17 that R(A + B) = N(A + B)⊥ = [N(A) ∩ N(B)]⊥ = N(A)⊥ + N(B)⊥ = R(A) + R(B). In the case where B is skew Hermitian, R(A + B) = N(A − B)⊥ = [N(A) ∩ N(B)]⊥ = N(A)⊥ + N(B)⊥ = R(A) + R(B). In both cases, ( [ ]) R ([A B]) = R [A B] BA∗ = R(A2 + BB∗ ) = R(A2 ) + R(BB∗ ) = R(A) + R(B) = R(A + B). Related: Fact 3.12.16, Fact 4.10.31, and Fact 4.18.11. Fact 10.7.4. Let A, B ∈ Fn×n, and assume that A and B are positive semidefinite. Then, (A +

B)(A + B)+ is the projector onto the subspace R(A + B) = R(A) + R(B) = R([A B]) = span[R(A) ∪ R(B)]. Source: Fact 10.7.3. Related: Fact 8.8.20. Fact 10.7.5. Let A ∈ Fn×n, and assume that A + A∗ ≥ 0. Then, the following statements hold: i) N(A) = N(A + A∗ ) ∩ N(A − A∗ ). ii) R(A) = R(A + A∗ ) + R(A − A∗ ). iii) rank A = rank [A + A∗ A − A∗ ]. Source: Fact 10.7.3.

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POSITIVE-SEMIDEFINITE MATRICES

Fact 10.7.6. Let A1 , . . . , Ar ∈ Fn×n, and assume that A1 , . . . , Ar are positive semidefinite. Then,

 r  r ∑  ∑   R  Ai  = R(Ai ) = R([A1 · · · Ar ]). i=1

i=1

Source: Theorem 3.5.3, Fact 3.14.4, and Fact 10.7.2. Fact 10.7.7. Let A, B ∈ Fn×n, and assume that A and B are positive semidefinite. Then,

[

rank

A 0

] [ B A = rank A 0

] A+B = rank A + rank(A + B). A

Source: Use Theorem 10.3.5 to simultaneously diagonalize A and B. Fact 10.7.8. Let A, B ∈ Fn×n, assume that A and B are positive semidefinite, and let S ⊆

{1, . . . , n}. Then,

rank (AB)(S) ≤ rank (Ak )(S) = rank (A(S) )k = rank A(S) .

Source: [2991, p. 226]. Fact 10.7.9. Let A ∈ Fn×n, and assume that A is either positive semidefinite or an irreducible,

singular M-matrix. Then, the following statements hold: i) If α ⊂ {1, . . . , n}, then rank A ≤ rank A(α) + rank A(α∼ ) . ii) If α, β ⊆ {1, . . . , n}, then rank A(α∪β) ≤ rank A(α) + rank A(β) − rank A(α∩β) . iii) If 1 ≤ k ≤ n − 1, then ∑ ∑ k rank A(α) ≤ (n − k) rank A(α) . {α: card(α)=k+1}

{α: card(α)=k}

If, in addition, A is either positive definite, a nonsingular M-matrix, or totally positive, then all three inequalities hold as equalities. Source: [1896]. Related: Fact 10.16.39. Fact 10.7.10. Let A ∈ Fn×n and B ∈ Fn×m , assume that A is Hermitian, rank A = rank B = m, and R(B) = R(A). Then, B∗AB is nonsingular, and B = AB(B∗AB)−1 B∗B. Source: [278].

10.8 Facts on Unitary Matrices and the Polar Decomposition Fact 10.8.1. Let A ∈ Fn×m. Then, A⟨A⟩ = ⟨A∗ ⟩A. Fact 10.8.2. Let A ∈ Fn×m. Then, the following statements hold:

i) If rank A = m, then A⟨A⟩−1 is left inner. ii) If rank A = n, then ⟨A∗ ⟩−1A is right inner. iii) If n = m and A is nonsingular, then A⟨A⟩−1 and ⟨A∗ ⟩−1A are unitary. Fact 10.8.3. Let A ∈ Fn×m, where m ≤ n. Then, there exist M ∈ Fm×m and S ∈ Fn×m such that M is positive semidefinite, S satisfies S ∗S = Im , and A = SM. Furthermore, M is given uniquely by M = ⟨A⟩. If, in addition, rank A = m, then M is positive definite and S is given uniquely by ∫ ∞ 2 S = A⟨A⟩−1 = A (t2I + A∗A)−1 dt. π 0 Source: [1391, Chapter 8]. Fact 10.8.4. Let A ∈ Fn×m, where n ≤ m. Then, there exist M ∈ Fn×n and S ∈ Fn×m such that

M is positive semidefinite, S satisfies SS ∗ = In , and A = MS. Furthermore, M is given uniquely by M = ⟨A∗ ⟩. If, in addition, rank A = n, then S is given uniquely by ∫ ∞ 2 S = ⟨A∗ ⟩−1A = A∗ (t2I + AA∗ )−1 dt. π 0

Source: [1391, Chapter 8].

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CHAPTER 10

Fact 10.8.5. Let A ∈ Fn×n, where A is nonsingular. Then, there exist unique matrices M, S ∈

such that A = SM, M is positive definite, and S is unitary. In particular, M = ⟨A⟩ and S = A⟨A⟩−1 . Fact 10.8.6. Let A ∈ Fn×n, where A is nonsingular. Then, there exist unique matrices M, S ∈ n×n F such that A = MS, M is positive definite, and S is unitary. In particular, M = ⟨A∗ ⟩ and S = ⟨A∗ ⟩−1A. Related: Fact 10.8.1 and Fact 10.8.5. Fact 10.8.7. Let M1, M2 ∈ Fn×n, and assume that M1, M2 are positive definite. Furthermore, let S 1 , S 2 ∈ Fn×n, assume that S 1 , S 2 are unitary, and assume that M1 S 1 = S 2 M2 . Then, S 1 = S 2 . Source: Let A = M1 S 1 = S 2 M2 . Then, S 1 = (S 2 M22 S 2∗ )−1/2S 2 M2 = S 2 . Fact 10.8.8. Let A ∈ Fn×n. Then, there exist a unitary matrix S ∈ Fn×n and unique matrices M1, M2 ∈ Nn such that A = M1 S = SM2 . In particular, M1 = ⟨A∗ ⟩ and M2 = ⟨A⟩. Remark: If A is singular, then S is not uniquely determined. Fact 10.8.9. Let A, M, S ∈ Fn×n, and assume that M is positive semidefinite, S is unitary, and A = MS. Then, the following statements are equivalent: i) A is normal. ii) MS = SM. iii) AS = SA. iv) AM = MA. Source: [1448, p. 414] and [2991, p. 295]. Fact 10.8.10. Let A, B ∈ Fn×n, assume that A and B are unitary, and assume that A + B is nonsingular. Then, the unitary factor in the polar decomposition of A + B is A(A∗B)1/2. Source: [1391, p. 216] and [2064]. Remark: (A∗B)1/2 is the principal square root. Fact 10.8.11. Let A ∈ Fn×n be semicontractive, and define B ∈ F2n×2n by    A (I − AA∗ )1/2  △  B =   . (I − A∗A)1/2 −A∗ F

n×n

Then, B is unitary. Source: [1084, p. 180] and [2991, p. 191]. Fact 10.8.12. Let A ∈ Fn×m, and define B ∈ F(n+m)×(n+m) by    (I + A∗A)−1/2 −A∗ (I + AA∗ )−1/2  △   . B =  (I + AA∗ )−1/2A (I + AA∗ )−1/2 △

˜ I,˜ where I˜ = diag(Im , −In ). Source: [1324]. Then, B is unitary, det B = 1, and B∗ = IB n×m Fact 10.8.13. Let A ∈ F , assume that A is contractive, and define B ∈ F(n+m)×(n+m) by    (I − A∗A)−1/2 A∗ (I − AA∗ )−1/2  △  B =   . (I − AA∗ )−1/2A (I − AA∗ )−1/2 △ ˜ = I,˜ where I˜ = Then, B is Hermitian, det B = 1, and B∗IB diag(Im , −In ). Source: [1324].

10.9 Facts on Structured Positive-Semidefinite Matrices Fact 10.9.1. Let z ∈ C, assume that z is not real, and define the Hermitian, Toeplitz matrix

A ∈ Cn×n by

  1 z  z¯ 1   △  A =  z¯ z¯  . .  .. ..  z¯ z¯

z ··· z ··· 1 ··· .. . . . . z¯ · · ·

 z   z   z  . ..  .  1

725

POSITIVE-SEMIDEFINITE MATRICES

Then, the following statements hold: i) A is singular if and only if

(

1−z 1 − z¯

)n

z = . z¯

ii) Assume that Im z > 0. Then, A is positive semidefinite if and only if arg z (n − 1)π + ≤ arg(z − 1). n n Source: [1789]. Example: Let n = 2 and z = αȷ, where α > 0. Then, A is positive semidefinite if and only if 3π/4 < π/2 + atan 1/α; that is, if and only if α < 1. Related: Fact 6.10.21. Fact 10.9.2. Let ϕ: R 7→ C, and assume that, for all x1 , . . . , xn ∈ R, the matrix A ∈ Cn×n, △ where A(i, j) = ϕ(xi − x j ), is positive semidefinite. (The function ϕ is positive semidefinite.) Then, the following statements hold: i) For all x1 , x2 ∈ R, it follows that |ϕ(x1 ) − ϕ(x2 )|2 ≤ 2ϕ(0)Re[ϕ(0) − ϕ(x1 − x2 )]. △ ii) The function ψ : R 7→ C, where, for all x ∈ R, ψ(x) = ϕ(x), is positive semidefinite. △ iii) For all α ∈ R, the function ψ : R 7→ C, where, for all x ∈ R, ψ(x) = ϕ(αx), is positive semidefinite. △ iv) The function ψ : R 7→ C, where, for all x ∈ R, ψ(x) = |ϕ(x)|2 , is positive semidefinite. △ v) The function ψ : R 7→ C, where, for all x ∈ R, ψ(x) = Re ϕ(x), is positive semidefinite. vi) If ϕ1 : R 7→ C and ϕ2 : R 7→ C are positive semidefinite, then ϕ3 : R 7→ C, where, for all △ x ∈ R, ϕ3 (x) = ϕ1 (x)ϕ2 (x), is positive semidefinite. vii) If ϕ1 : R 7→ C and ϕ2 : R 7→ C are positive semidefinite and α1 , α2 are positive numbers, △ then ϕ3 : R 7→ C, where, for all x ∈ R, ϕ3 (x) = α1 ϕ1 (x) + α2 ϕ2 (x), is positive semidefinite. viii) Let ϕ : R 7→ C, and assume that ϕ is bounded and continuous. Furthermore, for all x, y ∈ R, △ define K: R × R 7→ C by K(x, y) = ϕ(x − y). Then, ϕ is positive semidefinite if and only if, for every continuous integrable function f : R 7→ C, it follows that ∫ K(x, y) f (x) f (y) dx dy ≥ 0. R2

Source: [454, pp. 141–144]. Remark: K is a kernel function associated with a reproducing kernel

space. See [1160] for extensions to vector arguments, and [2400] and Fact 10.9.7 for applications. n×n Fact 10.9.3. Let A ∈ Cn×n √ , assume that A is positive definite, and let B ∈ C , where, for all △ i, j ∈ {1, . . . , n}, B(i, j) = A(i, j) / A(i,i) A( j, j) . Then, B is positive definite. Source: [2991, p. 204]. △

Fact 10.9.4. Let a1 < · · · < an be positive numbers, and define A ∈ Rn×n by A(i, j) = min {ai , a j }.

Then, A is positive definite, det A =

n ∏

(ai − ai−1 ),

i=1

and, for all x ∈ Rn, xTA−1x =

n ∑ (x(i) − x(i−1) )2 i=1

△

△

ai − ai−1

,

where a0 = 0 and x(0) = 0. Remark: A is a covariance matrix arising in Brownian motion. See [1379, p. 132] and [2911, p. 50]. Related: Fact 10.9.5. Fact 10.9.5. Let a1 , . . . , an > 0, and define A ∈ Rn×n by the following expressions: △ i) A(i, j) = min {ai , a j }.

726

CHAPTER 10 △

ii) A(i, j) =

1 . max {ai ,a j }

△ ai

iii) A(i, j) = △

iv) A(i, j) = △

v) A(i, j) = vi) A(i, j)

aj ,

where a1 ≤ · · · ≤ an .

1 ai +a j .

√

ai a j . √ = 1/ ai a j . △

△

vii) A(i, j) =

ai a j ai +a j . p

p

△ ai −a j

viii) A(i, j) =

ai −a j p

ix) x)

, where p ∈ [0, 1].

p

△ a +a A(i, j) = aii +a jj , where △ log a −log a A(i, j) = aii −a j j .

p ∈ [−1, 1].

Then, A is positive semidefinite. If, in addition, α > 0, then A⊙α is positive semidefinite. Source: [451, 452], [454, pp. 153, 178, 189], [455], and [922, p. 90]. Remark: A in iii) is the Schur product of the matrices defined in i) and ii). Related: Fact 4.23.4, Fact 10.9.4, and Fact 10.9.6. Fact 10.9.6. Let A ∈ Rn×n , and assume that, for all i, j ∈ {1, . . . , n}, A(i, j) = min {i, j}; that is,   1 1   1 1 1 · · ·    1 2 2 · · · 2 2    ..   . 3 3   1 2 3 △   A =  . . . ..  . ..  .. .. .. ...  . .    1 2 3 . . . n − 1 n − 1      1 2 3 ··· n − 1 n Then, det A = 1, A is positive definite,   2 −1 0 · · ·   −1 2 −1 · · ·  ..  .  0 −1 2 −1 A =  . . . . .. .. ..  ..  ..  0 . 0 0   0 0 0 ··· and

spec(A) = { 41 csc2

π 1 2(2n+1) , 4

csc2

 0   0 0    0 0   ..  , .. . .   2 −1   −1 1 0

3π 1 2(2n+1) , . . . , 4

csc2

(2n−1)π 2(2n+1) }.

Source: Six proofs that A is positive definite are given in [455]. Remark: tr A and tr A−1 yield equalities in Fact 2.16.10. Evaluating det A yields an equality in Fact 2.16.15. Related: Fact

10.9.5. Fact 10.9.7. Let a1 , . . . , an ∈ R, and define A ∈ Cn×n by either of the following expressions:

i) A(i, j) =

△

1 1+ ȷ(ai −a j ) .

ii) A(i, j) =

△

1 1− ȷ(ai −a j ) .

△

1 . 1+(ai −a j )2

iii) A(i, j) =

727

POSITIVE-SEMIDEFINITE MATRICES △

iv) A(i, j) =

1 1+|ai −a j | .

△

v) A(i, j) = e ȷ(ai −a j ) . △

vi) A(i, j) = cos(ai − a j ). △ sin[(ai −a j )]

ai −a j . ai −a j A(i, j) = sinh[(a . i −a j )] △ sinh p(ai −a j ) A(i, j) = sinh(ai −a j ) , where p ∈ (0, 1). △ tanh[(a −a )] A(i, j) = ai −ai j j . sinh[(ai −a j )] △ , where p ∈ A(i, j) = (ai −a j )[cosh(a i −a j )+p]

vii) A(i, j) = viii) ix) x) xi)

△

(−1, 1].

△

xii) A(i, j) = xiii)

1 cosh(ai −a j )+p , where p ∈ (−1, 1]. △ cosh p(ai −a j ) A(i, j) = cosh(ai −a , where p ∈ [−1, 1]. j) △

2

△

p

xiv) A(i, j) = e−(ai −a j ) . xv) A(i, j) = e−|ai −a j | , where p ∈ [0, 2]. △ 1+p(ai −a j )2

xvi) A(i, j) =

1+q(ai −a j )2

, where 0 ≤ p ≤ q.

△

xvii) A(i, j) = tr eB+ ȷ(ai −a j )C, where B, C ∈ Hn and BC = CB. △

xviii) A(i, j) = 1/ det(Ai + A j ), where A1 , . . . , An ∈ Pn . Then, A is positive semidefinite. Finally, if, α is a nonnegative number and A is defined by either iv), ix), x), xi), xiii), or xvi), then A⊙α is positive semidefinite. Source: [454, pp. 141–144, 153, 177, 188], [477], [922, p. 90], and [1448, pp. 400, 401, 456, 457, 462, 463]. viii) is given in [2512]. Remark: xv) is related to the Bessis-Moussa-Villani conjecture in Fact 10.14.45 and Fact 10.14.46. Related: Fact 10.9.2. Problem: In each case, determine rank A. Fact 10.9.8. Define A ∈ Rn×n by either of the following expressions: ( ) △ i) A(i, j) = i+i j . △

ii) A(i, j) = (i + j)!. △

iii) A(i, j) = min {i, j}. △

iv) A(i, j) = gcd {i, j}. △

v) A(i, j) = ij . △

vi) A(i, j) =

1 i+ j−1 .

Then, A is positive semidefinite. If, in addition, α ≥ 0, then A⊙α is positive semidefinite. Remark: Fact 10.25.2 implies that A⊙α is positive semidefinite for all α ∈ [0, n − 2]. Remark: i) is the Pascal matrix. See [9, 451, 973] and xix) of Fact 1.16.13. Remark: The determinant of iv) can be expressed in terms of the Euler totient function. See [155, 552] and Fact 1.20.4. Remark: v) is a special case of iii) of Fact 10.9.5. This is the Lehmer matrix. Remark: vi) is the Hilbert matrix, which is positive definite. See Fact 4.23.4. Related: Fact 3.16.28. Fact 10.9.9. Let a1 , . . . , an ≥ 0, let p ∈ R, assume that either a1 , . . . , an > 0 or p > 0, and, for △ all i, j ∈ {1, . . . , n}, define A ∈ Rn×n by A(i, j) = (ai a j ) p. Then, A is positive semidefinite. Source: △ △ Let a = [a1 · · · an ]T and A = a⊙pa⊙pT. Fact 10.9.10. Let a ∈ Cn , assume that, for all i ∈ {1, . . . , n}, a(i) + a( j) , 0, and, for all

728

CHAPTER 10

i, j ∈ {1, . . . , n}, define A ∈ Cn×n by △

A(i, j) =

1 . a(i) + a( j)

Then, A is positive definite if and only if a >> 0 and the components of a are distinct. Furthermore, A is totally positive if and only if A is positive definite and either a = a↓ or a = a↑ . Remark: See [1042]. Remark: A is a Cauchy matrix. See Fact 10.9.20. Related: Fact 15.19.23. Fact 10.9.11. Let a1 , . . . , an > 0, let α > 0, and, for all i, j ∈ {1, . . . , n}, define A ∈ Rn×n by 1 △ A(i, j) = . (ai + a j )α Then, A is positive semidefinite. Source: [451], [454, pp. 24, 25], and [2251]. Remark: For α = 1, A is a Cauchy matrix. See Fact 4.27.5. Related: Fact 7.12.15 and Fact 10.9.10. Fact 10.9.12. Let a1 , . . . , an > 0, let r ∈ [−1, 1], and, for all i, j ∈ {1, . . . , n}, define A ∈ Rn×n by △

A(i, j) =

ari + arj ai + a j

.

Then, A is positive semidefinite. Source: [2977, p. 74]. Fact 10.9.13. Let a1 , . . . , an > 0, let q > 0, let p ∈ [−q, q], and, for all i, j ∈ {1, . . . , n}, define A ∈ Rn×n by aip + a pj △ . A(i, j) = q ai + aqj Then, A is positive semidefinite. Source: Let r = p/q and bi = aqi . Then, A(i, j) = (bri + brj )/(bi + b j ). Now, use Fact 10.9.12. See [1988] for the case where q ≥ p ≥ 0. Remark: The case where q = 1 and p = 0 yields a Cauchy matrix. In the case where n = 2, A ≥ 0 yields Fact 2.2.31. Problem: Under what conditions is A positive definite? Fact 10.9.14. Let a1 , . . . , an > 0, let p ∈ (−1, ∞), and define A ∈ Rn×n by △

A(i, j) =

a3i

+

p(a2i a j

1 . + ai a2j ) + a3j

Then, A is positive semidefinite. Source: [459]. Fact 10.9.15. Let a1 , . . . , an > 0, p ∈ [−1, 1], q ∈ (−2, 2], and, for all i, j ∈ {1, . . . , n}, define A ∈ Rn×n by aip + a pj △ . A(i, j) = 2 ai + qai a j + a2j Then, A is positive semidefinite. Source: [459, 2974] and [2977, p. 76]. Fact 10.9.16. Let A ∈ Fn×n, and√assume that A is Hermitian, for all i ∈ {1, . . . , n}, A(i,i) > 0, and, 1 for all i, j ∈ {1, . . . , n}, |A(i, j) | < n−1 A(i,i) A( j, j) . Then, A is positive definite. Source: Note that  ]∗  1 n−1 ∑ n [ ∑ A(i, j)  [ x ] x(i)  n−1 A(i,i) ∗  (i) .  x Ax =  x( j) x( j)  A 1 (i, j) i=1 j=i+1 n−1 A( j, j) Credit: A. Roup. Fact 10.9.17. Let A ∈ Fn×n. Then, for all i, j ∈ {1, . . . , n}, |A(i, j) |2 < ⟨A∗ ⟩(i,i) ⟨A⟩( j, j) . Now, let

B ∈ Fn×n, and assume that B is semicontractive. Then, n ∏ |det(A ⊙ B)|2 ≤ ⟨A∗ ⟩(i,i) ⟨A⟩(i,i) . i=1

729

POSITIVE-SEMIDEFINITE MATRICES

Source: [1460]. Fact 10.9.18. Let A ∈ Rn×n, assume that A is positive semidefinite, assume that A(i,i) > 0 for all

i ∈ {1, . . . , n}, and define B ∈ Rn×n by

△

B(i, j) =

A(i, j) , µα (A(i,i) , A( j, j) )

where, for positive scalars α, x, y, △

µα (x, y) =

[

1 α 2 (x

+ yα )

]1/α .

Then, B is positive semidefinite. If, in addition, A is positive definite, then B is positive definite. In particular, letting α ↓ 0, α = 1, and α → ∞, the respective matrices C,D, E ∈ Rn×n defined by △

C(i, j) = √

A(i, j) A(i,i) A( j, j)

,

△

D(i, j) =

2A(i, j) , A(i,i) + A( j, j)

△

E(i, j) =

A(i, j) max {A(i,i) , A( j, j) }

are positive semidefinite. Finally, if A is positive definite, then C, D, and E are positive definite. Source: [2350]. Remark: The assumption that the diagonal entries of A are positive can be weakened. See [2350]. Related: Fact 2.2.58. Fact 10.9.19. Let α, β, γ ∈ [0, π], and define A ∈ R3×3 by

   1 cos α cos γ    1 cos β  . A =  cos α   cos γ cos β 1

Then, det A = 4[sin 12 (α + β + γ)][sin 21 (−α + β + γ)][sin 12 (α − β + γ)] sin 21 (α + β − γ). Furthermore, A is positive semidefinite if and only if α ≤ β+γ, β ≤ α+γ, γ ≤ α+β, and α+β+γ ≤ 2π. Finally, A is positive definite if and only if all of these inequalities are strict. Source: [305, 955]. Fact 10.9.20. Let λ1 , . . . , λn ∈ C, assume that, for all i ∈ {1, . . . , n}, Re λi < 0, and, for all i, j ∈ {1, . . . , n}, define A ∈ Cn×n by −1 △ A(i, j) = . λi + λ j Then, A is positive definite. Source: Note that A = 2B⊙(1n×n − C)⊙−1, where B(i, j) = (λi +1)(λ j +1) . (λi −1)(λ j −1)

1 (λi −1)(λ j −1)

and

Then, B is positive semidefinite and (1n×n − C)⊙−1 = 1n×n + C + C ⊙2 + C ⊙3 + · · · . ) Remark: A is the solution of a Lyapunov equation. See Fact 16.22.18 and Fact 16.22.19. Remark: A is a Cauchy matrix. See Fact 4.23.4, Fact 4.27.5, Fact 4.27.6, and Fact 10.9.20. Remark: A is the Gram matrix whose entries are inner products of the functions fi (t) = e−λi t. See [454, p. 3]. Fact 10.9.21. Let λ1 , . . . , λn ∈ OIUD and w1, . . . , wn ∈ C. Then, there exists an analytic function ϕ: OIUD 7→ OIUD such that ϕ(λi ) = wi for all i ∈ {1, . . . , n} if and only if A ∈ Cn×n is positive semidefinite, where, for all i, j ∈ {1, . . . , n},

C(i, j) =

△

A(i, j) = Source: [1996]. Remark: A is a Pick matrix.

1 − wi w j 1 − λi λ j

.

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CHAPTER 10

Fact 10.9.22. Let α0 , . . . , αn > 0, and define the tridiagonal matrix A ∈ Rn×n by

  α0 + α1  −α1  △  A =  0  ..  .  0

−α1 α1 + α2 −α2 .. .

0 −α2 α2 + α3 .. .

0 0 −α3 .. .

0

0

0

Then, for all k = 2, . . . , n, det A({1,...,k})

··· ··· ··· . · .· · ···

0 0 0 .. .

αn−1 + αn

      .  

  k  ∑ −1  =  αi  α0 α1 · · · αk . i=0

Furthermore, A is positive definite. Source: [302, p. 115]. Remark: A is a stiffness matrix arising in structural analysis. Related: Fact 4.24.4. Fact 10.9.23. Let A ∈ Rn×n, where n ≥ 1, be the tridiagonal, Toeplitz matrix    b a 0 · · · 0 0     a b a · · · 0 0     0 a b . . . 0 0   △  . A =  . . . . . . . . . . . ...   .. ..     0 0 0 . . . b a      0 0 0 ··· a b { } iπ spec(A) = b + 2|a| cos : i ∈ {1, . . . , n} . n+1 nπ Furthermore, A is positive semidefinite if and only if b + 2|a| cos n+1 ≥ 0. Finally, A is positive nπ definite if and only if b + 2|a| cos n+1 > 0. Related: Fact 7.12.46. Then,

10.10 Facts on Equalities and Inequalities for One Matrix Fact 10.10.1. Let A ∈ Hn . Then, the following statements are equivalent:

i) Neither A nor −A is positive semidefinite. ii) At least one of the following statements holds: a) There exists an even integer m ≤ n and a principal submatrix B ∈ Fm×m such that det B < 0. b) There exist odd integers k, l ≤ n and principal submatrices C ∈ Fk×k and D ∈ Fl×l such that (det C) det D < 0. Source: [2991, p. 206]. Fact 10.10.2. Let A ∈ Fn×n, and assume that A is positive semidefinite. Then, A⊙2 , A⊙3 , and ⊙2 |A| are positive semidefinite. If, in addition, n ≤ 3, then, |A| is positive semidefinite. Source: [1953]. Remark: If n ≥ 4, then this result does not hold. Let   √1 0 − √13   1 3   1  √1 1 0  √3 3  . A =   √1 √1   0 1  3 3   1  √1 − √3 0 1 3 √ √ √ √ Then, mspec(A) = {1 − 6/3, 1 − 6/3, 1 + 6/3, 1 + 6/3}ms , whereas mspec(|A|) = {1, 1, 1 −

731

POSITIVE-SEMIDEFINITE MATRICES

√

√ 12/3, 1+ 12/3}ms . Related: Fact 11.12.7. Fact 10.10.3. Let A ∈ Fn×n, assume that A is positive semidefinite, and let S ⊆ {1, . . . , n}. Then, (A(S) )2 ≤ (A2 )(S) ,

(A1/2 )(S) ≤ (A(S) )1/2 .

Now, assume that A is positive definite. Then, (A(S) )−1 ≤ (A−1 )(S) ,

(A(S) )−1/2 ≤ (A−1/2 )(S) .

Source: [2991, p. 219]. Fact 10.10.4. Let x ∈ Fn. Then, xx∗ ≤ x∗xI. △ Fact 10.10.5. Let x ∈ Fn, assume that x is nonzero, and define A ∈ Fn×n by A = x∗ xI − xx∗.

Then, A is positive semidefinite, mspec(A) = {x∗ x, . . . , x∗ x, 0}ms , and rank A = n − 1. Fact 10.10.6. Let x, y ∈ Fn, assume that x and y are linearly independent, and define A ∈ Fn×n by △ A = (x∗ x + y∗ y)I − xx∗ − yy∗. Then, A is positive definite. Now, let F = R. Then, √ { mspec(A) = xTx + yTy, . . . , xTx + yTy, 21 (xTx + yTy) + 14 (xTx − yTy)2 + (xTy)2 , √ } T 1 T 1 T T 2 T 2 (x x + y y) − 2 4 (x x − y y) + (x y) △

ms

.

△

Source: To show that A is positive definite, write A = B+C, where B = x∗ xI−xx∗ and C = y∗ yI−yy∗.

Then, using Fact 10.10.5 it follows that N(B) = span {x} and N(C) = span {y}. Now, it follows from Fact 10.7.3 that N(A) = N(B) ∩ N(C) = {0}. Therefore, A is nonsingular and thus positive definite. The expression for mspec(A) follows from Fact 6.9.15. Fact 10.10.7. Let x1 , . . . , xn ∈ R3, assume that span {x1 , . . . , xn } = R3, and define A ∈ R3×3 by △

A=

n ∑ (xiT xi I − xi xiT ). i=1

Then, A is positive definite. Furthermore, λ1 (A) < λ2 (A) + λ3 (A),

d1 (A) < d2 (A) + d3 (A). ∑ 2 > 0. Now, let the Source: Suppose that d1 (A) = A(1,1) . Then, d2 (A) + d3 (A) − d1 (A) = 2 ni=1 xi(3) ∑n T T 3×3 T matrix S ∈ R be such that SAS = i=1 ( xˆi xˆi I − xˆi xˆi ) is diagonal, where, for all i ∈ {1, . . . , n}, △ xˆi = Sxi . Then, for all i ∈ {1, 2, 3}, di (A) = λi (A). Remark: A is the inertia matrix for a rigid body consisting of n discrete particles. For a homogeneous continuum body B whose density is ρ, the inertia matrix is given by ∫∫∫ I=ρ (rTrI − rrT ) dxdydz, △

[ x]

B

where r = yz . Remark: The eigenvalues and diagonal entries of A represent the lengths of the sides of triangles. See Fact 5.2.14 and [2196, p. 220]. Fact 10.10.8. Let A ∈ F2×2 , assume that A is positive semidefinite and nonzero, and define B ∈ F2×2 by √ √ △ B = (tr A + 2 det A)−1/2 (A + det AI). Then, B = A1/2. Source: [1304, pp. 84, 266, 267]. Related: Fact 3.15.34. Fact 10.10.9. Let A ∈ Fn×n, and assume that A is Hermitian. Then, rank A = ν− (A) + ν+ (A),

def A = ν0 (A).

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Fact 10.10.10. Let A ∈ Fn×n, and assume that A is positive semidefinite. Then, for all i ∈

{1, . . . , n}, A(i,i) ≥ 0, and, for all i, j ∈ {1, . . . , n}, |A(i, j) |2 ≤ A(i,i) A( j, j) . Fact 10.10.11. Let A ∈ Fn×n, assume that A is positive semidefinite, and assume that there exists i ∈ {1, . . . , n} such that A(i,i) = 0. Then, rowi (A) = 0 and coli (A) = 0. Fact 10.10.12. Let A ∈ Fn×n. Then, A ≥ 0 if and only if A ≥ −A. Fact 10.10.13. Let A ∈ Fn×n, and assume that A is Hermitian. Then, A2 ≥ 0. Fact 10.10.14. Let A ∈ Fn×n, and assume that A is Hermitian. Then, A ≤ 21 [(A2 )1/2 + A]. Fact 10.10.15. Let A ∈ Fn×n, and assume that A is skew Hermitian. Then, A2 ≤ 0. Fact 10.10.16. Let A ∈ Fn×n and α > 0. Then, A2 + A2∗ ≤ αAA∗ + α1 A∗A. Furthermore, equality holds if and only if αA = A∗. Fact 10.10.17. Let A ∈ Fn×n. Then, (A − A∗ )2 ≤ 0 ≤ (A + A∗ )2 ≤ 2(AA∗ + A∗A). Fact 10.10.18. Let A ∈ Fn×n and α > 0. Then,

A + A∗ ≤ αI + α−1AA∗. Equality holds if and only if A = αI. Fact 10.10.19. Let A ∈ Fn×n, and assume that A is positive definite. Then, 2I ≤ A + A−1. Equality holds if and only if A = I. Furthermore, 2n ≤ tr A + tr A−1. Fact 10.10.20. Let A ∈ Fn×n, and assume that A is positive definite. Then,

(11×n A−11n×1 )−11n×n ≤ A. Source: Set B = 1n×n in Fact 10.25.21. See [2985]. [ Fact 10.10.21. Let A ∈ Fn×n, and assume that A is positive definite. Then, AI

I A−1

]

is positive

semidefinite. Fact 10.10.22. Let A ∈ Fn×n, and assume that A is Hermitian. Then, A2 ≤ A if and only if

0 ≤ A ≤ I. Furthermore, A2 < A if and only if 0 < A < I. Fact 10.10.23. Let A ∈ Fn×n, and assume that A is Hermitian. Then, A + αI ≥ 0 if and only if α ≥ −λmin (A). Furthermore, A2 + A + 41 I ≥ 0. Fact 10.10.24. Let A ∈ Fn×m. Then, the following statements are equivalent:

σmax (A) ≤ 1. σmax (A∗ ) ≤ 1. σmax (AA∗ ) ≤ 1. σmax (A∗A) ≤ 1. AA∗ ≤ In . A∗A ≤ Im . Source: Fact 7.12.38. Fact 10.10.25. Let A ∈ Fn×n, and assume that either AA∗ ≤ A∗A or A∗A ≤ AA∗. Then, A is normal. Source: ii) of Corollary 10.4.10. i) ii) iii) iv) v) vi)

733

POSITIVE-SEMIDEFINITE MATRICES

Fact 10.10.26. Let A ∈ Fn×n, and assume that A is a projector. Then, the following statements

hold: i) 0 ≤ A ≤ I. ii) For all i ∈ {1, . . . , n}, 0 ≤ A(i,i) ≤ 1. √ ∑ iii) For all i ∈ {1, . . . , n}, nj=1 |A(i, j) | ≤ n(rank A)A(i,i) . √ ∑ iv) ni, j=1 |A(i, j) | ≤ n rank A. ∑ v) For all i ∈ {1, . . . , n}, ni, j=1, j,i |A(i, j) |2 ≤ 14 . vi) For all i, j ∈ {1, . . . , n}, |A(i, j) |2 ≤ min {A(i,i) A( j, j) , (1 − A(i,i) )(1 − A( j, j) )}. vii) For all distinct i, j ∈ {1, . . . , n}, |A(i, j) | ≤ 21 . Source: [260]. Fact 10.10.27. Let A ∈ Fn×n, assume that A is (semisimple, Hermitian), and assume that there exists a nonnegative integer k such that Ak = Ak+1. Then, A is (idempotent, a projector). Fact 10.10.28. Let A ∈ Fn×n, and assume that A is nonsingular. Then, ⟨A−1 ⟩ = ⟨A∗ ⟩−1. Fact 10.10.29. Let A ∈ Fn×m, and assume that A∗A is nonsingular. Then, ⟨A∗ ⟩ = A⟨A⟩−1A∗. Fact 10.10.30. Let A ∈ Fn×n. Then, A is unitary if and only if there exists a nonsingular matrix B ∈ Fn×n such that A = ⟨B∗ ⟩−1B. If, in addition, A is real, then det A = sign(det B). Source: For necessity, set B = A. Related: Fact 4.14.6. Fact 10.10.31. Let A ∈ Fn×n. Then, the following statements hold: i) A is positive semidefinite if and only if A = ⟨A⟩. ii) A is normal if and only if ⟨A⟩ = ⟨A∗ ⟩. iii) A is normal if and only if [A, ⟨A⟩] = 0. iv) If A is normal, then ⟨A + A∗ ⟩ = ⟨A⟩ + ⟨A∗ ⟩. v) ⟨A⟩ and ⟨A∗ ⟩ are similar. Source: [987, 1242] and [2991, pp. 290, 291]. Related: Fact 4.10.12. Fact 10.10.32. Let A ∈ Fn×n. Then, −⟨A⟩ − ⟨A∗ ⟩ ≤ A + A∗ ≤ ⟨A⟩ + ⟨A∗ ⟩. Source: Fact 10.12.27 and [1783]. Fact 10.10.33. Let A ∈ Fn×n, assume that A is normal, and let α, β ∈ (0, ∞). Then,

−α⟨A⟩ − β⟨A∗ ⟩ ≤ ⟨αA + βA∗ ⟩ ≤ α⟨A⟩ + β⟨A∗ ⟩. In particular,

−⟨A⟩ − ⟨A∗ ⟩ ≤ ⟨A + A∗ ⟩ ≤ ⟨A⟩ + ⟨A∗ ⟩.

Source: [1783, 2987]. Related: Fact 10.12.27. Fact 10.10.34. Let A ∈ Fn×n. Then, there exists a unitary matrix S ∈ Fn×n such that 1 2 (A

+ A∗ ) ≤ S ∗ ⟨A⟩S .

Source: Fact 10.11.13, Fact 10.21.9, and [2991, p. 289]. Fact 10.10.35. Let A ∈ Fn×n. The following statements hold:

i) If A ∈ Fn×n is positive definite, then I + A is nonsingular and the matrices I − B and I + B △ are positive definite, where B = (I + A)−1(I − A). △ ii) If I + A is nonsingular and the matrices I − B and I + B are positive definite, where B = −1 (I + A) (I − A), then A is positive definite. Source: [1009]. Related: Fact 4.13.24, Fact 4.13.25, Fact 4.13.26, Fact 4.28.12, Fact 15.22.10.

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Fact 10.10.36. Let A ∈ Fn×n, assume that

1 2 ȷ (A

− A∗ ) is positive definite, and define

△

B = [ 21ȷ (A − A∗ )]1/2A−1A∗ [ 21ȷ (A − A∗ )]−1/2 . Then, B is unitary. Source: [1012]. Remark: A is strictly dissipative if 21ȷ (A − A∗ ) is negative definite. A is strictly dissipative if and only if − ȷA is dissipative. See [1010, 1011]. Remark: A−1A∗ is similar to a unitary matrix. See Fact 4.13.6. Related: Fact 10.15.2 and Fact 10.21.18. Fact 10.10.37. Let A ∈ Rn×n, assume that A is positive definite, assume that A ≤ I, and define △ △ 1 2 1/2 the sequence (Bk )∞ k=0 by B0 = 0 and Bk+1 = Bk + 2 (A − Bk ). Then, limk→∞ Bk = A . Source: [353, p. 181]. Related: Fact 7.17.23. Fact 10.10.38. Let A ∈ Rn×n, assume that A is nonsingular, and define the sequence (Bk )∞ k=0 by △ △ B0 = A andBk+1 = 21 (Bk + B−k T ). Then, limk→∞ Bk = (AAT )−1/2A. Remark: The limit is a unitary matrix. See Fact 10.10.30 and [300, p. 224]. △ Fact 10.10.39. Let a, b ∈ R, and define the symmetric, Toeplitz matrix A ∈ Rn×n by A = aIn + b1n×n . Then, A is positive definite if and only if a + nb > 0 and a > 0. Related: Fact 3.16.18 and Fact 6.10.21. Fact 10.10.40. Let x1 , . . . , xn ∈ Rm , and define n n ∑ ∑ △ 1 △ 1 x j, S = (x j − x)(x j − x)T. x= n j=1 n j=1 Then, for all i ∈ {1, . . . , n},

(xi − x)(xi − x)T ≤ (n − 1)S.

Furthermore, equality holds if and only if all of the elements of {x1 , . . . , xn }\{xi } are equal. Source: [1528, 2132, 2720]. Related: This result extends the Laguerre-Samuelson inequality given by Fact 2.11.37. △ Fact 10.10.41. Let x1 , . . . , xn ∈ Fn, and define A ∈ Fn×n by A(i, j) = xi∗ x j for all i, j ∈ {1, . . . , n}, △ and B = [x1 · · · xn ]. Then, A = B∗B. Consequently, A is positive semidefinite and rank A = rank B. Conversely, let A ∈ Fn×n, and assume that A is positive semidefinite. Then, there exist x1 , . . . , xn ∈ Fn such that A = B∗B, where B = [x1 · · · xn ]. Source: The converse follows from Corollary 7.5.5. Remark: A is the Gram matrix of x1 , . . . , xn . Fact 10.10.42. Let A ∈ Fn×n, and assume that A is positive semidefinite. Then, there exists B ∈ Fn×n such that B is lower triangular, B has nonnegative diagonal entries, and A = BB∗. If, in addition, A is positive definite, then B is unique and has positive diagonal entries. Remark: This is the Cholesky decomposition. Related: Fact 7.17.12. Fact 10.10.43. Let A ∈ Fn×m, and assume that rank A = m. Then, 0 ≤ A(A∗A)−1A∗ ≤ I. Fact 10.10.44. Let A ∈ Fn×m. Then, I − A∗A is positive definite if and only if I − AA∗ is positive definite. If these conditions hold, then (I − A∗A)−1 = I + A∗(I − AA∗ )−1A. △

Fact 10.10.45. Let A ∈ Fn×n, let α ∈ (0, ∞), and define Aα ∈ Fn×n by Aα = (αI + A∗A)−1A∗. Then,

the following statements are equivalent: i) AAα = Aα A. ii) AA∗ = A∗A. Furthermore, the following statements are equivalent: iii) Aα A∗ = A∗Aα . iv) AA∗A2 = A2A∗A.

735

POSITIVE-SEMIDEFINITE MATRICES

Source: [2653]. Remark: Aα is a regularized Tikhonov inverse. △ △ Fact 10.10.46. Let A ∈ Pn×n, and define α = λmax (A) and β = λmin (A). Then,

A−1 ≤

1 (α + β)2 −1 α+β I− A≤ A . αβ αβ 4αβ

Source: [1970]. Fact 10.10.47. Let A ∈ Nn. Then, the following statements hold:

i) If α ∈ [0, 1], then Aα ≤ αA + (1 − α)I. ii) If α ∈ [0, 1] and A is positive definite, then [αA−1 + (1 − α)I]−1 ≤ Aα ≤ αA + (1 − α)I. iii) If α ≥ 1, then αA + (1 − α)I ≤ Aα. iv) If A is positive definite and either α ≥ 1 or α ≤ 0, then αA + (1 − α)I ≤ Aα ≤ [αA−1 + (1 − α)I]−1. Source: [1124, pp. 122, 123] and [2991, p. 206]. Remark: This is a special case of Young’s inequality. See Fact 2.2.53, Fact 2.2.50, Fact 10.11.73, Fact 10.14.8, Fact 10.14.33, Fact 10.14.34, and Fact 10.11.38. ∑ Fact 10.10.48. Let A ∈ Nn, let α1 , . . . , αn > 0, and assume that ni=1 α1i = 1. Then,

A ≤ diag(α1 A(1,1) , . . . , αn A(n,n) ). △

△

Source: Assume that A is positive definite, and define B = diag(α1 A(1,1) , . . . , αn A(n,n) ). Then, C =

B−1/2AB−1/2 is positive definite. Furthermore, λ1 (C) ≤ tr C = 1. Hence, A ≤ B. Fact 10.10.49. Let A ∈ Pn. Then, ∫ ∞ ∫ 1 ∫ ∞ 1 1 log A = I −(xI + A)−1 dx = (A− I)[x(A− I)+ I]−1 dx = (A− I)[(x −1)I + A]−1 dx. x+1 x 0 0 1

Source: [1381, pp. 115, 116, 135]. Related: Fact 14.2.19. Fact 10.10.50. Let A ∈ Pn. Then, I − A−1 ≤ log A ≤ A − I. Furthermore, if A ≥ I, then log A is positive semidefinite, and, if A > I, then log A is positive definite. Source: Fact 2.15.2.

10.11 Facts on Equalities and Inequalities for Two or More Matrices Fact 10.11.1. Let A, B ∈ Fn×n , and assume that A is positive semidefinite. Then, AB = BA if and only if A1/2B = BA1/2 . Source: [2991, p. 205]. Fact 10.11.2. Let A ∈ Fn×m, B ∈ Fn×l, and α > 0. Then, the following statements are equivalent:

i) There exists C ∈ Fl×m such that A = BC and σ2max (C) ≤ α. ii) AA∗ ≤ αBB∗. If these statements hold, then R(A) ⊆ R(B). Related: Theorem 10.6.2. See [2991, p. 187]. n ∞ n Fact 10.11.3. Let (Ai )∞ i=1 ⊂ H and (Bi )i=1 ⊂ H , assume that, for all i ∈ P, Ai ≤ Bi , and assume △ △ that A = limi→∞ Ai and B = limi→∞ Bi exist. Then, A ≤ B. Fact 10.11.4. Let A, B ∈ Nn, and assume that A ≤ B. Then, R(A) ⊆ R(B) and rank A ≤ rank B. Furthermore, R(A) = R(B) if and only if rank A = rank B. Fact 10.11.5. Let A, B ∈ Pn and S ⊆ {1, . . . , n}. Then, { } [(A + B)−1 ](S) [(A + B)(S) ]−1 ≤ ≤ (A−1 )(S) + (B−1 )(S) . (A(S) )−1 + (B(S) )−1 Source: [2991, p. 223].

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CHAPTER 10

Fact 10.11.6. Let A, B ∈ Fn×n and S ⊆ {1, . . . , n}. Then,

(A∗ )(S) A(S) ≤ (A∗A)(S) , (AB)(S) (B∗A∗ )(S) ≤ (ABB∗A∗ )(S) , A(S) B(S) (B∗ )(S) (A∗ )(S) ≤ A(S) (BB∗ )(S) (A∗ )(S) . Source: [2991, pp. 223, 318]. Fact 10.11.7. Let A ∈ Fn×m and B ∈ Fm×l . Then, (AB)+AB ≤ B+B. Source: [2238, p. 312]. Fact 10.11.8. Let A ∈ Fn×m, let B ∈ Fm×m, and assume that B is a projector. Then, AB = A if and only if A+A ≤ B. Source: [2238, p. 312]. Fact 10.11.9. Let A, B ∈ Fn×n, and assume that A and B are projectors. Then, the following

statements are equivalent: i) A ≤ B. ii) For all x ∈ Fn, ∥Ax∥2 ≤ ∥Bx∥2 . iii) R(A) ⊆ R(B). iv) AB = A. v) BA = A. vi) B − A is a projector. Source: [1133, p. 43] and [2418, p. 24]. Related: Fact 4.18.3 and Fact 4.18.5. Fact 10.11.10. Let A, B ∈ Nn, and assume that A and B are semicontractive. Then, − 41 I ≤ AB + BA. Source: [2990, p. 81]. If A and B are projectors, then AB + BA + 14 I = (A + B − 21 I)2 ≥ 0. See [246]. Related: Fact 7.13.4, Fact 10.11.24, and Fact 10.22.32. Fact 10.11.11. Let A, B ∈ Cn×n, assume that A and B are nonsingular, and assume that ȷ(A∗B − ∗ B A) is positive semidefinite. Then, A + ȷB is nonsingular. Source: [1158, p. 71]. Fact 10.11.12. Let A, B ∈ Hn . Then, the following statements hold:

i) λmin (A) ≤ λmin (B) if and only if λmin (A)I ≤ B. ii) λmax (A) ≤ λmax (B) if and only if A ≤ λmax (B)I. Fact 10.11.13. Let A, B ∈ Hn, and consider the following statements: i) A ≤ B. ii) There exists a unitary matrix S ∈ Fn×n such that A ≤ SBS ∗. iii) For all i ∈ {1, . . . , n}, λi (A) ≤ λi (B). Then, i) =⇒ ii) ⇐⇒ iii). Source: ii) =⇒ iii) follows from the monotonicity theorem given by Theorem 10.4.9 and the fact that, for all i ∈ {1, . . . , n}, λi (B) = λi (SBS ∗ ). iii) =⇒ ii) is given in [2991, pp. 287, 288]. Fact 10.11.14. Let A, B ∈ Fn×n, and assume that A is positive semidefinite and B is positive definite. Then, 0 ≤ A < B if and only if ρmax (AB−1 ) < 1. Fact 10.11.15. Let A, B ∈ Fn×n , and assume that A and B are positive semidefinite. Then, the following statements hold: i) I + AB and I + BA are nonsingular. ii) (I + AB)−1A = A(I + BA)−1 . iii) (I + AB)−1A is Hermitian. Related: Fact 3.20.6.

737

POSITIVE-SEMIDEFINITE MATRICES

Fact 10.11.16. Let A, B ∈ Fn×n, and assume that A is positive definite and B is Hermitian. Then,

2B ≤ BAB + A−1 . Fact 10.11.17. Let A, B ∈ Pn . Then,

(A−1 + B−1 )−1 = A(A + B)−1B = B(A + B)−1A. Remark: This equality holds for all nonsingular A, B ∈ Fn×n such that A + B is nonsingular. Fact 10.11.18. Let A, B ∈ Pn . Then,

(A + B)−1 ≤ 14 (A−1 + B−1 ). Equivalently,

A + B ≤ AB−1A + BA−1B.

In both inequalities, equality holds if and only if A = B. Source: [2983, p. 168]. Related: Fact 2.2.12. Fact 10.11.19. Let A, B ∈ Fn×n, and assume that A is positive definite, B is Hermitian, and A+ B is nonsingular. Then, (A + B)−1 + (A + B)−1B(A + B)−1 ≤ A−1. If, in addition, B is nonsingular, then the inequality is strict. Source: This inequality is equivalent to BA−1B ≥ 0. See [2164]. Fact 10.11.20. Let A, B ∈ Pn and α ∈ [0, 1]. Then, β[αA−1 + (1 − α)B−1 ] ≤ [αA + (1 − α)B]−1, where

△

β=

min

µ∈mspec(A−1B)

4µ . (1 + µ)2

Source: [2072]. Remark: This is a reverse form of an inequality based on convexity. Fact 10.11.21. Let A ∈ Fn×m and B ∈ Fm×m, and assume that B is positive semidefinite. Then,

ABA∗ = 0 if and only if AB = 0. Fact 10.11.22. Let A, B ∈ Nn. Then, AB is positive semidefinite if and only if AB is normal. Source: [531, p. 1456]. Fact 10.11.23. Let A, B ∈ Hn, and assume that either i) either A or B is positive definite, or ii) A and B are positive semidefinite. Then, AB is semisimple. Source: Theorem 10.3.2 and Theorem 10.3.6. Fact 10.11.24. Let A, B ∈ Hn, and assume that A⊙I is positive definite and AB + BA is (positive semidefinite, positive definite). Then, B is (positive semidefinite, positive definite). Source: [454, p. 8], [1768, p. 120], [2877], and [2991, p. 214]. Alternatively, use Corollary 15.10.4. Related: Fact 7.13.4, Fact 10.11.10, and Fact 10.22.32. Fact 10.11.25. Let A, B, C ∈ Nn, and assume that A = B + C. Then, the following statements are equivalent: i) rank A = rank B + rank C. ii) There exists S ∈ Fm×n such that rank S = m, R(S ) ∩ N(A) = {0}, and either B = AS ∗ (SAS ∗ )−1SA or C = AS ∗ (SAS ∗ )−1SA. Source: [627, 735]. Fact 10.11.26. Let A, B ∈ Fn×n, and assume that A and B are Hermitian and nonsingular. Then, the following statements hold: i) If every eigenvalue of AB is positive, then In A = In B.

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CHAPTER 10

ii) In A − In B = In(A − B) + In(A−1 − B−1 ). iii) If In A = In B and A ≤ B, then B−1 ≤ A−1. Source: [97, 228, 2141]. Remark: An extension to singular A and B is given by Fact 10.24.15. Credit: The equality ii) is due to G. P. H. Styan. See [2141]. Fact 10.11.27. Let A, B ∈ Hn, and assume that A ≤ B. Then, A(i,i) ≤ B(i,i) for all i ∈ {1, . . . , n}. Fact 10.11.28. Let A, B ∈ Hn, and assume that A ≤ B. Then, sig A ≤ sig B. Source: [860, p. 148]. Fact 10.11.29. Let A, B ∈ Hn, and assume that ⟨A⟩ ≤ B. Then, either A ≤ B or −A ≤ B. Source: [2986]. Fact 10.11.30. Let A, B ∈ Fn×n, and assume that A is positive semidefinite and B is positive definite. Then, A ≤ B if and only if AB−1A ≤ A. Fact 10.11.31. Let A, B ∈ Nn, and assume that A ≤ B. Then, there exists S ∈ Fn×n such that A = S ∗BS and S ∗S ≤ I. Source: [970, p. 269]. Fact 10.11.32. Let A, B ∈ Fn×m, and assume that AA∗ ≤ BB∗ . Then, for all i ∈ {1, . . . , min{n, m}}, σi (AA∗ ) ≤ σi (BB∗ ) and σi (A) ≤ σi (B). In particular, σmax (A) ≤ σmax (B). Source: Use Fact 7.12.38 and either Theorem 10.4.9 or Fact 11.10.30. Related: Fact 11.10.31. Fact 10.11.33. Let A, B, C, D ∈ Nn, and assume that 0 < D ≤ C and BCB ≤ ADA. Then, B ≤ A. Source: [186, 669]. Fact 10.11.34. Let A ∈ Fn×m and B ∈ Fk×m. Then, there exist unitary matrices S 1 , S 2 ∈ Fm×m such that ⟨A + B⟩ ≤ S 1 ⟨A⟩S 1∗ + S 2 ⟨B⟩S 2∗ . Source: [2991, pp. 289, 290]. Remark: This is a matrix version of the triangle inequality. See [93, 2613]. Remark: There exist A ∈ Fn×m and B ∈ Fk×m such that ⟨A + B⟩ ≤ ⟨A⟩ + ⟨B⟩ does not hold. See [2991, p. 291]. Related: Fact 10.14.50, Fact 10.25.52, and Fact 11.10.9. Fact 10.11.35. Let A, B ∈ Fn×n, and let p, q ∈ (0, ∞) satisfy 1/p + 1/q = 1. Then, ⟨√ √ ⟩2 q p A + = p⟨A⟩2 + q⟨B⟩2 . ⟨A − B⟩2 + q pB Source: [2991, p. 291]. √ √ Fact 10.11.36. Let A, B ∈ Fn×n, a, b ∈ (0, ∞), and c ∈ (− ab, ab.) Then,

a⟨A⟩2 + b⟨B⟩2 + c(A∗B + B∗A) ≥ 0. Source: [2991, p. 291]. ∑ Fact 10.11.37. Let A1 , . . . , Ak ∈ Fn×n, let a1 , . . . , ak ∈ (0, ∞), and assume that ki=1 ai = 1.

Then,

⟨∑ k

⟩2 ≤

ai Ai

i=1

k ∑

ai ⟨Ai ⟩2 .

i=1

Source: [2991, p. 292]. Fact 10.11.38. Let A ∈ Fn×m and B ∈ Fl×m, and let p, q > 1 satisfy 1/p + 1/q = 1. Then, there

exists a unitary matrix S ∈ Fm×m such that

⟨AB∗ ⟩ ≤ S ∗ Furthermore,

tr ⟨AB∗ ⟩ ≤

(

1 p

p 1 p ⟨A⟩

) + 1q ⟨B⟩q S.

tr ⟨A⟩ p + 1q tr ⟨B⟩q .

Source: [93, 95, 1418] and [2977, p. 28]. Remark: This is a matrix version of Young’s inequality. Related: Fact 2.2.50, Fact 2.2.53, Fact 10.11.39 Fact 10.11.73, Fact 10.14.8, Fact 10.14.33, Fact

739

POSITIVE-SEMIDEFINITE MATRICES

10.14.34, and Fact 11.16.30. Fact 10.11.39. Let A, B ∈ Fn×n, and assume that A and B are positive definite. Then, there exists a unitary matrix S ∈ Fn×n such that ⟨AB⟩ ≤ 21 S (A2 + B2 )S ∗. Source: [196, 468]. Fact 10.11.40. Let A, B ∈ Fn×n, and assume that A and B are projectors. Then, ABA ≤ B if and only if AB = BA. Source: [2709]. Fact 10.11.41. Let A, B ∈ Fn×n, assume that A is positive semidefinite, 0 ≤ A ≤ I, and B is △ △ positive definite, and define α = λmin (B) and β = λmax (B). Then, ABA ≤

(α + β)2 B. 4αβ

Source: [542]. Related: Fact 2.12.10. △ Fact 10.11.42. Let A ∈ Fn×n , define r = rank A, assume that A is positive semidefinite, let △

△

P ∈ Fn×n be the projector onto R(A), define α = λmin (A) and β = λr (A), and let B ∈ Fn×m . Then, B∗AB ≤

(α + β)2 ∗ B PB(B∗A+B)+B∗PB, 4αβ

(β − α)2 ∗ + B∗AB − B∗PB(B∗A+B)+B∗PB ≤ B A B, 4 √ √ B∗AB − B∗PB(B∗A+B)+B∗PB ≤ ( β − α)2 B∗PB,

(α + β)2 ∗ B AB(B∗PB)+B∗AB, 4αβ

B∗A2B ≤

(β − α)2 ∗ B PB, 4 √ √ B∗A2B − B∗AB(B∗PB)+B∗AB ≤ ( β − α)2 B∗AB. B∗A2B − B∗AB(B∗PB)+B∗AB ≤

Now, assume that A is positive definite. Then, B∗AB ≤

(α + β)2 ∗ B B(B∗A−1B)−1B∗B, 4αβ

B∗A2B ≤

(α + β)2 ∗ B AB(B∗B)−1B∗AB, 4αβ

(β − α)2 ∗ −1 (β − α)2 ∗ B∗AB − B∗B(B∗A−1B)−1B∗B ≤ B A B, B∗A2B − B∗AB(B∗B)−1B∗AB ≤ B B, 4 4 √ √ √ √ B∗AB − B∗B(B∗A−1B)−1B∗B ≤ ( β − α)2 B∗B, B∗A2B − B∗AB(B∗PB)−1B∗AB ≤ ( β − α)2 B∗AB, det(B∗AB) det(B∗A−1B) ≤

min ∏ {m,n−m} i=1

Finally, if 2m ≤ n, then

[λi (A) + λn−i+1 (A)]2 . 4λi (A)λn−i+1 (A)

 ∑m   i=1 [λi (A) + λn−i+1 (A)] 2 tr B∗AB  . ≤  ∑m √ tr (B∗A−1B)−1 2 i=1 λi (A)λn−i+1 (A)

Source: [1357]. The last two inequalities are given in [2826]. Remark: These are matrix extensions of the Kantorovich inequality. Related: Fact 2.11.134, Fact 10.18.8, and Fact 10.11.43. △ Fact 10.11.43. Let A ∈ Fn×n , let B ∈ Fn×m , assume that A is positive definite, define α = λmin (A) △

and β = λmax (A), and assume that B is left inner. Then, B∗AB ≤

(α + β)2 ∗ −1 −1 (B A B) , 4αβ

B∗A2B ≤

(α + β)2 ∗ 2 (B AB) , 4αβ

(β − α)2 ∗ −1 (β − α)2 B∗AB − (B∗A−1B)−1 ≤ B A B, B∗A2B − (B∗AB)2 ≤ I, 4 4 √ √ √ √ B∗AB − (B∗A−1B)−1 ≤ ( β − α)2 I, B∗A2B − (B∗AB)2 ≤ ( β − α)2 B∗AB. Source: [1357]. Remark: These are matrix extensions of the Kantorovich inequality. Related: Fact 2.11.134, Fact 10.18.8, and Fact 10.11.42.

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CHAPTER 10

Fact 10.11.44. Let A, B ∈ Fn×n, and assume that A and B are projectors. Then, (A+B)1/2 ≤ A+B if and only if AB = BA. Source: [2696, p. 30]. Fact 10.11.45. Let A, B ∈ Nn, and assume that 0 ≤ A ≤ B. Then,

(A + 41 A2 )1/2 ≤ (B + 14 B2 )1/2. Source: [2063]. Fact 10.11.46. Let A ∈ Nn, let B ∈ Fl×n, let p, q ∈ R[s], and assume that p , 0, q , 0, p(0) =

q(0) = 0, and all of the coefficients of p and q are nonnegative. Then, rank Bp(A)B∗ = rank Bq(A)B∗ . Source: Note that rank Bp(A)1/2 = rank Bq(A)1/2 . Fact 10.11.47. Let A, B ∈ Nn, and assume that A ≤ B and AB = BA. Then, A2 ≤ B2. Source: [229]. Fact 10.11.48. Let A, B ∈ Nn, and assume that BA2 B ≤ I. Then, B1/2AB1/2 ≤ I. Source: [2991, p. 214]. Fact 10.11.49. Let A ∈ Pn and B, C ∈ Nn. Then, 2tr ⟨B1/2C 1/2 ⟩ ≤ tr(AB + A−1C). Furthermore, there exists A such that equality holds if and only if rank B = rank C = rank B1/2C 1/2. Source: [80, 1057]. Remark: A matrix A for which equality holds is given in [80]. Remark: Applications to linear systems are given in [2896]. Fact 10.11.50. Let A, B ∈ Pn, let S ∈ Fn×n , assume that S is nonsingular and SAS ∗ = diag(α1 , . . . , αn ) and SBS ∗ = diag(β1 , . . . , βn ), and define △

Cl = S −1 diag(min {α1 , β1 }, . . . , min {αn , βn })S −∗ , △

Cu = S −1 diag(max {α1 , β1 }, . . . , max {αn , βn })S −∗. ˆ Sˆ ∗ , SBS ∗ Then, Cl ≤ A ≤ Cu and Cl ≤ B ≤ Cu . Furthermore, if Sˆ ∈ Fn×n is nonsingular, SAS ∗ = SA ∗ ˆ ˆ ˆ ˆ ˆ ˆ = SBS , and Cl and Cu are defined as Cl and Cu with S replaced by S , then Cl = Cl and Cˆ u = Cu . Source: [1805]. Fact 10.11.51. Let A, B ∈ Hn, and define “glb” with respect to Hn . Then, the following statements hold: i) If glb({A, B}) exists, then either A ≤ B or B ≤ A. ii) If A, B ∈ Nn are projectors, then glb({A, B}) = 2A(A + B)+B, which is the projector onto R(A) ∩ R(B). iii) glb({A, B}) exists if and only if glb({A, glb({AA+, BB+ })}) and glb({B, glb({AA+, BB+ })}) are comparable. If these conditions hold, then glb({A, B}) = min {glb({A, glb({AA+, BB+ })}), glb{B, glb({AA+, BB+ })})}. △

iv) glb({A, B}) exists if and only if sh(A, B) and sh(B, A) are comparable, where sh(A, B) = limα→∞ αB(αB + A)+A. If these conditions hold, then glb({A, B}) = min {sh(A, B), sh(B, A)}. Source: [96, 957, 1169, 1570,[ 2078]. Remark: ] [ ]The shorted operator “sh” is defined in Fact 10.24.21. Remark: Let A = 10 00 and B = 00 01 . Then, C = 0 is a lower bound for {A, B}.

[ √ ] √ √ 2 √ Furthermore, D = −1 , which has eigenvalues −1 − 2 and −1 + 2, is also a lower bound 2 −1 for {A, B} but is not comparable with C. Hence, neither C nor D is the greatest lower bound for A and B. Finally, since C is the unique positive-semidefinite lower bound for A and B, it follows that glb({A, B}) does not exist. Consequently, Hn with the ordering “≤” is not a lattice.

741

POSITIVE-SEMIDEFINITE MATRICES △

Fact 10.11.52. Let A, B ∈ Nn, and define “glb” with respect to Nn , and define A+ = 12 (A + ⟨A⟩).

Then,

glb({A, B}) = A − (A − B)+ = B − (B − A)+ . [ ] [ ] Source: [2623]. Remark: Let A = 10 00 and B = 00 01 , and suppose that Z is the least upper bound [ 2/3 ] △ for A and B. Hence, A ≤ Z ≤ I and B ≤ Z ≤ I, and thus Z = I. Next, note that X = 4/3 2/3 4/3 satisfies A ≤ X and B ≤ X. However, Z ≤ X is false, and thus {A, B} does not have a least upper bound. Therefore, Nn with the ordering “≤” is not a lattice. See [523, p. 11] and [1805]. Remark: The cone Nn is a partially ordered set under the spectral ordering. See Fact 10.11.57. Fact 10.11.53. Let A1 , . . . , Ak ∈ Pn. Then, −1  k k ∑ ∑  2  A−1 Ai  ≤ n  i . i=1

i=1

Remark: This is an extension of Fact 2.11.135. Fact 10.11.54. Let A, B ∈ Hn. Then, [ 12 (A + B)]2 ≤ 21 (A2 + B2 ). Source: [2991, p. 292]. Fact 10.11.55. Let A, B ∈ Nn, let p be a real number, and assume that either p ∈ [1, 2] or A and B are positive definite and p ∈ [−1, 0] ∪ [1, 2]. Then, [ 21 (A + B)] p ≤ 12 (A p + B p ). Source: [1716]. Fact 10.11.56. Let A1 , . . . , Ak ∈ Nn, and let p, q ∈ R satisfy 1 ≤ p ≤ q. Then,

1/q 1/p  k  k  1 ∑ q   1 ∑  p   Ai  ≤  Ai  . k k i=1 i=1

Source: [443]. Fact 10.11.57. Let A, B ∈ Nn, and let p, q ∈ R satisfy 1 ≤ p ≤ q. Then,

[ 12 (Ap + B p )]1/p ≤ [ 12 (Aq + Bq )]1/q. Furthermore,

△

µ(A, B) = lim [ 21 (Ar + Br )]1/r r→∞

exists and satisfies A ≤ µ(A, B) and B ≤ µ(A, B). Finally, 1

lim [ 12 (Ar + Br )]1/r = e 2 (log A+log B) . r→0

Source: [354, 443]. Remark: µ(A, B) is the least upper bound of A and B with respect to the spectral ordering. See [92, 102, 1589] and Fact 10.23.3. Remark: This result does not hold for

[ ]3 [ ] [ ] 8 10 p = 1 and q = 1/3. For example, let A = 21 12 = 13 8 5 and B = 0 0 . Related: Fact 11.10.13 and Fact 12.17.11. Fact 10.11.58. Let A, B ∈ Fn×n, assume that A and B are positive semidefinite, and assume that A > B ≥ 0. If r ∈ (0, 1], then 0 < (σrmax (A) − [σmax (A) − σmin (A − B)]r )I ≤ Ar − Br , 0 < (σrmax (A) − [σmax (A) − σmin (A − B)]r )I ≤ ([σmax (B) + σmin (A − B)]r − σrmax (B))I ≤ Ar − Br . In particular, 0 < σmin (A − B)I ≤ A − B. If A > B > 0, then 0
1, then q > 1 and the following statements hold: xii) ⟨A − B⟩2 ≤ p⟨A⟩2 + q⟨B⟩2 . xiii) In xii), equality holds if and only if (1 − p)A = B. Finally, the following statements hold: xiv) If pq > 0, then ⟨A − B⟩ ≤ p⟨A⟩2 + q⟨B⟩2 . xv) If pq < 0, then p⟨A⟩2 + q⟨B⟩2 ≤ ⟨A − B⟩. xvi) In xiv) and xv), equality holds if and only if (1 − p)A = B. Source: [7, 1095, 1416, 3018]. Remark: This extends Bohr’s inequality given by Fact 2.21.8. Related: Fact 10.11.84. Fact 10.11.84. Let A, B ∈ Fn×n and α ∈ R. If α , 0, then ⟨A − B⟩2 +

1 ⟨αA + B⟩2 = (1 + α)⟨A⟩2 + (1 + α1 )⟨B⟩2 . α

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CHAPTER 10

In particular,

⟨A − B⟩2 + ⟨A + B⟩2 = 2⟨A⟩2 + 2⟨B⟩2 .

If α ∈ [0, 1], then

⟨A − B⟩2 + ⟨αA + B⟩2 ≤ (1 + α)⟨A⟩2 + (1 + α1 )⟨B⟩2 .

If either α < 0 or α ≥ 1, then (1 + α)⟨A⟩2 + (1 + α1 )⟨B⟩2 ≤ ⟨A − B⟩2 + ⟨αA + B⟩2 . Finally,

⟨A + B⟩2 − ⟨A − B⟩2 = 4 Re(A∗B).

Source: [1095, 2093]. Remark: The first inequality is the generalized parallelogram law, and the second inequality is a matrix version of the parallelogram law. Related: Fact 2.21.8, Fact 10.11.83,

and Fact 11.8.3. ∑k

Fact 10.11.85. Let A1 , . . . , Ak ∈ Fn×n, let α1 , . . . , αk be nonzero real numbers, and assume that

i=1

αi = 1. Then,

⟨∑ ⟨ ⟩2 ⟩2 ∑ k k ∑ αi α j 1 2 ⟨Ai ⟩ − Ai − A j , Ai = α α j αi i=1 i=1 i

where the last summation is taken over all i, j ∈ {1, . . . , k} such that i ≤ j. Source: [1095, 2989]. Related: Fact 2.21.14, Fact 10.11.86, and Fact 11.8.4. ∑ Fact 10.11.86. Let A1 , . . . , Ak ∈ Fn×n, let α1 , . . . , αk ∈ [0, 1], and assume that ki=1 αi = 1. Then, ⟨∑ k

⟩2 αi A i

i=1

≤

k ∑

αi ⟨Ai ⟩2 .

i=1

Source: [1095]. Remark: This is a convexity condition. Related: Fact 10.11.85. Fact 10.11.87. Let A, B ∈ Fn×n, assume that A and B are nonsingular, let p and q be positive

numbers, and assume that 1/p + 1/q = 1. Then,

⟨⟨A⟩−1 − B⟨B⟩−1 ⟩2 ≤ ⟨A⟩−1 [p⟨A − B⟩2 + q(⟨A⟩ − ⟨B⟩)2 ]⟨A⟩−1 . Equality holds if and only if (p − 1)(A − B)⟨A⟩−1 = B(⟨A⟩−1 − ⟨B⟩−1 ). Furthermore, ⟨A⟨A⟩−1 − B⟨B⟩−1 ⟩ ≤ (⟨A⟩−1 [2⟨A − B⟩2 + 2(⟨A⟩ − ⟨B⟩)2 ]⟨A⟩−1 )1/2 . Source: [2212, 2360]. Remark: This is a matrix version of the Dunkl-Williams inequality given by Fact 11.7.11. Fact 10.11.88. Let A, B ∈ Fn×n, assume that A and B are nonsingular, let r ∈ R, let p, q ∈ (0, ∞), and assume that 1/p + 1/q = 1. Then,

⟨⟨A⟩r−1 − B⟨B⟩r−1 ⟩2 ≤ ⟨A⟩r−1 (p⟨A − B⟩2 + q⟨⟨B⟩r ⟨A⟩1−r − ⟨B⟩⟩2 )⟨A⟩r−1 . Equality holds if and only if (p − 1)(A − B)⟨A⟩r−1 = B(⟨A⟩r−1 − ⟨B⟩r−1 ). Source: [810]. Remark: Setting r = 0 yields Fact 10.11.87. Related: Fact 11.7.11. Fact 10.11.89. Each of the following functions ϕ: (0, ∞) 7→ (0, ∞) yields an increasing function ϕ: Pn 7→ Pn : p+1/2 i) ϕ(x) = xx2p +1 , where p ∈ [0, 1/2]. ii) ϕ(x) = x(1 + x) log(1 + 1/x). 1 iii) ϕ(x) = (1+x) log(1+1/x) . iv) ϕ(x) =

x−1−log x . (log x)2

POSITIVE-SEMIDEFINITE MATRICES

v) ϕ(x) = vi) ϕ(x) = vii) ϕ(x) = viii) ϕ(x) = ix) ϕ(x) = x) ϕ(x) = xi) ϕ(x) = xii) ϕ(x) = xiii) ϕ(x) = xiv) ϕ(x) =

749

x(log x)2 x−1−log x . x(x+2) log(x+2) . (x+1)2 x(x+1) (x+2) log(x+2) . (x2 −1) log(1+x) . x2 x(x−1) (x+1) log(x+1) . (x−1)2 (x+1) log x . ) ( p−1 x p −1 p x p−1 −1 , x−1 log x .

√

where p ∈ [−1, 2].

x.

x x+1 . x−1 x p −1 ,

xv) ϕ(x) = where p ∈ (0, 1]. Source: [1128, 2229]. To obtain xii), xiii), and xiv), set p = 1, 1/2, −1, respectively, in xi). Fact 10.11.90. Let A, B ∈ Fn×n, and assume that A and B are positive definite. Then, log AB−1 = A1/2 (log A1/2B−1A1/2 )A−1/2 . Source: [1381, p. 134].

10.12 Facts on Equalities and Inequalities for Partitioned Matrices Fact 10.12.1. Let A ∈ Fn×n, and assume that A is positive semidefinite. Then, the following

statements hold: [ ] [ A −A ] i) AA AA and −A A are positive semidefinite. [ αA βA ] [α β] ii) If β γ ∈ F2×2 is positive semidefinite, then βA γA is positive semidefinite. [ αA βA ] [α β] iii) If A and β γ are positive definite, then βA γA is positive definite. Source: Fact 9.4.22. ] [ Fact 10.12.2. Let A ∈ Fn×n, B ∈ Fn×m, C ∈ Fm×m, assume that BA∗ CB ∈ F(n+m)×(n+m) is positive [α β] semidefinite, and assume that β γ ∈ F2×2 is positive semidefinite. Then, the following statements hold: [ α1n×n β1n×m ] i) β1m×n is positive semidefinite. γ1 [ αA βB ] m×m ii) βB∗ γC is positive semidefinite. [ ] [ αA βB ] iii) If BA∗ CB is positive definite and α and γ are positive, then βB∗ γC is positive definite. Source: To prove i), use Proposition 10.2.5. ii) and iii) follow from Fact 10.25.16. [ ] △ Fact 10.12.3. Let A ∈ Fn×n, B ∈ Fn×m, C ∈ Fm×m, and assume that A = BA∗ CB ∈ F(n+m)×(n+m) is positive semidefinite. Then, the following statements are equivalent: i) rank A = rank A + rank C.  +  + + ∗ + + +  A + A B(A|A) B A −A B(A|A)  + ii) A =   . −(A|A)+B∗A+ (A|A)+ Source: [2179, Theorem 4.6] and [1458, pp. 44–46]. [A A ] 11 12 n×n Fact 10.12.4. Let A = A∗12 , and assume that A is positive semidefinite and A11 is A22 ∈ F positive definite. Then, λmin (A) ≤ λmin (A11 |A) ≤ λmin (A22 ).

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CHAPTER 10

Source: [2991, p. 273]. Fact 10.12.5. Let A, B ∈ Fn×n, assume[ that A] and B are[ positive ] semidefinite, and assume that A11 A12 A∗12 A22

A and B are partitioned identically as A =

and B =

B11 B12 B∗12 B22

. Then,

A22 |A + B22 |B ≤ (A22 + B22 )|(A + B). Now, assume that A22 and B22 are positive definite. Then, equality holds if and only if A12 A−1 22 = −1 B12 B22 . Source: xvii) of Proposition 10.6.17, Corollary 10.6.18, [1045, 2179], and [2991, p. 241]. Remark: The first inequality is an extension of Bergstrom’s inequality, which corresponds to the case where A11 is a scalar. See Fact 10.19.4. Fact 10.12.6. Let A, B ∈ Fn×n, assume ]positive definite, and assume that A and [ A A ]that A and[ BB are B B are partitioned identically as A = A∗1112 A1222 and B = B11∗12 B1222 . Then, 4(A11 + B11 )−1 ≤ (A22 |A)−1 + (B22 |B)−1 . Source: [2991, p. 232]. Fact 10.12.7. Let A, B ∈ Fn×n, assume positive semidefinite, assume that A ] ] A and B[are [ A Athat B B

and B are partitioned identically as A = positive definite. Then,

11

12

A∗12 A22

and B =

11

12

B∗12 B22

, and assume that A11 and B11 are

det A det B det(A + B) + ≤ = det[(A11 + B11 )|(A + B)], det A11 det B11 det(A11 + B11 ) ∗ −1 (A12 + B12 )∗ (A11 + B11 )−1 (A12 + B12 ) ≤ A∗12 A−1 11 A12 + B12 B11 B12 , ∗ −1 ∗ −1 −1 rank[A∗12 A−1 11 A12 + B12 B11 B12 − (A12 + B12 ) (A11 + B11 ) (A12 + B12 )] = rank(A12 − A11 B11 B12 ).

Source: [2991, p. 243]. Remark: The first inequality follows from Fact 10.16.16. △ Fact 10.12.8. Let B, C ∈ Fn×n, define A = B + ȷC, assume [ assume ] that[ B and]C are Hermitian, [ ]

that A, B, and C are partitioned identically as A = AA1121 assume that A22 , B22 , and C22 are nonsingular. Then,

A12 A22

,B=

B11 B12 B∗12 B22

, and C =

C11 C12 ∗ C12 C22

, and

−1 −1 −1 −1 −1 −1 ∗ A22 |A = B22 |B + ȷC22 |C + (B12 B−1 22 − C 12 C 22 )(B22 − ȷC 22 ) (B12 B22 − C 12 C 22 ) .

Source: [1830]. [ ] △ Fact 10.12.9. Let A ∈ Fn×n, B ∈ Fn×m, and C ∈ Fm×m, define A = BA∗ CB , and assume that A is

positive semidefinite. Then,

0 ≤ BC +B∗ ≤ A.

If, in addition, A is positive definite, then C is positive definite and 0 ≤ BC −1B∗ < A. Source: Proposition 10.2.5. △

Fact 10.12.10. Let A, B, C ∈ Fn×n, define A =

[

A B B∗ C

]

, and assume that A is positive semidefi-

nite. Then, −A − C ≤ B + B∗ ≤ A + C,

| tr(B + B∗ )| ≤ tr(A + C),

| det(B + B∗ )| ≤ det(A + C).

Now, assume that A is positive definite. Then, −A − C < B + B∗ < A + C,

|tr(B + B∗ )| < tr(A + C), △

△

| det(B + B∗ )| < det(A + C).

Source: Use Fact 10.12.9 with SAS T , S = [I I], and S = [I −I]. Related: Fact 10.12.11 and Fact

10.25.60.

751

POSITIVE-SEMIDEFINITE MATRICES

Fact 10.12.11. Let A, B, C ∈ Fn×n, and assume that

[

A B B∗ C

]

∈ F2n×2n is positive semidefinite. Then,

√ tr (B + B∗ )2 ≤ tr(A + C).

Source: [2991, p. 244]. Related: Fact 10.12.10.

[

]

Fact 10.12.12. Let A, B, C ∈ Fn×n, assume that BA∗ CB ∈ F2n×2n is positive semidefinite, and assume that AB = BA. Then, B∗B ≤ A1/2CA1/2. Source: [2985] and [2991, p. 224]. Fact 10.12.13. Let A, B, C ∈ Fn×n, and assume that

[

A B B∗ C

]

∈ F2n×2n is positive semidefinite. Then,

 | tr B|2    √   ≤ (tr A) tr C. | tr B2 | ≤ tr B∗B ≤ (tr A2 ) tr C 2     2 ∗  tr AC + | tr B| − tr B B

[ ∗] If, in addition, AB BC ∈ F2n×2n is positive semidefinite, then tr B∗B ≤ tr AC. Source: The uppermost inequality follows √ from Fact 10.12.64. In the middle string, the first inequality is given by Fact 11.13.2. | tr B2 | ≤ (tr A2 ) tr C 2 is given in [1953]. tr B∗B ≤ (tr A) tr C is given in [2206]. Alternatively, use Fact 10.14.54 with P = A, Q = B, and R = C. The lowermost inequality and the last inequality are given in [439, 1834, 1840]. [ ] △ A A Fact 10.12.14. Let A11 ∈ Rn×n, A12 ∈ Rn×m, and A22 ∈ Rm×m, define A = A11T A12 ∈ R(n+m)×(n+m), 22 12 and assume that A is symmetric. Then, A is positive semidefinite if and only if, for all B ∈ Rn×m, T 1/2 1/2 tr BAT12 ≤ tr (A1/2 11 BA22 B A11 ) .

Source: [347]. [ ] △ Fact 10.12.15. Let A ∈ Fn×n, B ∈ Fn×m, and C ∈ Fm×m, and define A = BA∗ CB . Then, A is posi-

tive semidefinite if and only if A and C are positive semidefinite and there exists a semicontractive matrix Y ∈ Fn×m such that B = A1/2YC 1/2. Source: [1465] and [2991, p. 185]. Related: xiii) and xx) of Fact 10.11.68. Also, Fact 10.12.16 and Fact 10.24.25. n×n n×m m×m Fact 10.12.16. [ ]Let A ∈ F , B ∈ F , and C ∈ F , and assume that A and C are positive (n+m)×(n+m) A B definite. Then, B∗ C ∈ F is positive semidefinite if and only if

Furthermore,

[

A B B∗ C

]

σmax (A−1/2BC −1/2 ) ≤ 1. ∈ F(n+m)×(n+m) is positive definite if and only if σmax (A−1/2BC −1/2 ) < 1.

Source: [1953]. Related: Fact 10.12.15. Fact 10.12.17. Let A ∈ Fn×n, B ∈ Fn×m, and C ∈ Fm×m, assume that A and C are positive definite,

and assume that Then,

then

[

[

A B B∗ C

A B B∗ C

]

]

2 σmax (B) ≤ σmin (A)σmin (C).

∈ F(n+m)×(n+m) is positive semidefinite. If, in addition, 2 σmax (B) < σmin (A)σmin (C),

∈ F(n+m)×(n+m) is positive definite. Source: Note that

2 (A−1/2BC −1/2 ) ≤ λmax (A−1/2BC −1B∗A−1/2 ) ≤ σmax (C −1 )λmax (A−1/2BB∗A−1/2 ) σmax

≤

σmax (A−1 ) 1 1 λmax (B∗A−1B) ≤ λmax (B∗B) = σ2 (B) ≤ 1. σmin (C) σmin (C) σmin (A)σmin (C) max

752

CHAPTER 10

The result now follows from Fact 10.12.16. Fact 10.12.18. Let A, B ∈ Fn×n, and assume that A and B are Hermitian. Then, −A ≤ B ≤ A if [ ] [ ] and only if AB AB is positive semidefinite. Furthermore, −A < B < A if and only if AB AB is positive definite. Source: Note that [ ][ ] [ ] [ ] 1 I −I A B 1 I I A−B 0 = . √ √ 0 A+B 2 I I B A 2 −I I Related: Fact 10.16.7. [ ] Fact 10.12.19. Let A ∈ Fn×n, B ∈ Fn×m, and C ∈ Fm×m, assume that BA∗ CB is positive semidefi△

nite, and let r = rank B. Then, for all k ∈ {1, . . . , r}, k ∏

σi (B) ≤

i=1

k ∏

max {λi (A), λi (C)},

i=1

k ∑

σi (B) ≤

k ∑

i=1

max {λi (A), λi (C)}.

i=1

Source: [2985]. [ ] Fact 10.12.20. Let A, B, C ∈ Fn×n, and assume that BA∗ CB is positive semidefinite. Then, for all

k ∈ {1, . . . , n},

k ∏

σi (B) ≤

k √ ∏

λi (A)λi (C),

k ∑

i=1

i=1

i=1

k ∏

k ∏

k ∑

ρi (B) ≤

i=1

√

λi (A)λi (C),

σi (B) ≤

i=1

ρi (B) ≤

i=1

i=1

k √ ∑ λi (A)λi (C), k √ ∑ λi (A)λi (C). i=1

Source: Fact 3.25.15 and [2991, p. 352]. △

Fact 10.12.21. Let A, B, C ∈ Fn×n, and assume that A =

[

A B B∗ C

]

∈ F2n×2n is Hermitian. Then,

   A(1,1)  [ ]  .  s λ(A) s  ..  ≺ ≺ λ(A). λ(C)   A(2n,2n) Now, assume that B is either Hermitian or skew Hermitian. Then, [ ]↓ [ ]↓ s λ(A + C) λ(A + C) ↓ s − [λ(A)] ≺ λ(A) ≺ − [λ(A)]↑ , λ(A + C) λ(A + C) [ ] [ ]↓ s λ(A + C) ↓ ↑ s λ(A + C) s [λ(A)] + [λ(A)] ≺ ≺ 2λ(A) ≺ + [λ(A)]↓ − [λ(A)]↑ . λ(A + C) λ(A + C) In addition, assume that A is positive semidefinite. Then,    A(1,1)  [ ] [ ] s λ(A + C)  ..  s λ(A) s .  .  ≺ λ(C) ≺ λ(A) ≺ 0   A(2n,2n) Source: [1850], [1969, p. 225], [1971, p. 308], and [2750]. △

Fact 10.12.22. Let A ∈ Fn×n, B ∈ Fn×m, and C ∈ Fm×m, define A =

positive definite. Then,

tr A−1 + tr C −1 ≤ tr A−1.

Furthermore, B is nonzero if and only if tr A−1 + tr C −1 < tr A−1.

[

A B B∗ C

]

, and assume that A is

753

POSITIVE-SEMIDEFINITE MATRICES

Source: Proposition 10.2.6 and [2030]. Fact 10.12.23. Let A ∈ Fn×n, assume that A is positive semidefinite, let α > 0, and define [ ] △ αA A A = A α−1A . Then, A is positive semidefinite, and rank A = rank A. Source: [2991, p. 222]. △

Fact 10.12.24. Let A ∈ Fn×n, assume that A is positive definite, and define A = A is positive semidefinite, and rank A = n. Source: [2991, p. 222]. Fact 10.12.25. Let A ∈ Fn×m, and define △

A=

[

σmax (A)Im A

[

A I I A−1

]

. Then,

] A∗ . σmax (A)In

Then, A is positive semidefinite, and rank A = n + m − amult⟨A⟩ [σmax (A)]. Source: [2991, p. 222]. Fact 10.12.26. Let A ∈ Fn×n, assume that A is positive semidefinite, and define [ ] A △ λ1 (A)In A= . A λ1 (A)In Then, A is positive semidefinite, and rank A = 2n − amultA [λ1 (A)]. Fact 10.12.27. Let A ∈ Fn×m, and define [ ] A∗ △ ⟨A⟩ A= . A ⟨A∗ ⟩ Then, A is positive semidefinite, and rank A = rank A. If, in addition, n = m, then −⟨A⟩ − ⟨A∗ ⟩ ≤ A + A∗ ≤ ⟨A⟩ + ⟨A∗ ⟩. Source:[ Fact 10.12.9. The rank equality follows from Proposition 10.2.4 and Fact 10.24.4. Re] ⟨A∗ ⟩ A∗ mark: A ⟨A⟩ is not necessarily positive semidefinite. See [2991, p. 317] and Fact 10.12.52. Related: Fact 10.10.32, Fact 10.24.4, and [2991, p. 222]. Fact 10.12.28. Let A ∈ Fn×m, let α ∈ [0, 1], and define

[

⟨A⟩2α A= A △

] A∗ . ⟨A∗ ⟩2(1−α)

Then, A is positive semidefinite. Source: [2991, p. 318]. Fact 10.12.29. Let A ∈ Fn×n, assume that A is normal, and define [ ] A △ ⟨A⟩ A= . A∗ ⟨A⟩ Then, A is positive semidefinite. Source: Use Fact 10.12.27 and ⟨A⟩ = ⟨A∗ ⟩. See [1450, p. 213]. Fact 10.12.30. Let A ∈ Fn×n, and define [ ] I A △ A= ∗ . A I Then, A is positive semidefinite if and only if σmax (A) ≤ 1. Furthermore, A is positive definite if and only if σmax (A) < 1. Source: Note that [ ] [ ][ ][ ] I A I 0 I 0 I A = ∗ . A∗ I A I 0 I − A∗A 0 I Fact 10.12.31. Let A ∈ Fn×m, and define △

A=

[

In A∗

] A . Im

754

CHAPTER 10

Then, A is (positive semidefinite, positive definite) if and only if A is (semicontractive, contractive). Furthermore,      m   n      ∗ In A =  0  + In(Im − A A) =  0  + In(In − AA∗ ).     0 0 [ I A ]∗ [ I 0 ] [ I A ] n n Source: Note that A = 0n Im 0 Im . See [2991, p. 259]. 0 Im −A∗A n×n Fact 10.12.32. Let A ∈ F , assume that A is positive semidefinite, assume that spec(A) ⊂ [0, 1], and define A ∈ F2n×2n by [ ] A (A − A2 )1/2 △ A= . (A − A2 )1/2 I−A Then, A is a projector, and rank A = n. Source: [2648] and Fact 4.17.13. Fact 10.12.33. Let A ∈ Fn×m and B ∈ Fn×l, and define [ ∗ ] A A A∗B △ A= ∗ . B A B∗B Then, A is positive semidefinite, and 0 ≤ A∗B(B∗B)+B∗A ≤ A∗A. If m = l, then −A∗A − B∗B ≤ A∗B + B∗A ≤ A∗A + B∗B, If m = l = 1, then

(A + B)∗ (A + B) ≤ 2(A∗A + B∗B).

|A∗B|2 ≤ A∗AB∗B.

Remark: The last inequality is the Cauchy-Schwarz inequality. See Fact 10.16.25. Related: Fact

10.25.57. Fact 10.12.34. Let A, B ∈ Fn×m, and define

[

I + A∗A A= A+B △

] [ A∗ + B∗ I = I + BB∗ B

A∗ I

][

I A

] B∗ . I

Then, A is positive semidefinite. Furthermore, there exists a semicontractive matrix C ∈ Fm×n such that A + B = (I + A∗A)1/2C(I + BB∗ )1/2 . Source: Fact 10.12.15 and [2991, pp. 187, 222, 228]. Related: Fact 10.16.27. Fact 10.12.35. Let A, B ∈ Fn×m, and define

[

I + A∗A A= I − B∗A △

] I − A∗B , I + B∗B

[

I + A∗A B= I + B∗A △

] I + A∗B . I + B∗B

Then, A and B are positive semidefinite, and 0 ≤ (I − A∗B)(I + B∗B)−1 (I − B∗A) ≤ I + A∗A, 0 ≤ (I + A∗B)(I + B∗B)−1 (I + B∗A) ≤ I + A∗A. Related: Fact 10.16.28. Fact 10.12.36. Let A, B ∈ Fn×m. Then,

(A + B)(I + B∗B)−1 (A + B)∗ ≤ (A + B)(I + B∗B)−1 (A + B)∗ + (I − AB∗ )(I + BB∗ )−1 (I − BA∗ ) = I + AA∗ .

755

POSITIVE-SEMIDEFINITE MATRICES

Source: Set C = A in Fact 3.20.19. See also [2983, p. 185] and [2991, pp. 228–231]. Fact 10.12.37. Let A, B ∈ Fn×n, assume that A and B are positive semidefinite, and define

[

A A = 1/2 1/2 B A △

] [ 1/2 ] A1/2B1/2 A = 1/2 [A1/2 B1/2 ]. B B

Then, A is positive semidefinite Fact 10.12.38. Let A ∈ Fn×n and B ∈ Fn×m, assume that A is positive semidefinite, and define [ ] [ 1/2 ] A AB A △ A= ∗ = ∗ 1/2 [A1/2 A1/2B]. B A B∗AB BA Then, A is positive semidefinite, and 0 ≤ AB(B∗AB)+B∗A ≤ A. If, in addition n = m, then

−A − B∗AB ≤ AB + B∗A ≤ A + B∗AB.

Now, let α ∈ (0, ∞). Then,

0 ≤ AB(αI + B∗AB)−1B∗A ≤ A.

Fact 10.12.39. Let A ∈ Fn×n and B ∈ Fn×m, assume that A is positive definite, and define △

A= [

Then, A=

[

A B∗

] B . B∗A−1B

] [ A1/2 I 1/2 −1/2 [A A B] = B∗A−1/2 0

0 B∗

][

A I

I A−1

][

I 0

] 0 , B

and thus A is positive semidefinite. Furthermore, 0 ≤ B(B∗A−1B)+B∗ ≤ A. Furthermore, if rank B = m, then rank[A − B(B∗A−1B)−1B∗ ] = n − m. Now, assume that n = m. Then, −A − B∗A−1B ≤ B + B∗ ≤ A + B∗A−1B. Source: Fact 10.12.9 and [2991, p. 223]. Remark: I − A−1/2B(B∗A−1B)+B∗A−1/2 is a projector. Related: Fact 10.25.58. Fact 10.12.40. Let A ∈ Fn×n , B ∈ Fn×m, and C ∈ [ Fm×m] , assume that A is positive definite, △

assume that C is positive semidefinite, and define A = BA∗ CB . Then, A is positive semidefinite if and only if B∗A−1B ≤ C. Source: [2991, p. 220]. Related: Fact 10.12.65 and Fact 10.12.66. [ ] [ ∗] Fact 10.12.41. Let A, B ∈ Fn×n, and assume that BA∗ AB is positive semidefinite. Then, AB BA [ ] is positive semidefinite. Source: Consider a congruence transformation with 0I 0I . See [2991, p. [ ] [ ] B is positive semidefinite, then A B∗ is positive semidefinite. Now, let 317]. Remark: If BA∗ αI B αI [ ] [ ∗] C ∈ Fn×n, and assume that BA∗ CB is positive semidefinite. Then, AB BC is not necessarily positive semidefinite. See [1840], [2991, p. 317], and Fact 10.12.52. [ xx∗ +x∗ xI yx∗ +x∗ yI ] Fact 10.12.42. Let x, y ∈ Fn . Then, xy∗ +y∗ xI yy∗ +y∗ yI is positive semidefinite. Source: [1834, [ xx∗ xy∗ ] [ x∗ xI x∗ yI ] [ ∗ yx∗ ] 1837]. Remark: yx∗ yy∗ and y∗ xI y∗ yI are positive semidefinite, but xx xy∗ yy∗ is not necessarily positive semidefinite. See Fact 10.12.41.

756

CHAPTER 10

n×n n×m Fact 10.12.43. ] A ∈ R and B ∈ R , assume that A is symmetric, m < n, and rank B = m, [ Let △

and define A =

0 BT B A

. Then, the following statements are equivalent:

i) For all x ∈ N(BT ), xTAx > 0. ii) For all i ∈ {2m + 1, . . . , n + m}, sign det A({1,...,i}) = (−1)m . Source: [1895, p. 48]. Related: Fact 10.12.44. Fact 10.12.44.[ Let]A ∈ Fn×n and B ∈ Fn×m, assume that A is positive semidefinite and rank B = △ m, and define A = BA∗ B0 . Then, the following statements are equivalent: i) A is nonsingular. ii) N(A) ∩ N(B∗ ) = {0}. iii) For all x ∈ N(B∗ ), x∗Ax > 0. iv) A + BB∗ is positive definite. Source: [558, p. 523]. Remark: A is the KKT matrix. Related: Fact 10.12.43. n×n n×m Fact 10.12.45. [ Let ] A ∈ F and B ∈ F , assume that A is positive definite and rank B = m, △ ∗ −1 A B and define A = B∗ 0 . Then, B A B is positive definite, det A = (−1)m (det A) det B∗A−1B, A is nonsingular, and   −1  A − A−1B(B∗A−1B)−1B∗A−1 A−1B(B∗A−1B)−1   . A−1 =  (B∗A−1B)−1B∗A−1 −(B∗A−1B)−1 Furthermore,

[

A2 + BB∗ A = B∗A

AB B∗B

2

]

is positive definite, and AB(B∗B)−1B∗A < A2 + BB∗ . Source: Proposition 3.9.3 and Proposition 3.9.7. Fact 10.12.46. Let A, B ∈ Fn×n, and define [ ] A∗ + B∗ △ ⟨A⟩ + ⟨B⟩ A= . A+B ⟨A∗ ⟩ + ⟨B∗ ⟩ Then, A is positive semidefinite. Furthermore, (det ⟨A + B⟩)2 ≤ det(⟨A⟩ + ⟨B⟩) det(⟨A∗ ⟩ + ⟨B∗ ⟩). In particular,

det ⟨A + A∗ ⟩ ≤ det(⟨A⟩ + ⟨A∗ ⟩).

Source: [2991, p. 317].

[

]

A +B AB+BA is positive semidefinite. Source: [1840]. Fact 10.12.47. Let A, B ∈ Hn . Then, AB+BA A2 +B2 n×n Fact 10.12.48. Let A, B ∈ F , assume that A is positive definite, and let S ⊆ {1, . . . , n}. Then, 2

2

B∗(S) (A−1 )(S) B(S) ≤ (B∗A−1B)(S) . Source: [2991, p. 221]. Fact 10.12.49. Let A ∈ Fn×n and B ∈ Fn×m, assume that A is positive definite, and define

[

B∗AB A= B∗B △

[

Then, A=

] B∗B . B∗A−1B

] B∗A1/2 [A1/2B A−1/2B], B∗A−1/2

757

POSITIVE-SEMIDEFINITE MATRICES

and thus A is positive semidefinite. Furthermore, 0 ≤ B∗B(B∗A−1B)+B∗B ≤ B∗AB. Now, assume in addition that n = m. Then, −B∗AB − B∗A−1B ≤ 2B∗B ≤ B∗AB + B∗A−1B. Source: Fact 10.12.9. Related: Fact 10.16.26 and Fact 10.25.58. Fact 10.12.50. Let A, B, C ∈ Fn×m, assume that A and B are positive definite, and assume that [ −1 −1 −1 A ] C is Hermitian. Then, A+B ≤ C. Source: A A+C is positive semidefinite if and only if −(A + B )

[2991, p. 222]. Fact 10.12.51. Let A, B ∈ Fn×m, let α, β ∈ (0, ∞), and define

] β −1I + αA∗A (A + B)∗ A= . A+B α−1I + βBB∗ △

Then,

[ A=

β−1/2I β1/2B

α1/2A∗ α−1/2I

][

[

β−1/2I α1/2A

] [ ∗ ] [ −1 β1/2B∗ αA A A∗ β I = + α−1/2I A α−1I B

] B∗ , βBB∗

and thus A is positive semidefinite. Furthermore, (A + B)∗ (α−1I + βBB∗ )−1 (A + B) ≤ β −1I + αA∗A. Now, assume in addition that n = m. Then, −(β−1/2 + α−1/2 )I − αA∗A − βBB∗ ≤ A + B + (A + B)∗ ≤ (β−1/2 + α−1/2 )I + αA∗A + βBB∗. Related: Fact 10.16.29 and Fact 10.25.61. Fact 10.12.52. Let A, B ∈ Fn×m, and assume that I −A∗A and thus I −AA∗ are nonsingular. Then,

I − B∗B = (I − B∗A)(I − A∗A)−1(I − A∗B) − (A − B)∗ (I − AA∗ )−1 (A − B). Now, assume that I − A∗A is positive definite. Then, I − B∗B ≤ (I − B∗A)(I − A∗A)−1(I − A∗B). Now, assume that I − B∗B is positive definite. Then, I − A∗B is nonsingular. Next, define [ ] [ ] ∗ −1 (I − B∗A)−1 (I − A∗A)−1 (I − A∗B)−1 △ (I − A A) △ A= , A0 = . (I − A∗B)−1 (I − B∗B)−1 (I − B∗A)−1 (I − B∗B)−1 Then, A and A0 are positive semidefinite. Furthermore, −(I − A∗A)−1 − (I − B∗B)−1 ≤ (I − B∗A)−1 + (I − A∗B)−1 ≤ (I − A∗A)−1 + (I − B∗B)−1, tr (I − A∗B)−1 (I − B∗A)−1 ≤ tr (I − A∗A)−1 (I − B∗B)−1 . Source: For the first equality, set D = −B∗ and C = −A∗, and replace B with −B in Fact 3.20.16.

See [93, 2184] and [2991, p. 231]. The penultimate string follows from Fact 10.12.9. The last inequality is given in [1840]. Remark: The equality is Hua’s matrix equality. This result does not assume that either I − A∗A or I − B∗B is positive semidefinite. The inequality and Fact 10.16.28 are Hua’s inequalities. See [2184, 2933]. A generalization is given in Fact 3.20.17. Remark: The fact that A0 is positive semidefinite is noted in [1844]. See Fact 10.12.27. Remark: Extensions to the case where I − A∗A is singular are considered in [2184]. Related: Fact 10.10.44 and Fact 10.16.28.

758

CHAPTER 10

Fact 10.12.53. Let X ∈ Fn×m, define U ∈ F(n+m)×(n+m) by

[

△

U=

(I + X∗X)−1/2 (I + XX∗ )−1/2X

] −X∗ (I + XX∗ )−1/2 , (I + XX∗ )−1/2

and let A ∈ Fn×n, B ∈ Fn×m, C ∈ Fm×n, and D ∈ Fm×m. Then, the following statements hold: i) Assume that D is nonsingular and X = D−1C. Then, [ ] [ ] A B (A − BX)(I + X∗X)−1/2 (B + AX ∗ )(I + XX∗ )−1/2 = U. C D 0 D(I + XX∗ )1/2 ii) Assume that A is nonsingular and X = CA−1. Then, [ ] [ ] A B (I + X∗X)1/2A (I + X∗X)−1/2 (B + X∗D) =U . C D 0 (I + XX∗ )−1/2 (D − XB) Source: [1324]. Remark: See Proposition 3.9.3 and Proposition 3.9.4. Fact 10.12.54. Let X ∈ Fn×m, assume that σmax (X) < 1, define U ∈ F(n+m)×(n+m) by △

U=

[

(I − X∗X)−1/2 (I − XX∗ )−1/2X

] X∗ (I − XX∗ )−1/2 , (I − XX∗ )−1/2

and let A ∈ Fn×n, B ∈ Fn×m, C ∈ Fm×n, and D ∈ Fm×m. Then, the following statements hold: i) Assume that D is nonsingular and X = D−1C. Then, [ ] [ ] A B (A − BX)(I − X∗X)−1/2 (B + AX ∗ )(I − XX∗ )−1/2 = U. C D 0 D(I − XX∗ )1/2 ii) Assume that A is nonsingular and X = CA−1 . Then, [ ] [ ] A B (I − X∗X)1/2A (I − X∗X)−1/2 (B − X∗D) =U . C D 0 (I − XX∗ )−1/2 (D − XB) Source: [1324]. Related: Proposition 3.9.3 and Proposition 3.9.4. Fact 10.12.55. Let A, B ∈ Fn×m and C, D ∈ Fm×m, assume that C and D are positive definite,

and define △

A=

[

AC −1A∗ + BD−1B∗ (A + B)∗

] A+B . C+D

Then, A is positive semidefinite, and (A + B)(C + D)−1(A + B)∗ ≤ AC −1A∗ + BD−1B∗. Now, assume that n = m. Then, −AC −1A∗ − BD−1B∗ − C − D ≤ A + B + (A + B)∗ ≤ AC −1A∗ + BD−1B∗ + C + D. Source: [1355, 1819] and [2263, p. 151]. Remark: Replacing A, B, C, D by αB1, (1 − α)B2 , αA1 , (1 − α)A2 yields xiv) of Proposition 10.6.17. Fact 10.12.56. Let A ∈ Rn×n, assume that A is positive definite, and let S ⊆ {1, . . . , n}. Then,

(A(S) )−1 ≤ (A−1 )(S) . Source: [1448, p. 474]. Remark: Generalizations of this result are given in [727]. Fact 10.12.57. Let A, D ∈ Fn×n, B ∈ Fn×m, and C ∈ Fm×m, and assume that

positive semidefinite, C is positive definite, and D is positive definite. Then, definite.

[

A B ∗ [ B C] A+D B B∗ C

]

∈ Fn×n is

is positive

759

POSITIVE-SEMIDEFINITE MATRICES

Fact 10.12.58. Let A ∈ Fn×n , B ∈ Fn×m, and C ∈ Fm×m , and assume that

[

A B B∗ C

]

is positive

semidefinite. Then, there exist unitary matrices U, V ∈ F such that [ ] [ ] [ ] A B A 0 ∗ 0 0 ∗ =U U +V V . B∗ C 0 0 0 C (n+m)×(n+m)

Now, let f : [0, ∞) 7→ R, and assume that f is concave. Then, ([ ]) A B tr f ≤ tr f (A) + tr f (C). B∗ C Source: [544, 546, 548]. Related: Fact 10.12.60 and Fact [ 11.12.6. ] n×n Fact 10.12.59. Let A, B, C ∈ F , and assume that BA∗ CB is positive semidefinite. Then, there

exist unitary matrices U, V ∈ F(n+m)×(n+m) such that [ ] [ ] [ A B A + C + ȷ(B∗ − B) 0 ∗ 1 0 1 = U U + V 2 2 B∗ C 0 0 0

] 0 V ∗. A + C + ȷ(B − B∗ )

Source: [548]. Related: Fact 10.12.58. [ ] Fact 10.12.60. Let A, B, C ∈ Cn×n , and assume that B is Hermitian and AB CB is positive semidef-

inite. Then, there exist left-inner matrices U, V ∈ F2n×n such that [ ] A B = 12 [U(A + C)U ∗ + V(A + C)V ∗ ]. B C

Source: [546, 549]. Related: Fact 10.12.58. Fact 10.12.61. Let A ∈ F(n+m+l)×(n+m+l), assume that A is Hermitian, assume that A is partitioned

  A11  A =  A∗12  ∗ A13

as

△

and assume that B =

[

A11 A∗12

A12 A22 A∗23

 A13   A23  ,  A33

] A12 is positive definite. Then, A22

∗ ∗ −1 A∗23 A−1 22 A23 ≤ [A13 A23 ]B

Equivalently,

[ A B|A ≤ A22 ∗22 A23

[

] A13 . A23

] A23 . A33

Source: [3007]. Fact 10.12.62. Let A ∈ F(n+m+l)×(n+m+l), assume that A is positive semidefinite, and assume that

A has the form

  A11  A =  A∗12  0

A12 A22 A∗23

 0   A23  .  A33

Then, there exist positive-semidefinite matrices B, C ∈ F(n+m+l)×(n+m+l) such that A = B + C and such that B and C have the form      B11 B12 0   0 0 0      B =  B∗12 B22 0  , C =  0 C22 C23  .     ∗ 0 0 0 0 C23 C33 Source: [1373].

760

CHAPTER 10

Fact 10.12.63. For all i, j ∈ {1, . . . , k}, let Ai j ∈ Fni ×n j , define

  A11  . △  A =  ..  A1k

··· . · .· · ···

 A1k   ..  .  ,  Akk

and assume that A is square and (positive semidefinite, positive definite). Furthermore, define  ˆ  A11 · · · Aˆ 1k   . ..   . △  · .· · Aˆ =  .. .  ,   Aˆ 1k · · · Aˆ kk where Aˆ i j = 11×ni Ai j 1n j ×1 is the sum of the entries of Ai j for all i, j ∈ {1, . . . , k}. Then, Aˆ is (positive ∑k semidefinite, positive definite). Source: Aˆ = BABT, where the entries of B ∈ Rk× i=1 ni are 0’s and 1’s. See [87] and [2991, p. 224]. Fact 10.12.64. For all i, j ∈ {1, . . . , k}, let Ai j ∈ Fn×n , define A ∈ Fkn×kn by    A11 · · · A1k   . ..   . △  · .· · A =  .. .  ,   ∗ A1k · · · Akk and assume that A is positive semidefinite. Then,    2  tr A11  tr A11 · · · tr A1k       .. .. . ..  ≥ 0,  · .· ·   . . .   ∗   ∗ tr A1k A1k tr A1k · · · tr Akk Furthermore,

  tr A11 1  det A ≤ kn det  ...  k tr A∗1k

··· . · .· · ···

··· . · .· · ···

 tr A∗1k A1k   ..  ≥ 0. .  tr A2kk

k tr A1k   ..  . .   tr Akk

Source: [852, 1953, 2206] and [2991, p. 237]. The last inequality is given in [1853]. Related:

Fact 10.16.50. Fact[ 10.12.65. Let A ∈ Fn×n, B ∈ Fn×m, and C ∈ Fm×m, and assume that A ∈ F(n+m)×(n+m) defined ] △

by A =

A B B∗ C

is positive semidefinite. Then, for all i ∈ {1, . . . , min {n, m}}, 2σi (B) ≤ σi (A).

Source: [474, 2587] and [2991, p. 355]. Related: Fact 10.12.40 and Fact 10.12.66. △

Fact 10.12.66. Let A ∈ Fn×n, B ∈ Fn×m, and C ∈ Fm×m, and assume that A =

[

A B B∗ C

F(n+m)×(n+m) is positive semidefinite. Then, max{σmax (A), σmax (B)} ≤ σmax (A) ≤ 21 [σmax (A) + σmax (B) +

√

[σmax (A) − σmax (B)]2 + 4σ2max (C)]

≤ σmax (A) + σmax (B), max{σmax (A), σmax (B)} ≤ σmax (A) ≤ max {σmax (A), σmax (B)} + σmax (C).

]

∈

761

POSITIVE-SEMIDEFINITE MATRICES

[ (A) σ (B) ] [ (A) ρ (B) ] max max Furthermore, if n = m, then σσmax and ρρmax are positive semidefinite. Source: max (B) σmax (C) max (B) ρmax (C) [1465]. Related: Fact 10.12.40, Fact 10.12.65, and Fact 11.16.15.

10.13 Facts on the Trace for One Matrix Fact 10.13.1. Let A ∈ Fn×n, and assume that A is positive semidefinite. Then, A = 0 if and only

if tr A = 0.

Fact 10.13.2. Let A ∈ Fn×n. Then,

| Re tr A| ≤ | tr A| ≤

n ∑

ρi (A) ≤ tr ⟨A⟩ =

i=1

n ∑

σi (A).

i=1

Furthermore, consider the following statements: i) A is positive semidefinite ii) tr A = tr ⟨A⟩. iii) | tr A| = tr ⟨A⟩. iv) There exists α ∈ C such that |α| = 1 and αA is positive semidefinite. v) A is normal. ∑ vi) ni=1 ρi (A) = tr ⟨A⟩. Then, i) ⇐⇒ ii) =⇒ iii) ⇐⇒[iv) =⇒ ] v) =⇒ vi). Source: Fact 7.12.30, Fact 10.21.10, and [2991, 0 shows that v) does not imply iv). Related: Fact 10.21.10 states pp. 312, 313]. Remark: A = 10 −1 ∑ that A is normal if and only if ni=1 ρ2i (A) = tr ⟨A⟩2 . Additional necessary and sufficient conditions are given by Fact 7.12.30. See [1499]. Fact 10.13.3. Let A ∈ Fn×n. Then,  ∗    tr ⟨A⟩⟨A ⟩ 2 | tr A | ≤    tr ⟨A2 ⟩ ≤ tr ⟨A⟩2 = tr A∗A. Source: For the upper inequality, see [1783, 2987]. For the lower inequalities, use Fact 10.21.9 and Fact 11.13.2. Related: Fact 7.12.13, Fact 10.13.5, and Fact 10.13.6. Fact 10.13.4. Let A ∈ Fn×n. Then, tr ⟨A⟩ = tr ⟨A∗ ⟩. Fact 10.13.5. Let A ∈ Fn×n. Then, for all k ∈ {1, . . . , n}, k ∑

σi (A2 ) ≤

i=1

Hence,

k ∑

σ2i (A).

i=1

tr (A2∗A2 )1/2 ≤ tr A∗A.

Equivalently, tr ⟨A2 ⟩ ≤ tr ⟨A⟩2. Source: Let B = A and r = 1 in Proposition 11.6.2. See also Fact 11.13.2. Fact 10.13.6. Let A ∈ Fn×n, and let k denote the number of nonzero eigenvalues of A. Then,  | tr A2 | ≤ tr ⟨A2 ⟩       tr ⟨A⟩⟨A∗ ⟩  ≤ tr ⟨A⟩2.     1 | tr A|2  k

Source: The upper bound for | tr A | is given by Fact 11.13.2. The upper bound for tr ⟨A2 ⟩ is given by 2

Fact 10.13.5. To prove the next inequality, let A = S 1DS 2 denote the singular value decomposition △ of A. Then, tr ⟨A⟩⟨A∗ ⟩ = tr S 3∗ DS 3 D, where S 3 = S 1 S 2 , and tr A∗A = tr D2. The result follows from Fact 7.13.12. The last inequality is given by Fact 7.12.13. Related: Fact 7.12.13 and Fact 11.13.2.

762

CHAPTER 10

Fact 10.13.7. Let A ∈ Fn×n. Then, the following statements are equivalent:

i) A is Hermitian, and tr A > 0. ii) There exists matrices B, S ∈ Fn×n such that S is nonsingular, SBS −1 is positive definite, and A = B + B∗. Source: [2991, p. 265]. Fact 10.13.8. Let A ∈ Fn×n, assume that A is positive definite, let p and q be real numbers, and assume that p ≤ q. Then, ) ( )1/q ( p 1/p 1 ≤ n1 tr Aq . n tr A Furthermore, lim p↓0

(

) p 1/p 1 n tr A

= det A1/n.

Source: Fact 2.11.86. Fact 10.13.9. Let A ∈ Fn×n, and assume that A is positive definite. Then,

n2 ≤ (tr A) tr A−1. Finally, equality holds if and only if A = In . Related: Fact 10.25.23. Bounds on tr A−1 are given in [211, 682, 2170, 2318]. Fact 10.13.10. Let A ∈ Fn×n, assume that A is positive definite, and let α > 0. Then, n tr A ≤ (tr A1+α ) tr A−α . In particular,

n tr A ≤ (tr A3/2 ) tr A−1/2 .

Source: [2751]. Fact 10.13.11. Let A ∈ Fn×n, assume that A is Hermitian, and consider the following statements:

√ i) (n − 1) tr A2 ≤ tr A. ii) A is positive semidefinite. iii) tr A2 ≤ (tr A)2 ≤ n tr A2 . Then, i) =⇒ ii) =⇒ iii). Source: [2991, pp. 205, 213]. The second inequality in iii) follows from Fact 2.11.16 with k = 1. △ △ Fact 10.13.12. Define S = {A ∈ Nn : tr A ≤ 1}, and define f : S 7→ R by f (A) = (tr A) det(I −A). Then, f is concave. Source: [928]. Related: Fact 2.12.53. Fact 10.13.13. Let A ∈ Fn×n, and assume that A is positive semidefinite. Then, tr eA ≤ etr A + n − 1.

Furthermore, equality holds if and only if rank A ≤ 1. Source: [2991, p. 348]. Fact 10.13.14. Let A ∈ Fn×n, and assume that A is positive semidefinite. Then, the following statements hold: i) Let r ∈ [0, 1]. Then, for all k ∈ {1, . . . , n}, n n ∑ ∑ λri (A) ≤ dri (A). i=k

In particular, tr Ar ≤

i=k n ∑ [A(i,i) ]r. i=1

763

POSITIVE-SEMIDEFINITE MATRICES

ii) Let r ≥ 1. Then, for all k ∈ {1, . . . , n}, k ∑

dri (A) ≤

k ∑

i=1

In particular,

λri (A).

i=1

n ∑ [A(i,i) ]r ≤ tr Ar. i=1

iii) If either r = 0 or r = 1, then tr Ar =

n ∑ [A(i,i) ]r . i=1

iv) If r , 0 and r , 1, then tr Ar =

n ∑ [A(i,i) ]r i=1

if and only if A is diagonal. Source: Fact 3.25.8, Fact 10.21.11, [1922], and [1924, p. 217]. Related: Fact 10.21.11. Fact 10.13.15. Let A ∈ Fn×n and k ≥ 1. Then,

tr A∗kAk ≤ tr (A∗A)k ≤ (tr A∗A)k . In particular,

tr A∗2A2 ≤ tr (A∗A)2 ≤ (tr A∗A)2 .

Source: Fact 10.13.16 and [2991, pp. 197, 371]. Fact 10.13.16. Let A ∈ Fn×n and p, q ∈ [0, ∞). Then,

tr (A∗pA p )q ≤ tr (A∗A) pq. Furthermore, equality holds if and only if tr A∗pA p = tr (A∗A) p. Source: [2476] and [2991, p. 197]. Fact 10.13.17. Let A ∈ Fn×n, p ∈ [2, ∞), and q ∈ [1, ∞). Then, A is normal if and only if tr (A∗pA p )q = tr (A∗A) pq. Source: [987, 2476]. Related: Fact 4.10.12. △ Fact 10.13.18. Let A ∈ Fn×n, and define B = 21 (A + A∗ ). Then, the following statements hold:

i) If B is positive semidefinite and p ≥ 0, then tr B p ≤ tr ⟨A⟩ p . ii) If B is positive definite and p ≤ 0, then tr ⟨A⟩ p ≤ tr B p . iii) If k ≥ 0, then tr B2k+1 ≤ tr ⟨A⟩2k+1. Source: [1142]. Remark: Setting k = 0 in iii) yields tr 21 (A + A∗ ) ≤ tr ⟨A⟩, which is given by Fact 7.12.25.

10.14 Facts on the Trace for Two or More Matrices Fact 10.14.1. Let A, B ∈ Fn×n , assume that A and B are Hermitian, and assume that, for all

i ∈ {1, . . . , 3n}, tr (A + B)i = tr Ai + tr Bi . Then, the following statements hold: i) AB = 0. ii) R(A + B) = R(A) + R(B). iii) rank(A + B) = rank A + rank B. Source: [2236] and [2991, p. 286].

764

CHAPTER 10

Fact 10.14.2. Let A1 , . . . , Am ∈ Fn×n , assume that A1 , . . . , Am are normal, and assume that, for

∑ ∑m k k all k ∈ {1, . . . , (m + 1)n}, tr ( m i=1 Ai ) = i=1 tr Ai . Then, for all distinct i, j ∈ {1, . . . , m}, Ai A j = 0. Source: [1842]. Fact 10.14.3. Let A, B ∈ Fn×n, and assume that A and B are Hermitian. Then, √ √ √ √ tr (A + B)2 ≤ tr A2 + tr B2 . | tr AB| ≤ (tr A2 ) tr B2 ≤ 12 tr(A2 + B2 ), The first inequality is an equality if and only if A and B are linearly dependent. The second inequality is an equality if and only if tr A2 = tr B2. All three terms are equal if and only if A = B. Source: Corollary 11.3.9 and [2991, p. 264]. Related: Fact 10.14.22. Fact 10.14.4. Let A, B ∈ Fn×n, assume that A and B are Hermitian, and assume that −A ≤ B ≤ A. Then, tr B2 ≤ tr A2. Source: 0 ≤ tr[(A − B)(A + B)] = tr A2 − tr B2. See [2697]. Remark: For 0 ≤ B ≤ A, this result is

a special case of xxi) of Proposition 10.6.13. Fact 10.14.5. Let A, B ∈ Fn×n, and assume that A and B are positive semidefinite. Then, AB = 0 if and only if tr AB = 0. Fact 10.14.6. Let A, B ∈ Fn×n, assume that A and B are positive semidefinite, let p, q ∈ (1, ∞), and assume that 1/p + 1/q = 1. Then, tr AB ≤ tr ⟨AB⟩ ≤ (tr Ap )1/p (tr Bq )1/q ≤ tr ( 1p Ap + q1 Bq ). Equality holds in at least one inequality if and only if A p−1 and B are linearly dependent. Source: [1922], [1924, pp. 219, 222], and [2991, p. 370]. Remark: This is a matrix version of H¨older’s inequality. Related: Fact 10.14.8 and Fact 10.14.21. Fact 10.14.7. Let f : Pn 7→ (0, ∞) be given by one of the following definitions: i) f (A) = (det A)1/n . ii) p ∈ (0, 1) and f (A) = (tr A p )1/p . iii) p ∈ (0, 1) and f (A) = (tr A p−1 )1/(p−1) . p iv) p ∈ [0, 1] and f (A) = trtrAAp−1 . Then, for all A, B ∈ Pn , f (A) + f (B) ≤ f (A + B). Source: [1779]. Remark: f is an anti-norm on Pn . Remark: i) gives the Minkowski determinant theorem given by Corollary 10.4.15. Related: Fact 2.2.59, Fact 11.8.21, and Fact 11.10.4. Fact 10.14.8. Let A1 , . . . , Am ∈ Fn×n, assume that A1 , . . . , Am are positive semidefinite, let p1 , . . . , pm ∈ [1, ∞), and assume that p11 + · · · + p1m = 1. Then, tr ⟨A1 · · · Am ⟩ ≤

m ∏

(tr Aipi )1/pi ≤ tr

i=1

m ∑

1 pi pi Ai .

i=1

Furthermore, the following statements are equivalent: ∏ pi 1/pi i) tr ⟨A1 · · · Am ⟩ = m . i=1 (tr Ai ) ∑ m 1 pi ii) tr ⟨A1 · · · Am ⟩ = tr i=1 pi Ai . iii) A1p1 = · · · = Ampm. Source: [1940]. Remark: The first inequality is a matrix version of H¨older’s inequality. The first and third terms are a matrix version of Young’s inequality. See Fact 2.2.50, Fact 2.2.53, Fact 10.11.73, Fact 10.14.33, Fact 10.14.34, and Fact 10.11.38. Fact 10.14.9. Let A1 , . . . , Am ∈ Fn×n, assume that A1 , . . . , Am are positive semidefinite, let

765

POSITIVE-SEMIDEFINITE MATRICES

∑ α1 , . . . , αm be nonnegative numbers, and assume that m i=1 αi ≥ 1. Then, m m ∏ ∏ tr (tr Ai )αi. Aαi i ≤ i=1 i=1 ∑ Furthermore, if m α = 1, then equality holds if and only if A2 , . . . , Am are scalar multiples of A1 , ∑m i=1 i whereas, if i=1 αi > 1, then equality holds if and only if A2 , . . . , Am are scalar multiples of A1 and ∑ ∏m ∑m αi rank A1 = 1. Source: [709]. Remark: If m i=1 αi = 1, then i=1 (tr Ai ) ≤ i=1 αi tr Ai . See Fact 2.11.87. Related: Fact 10.14.6. Fact 10.14.10. Let A, B ∈ Fn×n, let p, q ∈ (1, ∞), assume that 1/p + 1/q = 1, and let r ∈ [1, ∞). Then, tr ⟨AB⟩r ≤ (tr ⟨A⟩rp )1/p (tr ⟨B⟩rq )1/q . Source: [2439]. Related: Fact 10.14.11. ∑ Fact 10.14.11. Let A1 , . . . , Am ∈ Fn×n, let p1 , . . . , pm ∈ (1, ∞), assume that m i=1 1/pi = 1, and

let r ∈ [1, ∞). Then,

tr ⟨A1 A2 · · · Am ⟩r ≤

m m ∏ ∑ rpi 1 (tr ⟨Ai ⟩rpi )1/pi ≤ tr pi ⟨Ai ⟩ . i=1

i=1

Now, assume that A1 , . . . , Am are positive semidefinite. Then, | tr A1 A2 · · · Am | ≤ m

Finally, if α1 , . . . , αm ∈ [0, 1] and

∑m i=1

m ∏

tr Am i .

i=1

αi = 1, then

tr ⟨Aα1 1 Aα2 2 · · · Aαmm ⟩r ≤

m ∏ i=1

(tr Ari )αi ≤

m ∑

αi tr Ari .

i=1

Source: [2439]. Related: Fact 10.14.10. Fact 10.14.12. Let A, B ∈ Fn×n, let p, q ∈ (1, ∞), assume that 1/p + 1/q = 1, and let r ∈ [1, ∞).

Then,

tr ⟨AB∗ ⟩r ≤ tr( 1p ⟨A⟩rp + q1 ⟨B⟩rq ), tr ⟨AB∗ ⟩r ≤ tr ( 1p ⟨A⟩ p + 1q ⟨B⟩q )r .

Furthermore,

tr ⟨AB∗ ⟩ ≤

1 2

tr(A∗A + B∗B),

tr ( 21 ⟨A⟩2 + 21 ⟨B⟩2 )r ≤

1 2

tr ⟨A⟩2r + 12 tr ⟨B⟩2r .

Source: [2439]. Fact 10.14.13. Let A1 , . . . , Am , B1 , . . . , Bm ∈ Nn×n, let p, q ∈ (1, ∞), and assume that 1/p+1/q =

1. Then, tr

m ∑ i=1

 m   m 1/p m m  ∑ p 1/p  ∑  1 ∑ p 1 ∑ q p    Ai Bi ≤ tr Ai  tr Bi  ≤ tr Ai + tr B. p i=1 q i=1 i i=1 i=1

Source: [2439]. Fact 10.14.14. Let A, B ∈ Fn×n. Then, | tr AB|2 ≤ (tr A∗A) tr BB∗. Source: [2983, p. 25] and Corollary 11.3.9. Related: Fact 10.14.15. Fact 10.14.15. Let A ∈ Fn×m and B ∈ Fm×n, and let k ≥ 1. Then,

Re tr (AB)2k ≤ | tr (AB)2k | ≤ tr (A∗ABB∗ )k ≤ tr (A∗A)k(BB∗ )k ≤ [tr (A∗A)k ] tr (BB∗ )k.

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CHAPTER 10

In particular,

| tr (AB)2 | ≤ tr A∗ABB∗ ≤ (tr A∗A) tr BB∗.

Source: See [2954] for the case where n = m. If n , m, then A and B can be augmented with zeros. Remark: The term | tr AB|2 in Fact 10.14.14 cannot be ordered with | tr (AB)2 | and tr A∗ABB∗ . Fact 10.14.16. Let A, B ∈ Fn×n, assume that A and B are Hermitian, and let k ≥ 1. Then,

 2 2 k    (tr A B ) √ tr (AB) ≤ | tr (AB) | ≤ tr ⟨(AB) ⟩ ≤ tr (A B ) ≤    tr A2kB2k ≤ (tr A4k ) tr B4k , 2k

2k

2k

2 2 k

| tr (AkBk )2 | ≤ tr A2kB2k . In particular, tr (AB)2 ≤ tr A2B2 . Furthermore, tr (AB)2 = tr A2B2 if and only if A and B commute. Finally, if k is even, then  2 2 k    (tr A B ) 2k 2k 2k 2 2 k √ tr (AB) ≤ | tr (AB) | ≤ tr ⟨(AB) ⟩ ≤ tr (A B ) ≤    | tr (Ak Bk )2 | ≤ tr A2kB2k ≤ (tr A4k ) tr B4k . Source: Fact 10.14.15, [103, 2954], and [2991, pp. 260, 261, 371]. Remark: Fact 7.13.11 implies that tr (AB)2k and tr (A2B2 )k are real. Fact 10.14.17. Let A, B ∈ Fn×n, assume that A and B are positive semidefinite, and let k ≥ 1. Then, √ tr (AB)k ≤ (tr A2k ) tr B2k ≤ (tr Ak ) tr Bk .

In particular,

tr AB ≤

√ (tr A2 ) tr B2 ≤ (tr A) tr B.

Source: [2951]. Fact 10.14.18. Let A, B ∈ Fn×n, assume that A and B are positive semidefinite, and let k, m ∈ P,

where m ≤ k. Then, In particular,

tr (AmBm )k ≤ tr (AkBk )m . tr (AB)k ≤ tr AkBk ,

tr AB ≤ (tr A) tr B.

If, in addition, k is even, then tr (AB)k ≤ tr (A2B2 )k/2 ≤ tr AkBk . Finally, tr (AB)k ≤ tr AkBk if and only if either n = 1 or [A, B] = 0. Source: Fact 10.22.27, Fact 10.22.37, and [2991, pp. 207, 368, 369, 371]. Remark: Fact 7.13.11 implies that tr (AB)k is real. Remark: tr (AB)k ≤ tr AkBk is the Lieb-Thirring inequality. See [449, p. 279]. tr (AB)k ≤ tr (A2B2 )k/2 follows from Fact 10.14.24. See [2932, 2954]. Fact 10.14.19. Let A, B ∈ Fn×n, assume that A and B are positive semidefinite, and let k ≥ 1. Then, tr (AB)k ≤ min {σkmax (A) tr Bk , σkmax (B) tr Ak }. Source: [2439]. Fact 10.14.20. Let A, B ∈ Fn×n, assume that A is positive semidefinite, assume that B is Hermi-

tian, and let α ∈ [0, 1]. Then,

tr (AB)2 ≤ tr A2αBA2−2αB ≤ tr A2B2 . Source: [1111].

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POSITIVE-SEMIDEFINITE MATRICES

Fact 10.14.21. Let A, B ∈ Fn×n, and assume that A and B are positive semidefinite. Then,

tr AB ≤ tr (AB2A)1/2 = tr ⟨AB⟩ ≤ 41 tr (A + B)2 , tr (AB)2 ≤ tr A2B2 ≤

1 16 tr (A

+ B)4.

Source: Fact 10.14.24 and Fact 11.10.45. Fact 10.14.22. Let A, B ∈ Fn×n, and assume that A and B are positive semidefinite. Then,

tr AB = tr A1/2BA1/2 = tr [(A1/2BA1/2 )1/2 (A1/2BA1/2 )1/2 ] ≤ [tr (A1/2BA1/2 )1/2 ]2 ≤ (tr A)(tr B) ≤ 41 (tr A + tr B)2 ≤ 21 [(tr A)2 + (tr B)2 ], tr AB ≤ tr (AB2A)1/2 ≤

√

√ √ √ tr A2 tr B2 ≤ 14 ( tr A2 + tr B2 )2 ≤ 12 (tr A2 + tr B2 ) ≤ 21 [(tr A)2 + (tr B)2 ].

Source: Fact 2.2.12. Note that

tr (A1/2BA1/2 )1/2 =

n ∑

λ1/2 i (AB).

i=1

The second inequality follows from Proposition 11.3.6 with p = q = 2, r = 1, and A and B replaced by A1/2 and B1/2. Related: Fact 3.15.19. Fact 10.14.23. Let A, B ∈ Fn×n, assume that A and B are positive semidefinite, and let p ≥ 1. Then, tr AB ≤ tr (A p/2B pA p/2 )1/p. Source: [1108]. Fact 10.14.24. Let A, B ∈ Fn×n, assume that A and B are positive semidefinite, and let p ≥ 0

and r ≥ 1. Then, In particular,

tr (A1/2BA1/2 ) pr ≤ tr (Ar/2BrAr/2 ) p . tr (A1/2BA1/2 )2p ≤ tr (AB2A) p ,

tr AB ≤ tr (AB2A)1/2 = tr ⟨AB⟩.

Source: Fact 10.22.27 and Fact 10.22.37. Remark: This is the Araki-Lieb-Thirring inequality. See [163, 192] and [449, p. 258]. Related: Fact 10.11.78, Fact 10.22.36, and Fact 11.10.50. Fact 10.14.25. Let A, B ∈ Fn×n, assume that A and B are positive semidefinite, and let q ≥ 0

and p ∈ [0, 1]. Then,

pq σ2pq ≤ tr (ApB pAp )q ≤ tr (ABA) pq. max (A) tr B

Source: [192]. Remark: The right-hand inequality is equivalent to the Araki-Lieb-Thirring in-

equality given by Fact 10.14.24, where p and q correspond to 1/r and pr, respectively. Fact 10.14.26. Let A, B ∈ Fn×n, and assume that A and B are positive semidefinite. Then, tr B2 (ABA)2 ≤ tr A4B4 . Source: [456]. Fact 10.14.27. Let A, B ∈ Fn×n, assume that A and B are positive semidefinite, and let p, q ≥ 0.

Then,

tr AB pABq ≤ tr A2B p+q .

If, in addition, p < q, then 2 tr AB pABq ≤ tr AB pABq + tr AB2pABq−p ≤ 2 tr A2B p+q . Source: [456, 1353].

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CHAPTER 10

Fact 10.14.28. Let A, B ∈ Fn×n, assume that A and B are positive semidefinite, and let p ≥ r ≥ 0.

Then,

[tr (A1/2BA1/2 ) p ]1/p ≤ [tr (A1/2BA1/2 )r ]1/r .

Furthermore,

[tr (AB)2 ]1/2 ≤ (tr A2B2 )1/2 = (tr AB2A)1/2 ≤ tr (AB2A)1/2 ,  2 1/2    tr (AB A) [tr (A1/2BA1/2 )2 ]1/2 ≤ tr AB ≤    [tr (A1/2BA1/2 )1/2 ]2 .

Source: [829], Fact 2.11.90, Fact 10.14.20, and Fact 10.14.23. Fact 10.14.29. Let A, B ∈ Fn×n, assume that A and B are positive semidefinite, assume that

B ≤ I, and let q ≥ p > 0. Then,

tr (BA pB)1/p ≤ tr (BAqB)1/q ,

tr (BA pB)1/p ≤ tr (B p/q A pB p/q )1/p .

Source: [2693]. Fact 10.14.30. Let A, B ∈ Fn×n, assume that A and B are positive semidefinite, assume that

A ≤ B, and let p, q ≥ 0. Then,

tr ApBq ≤ tr B p+q.

If, in addition, A and B are positive definite, then this inequality holds for all p, q ∈ R satisfying q ≥ −1 and p + q ≥ 0. Source: [537]. Fact 10.14.31. Let A, B ∈ Fn×n, assume that A and B are positive semidefinite, assume that A ≤ B, let f : [0, ∞) 7→ [0, ∞), and assume that f (0) = 0, f is continuous, and f is increasing. Then, tr f (A) ≤ tr f (B). Now, let p > 1 and q ≥ max {−1, −p/2}, and, if q < 0, assume that A is positive definite. Then, tr f (Aq/2B pAq/2 ) ≤ tr f (Ap+q ). Source: [1121]. Fact 10.14.32. Let A, B ∈ Fn×n , assume that A and B are positive semidefinite, and let α ∈ [0, 1].

Then,

tr(A + B) ≤ tr ⟨A − B⟩ + 2 tr AαB1−α .

Source: [1426]. Remark: This is the Powers-Stormer inequality. Related: Fact 2.2.45 and Fact

11.10.16. Fact 10.14.33. Let A, B ∈ Fn×n, assume that A and B are positive semidefinite, let α ∈ [0, 1], △

△

and define δ = min {α, 1 − α}. ρ = max {α, 1 − α}. Then, 1 2

tr(A + B − ⟨A − B⟩) ≤ tr AαB1−α ≤ tr ⟨AαB1−α ⟩ √ √ ≤ tr ⟨AαB1−α ⟩ + δ( tr A − tr B)2 ≤ (tr A)α (tr B)1−α ≤ tr[αA + (1 − α)B] = [(tr A2α ) tr B2(1−α) ]1/2 + ρ(tr A + tr B − tr ⟨A1/2B1/2 ⟩).

Furthermore, the first two inequalities are equalities if and only if A and B are linearly dependent, while the last inequality is an equality if and only if A = B. Source: Use Fact 10.14.6 or Fact 10.14.9 for the inequality involving the first and third terms, and use Fact 2.2.53 for the right-most inequality. The inequality involving the third and fifth terms is given in [1640]. The last inequality is given in [1356]. Related: Fact 2.2.50, Fact 2.2.53, Fact 10.11.38, Fact 10.11.73, Fact 10.14.8, Fact 10.14.33, and Fact 10.16.14.

769

POSITIVE-SEMIDEFINITE MATRICES

Fact 10.14.34. Let A, B ∈ Fn×n, assume that A and B are positive definite, and let α ∈ [0, 1].

Then,

tr A−αBα−1 tr [αA + (1 − α)B]−1 tr [αA + (1 − α)B]

−1

}

≤ (tr A−1 )α (tr B−1 )1−α ≤ tr[αA−1 + (1 − α)B−1 ],

  −1 α −1 1−α       (tr A ) (tr B ) ≤ tr[αA−1 + (1 − α)B−1 ]. ≤   −1 −1/2 −1/2 α−1   tr A (A BA ) 

Remark: In the first string of inequalities, the upper left inequality and right-hand inequality are

equivalent to Fact 10.14.33. The lower left inequality is given by xxxiii) of Proposition 10.6.17. The second string of inequalities combines the lower left inequality in the first string of inequalities with the third string of inequalities in Fact 10.11.73. Remark: These inequalities interpolate the convexity of ϕ(A) = tr A−1. See Fact 2.2.53. Related: Fact 2.2.50, Fact 2.2.53, Fact 10.11.38, Fact 10.11.73, Fact 10.14.8, and Fact 10.14.33. Fact 10.14.35. Let A, B ∈ Fn×n, and assume that B is positive semidefinite. Then, | tr AB| ≤ σmax (A)tr B. Source: Proposition 10.4.13 and σmax (A + A∗ ) ≤ 2σmax (A). Related: Fact 7.13.12. Fact 10.14.36. Let A, B ∈ Fn×n, assume that A and B are positive semidefinite, and let p ∈

[1, ∞). Then,

tr(Ap + B p ) ≤ tr (A + B) p ≤ [(tr Ap )1/p + (tr B p )1/p ] p ≤ 2 p−1 (tr A p + tr B p ). Furthermore, the second inequality is an equality if and only if A and B are linearly dependent. Source: [537] and [1922]. Remark: The first inequality is the McCarthy inequality. The second

inequality is a special case of the triangle inequality for the norm ∥ · ∥σp and a matrix version of Minkowski’s inequality. Remark: Fact 11.10.53 restates these inequalities in terms of the Schatten norm, and Fact 11.10.59 extends these inequalities to more than two matrices. Fact 10.14.37. Let A, B ∈ Fn×n, assume that A and B are positive semidefinite, and let p ∈ R. Then, the following statements hold: i) If p ∈ [0, 1], then 2 p−1 (tr Ap + tr B p ) ≤ [(tr A p )1/p + (tr B p )1/p ] p ≤ tr (A + B) p ≤ tr Ap + tr B p . ii) If p ∈ [0, 1] ∪ [2, 3], then tr Ap + tr B p + (2 p − 2) tr Ap/2B p/2 ≤ tr (A + B) p , tr Ap + tr B p + (2 p − 2) tr (A1/2BA1/2 ) p/2 ≤ tr (A + B) p . iii) If p ≥ 1, then 2−1/p tr(A + B) + (1 − 21−1/p ) tr A1/2B1/2 ≤ tr [ 12 (Ap + B p )]1/p . iv) If p < 0 and A and B are positive definite, then tr (A + B) p ≤ 2 p−1 (tr Ap + tr B p ). v) If p ∈ [0, 1] ∪ [2, 3], then tr(A p + B p ) + (2 p − 2) tr A p/2B p/2 ≤ tr (A + B) p . vi) If either p ∈ [1, 2] or both A and B are positive definite and p < 0, then tr (A + B) p ≤ tr(A p + B p ) + (2 p − 2) tr A p/2B p/2 .

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CHAPTER 10

vii) If p ≥ 1, then 1 21/p

tr(A + B) + (1 − 21−1/p ) tr A1/2B1/2 ≤ tr [ 21 (A p + B p )]1/p .

viii) If p ∈ [0, 1] ∪ [2, ∞), then tr(A p + B p ) + (2 p − 2) tr (A1/2BA1/2 ) p/2 ≤ tr (A + B) p . ix) If either p ∈ [1, 2] or both A and B are positive definite and p < 0, then tr (A + B) p ≤ tr(A p + B p ) + (2 p − 2) tr (A1/2BA1/2 ) p/2 . x) 12 tr (AB)2 ≤ 4 tr(A3B + A2B2 + AB3 ) + 2 tr (AB)2 . Source: [188, 194, 2750]. The following cases are conjectures: p ∈ (3, 4) ∪ (4, ∞) in viii); p ∈ (−2, 0) in ix). Fact 10.14.38. Let A, B ∈ Fn×n, and assume that A and B are positive semidefinite. If p ∈ [0, 1], then tr [A2 + (AB)k (BA)k ] p ≤ tr [A2 + (BA)k (AB)k ] p . If p ≥ 1, then

tr [A2 + (BA)k (AB)k ] p ≤ tr [A2 + (AB)k (BA)k ] p .

Source: [1850]. Fact 10.14.39. Let A, B ∈ Fn×n, and assume that A and B are positive semidefinite. If p ∈ [0, 1],

then

tr (I + A + B + AB) p ≤ tr (I + A + B + B1/2AB1/2 ) p .

If p ≥ 1, then

tr (I + A + B + B1/2AB1/2 ) p ≤ tr (I + A + B + AB) p .

Source: [1112] and [1381, pp. 266, 267]. Fact 10.14.40. Let A, B ∈ Fn×n, and assume that A and B are positive semidefinite. Then, s

λ(A2 + BA2B) ≺ λ(A2 + AB2A). If p ∈ [0, 1], then

tr (A2 + AB2A) p ≤ tr (A2 + BA2B) p .

If p ≥ 1, then

tr (A2 + BA2B) p ≤ tr (A2 + AB2A) p .

Now, assume that A is positive definite. Then, tr (A2 + BA2B)−1 ≤ tr (A2 + AB2A)−1 . Source: [1112]. The last statement is given in [1845]. Fact 10.14.41. Let A, B ∈ Fn×n, and assume that A and B are positive definite. Then,

tr A#B ≤ tr A1/2 B1/2 ,

tr (A#B)2 ≤ tr (A1/2 B1/2 )2 ≤ tr AB,

tr[A + B + 2(A#B)] ≤ tr (A1/2 + B1/2 )2 ,

tr A(A#B) ≤ tr A3/2B1/2 ,

tr [A + B + 2(A#B)]2 ≤ tr (A1/2 + B1/2 )4 ,

tr log (A1/2 + B1/2 )2 ≤ tr log[A + B + 2(A#B)], slog

λ(A#B) ≺ λ(A1/2B1/2 ),

slog

λ(A1/2 (A#B)A1/2 ) ≺ λ(A3/4B1/2A3/4 ),

σmax [A + B + 2(A#B)] ≤ σmax (A + B + A1/2B1/2 + B1/2A1/2 ).

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POSITIVE-SEMIDEFINITE MATRICES

Furthermore, if p ∈ [1, 2], then tr [A + B + 2(A#B)] p ≤ (2 − p) tr (A1/2 + B1/2 )2 + (p − 1) tr (A1/2 + B1/2 )4 . Source: [476, 3020]. Fact 10.14.42. Let A, B ∈ Fn×n, and assume that A and B are positive definite. Then, n ∑ i=1

∑ 1 1 ≤ tr (A + B)−1 ≤ . λi (A) + λn+1−i (B) λ (A) + λi (B) i=1 i n

Source: [2750] and [2991, p. 362]. Fact 10.14.43. Let A, B ∈ Fn×n, and assume that A is positive definite, B is positive semidefinite,

and B ≤ A. Then,

det B tr B ≤ . det A tr A

Source: [2991, p. 215]. Fact 10.14.44. Let A1 , . . . , Am ∈ Fn×n, and assume that A1 , . . . , Am are positive definite. Then,

tr

)2 m i=1 Ai ∑ tr m i=1 Ai (∑

≤

n ∑ tr A2 i

i=1

tr Ai

.

Source: [2998]. Fact 10.14.45. Let A, B ∈ Fn×n, assume that A and B are positive semidefinite, let m ≥ 1, and define p ∈ F[s] by p(s) = tr (A + sB)m. Then, all of the coefficients of p are nonnegative. Remark:

This is the Bessis-Moussa-Villani trace conjecture. See [1058, 1395, 1820] and Fact 10.14.46. Fact 10.14.46. Let A, B ∈ Fn×n, assume that A is Hermitian and B is positive semidefinite, and define f (t) = eA+tB . Then, for all k ≥ 0 and t ≥ 0, (−1)k+1 f (k) (t) ≥ 0. Source: Fact 10.17.24 and [1395, 1820]. Related: Fact 10.14.45. Fact 10.14.47. Let A, B ∈ Fn×n, assume that A and B are Hermitian, and let f : R 7→ R. Then, the following statements hold: i) If f is convex, then there exist unitary matrices S 1 , S 2 ∈ Fn×n such that f [ 12 (A + B)] ≤ 12 [S 1 ( 12 [ f (A) + f (B)])S 1∗ + S 2 ( 21 [ f (A) + f (B)])S 2∗ ]. ii) If f is convex and even, then there exist unitary matrices S 1 , S 2 ∈ Fn×n such that f [ 12 (A + B)] ≤ 12 [S 1 f (A)S 1∗ + S 2 f (B)S 2∗ ]. iii) If f is convex and increasing, then there exists a unitary matrix S ∈ Fn×n such that f [ 12 (A + B)] ≤ S ( 21 [ f (A) + f (B)])S ∗. iv) There exist unitary matrices S 1 , S 2 ∈ Fn×n such that ⟨A + B⟩ ≤ S 1⟨A⟩S 1∗ + S 2 ⟨B⟩S 2∗ . v) If f is convex, then

tr f [ 21 (A + B)] ≤ tr 12 [ f (A) + f (B)].

Source: [538, 539, 546]. Related: Fact 10.14.48. Fact 10.14.48. Let f : R 7→ R, and assume that f is convex. Then, the following statements

hold: i) If f (0) ≤ 0, A ∈ Hn , and S ∈ Fn×m is contractive, then tr f (S ∗AS ) ≤ tr S ∗ f (A)S.

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CHAPTER 10

ii) If A1 , . . . , Ak ∈ Fn×n are Hermitian and S 1 , . . . , S k ∈ Fn×m satisfy   k k ∑  ∑ ∗  S i∗ f (Ai )S i . tr f  S i Ai S i  ≤ tr iii) If A ∈ H and S ∈ F

n×n

i=1

S i∗S i = I, then

i=1

i=1

n

∑k

is a projector, then tr Sf (SAS )S ≤ tr Sf (A)S.

Source: [539] and [2128, p. 36]. Remark: Special cases are considered in [1571]. Remark: The

first result is due to L. G. Brown and H. Kosaki, the second result is due to F. Hansen and G. K. Pedersen, and the third result is due to F. A. Berezin. Related: ii) generalizes v) of Fact 10.14.47. Fact 10.14.49. Let A1 , . . . , Am ∈ Fn×n and r ∈ [1, ∞). Then, ⟨∑ ⟩r m m ∑ tr Ai ≤ mr−1 tr ⟨Ai ⟩r . i=1

i=1

Source: [2439]. Related: Fact 10.14.50. Fact 10.14.50. Let A, B ∈ Fn×m . Then, tr ⟨A + B⟩ ≤ tr ⟨A⟩ + tr ⟨B⟩. Source: Fact 10.11.34, [2439], and [2991, p. 291]. Related: Fact 10.14.49 and Fact 10.14.51. Fact 10.14.51. Let A, B ∈ Fn×n and r ∈ [1, ∞). Then,

tr ⟨A⟩r + tr ⟨B⟩r ≤ tr ⟨A + B⟩r + tr ⟨A − B⟩r . Source: [2439]. Related: Fact 10.14.50. Fact 10.14.52. Let A, B ∈ Fn×n, and assume that B is positive semidefinite and A∗A ≤ B. Then,

| tr A| ≤ tr B1/2 . Source: Corollary 10.6.11 with r = 2 implies (A∗A)1/2 ≤ tr B1/2. It then follows from Fact 10.13.2

∑ ρi (A) ≤ ni=1 σi (A) = tr (A∗A)1/2 ≤ tr B1/2. See [347]. Fact 10.14.53. Let A, B ∈ Fn×n, assume that A is positive definite and B is positive semidefinite, let α ∈ [0, 1], and let β ≥ 0. Then, that | tr A| ≤

∑n

i=1

α/(2−α) . tr(−BA−1B + βBα ) ≤ β(1 − α2 ) tr ( αβ 2 A)

If, in addition, either A and B commute or B is a scalar multiple of a projector, then α/(2−α) −BA−1B + βBα ≤ β(1 − α2 )( αβ . 2 A)

Source: [1312, 1313]. [ ] [ ] Fact 10.14.54. Let A, P ∈ Fn×n, B, Q ∈ Fn×m, and C, R ∈ Fm×m, and assume that BA∗ CB , QP∗ Q R ∈

F(n+m)×(n+m) are positive semidefinite. Then,

| tr BQ∗ |2 ≤ (tr AP) tr CR. Source: [1783, 2987]. Fact 10.14.55. Let A, B ∈ Fn×n, and assume that A and B are projectors. Then,

(tr AB)2 ≤ (rank AB) tr ABAB. Furthermore, equality holds if and only if there exists α > 0 such that αAB is idempotent. Source: [239]. Fact 10.14.56. Let A, B ∈ Fn×m, let X ∈ Fn×n, and assume that X is positive definite. Then, | tr A∗B|2 ≤ (tr A∗XA) tr B∗X −1A. ] [ ∗ ∗] [ AB Source: Use Fact 10.14.54 with XI XI−1 and AA BA∗ BB∗ . See [1783, 2987].

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POSITIVE-SEMIDEFINITE MATRICES

Fact 10.14.57. Let A, B, C ∈ Fn×n, and assume that A and B are Hermitian and C is positive

semidefinite. Then, |tr ABC 2 − tr ACBC| ≤ 14 [λ1 (A) − λn (A)][λ1 (B) − λn (B)] tr C 2. Source: [541]. Fact 10.14.58. Let A, B, C ∈ Fn×n, assume that A, B, C are positive semidefinite, and assume

that B ≤ C. Then,

tr (A + B)−1B ≤ tr (A + C)−1C.

Source: [2991, p. 213]. Fact 10.14.59. Let A, B ∈ Fn×n, and assume that A and B are positive definite. Then,

tr (A − B)(B−1 − A−1 ) ≥ 0. Source: [2991, p. 213]. Fact 10.14.60. Let A, B, C, D, X ∈ Fn×n, and assume that A and B are positive semidefinite, C

and D are positive definite, and X is Hermitian. Then, tr X(A + C)−1X(B + D)−1 ≤ tr XA−1XB−1 , 4 tr (C − D)[(A + C)−1 − (B + D)−1 ] ≤ tr (A − B)(B−1 − A−1 ), | tr (C − D)(B + D)−1 (A − B)(A + C)−1 | ≤ tr[(A − B)(B−1 − A−1 ) + (C − D)((B + D)−1 − (A + C)−1 )]. Source: [355, 1113]. Fact 10.14.61. Let A1 , . . . , Ak ∈ Pn and { j1 , . . . , jk } = {1, . . . , k}. Then,

kn2 ≤

k ∑ (tr Ai ) tr A−1 ji . i=1

Now, assume that Ak ≤ · · · ≤ A1 . Then, kn2 ≤

k k ∑ ∑ (tr Ai ) tr A−1 (tr Ai ) tr A−1 i ≤ ji , i=1

kn ≤

i=1

k ∑

tr Ai A−1 ji .

i=1

Source: [2681]. Related: Fact 10.14.62. Fact 10.14.62. For all i ∈ {1, . . . , k}, let Ai , Bi ∈ Hn , assume that Ak ≤ · · · ≤ A1 and Bk ≤ · · · ≤

B1 , and let { j1 , . . . , jk } = {1, . . . , k}. Then, k ∑

tr Ai Bk−i+1 ≤

i=1

k

k ∑

k ∑

i=1

k ∑

i=1

tr (Ai + Bk−i+1 )2 ≤

k ∑

i=1

i=1

tr (Ai + B ji )2 ≤

i=1

tr (Ai − Bi )2 ≤

tr Ai Bi ,

i=1

 k  k k ∑ ∑  ∑  ≤ tr  Ai  Bi ≤ k tr Ai Bi ,

i=1 k ∑

tr Ai B ji ≤

i=1

tr Ai Bk−i+1

i=1

k ∑

k ∑ i=1

tr (Ai − B ji )2 ≤

k ∑

tr (Ai + Bi )2 ,

i=1 k ∑ i=1

tr (Ai − Bk−i+1 )2 .

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Now, assume that Ak and Bk are positive semidefinite, and let l ≥ 0. Then, k k k ∑ ∑ ∑ (tr Ai Bk−i+1 )l ≤ (tr Ai B ji )l ≤ (tr Ai Bi )l . i=1

i=1

i=1

Source: [2681]. Related: Fact 10.14.61, Fact 10.16.36, and Fact 10.25.45.

10.15 Facts on the Determinant for One Matrix Fact 10.15.1. Let A ∈ Fn×n, and assume that A is positive semidefinite. Then, (n−1)/n (n−1)/n λmin (A) ≤ λ1/n (A) ≤ λn (A) ≤ λ1 (A) ≤ λ1/n max (A)λmin min (A)λmax (A) ≤ λmax (A), n−1 n λnmin (A) ≤ λmax (A)λn−1 min (A) ≤ det A ≤ λmin (A)λmax (A) ≤ λmax (A).

Source: Fact 7.12.33. Fact 10.15.2. Let A ∈ Rn×n, and assume that A + AT is positive semidefinite. Then,

[ 21 (A + AT )]A ≤ 12 (AA + AAT ). Now, assume that A + AT is positive definite. Then, [det 12 (A + AT )][ 21 (A + AT )]−1 ≤ (det A)[ 12 (A−1 + A−T )]. Furthermore,

[det 12 (A + AT )][ 21 (A + AT )]−1 < (det A)[ 21 (A−1 + A−T )]

if and only if rank(A − AT ) ≥ 4. Finally, if n ≥ 4 and A − AT is nonsingular, then (det A)[ 12 (A−1 + A−T )] < [det A − det 12 (A − AT )][ 21 (A + AT )]−1 . Source: [1011, 1536]. Remark: This result does not hold for complex matrices. Related: Fact

7.20.1, Fact 10.10.36, Fact 10.15.2, and Fact 10.21.18. Fact 10.15.3. Let A ∈ Fn×n, and assume that A + A∗ is positive semidefinite. Then, det 21 (A + A∗ ) ≤ | det A|. Furthermore, if A + A∗ is positive definite, then equality holds if and only if A is Hermitian. Source: The inequality follows from Fact 7.12.25 and Fact 7.12.32. See [2991, p. 205]. Remark: This is the Ostrowski-Taussky inequality. Related: Fact 10.15.3. Fact 10.15.4. Let A ∈ Fn×n, and assume that A + A∗ is positive semidefinite. Then, [det 12 (A + A∗ )]2/n + | det 21 (A − A∗ )|2/n ≤ | det A|2/n. Furthermore, if A + A∗ is positive definite, then equality holds if and only if every eigenvalue of (A + A∗ )−1(A − A∗ ) has the same absolute value. Finally, if n ≥ 2, then det 21 (A + A∗ ) ≤ det 12 (A + A∗ ) + | det 21 (A − A∗ )| ≤ | det A|. Source: [1012, 1537]. To prove the last result, use Fact 2.2.59. Remark: Setting A = 1 + ȷ shows that the last result can fail for n = 1. Remark: −A is semidissipative. Remark: The last result interpolates Fact 10.15.3. Remark: Extensions to the case where A + A∗ is not positive semidefinite

are considered in [2611]. Fact 10.15.5. Let A ∈ Fn×m. Then, | det(I + A)| ≤ det(I + ⟨A⟩). Source: [2420]. Related: Fact 10.16.17.

775

POSITIVE-SEMIDEFINITE MATRICES

Fact 10.15.6. Let A ∈ Fn×n, assume that A is positive semidefinite, and define

1 α = tr A, n △

∑ 1 β= |A(i, j) |. n(n − 1) i, j=1 n

△

i, j

Then,

| det A| ≤ (α − β)n−1[α + (n − 1)β].

Furthermore, if A = aIn + b1n×n , where a + nb > 0 and a > 0, then α = a + b, β = b, and equality holds. Source: [2119]. Related: Fact 3.16.18 and Fact 10.10.39. Fact 10.15.7. Let A ∈ Fn×n, assume that A is positive definite, and define △

β=

n ∑ |A(i, j) | 1 . √ n(n − 1) i, j=1 A(i,i) A( j, j) i, j

Then, | det A| ≤ (1 − β)n−1 [1 + (n − 1)β]

n ∏

A(i,i) ≤

i=1

n ∏

A(i,i) .

i=1

Source: [2119]. Remark: This result interpolates Hadamard’s inequality. See Fact 10.21.15 and

[910]. Fact 10.15.8. Let A ∈ Rn×n, and assume that A is positive definite. Then, n ∑

A(i,i) det A[i,i] ≤ det A + (n − 1)

i=1

n ∏

A(i,i) .

i=1

Source: [1836]. △ △ Fact 10.15.9. Let A ∈ Fn×n, assume that A is positive definite, and let λ1 = λ1 (A) and λn =

λn (A). Then,

(

n − λn tr (A + λn I)−1

)n

( ≤ det A ≤

)n n − λ 1 . tr (A + λ1 I)−1

Source: [673]. Fact 10.15.10. Let A ∈ Fn×n. Then,

1/2  n n n ∑ ∏ ∏    |A(i, j) |2  = ∥rowi (A)∥2 . | det A| ≤ i=1

j=1

i=1

Furthermore, equality holds if and only if AA∗ is diagonal. Now, let α > 0 be such that, for all i, j ∈ {1, . . . , n}, |A(i, j) | ≤ α. Then, | det A| ≤ αn nn/2. If, in addition, at least one entry of A has absolute value less than α, then | det A| < αn nn/2. Source: Replace A with AA∗ in Fact 10.21.15. See [2432, p. 34]. Related: Fact 3.16.24 and Fact

8.9.35. Fact 10.15.11. Let A ∈ Fn×n, and assume that A is positive definite. Then,

n + tr log A = n + log det A ≤ n(det A)1/n ≤ tr A ≤ (ntr A2 )1/2 , with equality if and only if A = I. Remark: (det A)1/n ≤ (tr A)/n follows from the arithmetic-mean– geometric-mean inequality.

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Fact 10.15.12. Let A ∈ Fn×n, assume that A is Hermitian, and assume that, for all i ∈ {1, . . . , n −

1}, det A({1,...,i}) > 0. Then, the following statements hold: i) For all i ∈ {1, . . . , n − 1}, A({1,...,i}) is positive definite. ii) λn−1 (A) is positive. iii) sign[λmin (A)] = sign(det A). iv) A is (positive semidefinite, positive definite) if and only if (det A ≥ 0, det A > 0). v) det A = 0 if and only if rank A = n − 1. Source: Note that det A = λmin (A)/[λ1 (A) · · · λn−1 (A)], and use Proposition 10.2.9 and Theorem 10.4.5. See [2403, p. 278]. Fact 10.15.13. Let A ∈ Rn×n, and assume that A is positive definite. Then, n ∑ ( )n [det A({1,...,i}) ]1/i ≤ 1 + n1 tr A < e tr A. i=1

Source: [70]. Fact 10.15.14. Let A ∈ Fn×n, assume that A is positive definite and Toeplitz, and, for all i ∈ △

{1, . . . , n}, define Ai = A({1,...,i}) ∈ Fi×i. Then,

(det A)1/n ≤ (det An−1 )1/(n−1) ≤ · · · ≤ (det A2 )1/2 ≤ det A1 . Furthermore,

det An−1 det A3 det A2 det A ≤ ≤ ··· ≤ ≤ . det An−1 det An−2 det A2 det A1

Source: [788] and [789, p. 682]. Fact 10.15.15. Let A ∈ Fn×n. Then, 0 ≤ det(I + AA) ≤ det(I + A∗A). Furthermore, the second inequality is an equality if and only if A is symmetric. Source: [914, 1838].

10.16 Facts on the Determinant for Two or More Matrices Fact 10.16.1. Let A, B ∈ Cn×n, and assume that spec(A) ⊂ [0, ∞) and spec(B) ⊂ [0, ∞). Then,

the following statements hold: s

i) If λ(A) ≺ λ(B), then det B ≤ det A. wlog

ii) If λ(A) ≺ λ(B), then det(I + A) ≤ det(I + B). Source: Fact 3.25.14, [1846], and [2991, p. 347]. Fact 10.16.2. Let A, B ∈ Fn×n, and assume that A is positive semidefinite and B is skew Hermitian. Then, (det A)2/n + | det B|2/n ≤ | det(A + B)|2/n. If A is positive definite, then equality holds if and only if every eigenvalue of A−1B has the same absolute value. Finally, if n ≥ 2, then det A ≤ det A + | det B| ≤ | det(A + B)|. Source: Replace A by A + B in Fact 10.15.4. Remark: An extension of the first inequality to the case where A is Hermitian is given in [714, 2611]. Setting A = 1 and B = ȷ shows that the second inequality in the last string does not hold in the case where n = 1. Fact 10.16.3. Let A, B ∈ Fn×n, and assume that A is positive semidefinite and B is skew Hermitian. Then, n √ ∏ det A ≤ | det(A + B)| = (det A) 1 + σ2i (A−1/2BA−1/2 ). i=1

777

POSITIVE-SEMIDEFINITE MATRICES

If A is positive definite, then equality holds if and only if B = 0. Finally, if A and B are real, then det A ≤ det(A + B). Source: [100, 714], [1343, p. 447], and [2263, pp. 146, 163]. Suppose that A and B are real. If

A is positive definite, then A−1/2BA−1/2 is skew symmetric, and thus det(A + B) = (det A) det(I + A−1/2BA−1/2 ) is positive. If A is positive semidefinite, then continuity implies that det(A + B) is nonnegative. Remark: Extensions are given in [480]. Fact 10.16.4. Let A, B ∈ Fn×n, and assume that A and B are positive definite. Then, n ∏

[λi (A) + λi (B)] ≤ det(A + B) ≤

i=1

n ∏

[λi (A) + λn+1−i (B)].

i=1

Source: [2991, p. 362]. Fact 10.16.5. Let A, B ∈ Fn×n, and assume that A and B are Hermitian. Then,

| det(A + ȷB)| ≤

n ∏

[σ2i (A) + σ2n−i+1 (B)]1/2 .

i=1

Now, assume that A and B are positive semidefinite. Then, | det(A + ȷB)| ≤ det(A + B) ≤ 2n/2 | det(A + ȷB)|, n n ∏ ∏ [λ2i (A) + λ2i (B)]1/2 ≤ | det(A + ȷB)| ≤ [λ2i (A) + λ2n−i+1 (B)]1/2 . i=1

i=1

Source: [100, 336, 714, 1832, 2285] and [2979, p. 99]. Related: Fact 10.16.6. Fact 10.16.6. Let A, B ∈ Cn×n, assume that A and B are Hermitian, and assume that A2 + B2 is

positive definite. Then, ȷ(AB − BA) is Hermitian,

spec[ ȷ(A2 + B2 )−1/2 (AB − BA)(A2 + B2 )−1/2 ] ⊂ (−1, 1), [

A det −B

] B = [det(A + ȷB)] det(A − ȷB) = det[A2 + B2 ± ȷ(AB − BA)] A = [det (A2 + B2 )] det[I ± ȷ(A2 + B2 )−1/2 (AB − BA)(A2 + B2 )−1/2 ] > 0.

[ A B] 2 2 Hence, −B A , A + ȷB, A − ȷB, and A + B ± ȷ(AB − BA) are nonsingular. Related: Fact 3.24.7 and Fact 10.16.5. Fact 10.16.7. Let A, B ∈ Fn×n, assume that A and B are Hermitian, and assume that −A ≤ B ≤ A. △ △ Then, | det B| ≤ det A. Source: Let S ∈ Fn×n be a nonsingular matrix such that D1 = SAS ∗ and D2 = ∗ SBS are diagonal. Then, −D1 ≤ D2 ≤ D1 , and thus |D2 | ≤ D1 and | det D2 | = det |D2 | ≤ det D1 . Therefore, | det B| = (det S −1 )(det S −∗ )| det D2 | ≤ (det S −1 )(det S −∗ ) det D1 = det A. Related: Fact 10.12.18. Fact 10.16.8. Let A, B ∈ Fn×n, assume that A and B are positive semidefinite, and assume that A ≤ B. Then, n det A + det B ≤ det(A + B). Source: [2263, pp. 154, 166]. Remark: Under weaker conditions, Corollary 10.4.15 implies that det A + det B ≤ det(A + B). Related: Fact 7.13.25. Fact 10.16.9. Let A, B ∈ Fn×n, and assume that A and B are positive semidefinite. Then, √ det A + det B ≤ det A + det B + (2n − 2) det AB ≤ det(A + B).

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If A and B are positive definite, then

√ det A + det B ≤ det A + det B + (2n − 2) det AB     n−1 n−1 ∑ ∑     √ det B det A (1,...,i) (1,...,i)  det B + (2n − 2n) det AB  det A + 1 + ≤ 1 +  det A(1,...,i) det B(1,...,i) i=1 i=1 ≤ det(A + B).

If B ≤ A, then

√ det A + (2n − 1) det B ≤ det A + det B + (2n − 2) det AB ≤ det(A + B).

Source: [1835, 2179] and [2418, p. 231]. Remark: The last inequality in the second string is due

to E. V. Haynsworth and D. J. Hartfiel. See [1839]. Fact 10.16.10. Let A, B ∈ Fn×n, and assume that A and B are positive semidefinite. Then, | det(A1/2B1/2 + B1/2A1/2 )| ≤ det(A + B). Source: [1843]. Fact 10.16.11. Let A, B ∈ Fn×n, assume that A and B are positive semidefinite, and let α, β ∈ C.

Then,

| det(αA + βB)| ≤ det(|α|A + |β|B).

Source: [2991, p. 215]. Fact 10.16.12. Let A, B ∈ Fn×n, assume that A and B are positive semidefinite, and let p ∈ [0, 2].

Then,

det(A2 + ⟨BA⟩ p ) ≤ det(A2 + A pB p ),

det(A2 + A pB p ) ≤ det(A2 + ⟨AB⟩ p ).

In particular, det(A2 + ⟨BA⟩) ≤ det(A2 + AB),

det(A2 + AB) ≤ det(A2 + ⟨AB⟩).

Source: [1846]. Remark: The second inequality is a conjecture given in [1846]. Fact 10.16.13. Let A, B ∈ Fn×n, assume that B is Hermitian, and assume that A∗BA < A + A∗.

Then, det A , 0.

Fact 10.16.14. Let A, B ∈ Fn×n, assume that A and B are positive semidefinite, and let α ∈ [0, 1].

Then,

(det A)α (det B)1−α = det AαB1−α ≤ det[αA + (1 − α)B].

Now, assume that A and B are positive definite. Then, equality holds if and only if A = B. In addition, det AαB1−α ≤ det AαB1−α + δn det(A + B − 2A#B) ≤ det[αA + (1 − α)B], △

where δ = min {α, 1 − α}. Source: The first inequality is xxxviii) of Proposition 10.6.17. See [1448, p. 467] and [1451, p. 488]. The last inequality is given in [1640]. Remark: A#B is the geometric mean of A and B. See Fact 10.11.68. Note that 2A#B ≤ A + B. Remark: α = 1/2 yields √ (det A) det B ≤ det 21 (A + B). Credit: H. Bergstrom. Related: Fact 10.14.33. Fact 10.16.15. Let A, B ∈ Fn×n, assume that A and B are positive semidefinite, assume that either A ≤ B or B ≤ A, let α ∈ [0, 1], and let p ≥ 1. Then, (det[αA + (1 − α)B]) p ≤ α(det A) p + (1 − α)(det B) p . Source: [830, 2840].

779

POSITIVE-SEMIDEFINITE MATRICES

Fact 10.16.16. Let A, B ∈ Fn×n, assume that A and B are positive definite, and let S ⊆ {1, . . . , n}.

Then,

det A det B det(A + B) + ≤ . det A(S) det B(S) det(A(S) + B(S) )

Source: xli) of Proposition 10.6.17, [2263, p. 145], and [2991, p. 243]. Related: Fact 10.12.7. Fact 10.16.17. Let A, B ∈ Fn×m. Then, there exist unitary matrices S 1 , S 2 ∈ Fn×n such that

I + ⟨A + B⟩ ≤ S 1 (I + ⟨A⟩)1/2S 2 (I + ⟨B⟩)S 2∗ (I + ⟨A⟩)1/2S 1∗. Therefore,

| det(I + A + B)| ≤ det(I + ⟨A + B⟩) ≤ det(I + ⟨A⟩)det(I + ⟨B⟩).

Source: [93, 2420, 2612]. Related: Fact 10.15.5 and Fact 10.16.18. Fact 10.16.18. Let A, B, C ∈ Fn×n, assume that A and C are positive semidefinite, assume that

B is Hermitian, and assume that

[A B]

is positive semidefinite. Then, ( [ ]) A B det(I + A + C) ≤ det I + ≤ det(I + A) det(I + C). B C BC

Source: [546]. Related: Fact 10.16.17. Fact 10.16.19. Let A, B ∈ Fn×m, assume that A and B are semicontractive, and let α ∈ [0, 1].

Then,

[det(I + ⟨A⟩)]α [det(I + ⟨B⟩)]1−α ≤ det(I + α⟨A⟩) det(I + (1 − α)⟨B⟩), [det(I − ⟨A⟩)]α [det(I − ⟨B⟩)]1−α ≤ det(I − α⟨A⟩) det(I − (1 − α)⟨B⟩).

Source: [1831]. Fact 10.16.20. Let A, B ∈ Fn×n, assume that A + A∗ > 0 and B + B∗ ≥ 0, and let α > 0. Then, det(αI + AB) , 0, and thus AB has no negative eigenvalues. Source: [1284]. Remark: Equivalently, −A is dissipative and −B is semidissipative. Problem: Find a positive lower bound

for |det(αI + AB)| in terms of α, A, and B. Fact 10.16.21. Let A1 , . . . , Ak ∈ Fn×n, assume that A1 , . . . , Ak are positive semidefinite, and let α1 , . . . , αk ∈ C. Then, k k ∑ ∑ det αi Ai ≤ det |αi |Ai . i=1 i=1

Source: [2263, p. 144] and [2991, p. 362]. Fact 10.16.22. Let A1 , . . . , Ak ∈ Fn×n, assume that A1 , . . . , Ak are contractive, let α1 , . . . , αk ∈

[0, 1], and assume that

∑k

i=1

αi = 1. Then, ∑ ]α k [ det(I + ki=1 αi Ai ) ∏ det(I + ⟨Ai ⟩) i . ∑ ≤ det(I − ki=1 αi Ai ) i=1 det(I − ⟨Ai ⟩)

Source: [1831]. △ Fact 10.16.23. Let A, B, C ∈ Rn×n, let D = A + ȷB, and assume that CB + BTCT < D + D∗. Then,

det A , 0.

Fact 10.16.24. Let A, B ∈ Fn×n, assume that A and B are positive semidefinite, and let m ≥ 1.

Then,

n(det AB)m/n ≤ tr AmBm ,

√n

n det A

≤ tr A−1 .

Source: [829]. Remark: Assuming det B = 1 and setting m = 1 yields Proposition 10.4.14.

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CHAPTER 10

Fact 10.16.25. Let A, B ∈ Fn×m. Then,

| det A∗B|2 ≤ (det A∗A)(det B∗B). [ ∗ ∗ ] Source: Use Fact 10.12.33 or apply Fact 10.16.50 to AA∗AB BB∗AB . Remark: This is a determinantal version of the Cauchy-Schwarz inequality. Fact 10.16.26. Let A ∈ Fn×n, assume that A is positive definite, and let B ∈ Fm×n. Then, (det BB∗ )2 ≤ (det BAB∗ )det BA−1B∗. Source: Fact 10.12.49. Fact 10.16.27. Let A, B ∈ Fn×n. Then,

| det(A + B)|2 + | det(I − AB∗ )|2 ≤ det(I + AA∗ )det(I + B∗B), | det(A − B)|2 + | det(I + AB∗ )|2 ≤ det(I + AA∗ )det(I + B∗B). Furthermore, the first inequality is an equality if and only if either n = 1, A + B = 0, or AB∗ = I. Source: Fact 10.12.36, [2983, p. 184], and [2991, pp. 228–230]. Related: Fact 10.12.34. Fact 10.16.28. Let A, B ∈ Fn×m, and assume that A and B are semicontractive. Then,   ∗ 2      | det(I − A B)|  ∗ ∗ ≤ det(I + A∗A) det(I + B∗B). 0 ≤ det(I − A A)det(I − B B) ≤      | det(I + A∗B)|2  Now, assume that n = m. Then, 0 ≤ det(I − A∗A)det(I − B∗B) ≤ | det(I − A∗B)|2 − | det(A − B)|2 ≤ | det(I − A∗B)|2 ≤ | det(I − A∗B)|2 + | det(A + B)|2 ≤ det(I + A∗A)det(I + B∗B), 0 ≤ det(I − A∗A)det(I − B∗B) ≤ | det(I + A∗B)|2 − | det(A + B)|2 ≤ | det(I + A∗B)|2 ≤ | det(I + A∗B)|2 + | det(A − B)|2 ≤ det(I + A∗A)det(I + B∗B). Furthermore, for all i ∈ {1, . . . , n}, σi [(I − A∗A)(I − B∗B)] ≤ σ2i (I − A∗B),

λi [(I − A∗A)(I − B∗B)] ≤ σ2i (I − A∗B),

σi [(I − A∗A)(I − B∗B)] ≤ σ2i (I − AB∗ ),

λi [(I − A∗A)(I − B∗B)] ≤ σ2i (I − AB∗ ),

λi [(I − A∗A)1/2 (I − B∗B)1/2 ] ≤ σi (I − A∗B),

λi [(I − A∗A)1/2 (I − B∗B)1/2 ] ≤ σi (I − AB∗ ).

Now, assume that A and B are contractive. Then,    det (I − A∗A)−1 det (I − A∗B)−1   ≥ 0.  det (I − B∗A)−1 det (I − B∗B)−1 Source: [93]. The third inequality follows from Fact 10.12.35. The second inequality for n = m is given in [2184]. Remark: The second inequality and Fact 10.12.52 are Hua’s inequalities. See [2564]. Remark: Extensions of the last inequality are given in [2933]. Related: Fact 10.12.52 and Fact 10.19.5. Fact 10.16.29. Let A, B ∈ Fn×n and α, β ∈ (0, ∞). Then,

| det(A + B)|2 ≤ det(β −1I + αA∗A)det(α−1I + βBB∗ ). √ √ Source: Use Fact 10.12.51 or Fact 10.16.27 with A and B replaced by αβA∗ and αβB∗ , respectively. See [2984].

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POSITIVE-SEMIDEFINITE MATRICES

Fact 10.16.30. Let A ∈ Fn×m, B ∈ Fn×l, C ∈ Fn×m, and D ∈ Fn×l. Then,

| det(AC ∗ + BD∗ )|2 ≤ det(AA∗ + BB∗ )det(CC ∗ + DD∗ ). ] △ [ Source: Use Fact 10.16.41 and AA∗ ≥ 0, where A = CA DB . Related: Fact 3.17.24. Fact 10.16.31. Let A ∈ Fn×m, B ∈ Fn×m, C ∈ Fk×m, and D ∈ Fk×m. Then, | det(A∗B + C ∗D)|2 ≤ det(A∗A + C ∗C) det(B∗B + D∗D). ] △ [ Source: Use Fact 10.16.41 and A∗A ≥ 0, where A = CA DB . Related: Fact 3.17.20. Fact 10.16.32. Let A, B, C ∈ Fn×n. Then,  ∗ ∗ ∗    det(A A + B B)det(I + CC ) 2 | det(B + CA)| ≤    det(BB∗ + CC ∗ ) det(I + A∗ A). Source: [1458, 3007]. Related: Fact 10.11.61. Fact 10.16.33. Let A, B, C ∈ Fn×n, and assume that C is semicontractive and B∗B ≤ A∗A. Then,

det(A∗A − B∗B) det(I − CC ∗ ) + | det(B − C ∗A)|2 ≤ | det(A − CB)|2 . Source: [3007] and Fact 10.11.61. Fact 10.16.34. Let A, B, C, D, X, Y ∈ Fn×n, and assume that X and Y are positive definite. Then,

| det(AXC ∗ + BY D∗ )|2 ≤ det(AXA∗ + BY B∗ ) det(CXC ∗ + DY D∗ ). In particular,

| det(XC ∗ + BY)|2 ≤ det(X + BY B∗ ) det(Y + CXC ∗ ), | det(B + C ∗ )|2 ≤ (I + BB∗ ) det(I + CC ∗ ), | det(AC ∗ + BD∗ )|2 ≤ det(AA∗ + BB∗ ) det(CC ∗ + DD∗ ).

Source: [3007]. Fact 10.16.35. Let A, B, C ∈ Fn×n, and assume that A, B, C are positive semidefinite. Then,

det(I + A + B) ≤ det (I + A)(I + B), (det A) det(A + B + C) ≤ det (A + B)(A + C), det(A + B) + det(B + C) ≤ det(A + B + C) + det B, det(A + B) + det(B + C) + det(C + A) ≤ det(A + B + C) + det A + det B + det C. Now, assume that A, B, C are positive definite. Then, det (A + B)−1 + det (A + C)−1 ≤ det A−1 + det (A + B + C)−1 . Source: [1835, 2186, 2681]. Remark: Note the analogy between the fourth equality and Hlawka’s identity given by Fact 2.21.11 and Fact 11.8.5. See [1835]. Related: Fact 2.13.3 and Fact 10.25.44. Fact 10.16.36. For all i ∈ {1, . . . , k}, let Ai , Bi ∈ Hn , assume that Ak ≤ · · · ≤ A1 and Bk ≤ · · · ≤

B1 , assume that Ak + Bk is positive semidefinite, and let { j1 , . . . , jk } = {1, . . . , k}. Then, k ∑

det(Ai + Bk−i+1 ) ≤

i=1 k ∏ i=1

k ∑

det(Ai + B ji ) ≤

i=1

det(Ai + Bi ) ≤

k ∏ i=1

det(Ai + B ji ) ≤

k ∑

det(Ai + Bi ),

i=1 k ∏ i=1

det(Ai + Bk−i+1 ).

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Now, assume that Ak + Bk is positive definite, and let l ∈ Z. Then, k ∑

det (Ai + Bk−i+1 )l ≤

i=1

k ∑

det (Ai + B ji )l ≤

i=1

k ∑

det (Ai + Bi )l .

i=1

Source: [2681]. Related: Fact 10.14.62. [ ] △ Fact 10.16.37. Let A ∈ Fn×n, B ∈ Fn×m, and C ∈ Fm×m, define A = BA∗ CB ∈ F(n+m)×(n+m), and

assume that A is positive definite. Then,

det A = (det A)det(C − B∗A−1B) ≤ (det A) det C ≤

n+m ∏

A(i,i) .

i=1

Now, suppose that n = m. Then, the first inequality is an equality if and only if B = 0. Source: The second inequality is obtained by successive application of the first inequality. See [2991, p. 216]. Remark: det A ≤ (det A) det C is Fischer’s inequality. The inequality comprised of the first and last terms is Hadamard’s inequality. See Fact 10.21.15. [A A ] △ △ 11 12 (n+m)×(n+m) Fact 10.16.38. Let A = A∗12 , assume that A is nonsingular, and define B = A22 ∈ F [B B ] AA∗ = B11∗12 B1222 ∈ P(n+m) . Then, 2 | det(A22 − B∗12 B−1 11 A12 )| det B11 ≤ det B.

Furthermore, equality holds if and only if A12 = 0. Source: [1841]. [ ] △ Fact 10.16.39. Let A ∈ Fn×n, B ∈ Fn×m, and C ∈ Fm×m, define A = BA∗ CB ∈ F(n+m)×(n+m), assume △

△

that A is positive definite, let k = min {m, n}, and, for all i ∈ {1, . . . , n + m}, let λi = λi (A). Then, n+m−k  k n+m ∏  ∏  ∏ [ 12 (λi + λn+m−i+1 )]2. λi  λi ≤ (det A) det C ≤  i=1

i=1

i=k+1

Source: The first inequality is given by Fact 10.16.37. The second inequality is given in [2103]. Remark: If n = m, then the second product is 1. Fact 10.16.40. Let A ∈ Fn×n, and assume that A is positive semidefinite. Then, the following

statements hold: i) If α ⊂ {1, . . . , n}, then det A ≤ [det A(α) ] det A[α] . ii) If α, β ⊆ {1, . . . , n}, then det A(α∪β) det A(α∩β) ≤ [det A(α) ] det A(β) . iii) If 1 ≤ k ≤ n − 1, then    

∏

(n−1   k−1)   det A(α)  ≤ 

{α: card(α)=k+1}

∏

(n−1  k ) det A(α)  .

{α: card(α)=k}

Source: [1448, p. 485], [1451, p. 507], and [1896]. Remark: i) is Fischer’s inequality, which is given in Fact 10.16.37; ii) is the Hadamard-Fischer inequality; iii) is Szasz’s inequality. See [789, 868, p. 680], [1448, p. 479], and [1896]. Related: Fact 10.16.39. [ ] △ Fact 10.16.41. Let A, B, C ∈ Fn×n, define A = BA∗ CB ∈ F2n×2n, and assume that A is positive semidefinite. Then, 0 ≤ det A ≤ (det A)det C − | det B|2 ≤ (det A)det C.

Furthermore, if A is positive definite, then | det B|2 < (det A)det C. Source: Assume that A is positive definite. Then, det A = (det A) det(C − B∗A−1B), and Minkowski’s determinant theorem

783

POSITIVE-SEMIDEFINITE MATRICES

Corollary 10.4.15 implies that (det A + | det B|2 )/det A = det A/det A + det B∗A−1B = det(C − B∗A−1B) + det B∗A−1B ≤ det C. Use continuity in the case where A is singular. If A is positive definite, then 0 ≤ B∗A−1B < C, and thus | det B|2/det A < det [C. See][1005, 1849]. Remark: If n = 1, then the second inequality is an equality. Remark: A = B is nonsquare, then A =

1000 0220 0220 [0 0 0 2] 111 121 111

shows that the converse of the last statement is false. Remark: If shows that it is not necessarily true that det B∗B ≤ (det A) det C. See

[2985]. Remark: | det B|2 ≤ (det A)det C is proved in [2263, p. 142]. Related: Fact 10.16.50. Fact 10.16.42. For all i, j ∈ {1, 2, 3}, let Ai j ∈ Fni ×n j , define A ∈ F(n1 +n2 +n3 )×(n1 +n2 +n3 ) by    A11 A12 A13    ∗ △  A =  A12 A22 A23  ,   ∗ ∗ A13 A23 A33 and assume that A is positive semidefinite. Then, ( [ A (det A) det A22 ≤ det ∗11 A12

A12 A22

])

[

A det ∗22 A23

] A23 . A33

If, in addition, A22 is positive definite, then equality holds if and only if A13 = A12 A−1 22 A23 . Now, assume that n1 = n2 = n3 . Then, ( [ ]) [ ] [ ] 2 A11 A12 A22 A23 A12 A13 (det A) det A22 ≤ det ∗ det ∗ − det . A12 A22 A23 A33 A22 A23 If, in addition, A22 is positive definite, then equality holds if and only if A13 = A12 A−1 22 A23 . Source: [1849]. The first result follows from the Hadamard-Fischer inequality given by Fact 10.16.39. [ ] A12 A13 Remark: In the equality case, Fact 3.17.13 implies that det = 0. A22 A23 Fact 10.16.43. Let B, C ∈ Fn×n , assume that B and C are positive definite, and define [ ] A11 A12 △ A= ∗ = B + ȷC, A12 A22 where A11 ∈ Fn1 ×n1 and A22 ∈ Fn2 ×n2 . Then, the following statements hold: i) Every principal submatrix of A is nonsingular. △ 1 1 ∗ ∗ ii) Define S = A22 − A21 A−1 22 A22 . Then, 2 (S + S ) and 2 ȷ (S − S ) are positive definite. iii) A is nonsingular, and A−1 = (B + CB−1C)−1 − ȷ(C + BC −1B)−1. { (B) λ (C) } △ △ iv) Let m = min {n1 , n2 } and κ = max λλmax , max . Then, min (B) λmin (C) (4κ)m | det A11 || det A22 | ≤ 2m | det A11 || det A22 |. (1 + κ)2m Source: [1500, 1807, 1830, 1832]. Remark: A is an accretive-dissipative[ matrix. ] △ Fact 10.16.44. Let A ∈ Fn×n, B ∈ Fn×m, and C ∈ Fm×m, define A = BA∗ CB ∈ F(n+m)×(n+m) , and

assume that A is positive semidefinite and A is positive definite. Then, ]2 [ λmax (A) − λmin (A) C. B∗A−1B ≤ λmax (A) + λmin (A) Source: [1783, 2987].

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CHAPTER 10 △

Fact 10.16.45. Let A, B, C ∈ Fn×n, define A =

semidefinite. Then,

[

[ | det B|2 ≤

△ Now, define Aˆ =

[ det A

det B det B∗ det C

]

A B B∗ C

λmax (A) − λmin (A) λmax (A) + λmin (A)

| det B|2 ≤ Hence,

[

]

∈ F2n×2n, and assume that A is positive

]2n (det A)det C.

]2 λmax (A) − λmin (A) (det A) det C. λmax (A) + λmin (A)

∈ F2×2. Then, [

| det B|2 ≤

ˆ − λmin (A) ˆ ]2 λmax (A) (det A) det C. ˆ + λmin (A) ˆ λmax (A)

Source: [1783, 2987]. Remark: These bounds cannot be ordered. [ ] △ Fact 10.16.46. Let A ∈ Fn×n, B ∈ Fn×m, and C ∈ Fm×m, define A = BA∗ CB ∈ F(n+m)×(n+m), assume

that A is positive semidefinite, and assume that A and C are positive definite. Then, det(A|A) det(C|A) ≤ det A.

Source: [1458]. Remark: This is the reverse Fischer inequality. [ ] △ Fact 10.16.47. Let A ∈ Fn×n, B ∈ Fn×m, C ∈ Fm×m, and define A = A0 CB ∈ F(n+m)×(n+m). Then,

| det(I + AA) det(I +CC)| ≤ det(I + A∗A) det(I +C ∗C) ≤ det(I +A∗A) ≤ det(I + A∗A) det(I + B∗B+C ∗C). Source: Fact 10.16.37 and [1838]. ∑ Fact 10.16.48. Let A1 , . . . , An ∈ Nn, let α1 , . . . , αn ∈ [0, 1], and assume that ni=1 αi = 1. Then n ∏

 n   ∑  (det Ai ) ≤ det  αi Ai  . αi

i=1

i=1

v t n ∏

In particular,

n

1∑ Ai . n i=1 n

det Ai ≤ det

i=1

Source: [1370]. Related: Fact 10.16.49. Fact 10.16.49. Let A1 , . . . , Ak ∈ Nn. Then,

v u t k

k ∏ i=1

∑ 1 det Ai . n k i=1 k

det Ai ≤

Source: [1853]. Related: If 1 ≤ k ≤ n, then this result follows from Fact 10.16.48 by setting

α1 = · · · = αk =

and αk+1 = · · · = αn = 0. Setting k = n yields the special case in Fact 10.16.48. Fact 10.16.50. Let Ai j ∈ Fn×n for all i, j ∈ {1, . . . , m}, define A ∈ Fmn×mn by    A11 · · · A1m   . ..   . △  · .· · A =  .. .  ,   ∗ A1m · · · Amm 1 k

785

POSITIVE-SEMIDEFINITE MATRICES

assume that A is positive semidefinite, let 1 ≤ k ≤ n, and define A˜ k ∈ Fm(k)×m(k) by  (k) (k)    A11 · · · A1m   △  . . .  ˜  .  . . Ak =  . · .· · .   ∗(k) A1m · · · A(k) mm n

Then, A˜ k is positive semidefinite. In particular,   det A11  ˜ An =  ...  det A∗1m

··· . · .· · ···

n

 det A1m   ..  .  det Amm

is positive semidefinite. Furthermore, det A ≤ det A˜ n . Now, assume that A is positive definite. Then, det A = det A˜ n if and only if, for all distinct i, j ∈ {1, . . . , m}, Ai j = 0. Source: The first statement is given in [852]. The inequality as well as the final statement are given in [2609]. See also [1848]. Remark: B(k) is the kth compound of B. See Fact 9.5.18. Remark: Every principal subdeterminant of A˜ n is lower bounded by the determinant of a principal submatrix of A, which is positive semidefinite and thus has nonnegative determinant. Hence, this inequality implies that A˜ n is positive semidefinite. Remark: A weaker result is given in [854] and quoted in [1950] in terms of[ elementary symmetric functions of the eigenvalues of ] each block. Remark: The example A =

1 0 1 0

0 1 0 0

1 0 1 0

0 0 0 1

shows that A˜ can be positive definite while A is

singular. Remark: The matrix whose (i, j) entry is det Ai j is a determinantal compression of A. See [853, 1953, 2609] and [2991, p. 221]. Related: Fact 10.12.64.

10.17 Facts on Convex Sets and Convex Functions Fact 10.17.1. Let f : Rn 7→ Rn, and assume that f is convex. Then, for all α ∈ R, the sets {x ∈ Rn : f (x) ≤ α} and {x ∈ Rn : f (x) < α} are convex. Source: [1059, p. 108]. Remark: The

converse is false. Let f (x) = x3. △ Fact 10.17.2. Let A ∈ Fn×n, assume that A is Hermitian, let α ≥ 0, and define the set S = {x ∈ n ∗ F : x Ax < α}. Then, the following statements hold: i) S is an open set. ii) S is a blunt cone if and only if α = 0. iii) S is a nonempty set if and only if either α > 0 or λmin (A) < 0. iv) S is a convex set if and only if A ≥ 0. v) S is a nonempty, convex set if and only if α > 0 and A ≥ 0. vi) The following statements are equivalent: a) S is a bounded set. b) S is a bounded, convex set. c) A > 0. vii) The following statements are equivalent: a) S is a nonempty, bounded set. b) S is a nonempty, bounded, convex set. c) α > 0 and A > 0. △ Fact 10.17.3. Let A ∈ Fn×n, assume that A is Hermitian, let α ≥ 0, and define the set S = {x ∈

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CHAPTER 10

Fn : x∗Ax ≤ α}. Then, the following statements hold: i) S is closed. ii) 0 ∈ S, and thus S is nonempty. iii) S is a pointed cone if and only if α = 0 or A ≤ 0. iv) S is a convex set if and only if A ≥ 0. v) The following statements are equivalent: a) S is a bounded set. b) S is bounded, convex set. c) A > 0. △ Fact 10.17.4. Let A ∈ Fn×n, assume that A is Hermitian, let α ≥ 0, and define the set S = {x ∈ n ∗ F : x Ax = α}. Then, the following statements hold: i) S is closed. ii) S is a nonempty set if and only if either α = 0 or λmax (A) > 0. iii) The following statements are equivalent: a) S is a pointed cone. b) 0 ∈ S. c) α = 0. iv) S = {0} if and only if α = 0 and either A > 0 or A < 0. v) S is a bounded set if and only if either A > 0 or both α > 0 and A ≤ 0. vi) S is a nonempty, bounded set if and only if A > 0. vii) The following statements are equivalent: a) S is a convex set. b) S is nonempty, convex set. c) α = 0 and either A > 0 or A < 0. viii) If α > 0, then the following statements are equivalent: a) S is a nonempty set. b) S is not a convex set. c) λmax (A) > 0. ix) The following statements are equivalent: a) S is a bounded, convex set. b) S is a nonempty, bounded, convex set. c) α = 0 and A > 0. △ Fact 10.17.5. Let A ∈ Fn×n, assume that A is Hermitian, let α ≥ 0, and define the set S = {x ∈ n ∗ F : x Ax ≥ α}. Then, the following statements hold: i) S is a closed set. ii) S is a pointed cone if and only if α = 0. iii) S is a nonempty set if and only if either α = 0 or λmax (A) > 0. iv) S is a bounded set if and only if S ⊆ {0}. v) The following statements are equivalent: a) S is a nonempty, bounded set. b) S = {0}. c) α = 0 and A < 0.

787

POSITIVE-SEMIDEFINITE MATRICES

vi) S is a convex set if and only if either S = ∅ or S = Fn. vii) S is convex and bounded if and only if S = ∅. viii) The following statements are equivalent: a) S is a nonempty, convex set. b) S = Fn. c) α = 0 and A ≥ 0. △ Fact 10.17.6. Let A ∈ Fn×n, assume that A is Hermitian, let α ≥ 0, and define the set S = {x ∈ Fn : x∗Ax > α}. Then, the following statements hold: i) S is an open set. ii) S is a blunt cone if and only if α = 0. iii) S is a nonempty set if and only if λmax (A) > 0. iv) The following statements are equivalent: a) S = ∅. b) λmax (A) ≤ 0. c) S is a bounded set. d) S is a convex set. △ △ Fact 10.17.7. Let f : Rn 7→ R, define g : Rn × Rn 7→ R by g(x, y) = xT y − f (y), define D = △ {x ∈ Rn : g(x, Rn ) is bounded}, and define h : D 7→ R by h(x) = supy∈Rn g(x, y). Then, the following statements hold: △ i) Define f : R 7→ R by f (x) = |x|. Then, h : [−1, 1] 7→ R is given by h(x) = 0. △ ii) Let p ∈ (1, ∞), and define f : R 7→ R by f (x) = 1p |x| p . Then, h : R 7→ R is given by h(x) = q1 |x|q , where q ∈ (0, ∞) satisfies 1/p + 1/q = 1. △

iii) Define f : R 7→ R by f (x) = e x . Then, h : [0, ∞) 7→ R is given by h(0) = 0 and, for all x > 0, h(x) = x log x − x. △ iv) Let a ∈ Rn and b ∈ R, and define f (x) = aTx + b. Then, h : {a} 7→ R is given by h(a) = −b. △ v) Let A ∈ Rn×n , assume that A is positive definite, and define f : Rn 7→ R by f (x) = 21 xTAx. 1 T −1 n Then, h : R 7→ R is given by h(x) = 2 x A x. Source: [523, pp. 49–63] and [1847]. Remark: h is the Legendre-Fenchel transform of f, and is also called the convex conjugate of f. Remark: h can be viewed as a mapping from Rn to R ∪ {∞}, △ where h(x) = ∞ for all x ∈ Rn \D. Fact 10.17.8. Let A ∈ Cn×n, and define the numerical range of A by △

Θ1 (A) = {x∗Ax: x ∈ Cn and x∗x = 1} and the set

△

Θ(A) = {x∗Ax: x ∈ Cn }.

Then, the following statements hold: i) Θ1 (A) is a closed, bounded, convex subset of C. ii) Θ(A) = {0} ∪ cone Θ1 (A). iii) Θ(A) is a pointed, closed, convex cone contained in C. iv) Θ(A) is a closed, bounded interval contained in R if and only if there exist α, β ∈ C and a Hermitian matrix B ∈ Cn×n such that A = αB + βI. v) If A is Hermitian, then Θ1 (A) is a closed, bounded interval contained in R. vi) If A is Hermitian, then Θ(A) is either (−∞, 0], [0, ∞), or R.

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CHAPTER 10

vii) Θ1 (A) satisfies conv spec(A) ⊆ Θ1 (A) ⊆ co{ν1 + µ1 ȷ, ν1 + µn ȷ, νn + µ1 ȷ, νn + µn ȷ}, where △

ν1 = λmax [ 21 (A + A∗ )], △

µ1 = λmax [ 21ȷ (A − A∗ )],

△

νn = λmin [ 12 (A + A∗ )], △

µn = λmin [ 21ȷ (A − A∗ )].

viii) If A is normal, then Θ1 (A) = conv spec(A). ix) If n ≤ 4 and Θ1 (A) = conv spec(A), then A is normal. x) Θ1 (A) = conv spec(A) if and only if either A is normal or there exist A1 ∈ Fn1 ×n1 and A2 [∈ Fn2]×n2 such that n1 + n2 = n, Θ1 (A1 ) ⊆ Θ1 (A2 ), A2 is normal, and A is unitarily similar to A01 A02 . xi) Let λ ∈ Θ1 (A) and assume that λ is the vertex of a triangle that contains Θ1 (A). Then, λ ∈ spec(A). xii) If Θ1 (A) is a line segment, then A is normal. xiii) If Θ1 (A) = [a, b] ⊂ R, then a, b ∈ spec(A). xiv) If λ ∈ spec(A) ∩ bd Θ1 (A), then AN(A − λI) ⊆ N(A − λI) and AR(A∗ − λI) ⊆ R(A∗ − λI). Source: [1280], [1450, pp. 11, 52], [2432, Chapter 9], [2979, pp. 19, 193–196], and [2991, pp. 108, 109]. Remark: Θ1 (A) is called the field of values in [1450, p. 5]. Remark: ix) is an example of the quartic barrier. See [787], Fact 10.19.24, and Fact 15.18.5. Related: Fact 6.10.30 and Fact 10.17.8. Fact 10.17.9. Let A ∈ Rn×n, and define the real numerical range of A by △

Ψ1 (A) = {xTAx: x ∈ Rn and xTx = 1} and the set

△

Ψ(A) = {xTAx: x ∈ Rn }.

Then, the following statements hold: i) Ψ1 (A) = Ψ1 [ 21 (A + AT )]. Ψ1 (A) = [λmin [ 21 (A + AT )], λmax [ 12 (A + AT )]]. If A is symmetric, then Ψ1 (A) = [λmin (A), λmax (A)]. Ψ(A) = {0} ∪ cone Ψ1 (A). Ψ(A) is either (−∞, 0], [0, ∞), or R. Ψ1 (A) = Θ1 (A) if and only if A is symmetric. Source: [1450, p. 83]. Remark: Θ1 (A) is defined in Fact 10.17.8. Fact 10.17.10. Let A, B ∈ Cn×n, assume that A and B are Hermitian, and define {[ ∗ ] } x Ax △ n ∗ Θ1 (A, B) = : x ∈ C and x x = 1 ⊆ R2. x∗Bx ii) iii) iv) v) vi)

Then, Θ1 (A, B) is a convex set. Source: [2248]. Related: This result follows from Fact 10.17.8. Fact 10.17.11. Let A, B ∈ Rn×n, assume that A and B are symmetric, and let α, β be real numbers. Then, the following statements are equivalent: i) There exists x ∈ Rn such that xTAx = α and xTBx = β. ii) There exists a positive-semidefinite matrix X ∈ Rn×n such that tr AX = α and tr BX = β. Source: [309, p. 84].

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POSITIVE-SEMIDEFINITE MATRICES

Fact 10.17.12. Let A, B ∈ Rn×n, assume that A and B are symmetric, and define △

Ψ1 (A, B) =

{[

] } xTAx n T : x ∈ R and x x = 1 ⊆ R2 , xTBx

△

Ψ(A, B) =

{[

} ] xTAx n ⊆ R2. : x ∈ R xTBx

Then, Ψ(A, B) is a pointed, convex cone. If, in addition, n ≥ 3, then Ψ1 (A, B) is a convex {[ set. ]} Source: [309, pp. 84, 89] and [903, 2245, 2248]. Remark: Ψ(A, B) = [cone Ψ1 (A, B)] ∪ 00 . Remark: The set Ψ(A, B) is not necessarily closed. See [903, 2188, 2189]. Fact 10.17.13. Let A, B ∈ Rn×n, assume that A and B are symmetric, let A, B be closed subsets of Rn , assume that A∪B = Rn , and assume that, for all x ∈ A, xTAx ≥ 0 and, for all x ∈ B, xTBx ≥ 0. Then, there exists α ∈ [0, 1] such that αA + (1 − α)B is positive semidefinite. Source: [2245]. Fact 10.17.14. Let A, B ∈ Rn×n, and assume that A and B are symmetric. Then, the following statements are equivalent: i) For all x ∈ Rn , either xTAx > 0 or xTBx > 0. ii) There exist α, β ∈ [0, ∞) such that αA + βB is positive definite. Now, assume that there exists y ∈ Rn such that yTBy < 0. Then, the following statements are equivalent: iii) For all x ∈ Rn , either xTAx ≥ 0 or xTBx > 0. iv) There exists α ∈ [0, ∞) such that A + αB is positive semidefinite. Source: [2245]. Remark: The equivalence of iii) and iv) is the S-lemma. Fact 10.17.15. Let n ≥ 2, let A, B ∈ Rn×n, assume that A and B are symmetric, let a, b ∈ Rn, let a0 , b0 ∈ R, assume that there exist real numbers α, β such that αA + βB > 0, and define   T      x Ax + aT x + a0   △  n ⊆ R2. Ψ=  : x ∈ R   T   T   x Bx + b x + b0 Then, Ψ is a closed, convex set. Source: [2248]. Fact 10.17.16. Let A, B, C ∈ Rn×n, where n ≥ 3, assume that A, B, and C are symmetric, and define   T    T       x Ax   x Ax          △  △  3  xTBx  : x ∈ Rn   xTBx  : x ∈ Rn and xTx = 1 ⊆ R3. ⊆ R , Φ(A, B, C) = Φ1 (A, B, C) =                   xTCx    xTCx  Then, Φ1 (A, B, C) is a convex set, and Φ(A, B, C) is a pointed, convex cone. Source: [573, 2245, 2248]. Fact 10.17.17. Let A, B, C ∈ Rn×n, where n ≥ 3, assume that A, B, and C are symmetric, and define  T        xTAx   △  n     x Bx Φ(A, B, C) =  : x ∈ R ⊆ R3.             xTCx  Then, the following statements are equivalent: i) There exist real numbers α, β, γ such that αA + βB + γC is positive definite. ii) Φ(A, B, C) is a pointed, one-sided, closed, convex cone, and {x ∈ Rn : xTAx = xTBx = xTCx = 0} ⊆ {0}. Source: [2248]. Fact 10.17.18. Let A ∈ Fn×n, assume that A is Hermitian, let b ∈ Fn and c ∈ R, and define f : Fn 7→ R by △ f (x) = x∗Ax + Re(b∗x) + c.

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Then, the following statements hold: i) f is convex if and only if A is positive semidefinite. ii) f is strictly convex if and only if A is positive definite. Now, assume that A is positive semidefinite. Then, f has a minimizer if and only if b ∈ R(A). If these conditions hold, then the following statements hold: iii) The vector x0 ∈ Fn is a minimizer of f if and only if x0 satisfies Ax0 = − 21 b. iv) x0 ∈ Fm minimizes f if and only if there exists a vector y ∈ Fm such that x0 = − 12 A+b + (I − A+A)y. v) If x0 ∈ Fm minimizes f , then f (x0 ) = c − x0∗ Ax0 = c − 41 b∗A+b. vi) If A is positive definite, then x0 = − 12 A−1b is the unique minimizer of f, and the minimum of f is given by f (x0 ) = c − x0∗ Ax0 = c − 41 b∗A−1b. Source: Use Proposition 8.1.8 and note that, for each x0 ∈ Fn satisfying Ax0 = − 12 b, it follows that

f (x0 ) = (x − x0 )∗A(x − x0 ) + c − x0∗ Ax0 = (x − x0 )∗A(x − x0 ) + c − 41 b∗A+b. Remark: This is the quadratic minimization lemma. Related: Fact 11.17.10. Fact 10.17.19. Let A ∈ Fn×n, assume that A is positive definite, and define ϕ: Fm×n 7→ R by △ ϕ(B) = tr BAB∗. Then, ϕ is strictly convex. Source: tr α(1 − α)(B1 − B2 )A(B1 − B2 )∗ > 0. △ Fact 10.17.20. Let p, q ∈ R, and define ϕ : (0, ∞) × (0, ∞) 7→ (0, ∞) by ϕ(a, b) = a p bq . Then,

the following conditions are equivalent: i) One of the following conditions holds: a) p ≥ 1, q ≤ 0, and p + q ≥ 1. b) p ≤ 0, q ≥ 1, and p + q ≥ 1. c) p ≤ 0 and q ≤ 0. △ ii) ϕ : (0, ∞) × (0, ∞) 7→ (0, ∞) defined by ϕ(a, b) = a p bq is convex. Furthermore, the following conditions are equivalent: iii) p ∈ [0, 1], q ∈ [0, 1], and p + q ≤ 1. iv) ϕ is concave. Source: [632]. △ Fact 10.17.21. Let n ≥ 1, let p, q ∈ R, and define ϕ: Pn × Pn → (0, ∞) by ϕ(A, B) = tr ApBq . Then, the following statements hold: i) Assume that one of the following conditions holds: a) p ∈ [1, 2], q ∈ [−1, 0], and p + q ≥ 1. b) p ∈ [−1, 0), q ∈ [1, 2], and p + q ≥ 1. c) p ∈ [−1, 0) and q ∈ [−1, 0). Then, ϕ is convex. ii) If p ∈ (0, 1), q ∈ (0, 1), and p + q ≤ 1, then ϕ is concave. iii) If p and q do not satisfy the assumptions of either i) or ii), then ϕ is neither convex nor concave. Source: [346].

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POSITIVE-SEMIDEFINITE MATRICES

Fact 10.17.22. Let n ≥ 1. Then, the following statements hold: △

i) ϕ: Pn × Pn 7→ Pn defined by ϕ(A, B) = A p/2 BqA p/2 is convex if and only if p = 2 and q ∈ [−1, 0). △ ii) ϕ: Pn × Pn × Pn 7→ Pn defined by ϕ(A, B, C) = A p/2BqA p/2C r is convex if and only if p = 2, q < 0, r < 0, and q + r ∈ [−1, 0). iii) If p ∈ [−1, 0), q ∈ [1, 2], p+q , 0, and s ∈ [min {1/(q−1), 1/(p+1)}, ∞), then ϕ: Pn ×Pn 7→ △ (0, ∞) defined by ϕ(A, B) = tr (A p/2BqA p/2 ) s is convex. △ iv) If p ∈ [−1, 0) and s ∈ [1/(p + 2), ∞), then ϕ: Pn × Pn 7→ (0, ∞) defined by ϕ(A, B) = tr (A p/2B2A p/2 ) s is convex. v) If p ∈ [0, 1], q ∈ [0, 1], and s ∈ [0, 1/(p + q], then ϕ: Pn × Pn 7→ (0, ∞) defined by △ ϕ(A, B) = tr (A p/2B2A p/2 ) s is convex. Source: [632]. Fact 10.17.23. Let A ∈ Fn×n , assume that A is Hermitian, let x ∈ Fn , and define f : R 7→ R by ∗ △ f (t) = x∗ eA+txx x. Then, f is increasing and strictly convex. Source: [2578]. △ Fact 10.17.24. Let B ∈ Fn×n, assume that B is Hermitian, let α1 , . . . , αk ∈ (0, ∞), define r = k n ∑k i=1 P → [0, ∞) by i=1 αi , assume that r ≤ 1, let q ∈ R, and define ϕ: ( )q ∑k △ ϕ(A1 , . . . , Ak ) = − tr eB+ i=1 αi log Ai . If q ∈ (0, 1/r], then ϕ is convex. Furthermore, if q < 0, then −ϕ is convex. Source: [1817, 1891, 2003]. Related: Fact 10.14.46. Fact 10.17.25. Let A1 , . . . , Am ∈ Rn×n , assume that A1 , . . . , Am are positive definite, define △ △ ∏ T −1 −1 f : Rn 7→ R by f (x) = m i=1 x Ai x, and, for all i, j ∈ {1, . . . , m}, define κi, j = λmax (Ai A j )/λmin (Ai A j ). Then, the following statements hold: i) f is convex if and only if, for all x, y ∈ Rn such that y , 0, m ∑ m m ∑ ∑ xTAi yxTA j y xTAi x + 2 ≥ 0. T y Ai y yTAi yyTA j y i=1 j=1, j,i i=1

ii) If, for all i ∈ {1, . . . , m},

) m ( √ ∑ κi, j − 1 2 1 ≤ , √ κi, j + 1 2 j=1

then f is convex. iii) If, for all distinct i, j ∈ {1, . . . , m},

√ 2  2m − 2 + 1   , κi, j ≤  √ 2m − 2 − 1

then f is convex. △ iv) Let A, B ∈ Fn×n , assume that A and B are positive definite, define g : Rn 7→ R by g(x) = △ xTAxxTBx, and define κ = λmax (AB−1 )/λmin (AB−1 ). Then, the following statements are equivalent: a) g is convex. √ √ κ−1 2 ≤ . b) √ 2 κ+1 √ c) κ ≤ 17 + 12 2.

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CHAPTER 10 △

v) Let A ∈ Fn×n , assume that A is positive definite, and define g : Rn 7→ R √by g(x) = △ xTAxxTA−1 x, and define κ = cond(A). Then, g is convex if and only if κ ≤ 3 + 2 2. √ √ Source: [1847]. Remark: See [1400]. Remark: (3 + 2 2)2 = 17 + 12 2.

10.18 Facts on Quadratic Forms for One Matrix Fact 10.18.1. Let G = (X, R) be a symmetric graph, where X = {x1 , . . . , xn }. Then, for all z ∈ Rn, the Laplacian matrix L of G satisfies 1∑ zTLz = (z(i) − z( j) )2 , 2 where the sum is taken over the set {(i, j): (xi , x j ) ∈ R}. Source: [591, pp. 29, 30] and [2028]. Fact 10.18.2. Let A ∈ Fn×n, and assume that A is Hermitian. Then, N(A) ⊆ {x ∈ Fn : x∗Ax = 0}. Furthermore, N(A) = {x ∈ Fn : x∗Ax = 0} if and only if either A ≥ 0 or A ≤ 0. Fact 10.18.3. Let x, y ∈ Fn. Then, xx∗ ≤ yy∗ if and only if there exists α ∈ F such that |α| ∈ [0, 1] and x = αy. Fact 10.18.4. Let x, y ∈ Fn. Then, xy∗ + yx∗ ≥ 0 if and only if x and y are linearly dependent. Source: Evaluate the product of the nonzero eigenvalues of xy∗ + yx∗, and use the Cauchy-Schwarz inequality |x∗y|2 ≤ x∗xy∗y. Fact 10.18.5. Let A ∈ Fn×n, assume that A is positive definite, let x ∈ Fn, and let a ∈ [0, ∞). Then, the following statements are equivalent: i) xx∗ ≤ aA. ii) x∗A−1x ≤ a. [ ] A x iii) ∗ ≥ 0. x a Source: Fact 3.17.3 and Proposition 10.2.5. Note that, if a = 0, then x = 0. Fact 10.18.6. Let A ∈ Fn×n, assume that A is positive definite, and let x, y ∈ Fn. Then,

2Re x∗y ≤ x∗Ax + y∗A−1y. Furthermore, if y = Ax, then equality holds. Therefore, x∗Ax = max{2Re x∗z − z∗A−1 z : z ∈ Fn }. Source: (A1/2 x − A−1/2 y)∗ (A1/2 x − A−1/2 y) ≥ 0. Credit: R. Bellman. See [1783, 2987]. Fact 10.18.7. Let A ∈ Fn×n, assume that A is positive definite, and let x, y ∈ Fn. Then,

|x∗y|2 ≤ (x∗Ax)(y∗A−1y). Source: Use Fact 10.12.33 with A replaced by A1/2 x and B replaced by A−1/2 y. Alternatively, use

Fact 10.12.24 and Fact 10.19.23. △ Fact 10.18.8. Let A ∈ Fn×n, assume that A is positive definite, let x ∈ Fn, and define α = λmin (A) △ and β = λmax (A). Then, (x∗x)2 ≤ (x∗Ax)(x∗A−1x) ≤

(α + β)2 ∗ 2 β ∗ 2 (x x) ≤ (x x) . 4αβ α

Source: [43], [1448, p. 443], [1451, p. 470], [1876], and [2991, pp. 249, 251]. Remark: The second inequality is the Kantorovich inequality. Related: Fact 2.11.134, Fact 10.11.42, and Fact

10.11.43. △

Fact 10.18.9. Let A ∈ Fn×n, assume that A is nonsingular, let x ∈ Fn, and define κ = ρ1 (A)/ρn (A).

Then,

|x∗Axx∗A−1 x| ≤ 41 (κ + κ−1 + 2)(x∗ x)2 ,

|x∗A2 xx∗A−2 x| ≤ 14 (κ + κ−1 )2 (x∗ x)2 .

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POSITIVE-SEMIDEFINITE MATRICES

Source: [1451, p. 474]. Remark: κ is the spectral condition number of A. Related: Fact 10.18.8. △ Fact 10.18.10. Let A ∈ Fn×n, assume that A is positive definite, and define κ = λ1 (A)/λn (A). If

x ∈ Fn, then

|x∗Axx∗A−1 x| ≤ 14 (κ + κ−1 + 2)(x∗ x)2 .

If x, y ∈ Fn and x∗ y = 0, then

(

κ−1 |x Ay| ≤ κ+1 ∗

2

)2

x∗Axy∗Ay.

Source: [1451, p. 474]. Remark: These are equivalent to the Kantorovich and Wielandt inequalities, respectively. Related: Fact 10.18.8 and Fact 10.18.21. △ Fact 10.18.11. Let A ∈ Fn×n, assume that A is positive definite, let x ∈ Fn, and define α = △

λmin (A) and β = λmax (A). Then,

(x∗x)1/2 (x∗Ax)1/2 − x∗Ax ≤

(α − β)2 ∗ x x, 4(α + β)

(x∗x)(x∗A2 x) − (x∗Ax)2 ≤ 41 (α − β)2 (x∗x)2 .

Source: [2214]. Remark: Extensions are given in [1510, 2214]. △ Fact 10.18.12. Let A ∈ Fn×n, assume that A is positive semidefinite, let r = rank A, let x ∈ Fn,

and assume that x < N(A). Then,

x∗Ax x∗ x 1/2 2 − ≤ [λ1/2 max (A) − λr (A)] . x∗ x x∗A+ x If, in addition, A is positive definite, then, for all nonzero x ∈ Fn, x∗ x x∗Ax 1/2 2 − ≤ [λ1/2 max (A) − λmin (A)] . x∗ x x∗A−1x Source: [2071, 2214]. The left-hand inequality in the last string is given by Fact 10.18.8. Fact 10.18.13. Let A ∈ Fn×n, assume that A is positive definite, let y ∈ Fn, let α > 0, and define △ f : Fn 7→ R by f (x) = |x∗y|2. Then, √ α A−1y x0 = y∗A−1y 0≤

minimizes f (x) subject to x∗Ax ≤ α. Furthermore, f (x0 ) = αy∗A−1y. Source: [73]. Fact 10.18.14. Let A ∈ Fn×n and x ∈ Fn. Then, |x∗Ax| ≤ x∗A∗Ax,

|x∗Ax|2 ≤ (x∗x)x∗A∗Ax.

Source: [2991, pp. 291, 302]. Fact 10.18.15. Let A ∈ Fn×n and x ∈ Fn. Then,

√ x∗ (A + A∗ )x ≤ 2 x∗xx∗A∗Ax.

Source: [1142]. Fact 10.18.16. Let A ∈ Fn×n. Then, the following statements are equivalent:

i) ii) iii) iv) v)

A is normal. For all x ∈ Fn, |x∗Ax| ≤ x∗ ⟨A⟩x. For all x ∈ Fn, |Ax| = |A∗ x|. There exists α ∈ [0, 12 ) ∪ ( 12 , 1] such that, for all x ∈ Fn, |x∗Ax| ≤ (x∗ ⟨A⟩x)α (x∗ ⟨A∗ ⟩x)1−α . For all x, y ∈ Fn, y∗A∗Ax = y∗AA∗ x. Source: [2991, pp. 294, 295, 303, 315, 318].

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Fact 10.18.17. Let A ∈ Fn×n, assume that A is positive semidefinite, and let x ∈ Fn. Then,

√ (x∗Ax)2 ≤ (x∗x)(x∗A2x) ≤ (x∗x) (x∗Ax)(x∗A3x),

(min 14 [λi (A) − λ j (A)]2 )(x∗x)2 ≤ (x∗x)(x∗A2x) − (x∗Ax)2 ≤ 14 [λ1 (A) − λn (A)]2 (x∗x)2 , where the minimum is taken over all distinct i, j ∈ {1, . . . , n}. Source: Use the Cauchy-Schwarz inequality Corollary 11.1.7. The second string is given in [1344]. Fact 10.18.18. Let A ∈ Fn×n, assume that A is positive semidefinite, and let x ∈ Fn. If α ∈ [0, 1], then x∗Aαx ≤ (x∗x)1−α (x∗Ax)α. Furthermore, if α > 1, then

(x∗Ax)α ≤ (x∗x)α−1 x∗Aαx.

Remark: The first inequality is the H¨older-McCarthy inequality, which is equivalent to Young’s

inequality. See [1124, p. 125] and [1126]. Matrix versions of the second inequality are given in [1421]. Related: Fact 10.10.47 and Fact 10.11.68. Fact 10.18.19. Let A ∈ Fn×n, assume that A is positive semidefinite, let x ∈ Fn, and let α, β ∈ [1, ∞), where α ≤ β. Then, (x∗Aα x)1/α ≤ (x∗Aβx)1/β. Now, assume that A is positive definite. Then, x∗ (log A)x ≤ log x∗Ax ≤ α1 log x∗Aα x ≤ β1 log x∗Aβx. Source: [1085]. Fact 10.18.20. Let A ∈ Fn×n, x, y ∈ Fn, and α ∈ (0, 1). Then,

|x∗Ay| ≤ ∥⟨A⟩α x∥2 ∥⟨A∗ ⟩1−αy∥2 . Consequently,

|x∗Ay| ≤ (x∗⟨A⟩x)1/2 (y∗⟨A∗ ⟩y)1/2.

Source: [1551] and [2991, p. 314]. Fact 10.18.21. Let A ∈ Fn×n, assume that A is positive semidefinite, let x, y ∈ Fn, and assume

that x∗y = 0. Then,

|x∗Ay|2 ≤

[

]2 λmax (A) − λmin (A) (x∗Ax)(y∗Ay). λmax (A) + λmin (A)

Furthermore, there exist x, y ∈ Fn satisfying x∗y = 0 and x∗ x = y∗y = 1 for which equality holds. Source: [1448, p. 443], [1451, p. 471], and [1783, 2987]. Remark: This is the Wielandt inequality. Related: Fact 10.19.23. Fact 10.18.22. Let A ∈ Fn×n, assume that A is Hermitian, let x, y ∈ Fn, and assume that x∗ x = ∗ y y = 1 and x∗y = 0. Then, 2|x∗Ay| ≤ λmax (A) − λmin (A). Furthermore, there exist x, y ∈ Fn satisfying x∗ x = y∗y = 1 and x∗y = 0 for which equality holds. Source: [1783, 2987] and [2991, p. 292]. Remark: λmax (A) − λmin (A) is the spread of A. Related: Fact 6.10.4, Fact 10.21.5, Fact 11.11.7, and Fact 11.11.8. Fact 10.18.23 provides a lower bound in the case where A is positive semidefinite. Fact 10.18.23. Let A ∈ Fn×n, assume that A is positive semidefinite, let x, y ∈ Fn, and assume that x∗y = 0. Then, √ √ (min 21 [λi (A) − λ j (A)]) x∗ xy∗y ≤ |x∗Ay| ≤ 12 [λ1 (A) − λn (A)] x∗ xy∗y,

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POSITIVE-SEMIDEFINITE MATRICES

where the minimum is taken over all i, j ∈ {1, . . . , n} such that i < j. Source: [1344]. Related: Fact 10.18.22 shows that the upper bound holds for Hermitian A. Fact 10.18.24. Let A ∈ Fn×n, assume that A is positive semidefinite, let x, y ∈ Fn, and assume that x∗ x = y∗y = 1. Then, √ |x∗Ax − y∗Ay| ≤ [λ1 (A) − λn (A)] 1 − (x∗ y)2 . Source: [1344]. Fact 10.18.25. Let A ∈ Fn×n, assume that A is positive semidefinite, let x, y ∈ Fn, and assume

that x∗ x = y∗y = 1 and x∗y = 0. Then, 1 4 [λn−1 (A)

+ λn (A)]2 ≤ x∗Axy∗Ay ≤ 14 [λ1 (A) + λ2 (A)]2 .

Source: [1344]. Fact 10.18.26. Let A ∈ Rn×n, assume that A is positive semidefinite, let x, y ∈ Rn, assume that

xTx = yTy = 1, and let θ ∈ [0, π] be the angle between x and y. Then,

xTAy ≤ 12 (cos θ)(xTAx + yTAy) + 12 (sin θ)δ(A). Source: [1344]. Problem: Extend this result to complex matrices. Fact 10.18.27. Let A ∈ Rm×m , assume that A is positive semidefinite, and assume that every en-

try of A is an integer. If every off-diagonal entry of A is even and, for all k ∈ {1, 2, 3, 5, 6, 7,10,14,15}, there exists x ∈ Rm such that k = xTAx, then, for all n ≥ 0, there exists x ∈ Rm such that n = xTAx. Furthermore, if, for all k ∈ {1, 2, 3, 5, 6, 7, 10, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 34, 35, 37, 42, 58, 93, 110, 145, 203, 290}, there exists x ∈ Rm such that k = xTAx, then, for all n ≥ 0, there exists x ∈ Rm such that n = xTAx. Source: [2887]. Related: Fact 1.11.27.

10.19 Facts on Quadratic Forms for Two or More Matrices Fact 10.19.1. Let A, B ∈ Fn×n, assume that A and B are Hermitian, assume that A + B is △

nonsingular, let x, a, b ∈ Fn , and define c = (A + B)−1 (Aa + Bb). Then,

(x − a)∗A(x − a) + (x − b)∗B(x − b) = (x − c)∗ (A + B)(x − c) + (a − b)∗A(A + B)−1B(a − b). Source: [2418, p. 278]. Fact 10.19.2. Let A, B ∈ Rn×n, assume that A is symmetric and B is skew symmetric, and let

x, y ∈ Rn . Then,

[ ]T [ x A y BT

B A

][ ] x = (x + ȷy)∗ (A + ȷB)(x + ȷy). y

Related: Fact 6.10.32. Fact 10.19.3. Let A, B ∈ Fn×n, assume that A is positive semidefinite, assume that AB is Hermi-

tian, and let x ∈ Fn. Then,

|x∗ABx| ≤ ρmax (B)x∗Ax.

Source: [1825]. Remark: This is an improvement of Reid’s inequality. Related results are given in [1826]. Credit: P. R. Halmos. Fact 10.19.4. Let A, B ∈ Fn×n, assume that A and B are positive definite, and let x ∈ Fn. Then,

x∗ (A + B)−1 x ≤ In particular,

1 (A−1 )

x∗A−1 xx∗B−1 x ≤ 1 (x∗A−1 x + x∗B−1 x). x∗A−1x + x∗B−1 x 4 +

(i,i)

1 (B−1 )

≤ (i,i)

1 . [(A + B)−1 ](i,i)

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Source: [1924, p. 201]. The right-hand inequality follows from Fact 2.2.12. Remark: This is Bergstrom’s inequality. Remark: This is a special case of Fact 10.12.5, which is a special case of xvii) of Proposition 10.6.17. Fact 10.19.5. Let A, B ∈ Fn×m, assume that I − A∗A and I − B∗B are positive semidefinite, and let x ∈ Cn. Then, x∗(I − A∗A)xx∗(I − B∗B)x ≤ |x∗(I − A∗B)x|2. Credit: M. Marcus. See [2184]. Related: Fact 10.16.28. Fact 10.19.6. Let A, B ∈ Fn, and assume that A is Hermitian and B is positive definite. Then,

x∗Ax , x∈F \{0} x∗Bx x∗Ax . λmax (AB−1 ) = max {λ ∈ R: det(A − λB) = 0} = max x∈Fn \{0} x∗Bx λmin (AB−1 ) = min {λ ∈ R: det(A − λB) = 0} = min n

Source: Lemma 10.4.3 and [2991, pp. 273, 285]. Fact 10.19.7. Let w, x, y, z ∈ Fn , let A, B ∈ Fn×n , and assume that A and B are positive definite.

Then,

2w∗ yx∗ z ≤ w∗A−1 wx∗Bx + y∗Ayz∗B−1 z.

Source: [241]. Fact 10.19.8. Let A ∈ Fn×n, B ∈ Fn×m, x ∈ Fn , and y ∈ Fm , and assume that A is positive definite.

Then,

|x∗By|2 ≤ x∗Axy∗B∗A−1By.

Source: [2991, p. 250]. Fact 10.19.9. Let A, B ∈ Fn×m, x ∈ Fn , and y ∈ Fm . Then,

|x∗ (A + B)y|2 ≤ x∗ (I + AA∗ )xy∗ (I + B∗B)y. Source: [2991, p. 250]. Fact 10.19.10. Let A, B ∈ Rn×n, assume that A is positive definite and B is symmetric, let

x, y ∈ Rn , and assume that xTx = yTy = 1. Then,

|xTABy − xTAxxTBy| ≤ 41 [λ1 (A) − λn (A)][λ1 (B) − λn (B)]. Source: [1344]. Fact 10.19.11. Let A, B ∈ Fn×n, and assume that A is positive definite and B is positive semidef-

inite. Then, for all nonzero x ∈ Fn , 4(x∗x)(x∗Bx) < (x∗Ax)2 if and only if there exists α > 0 such that αI + α−1B < A. If these conditions hold, then 4B < A2, and hence 2B1/2 < A. Source: Sufficiency follows from αx∗x + α−1x∗Bx < x∗Ax. Necessity follows from Fact 10.19.12. The last statement follows from (A − 2αI)2 ≥ 0 and 2B1/2 ≤ αI + α−1B. Fact 10.19.12. Let A, B, C ∈ Fn×n, assume that A, B, C are positive semidefinite, and assume that, for all nonzero x ∈ Fn, 4(x∗Cx)(x∗Bx) < (x∗Ax)2 . Then, there exists α > 0 such that αC + α−1B < A. Source: [2224]. Fact 10.19.13. Let A, B ∈ Fn×n, where A is Hermitian and B is positive semidefinite. Then, ∗ x Ax < 0 for all nonzero x ∈ Fn such that Bx = 0 if and only if there exists α > 0 such that A < αB. Source: To prove necessity, suppose that, for every α > 0, there exists a nonzero vector x such that x∗Ax ≥ αx∗Bx. Then, Bx = 0 implies that x∗Ax ≥ 0. Sufficiency is immediate. Fact 10.19.14. Let A, B ∈ Cn×n, and assume that A and B are Hermitian. Then, the following statements are equivalent: i) There exist α, β ∈ R such that αA + βB is positive definite.

797

POSITIVE-SEMIDEFINITE MATRICES

ii) {x ∈ Cn : x∗Ax = x∗Bx = 0} = {0}. Remark: This is Finsler’s lemma. See [185, 341, 1400, 1737, 2738, 2759]. Related: Fact 10.19.15, Fact 10.20.7, and Fact 10.20.8. Fact 10.19.15. Let A, B ∈ Rn×n, and assume that A and B are symmetric. Then, the following statements are equivalent: i) There exist α, β ∈ R such that αA + βB is positive definite. ii) Either xTAx > 0 for all nonzero x ∈ {y ∈ Fn : yTBy = 0} or xTAx < 0 for all nonzero x ∈ {y ∈ Fn : yTBy = 0}. Now, assume that n ≥ 3. Then, the following statement is equivalent to i) and ii): iii) {x ∈ Rn : xTAx = xTBx = 0} = {0}. Remark: This result is related to Finsler’s lemma. See [185, 341, 2759]. Related: Fact 10.19.14, Fact 10.20.7, and Fact 10.20.8. Fact 10.19.16. Let A, B ∈ Fn×n, assume that A and B are positive definite, and let C, D ∈ Fn×m . Then, (C + D)∗ (A + B)−1 (C + D) ≤ C ∗A−1C + D∗B−1D. In particular, if x, y ∈ Fn , then (x + y)∗ (A + B)−1 (x + y) ≤ x∗A−1x + y∗B−1 y. Source: Use xiv) of Lemma 10.6.16. See [1451, p. 475], [2142], and [2991, p. 243]. Related:

Fact 10.19.17 and Fact 10.25.59. Fact 10.19.17. Let A, B ∈ Fn×n, assume that A and B are positive semidefinite, let C, D ∈ Fn×m , and assume that R(C) ⊆ R(A) and R(D) ⊆ R(B). Then, (C + D)∗ (A + B)+ (C + D) ≤ C ∗A+C + D∗B+D. In particular, if x ∈ R(A) and y ∈ R(B), then (x + y)∗ (A + B)+ (x + y) ≤ x∗A+x + y∗B+ y. Source: [2142]. Related: Fact 10.19.16. Fact 10.19.18. Let A, B ∈ Fn×n, assume that A and B are positive semidefinite, let C ∈ Fn×m ,

assume that R(C) ⊆ R(A) ∩ R(B), and let α ∈ [0, 1]. Then,

C ∗ [αA + (1 − α)B]+C ≤ αC ∗A+C + (1 − α)C ∗B+C. Source: [2142]. Related: Fact 10.19.17. Fact 10.19.19. Let A1 , . . . , Ak ∈ Fn×n, assume that A1 , . . . , Ak are positive semidefinite, let

B1 , . . . , Bk ∈ Fn×m , assume that, for all i ∈ {1, . . . , k}, R(Bi ) ⊆ R(Ai ), let α1 , . . . , αk ≥ 0, and assume ∑ that ki=1 αi = 1. Then,  k ∗  k +  k  k ∑  ∑  ∑  ∑  αi Bi   αi Ai   αi Bi  ≤ αi B∗i A+i Bi . i=1

i=1

i=1

i=1

In particular, if, for all i ∈ {1, . . . , k}, xi ∈ R(Ai ), then  k ∗  k +  k  k ∑  ∑  ∑  ∑  αi xi   αi Ai   αi xi  ≤ αi xi∗ A+i xi .       i=1

i=1

i=1

i=1

Source: [2142]. Credit: N. Gaffke and O. Krafft. Related: Fact 10.19.17.

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Fact 10.19.20. Let A, B ∈ Cn×n , where A and B are Hermitian, and assume that x∗ (A + ȷB)x , 0

for all nonzero x ∈ Cn. Then, there exists t ∈ [0, π) such that (sin t)A + (cos t)B is positive definite. Source: [793] and [2539, p. 282]. Fact 10.19.21. Let A ∈ Rn×n, assume that A is symmetric, and let B ∈ Rn×m. Then, the following statements are equivalent: i) xTAx > 0 for all nonzero x ∈ N(BT ). ([ ]) A B ii) ν+ = n. BT 0 Furthermore, the following statements are equivalent: iii) xTAx ≥ 0 for all x ∈ N(BT ). ([ ]) A B iv) ν− = rank B. BT 0 Source: [661, 1921]. Related: Fact 7.9.22 and Fact 10.19.22. Fact 10.19.22. Let A ∈ Rn×n, assume that A is symmetric, let B ∈ Rn×m, where m ≤ n, and assume that [Im 0m×(n−m) ]B is nonsingular. Then, the following statements are equivalent: i) xTAx > 0 for all nonzero x ∈ N(BT ). [ T] ii) For all i ∈ {m + 1, . . . , n}, sign det 0 B = (−1)m. B A ({1,...,i})

Source: [200, p. 20], [1895, p. 312], and [1942]. Related: Fact 10.19.21. △

Fact 10.19.23. Let A ∈ Fn×n, B ∈ Fn×m, and C ∈ Fm×m, define A =

[

]

, and assume that A and C are positive semidefinite. Then, the following statements are equivalent: i) A is positive semidefinite. ii) |x∗By|2 ≤ (x∗Ax)(y∗Cy) for all x ∈ Fn and y ∈ Fm. iii) 2|x∗By| ≤ x∗Ax + y∗Cy for all x ∈ Fn and y ∈ Fm. If, in addition, A and C are positive definite, then the following statement is equivalent to i)–iii): iv) ρmax (B∗A−1BC −1 ) ≤ 1. Finally, if A is positive semidefinite and nonzero, then, for all x ∈ Fn and y ∈ Fm, [ ]2 λmax (A) − λmin (A) ∗ 2 |x By| ≤ (x∗Ax)(y∗Cy). λmax (A) + λmin (A) A B B∗ C

Source: [1448, p. 473], [1783, 2987], and [2991, p. 246]. Related: Fact 10.18.7 and Fact

10.18.21. Fact 10.19.24. Let n ≤ 4, let A ∈ Rn×n, assume that A is symmetric, and assume that, for

all nonnegative vectors x ∈ Rn, xTAx ≥ 0. Then, there exist B, C ∈ Rn×n such that B is positive semidefinite, C is symmetric and nonnegative, and A = B + C. Remark: If n ≥ 5, then this result does not hold. Hence, this result is an example of the quartic barrier. See [787], Fact 10.17.8, and Fact 15.18.5. Remark: A is copositive. ∑ Fact 10.19.25. Let m > k ≥ n ≥ 1, let x1 , . . . , xm ∈ Fn , and assume that ki=1 xi xi∗ is positive definite. Then,  j −2  k −1 m ∑ ∑  ∑  ∗ ∗ ∗ x j  xi xi  x j ≤ tr  xi xi  . j=k+1

i=1

i=1

Source: [1833]. Credit: T. W. Anderson and J. B. Taylor.

799

POSITIVE-SEMIDEFINITE MATRICES

10.20 Facts on Simultaneous Diagonalization Fact 10.20.1. Let A, B ∈ Fn×n , assume that A and B are Hermitian, and assume that R(A) ⊆

R(B). Then, there exists a unitary matrix S ∈ Fn×n , such that [ ] [ A1 0 B ∗ ∗ SAS = , SBS = 1 0 0 0 △

△

] 0 , 0

where A1 ∈ Hr , B1 = diag(λ1 (B), . . . , λr (B)), and r = rank B. Fact 10.20.2. Let A, B ∈ Fn×n, assume that A and B are Hermitian. Then, the following statements are equivalent: i) There exists a unitary matrix S ∈ Fn×n such that SAS ∗ and SBS ∗ are diagonal. ii) AB = BA. iii) AB is Hermitian. If, in addition, A is nonsingular, then the following statement is equivalent to i)–iii): iv) A−1B is Hermitian. Source: [360, p. 208], [970, pp. 188–190], and [1448, p. 229]. Related: The equivalence of i) and ii) is given by Fact 7.19.8. Fact 10.20.3. Let A, B ∈ Fn×n, assume that A and B are Hermitian, and assume that A is nonsingular. Then, there exists a nonsingular matrix S ∈ Fn×n such that SAS ∗ and SBS ∗ are diagonal if and only if A−1B is diagonalizable over R. Source: [1448, p. 229] and [2263, p. 95]. Fact 10.20.4. Let A ∈ Fn×n. Then, the following statements are equivalent: i) A is diagonalizable, and spec(A) ⊂ R. ii) There exist positive-definite B ∈ Fn×n and Hermitian C ∈ Fn×n such that A = BC. Now, assume that i) and ii) hold. Then, the following statements hold: iii) In(A) = In(C). iv) A is positive definite if and only if A is Hermitian and C is positive definite. Source: [2991, p. 265]. Related: This result provides the converse of Corollary 10.3.3. Fact 10.20.5. Let A, B ∈ Fn×n, assume that A and B are symmetric, and assume that A is nonsingular. Then, there exists a nonsingular matrix S ∈ Fn×n such that SAS T and SBS T are diagonal if and only if A−1B is diagonalizable. Source: [1448, p. 229] and [2759]. Remark: If F = C, then A and B may be complex symmetric. Fact 10.20.6. Let A, B ∈ Fn×n, and assume that A and B are Hermitian. Then, there exists a nonsingular matrix S ∈ Fn×n such that SAS ∗ and SBS ∗ are diagonal if and only if there exists a positive-definite matrix M ∈ Fn×n such that AMB = BMA. Source: [185]. Fact 10.20.7. Let A, B ∈ Fn×n, assume that A and B are Hermitian, and assume that there exist α, β ∈ R such that αA+βB is positive definite. Then, there exists a nonsingular matrix S ∈ Fn×n such that SAS ∗ and SBS ∗ are diagonal. Source: [1448, p. 465]. Remark: This result extends a result of K. Weierstrass. See [2759]. Remark: Suppose that B is positive definite. Then, by necessity of Fact 10.20.3, it follows that A−1B is diagonalizable over R, which proves iii) =⇒ i) of Proposition 7.7.13. Related: Fact 10.20.8. Fact 10.20.8. Let A, B ∈ Fn×n, assume that A and B are Hermitian, assume that {x ∈ Fn : x∗Ax = ∗ x Bx = 0} = {0}, and, if F = R, assume that n ≥ 3. Then, there exists a nonsingular matrix S ∈ Fn×n such that SAS ∗ and SBS ∗ are diagonal. Source: This result follows from Fact 7.19.10. See [1929] and [2263, p. 96]. Remark: For F = R, this result is due to E. Pesonen and J. Milnor. See [2759]. Related: Fact 7.19.10, Fact 10.19.14, Fact 10.19.15, and Fact 10.20.7.

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CHAPTER 10

10.21 Facts on Eigenvalues and Singular Values for One Matrix Fact 10.21.1. Let A =

where λ2 ≤ λ1 . Then,

[a b] b c

∈ F2×2, assume that A is Hermitian, and let mspec(A) = {λ1 , λ2 }ms , 2|b| ≤ λ1 − λ2 .

Now, assume that A is positive semidefinite. Then, (√ √ )√ √ 2|b| ≤ λ1 − λ2 λ1 + λ2 . If c > 0, then

√ √ |b| √ ≤ λ1 − λ2 . c

If a > 0 and c > 0, then

|b| λ1 − λ2 . √ ≤ ac λ1 + λ2

Finally, if A is positive definite, then |b| λ1 − λ2 , ≤ √ a 2 λ1 λ2

λ2 − λ22 4|b| ≤ √1 . λ1 λ2

Source: [1783, 2987]. Related: Fact 10.21.4. Fact 10.21.2. Let A ∈ Hn, and let χA (s) = sn + βn−1 sn−1 + · · · + β1 s + β0 . Then, the following

statements are equivalent: i) A is negative semidefinite. ii) For all i ∈ {1, . . . , n − 1}, βi ≥ 0. Furthermore, the following statements are equivalent: iii) A is negative definite. iv) For all i ∈ {1, . . . , n − 1}, βi > 0. Related: Fact 15.18.3. Fact 10.21.3. Let A ∈ Fn×m . If m ≤ n, then msval(A) = msval(⟨A⟩) = mspec(⟨A⟩). If n ≤ m, then

msval(A) = msval(⟨A∗ ⟩) = mspec(⟨A∗ ⟩).

Fact 10.21.4. Let A ∈ Fn×n, and assume that A is positive semidefinite. Then, the following

statements hold: i) If i, j ∈ {1, . . . , n}, then

λ1 (A)λn (A) [A(i,i) − A( j, j) ]2 [λ1 (A)+λn (A)]2 |λi (A) − 1n tr A| ≤ δ(A).

≤ A(i,i) A( j, j) − |A(i, j) |2 .

If i ∈ {1, . . . , n}, then If i, j ∈ {1, . . . , n}, then |λi (A) − A( j, j) | ≤ δ(A). If i, j ∈ {1, . . . , n} and i , j, then 2|A(i, j) | ≤ δ(A). If i, j ∈ {1, . . . , n}, then |A(i,i) − A( j, j) | ≤ δ(A). If i, j ∈ {1, . . . , n}, then 4|A(i,i) A( j, j) | ≤ [λ1 (A) + λ2 (A)]2 . tr A3 − n1 (tr A2 ) tr A ≤ n4 [λ1 (A) + λn (A)][λ1 (A) − λn (A)]2 . Source: [1344]. Remark: δ(A) = λmax (A) − λmin (A) is the spread of A. Related: Fact 10.21.1. ii) iii) iv) v) vi) vii)

801

POSITIVE-SEMIDEFINITE MATRICES

Fact 10.21.5. Let A ∈ Fn×m, and assume that A is Hermitian. Then, 2 n √2 n2 −1

 √  n tr A2 − (tr A)2 , n even   2√  √ ≤ δ(A) ≤ n tr A2 − (tr A)2 .   n  n tr A2 − (tr A)2 , n odd  √

Source: [674, 1344]. Remark: δ(A) = λmax (A) − λmin (A) is the spread of A. Related: Fact 6.10.4, Fact 10.18.22, Fact 11.11.7, and Fact 11.11.8. △ Fact 10.21.6. Let A ∈ Fn×n , assume that A is positive definite, and define κ = λmax (A)/λmin (A). Then, for all distinct i, j ∈ {1, . . . , n},

κ−1 (A(i,i) + A( j, j) ) ≤ A(i,i) + A( j, j) , max {|A(i,i) − A( j, j) |, |2 Re A(i, j) |} ≤ κ+1 { } A(i,i) + 2 Re A(i, j) + A( j, j) A(i,i) − 2 Re A(i, j) + A( j, j) A(i,i) max , , ≤ κ. A(i,i) − 2 Re A(i, j) + A( j, j) A(i,i) + 2 Re A(i, j) + A( j, j) A( j, j) Source: [1344]. Fact 10.21.7. Let A ∈ Fn×m. Then, for all i ∈ {1, . . . , min {n, m}}, λi (⟨A⟩) = σi (A). Hence,

tr ⟨A⟩ =

min {n,m} ∑

σi (A).

i=1

Fact 10.21.8. Let A ∈ Fn×n, and define △

[

A=

σmax (A)I A

] A∗ . σmax (A)I

Then, A is positive semidefinite. Furthermore, { } ⟨A⟩ + ⟨A∗ ⟩ ≤ 2σmax (A)I 2 ⟨A + A∗ ⟩ ≤ ≤ [σmax (A) + 1]I. A∗A + I Source: [2985]. Fact 10.21.9. Let A ∈ Fn×n. Then, for all i ∈ {1, . . . , n},

λi [ 21 (A + A∗ )] ≤ σi (A). Hence, Re tr A ≤ tr ⟨A⟩. Source: [2479] and [2991, pp. 288, 289]. Related: Fact 7.12.25. Fact 10.21.10. Let A ∈ Fn×n. If p > 0, then, for all k ∈ {1, . . . , n}, k ∑

ρip (A) ≤

i=1

k ∑

σip (A).

i=1

In particular, for all k ∈ {1, . . . , n}, k ∑

ρi (A) ≤

i=1

Hence, | tr A| ≤

n ∑

k ∑

σi (A).

i=1

ρi (A) ≤

i=1

n ∑

σi (A) = tr ⟨A⟩.

i=1

Furthermore, for all k ∈ {1, . . . , n}, k ∑ i=1

ρ2i (A) ≤

k ∑ i=1

σ2i (A).

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CHAPTER 10

Hence, Re tr A2 ≤ | tr A2 | ≤

n ∑

ρ2i (A) ≤

n ∑

i=1

σi (A2 ) = tr ⟨A2 ⟩ ≤

i=1

Furthermore,

n ∑

n ∑

σ2i (A) = tr A∗A.

i=1

ρ2i (A) = tr A∗A

i=1

if and only if A is normal. Finally, tr A = tr A∗A if and only if A is Hermitian. Source: Fact 3.25.15, Fact 7.12.32, [449, p. 42], [1450, p. 176], [2977, p. 19], and [2991, pp. 86, 312, 355]. See Fact 10.13.5 for tr ⟨A2 ⟩ = tr (A2∗A2 )1/2 ≤ tr A∗A. See Fact 4.10.13 and Fact 7.15.16. Remark: The first ∑ result is Weyl’s inequalities. ni=1 ρ2i (A) ≤ tr A∗A is Schur’s inequality. Related: Fact 7.12.26, Fact 7.12.30, Fact 10.13.2, Fact 10.13.14, Fact 10.21.11, and Fact 11.13.2. Fact 10.21.11. Let A ∈ Fn×n, and assume that A is Hermitian. Then, for all k ∈ {1, . . . , n}, 2

k ∑

di (A) ≤

i=1

k ∑

λi (A)

i=1

with equality for k = n; that is, tr A =

n ∑

di (A) =

n ∑

i=1

λi (A).

i=1

Hence,

s

[d1 (A) · · · dn (A)] ≺ [λ1 (A) · · · λn (A)], and thus, for all k ∈ {1, . . . , n},

n ∑

λi (A) ≤

i=k

In particular,

n ∑

di (A).

i=k

λmin (A) ≤ dmin (A) ≤ dmax (A) ≤ λmax (A).

Furthermore, [d1 (A) · · · dn (A)]T is an element of the convex hull of the n! vectors obtained by permuting the components of [λ1 (A) · · · λn (A)]T . Source: [449, p. 35], [1448, p. 193], [1969, p. 218], [1969, p. 300], and [2977, p. 18]. The last statement follows from Fact 4.11.6. Remark: This is Schur’s theorem. Related: Fact 7.12.26, Fact 10.13.2, Fact 10.13.14, Fact 10.21.10, and Fact 11.13.2. Fact 10.21.12. Let A ∈ Fn×n. Then, w

[ρ21 (A) · · · ρ2n (A)] ≺ [σ21 (A) · · · σ2n (A)], w

w

[d21 (|A|) · · · d2n (|A|)] ≺ [d1 (⟨A⟩)d1 (⟨A∗ ⟩) · · · dn (⟨A⟩)dn (⟨A∗ ⟩)] ≺ [σ21 (A) · · · σ2n (A)], wlog

[d21 (|A|) · · · d2n (|A|)] ≺ [d1 (⟨A⟩)d1 (⟨A∗ ⟩) · · · dn (⟨A⟩)dn (⟨A∗ ⟩)], w

[ρ1 (A + A∗ ) · · · ρn (A + A∗ )] ≺ [λ1 (⟨A⟩ + ⟨A∗ ⟩) · · · λn (⟨A⟩ + ⟨A∗ ⟩)], w

[ρ1 (A ⊙ A∗ ) · · · ρn (A ⊙ A∗ )] ≺ [λ1 (⟨A⟩⟨A∗ ⟩) · · · λn (⟨A⟩⟨A∗ ⟩)]. Source: [2991, p. 355].

803

POSITIVE-SEMIDEFINITE MATRICES

Fact 10.21.13. Let A ∈ Fn×n, assume that A is Hermitian, let k denote the number of positive

diagonal entries of A, and let l denote the number of positive eigenvalues of A. Then, k ∑

d2i (A) ≤

i=1

l ∑

λ2i (A).

i=1

Source: Write A = B+C, where B is positive semidefinite, C is negative semidefinite, and mspec(A)

= mspec(B) ∪ mspec(C). Furthermore, without loss of generality, assume that A(1,1) , . . . , A(k,k) are the positive diagonal entries of A. Then, k ∑

d2i (A) =

i=1

k ∑ i=1

A2(i,i) ≤

k k n l ∑ ∑ ∑ ∑ (A(i,i) − C(i,i) )2 = B2(i,i) ≤ B2(i,i) ≤ tr B2 = λ2i (A). i=1

i=1 ⊙2

i=1

i=1

Remark: This can be written as tr (A + |A|) ≤ tr (A + ⟨A⟩) . Credit: Y. Li. Fact 10.21.14. Let x, y ∈ Rn, where n ≥ 2. Then, the following statements are equivalent: 2

s

i) x ≺ y. ii) x is an element of the convex hull of the vectors y1 , . . . , yn! ∈ Rn, where each of these n! vectors is formed by permuting the components of y. iii) There exists a Hermitian matrix A ∈ Cn×n such that [A(1,1) · · · A(n,n) ]T = x and mspec(A) = {y(1) , . . . , y(n) }ms . Remark: This is the Schur-Horn theorem. Schur’s theorem given by Fact 10.21.11 is iii) =⇒ i), while the result i) =⇒ iii) is given in [1446]. The equivalence of ii) is given by Fact 4.11.6. This result is discussed in [309, 450, 575]. An equivalent version is given by Fact 4.13.16. Fact 10.21.15. Let A ∈ Fn×n, and assume that A is positive semidefinite. Then, for all k ∈ {1, . . . , n}, n n ∏ ∏ λi (A) ≤ di (A). i=k

i=k

In particular, det A ≤

n ∏

A(i,i) .

i=1

∏ Furthermore, equality holds if and only if either ni=1 A(i,i) = 0 or A is diagonal. Source: [1124, pp. 21–24], [1448, pp. 200, 477], [2977, p. 18], and [2991, pp. 218, 355]. Remark: The case k = 1 is Hadamard’s inequality. Remark: A geometric interpretation is given in [1138]. Related: Fact 10.15.10 and Fact 11.13.1. A refinement is given by Fact 10.15.7. Fact 10.21.16. Let A ∈ Fn×n, and assume that A is positive definite. Then, for all k ∈ {1, . . . , n}, n ∑ i=k

∑ 1 1 ≤ . di (A) i=k λi (A) n

Source: [2991, p. 355]. Fact 10.21.17. Let A ∈ Fn×n and α > 0. Then, for all k ∈ {1, . . . , n}, k ∏ i=1

[1 + αρi (A)] ≤

k ∏

[1 + ασi (A)].

i=1

Source: [970, p. 222]. △ △ Fact 10.21.18. Let A ∈ Fn×n, define H = 21 (A + A∗ ) and S = 21 (A − A∗ ), and assume that H is

positive definite. Then, the following statements hold:

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CHAPTER 10

i) ii) iii) iv)

A is nonsingular. 1 −1 −∗ ∗ −1 −1 2 (A + A ) = (H + S H S ) . −1 −1 σmax (A ) ≤ σmax (H ). σmax (A) ≤ σmax (H + S ∗H −1S ). Source: [1987]. Related: Fact 7.20.1, Fact 10.10.36, and Fact 10.15.2. Fact 10.21.19. Let A ∈ Fn×n, and assume that A is Hermitian. Then, A is diagonal if and only if mspec(A) = {A(1,1) , . . . , A(n,n) }ms . Source: Use Fact 10.21.15 with A + βI > 0. Fact 10.21.20. Let A ∈ Fn×n, and assume that A is Hermitian. Then, for all k ∈ {1, . . . , n}, n ∑ λi (A) = min {tr S ∗AS : S ∈ Fn×k and S ∗S = Ik } k ∑ i=n+1−k ≤ max {tr S ∗AS : S ∈ Fn×k and S ∗S = Ik } = λi (A). i=1

Now, assume that A is positive definite. Then, k ∑ [λi (A)]−1 = min {tr (S ∗AS )−1 : S ∈ Fn×k and S ∗S = Ik } i=1

≤ max {tr (S ∗AS )−1 : S ∈ Fn×k and S ∗S = Ik } =

n ∑

1 . λ (A) i=n+1−k i

Source: [1448, p. 191] and [2991, p. 273]. Remark: This is the minimum principle. Fact 10.21.21. Let A ∈ Fn×n, assume that A is Hermitian, and let S ∈ Fn×k satisfy S ∗S = Ik .

Then, for all i ∈ {1, . . . , k},

λi+n−k (A) ≤ λi (S ∗AS ) ≤ λi (A).

Consequently, k ∑ i=1

λi+n−k (A) ≤ tr S ∗AS ≤

k ∑

k ∏

λi (A),

i=1

λi+n−k(A) ≤ det S ∗AS ≤

i=1

k ∏

λi (A).

i=1

Source: [1448, p. 190], [2403, p. 111], and [2991, p. 273]. Remark: This is the Poincar´e separation theorem. Fact 10.21.22. Let A ∈ Fn×n, and let S ∈ Fn×k satisfy S ∗S = Ik . Then, for all i ∈ {1, . . . , k},

σi (S ∗AS ) ≤ σi (A). Source: [2991, p. 273]. Fact 10.21.23. Let n ≥ 2, let A ∈ Cn×n, and assume that A is positive definite. Then,

( tr A )n−1 . n−1 Source: [1172, p. 29]. Related: Fact 11.9.15, Fact 11.15.16, and Fact 11.15.17. σmax (AA ) ≤

10.22 Facts on Eigenvalues and Singular Values for Two or More Matrices Fact 10.22.1. Let A ∈ Fn, assume that A is Hermitian, and let x ∈ Fn . Then, for all i ∈ {2, . . . ,

n − 2},

λi+2 (A) ≤ λi+1 (A + xx∗ ) ≤ λi (A) ≤ λi (A + xx∗ ).

Furthermore, s

[λ1 (A + xx∗ ) · · · λn (A + xx∗ )] ≺ [λ1 (A) + x∗ x λ2 (A) · · · λn (A)].

805

POSITIVE-SEMIDEFINITE MATRICES

Source: [2991, pp. 279, 361]. Fact 10.22.2. Let A, B ∈ Fn, and assume that A and B are Hermitian. Then, for all i ∈ {1, . . . , n},

|λi (A) − λi (B)| ≤ σmax (A − B). Source: [2979, p. 54]. Fact 10.22.3. Let A, B ∈ Fn, and assume that A and B are Hermitian. Then, s

s

λ(A) + [λ(B)]↑ ≺ λ(A + B) ≺ λ(A) + λ(B),  [  1 ] s  λ[ 2 (A + B)]  s λ(A) . 2λ(A) ≺ λ(A + B) + λ(A − B),  1  ≺ λ(B) λ[ (A + B)] 2

Now, let k ≥ 1. Then,

s

λ[A2 + (BA)k (AB)k ] ≺ λ[A2 + (AB)k (BA)k ].

Source: [449, p. 71] and [450, 843, 1850, 2750]. Related: Corollary 10.6.19. Fact 10.22.4. Let A, B ∈ Fn×n, and assume that A and B are Hermitian. Then, for all k ∈

{1, . . . , n},

k ∑

λi (A) +

i=1

n ∑

k ∑

λi (B) ≤

k ∑ [λi (A) + λi (B)],

λi (A + B) ≤

i=1

i=n−k+1

i=1

k k k n ∑ ∑ ∑ ∑ [λi (A) − λi (B)] ≤ λi (A − B) ≤ λi (A) − λi (B). i=1

i=1

i=1

i=n−k+1

Source: Corollary 10.6.19. Related: Fact 15.17.6. Fact 10.22.5. Let A, B ∈ Fn×n, assume that A and B are Hermitian, let k ∈ {1, . . . , n}, and let

1 ≤ i1 < · · · < ik ≤ n. Then, k ∑ j=1

λi j (A) +

n ∑

λ j (B) ≤

k ∑

λi j (A + B) ≤

j=1

j=n−k+1

k ∑ [λi j (A) + λ j (B)]. j=1

Source: [449, p. 69], [2403, pp. 115, 116], [2991, pp. 281–284], and Fact 10.22.3. Remark:

The spectrum of a sum of Hermitian matrices is given by a subsequently confirmed conjecture from [1447]. See [450, 843, 1363, 1644, 1653, 1804, 1816]. Fact 10.22.6. Let A, B ∈ Fn×n, and assume that A and B are positive semidefinite. Then, [ ] [ ] [ ] [ ] s λ(A + B) λ(A) s λ(A + B) λ(A) 0 ≺ , ≺ . λ(B) 0 0 λ(B) 0 Source: [547], [1969, p. 242], [1971, p. 330], [2750], and [2991, p. 362]. Fact 10.22.7. Let A, B ∈ Fn×n, assume that A and B are positive semidefinite, and let r ∈ R. If

either r ≥ 1 or both r ≤ 0 and A and B are positive definite, then w

w

[(λ(A) + λ↑ (B)]⊙r ≺ [λ(A + B)]⊙r ≺ [λ(A + B)]⊙r ,  1  [ ] ] [ ⊙r  (λ[ 2 (A + B)])  w [λ(A)]⊙r w [λ(A + B)]⊙r ≺ .  1  ≺ [λ(B)]⊙r 0 (λ[ 2 (A + B)])⊙r If r ∈ (0, 1), then

w

w

[λ(A + B)]⊙r ≺ [λ(A + B)]⊙r ≺ [(λ(A) + λ↑ (B)]⊙r ,  [ ] [ ]  1 ⊙r [λ(A + B)]⊙r w [λ(A)]⊙r w  (λ[ 2 (A + B)])  ≺ ≺  1  . 0 [λ(B)]⊙r (λ[ (A + B)])⊙r 2

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CHAPTER 10

Source: [2750]. Fact 10.22.8. Let A, B ∈ Fn×n. Then, s

2λ(AA∗ + BB∗ ) ≺ λ(A∗A + B∗B − A∗B − B∗A) + λ(A∗A + B∗B + A∗B + B∗A). Source: [2750]. Fact 10.22.9. Let A, B ∈ Fn×m, and assume that A∗B is either Hermitian or skew Hermitian. If

n < m, then

[

If m = n, then

] λ(AA∗ + BB∗ ) s ≺ λ(A∗A + B∗B). 0 s

λ(AA∗ + BB∗ ) ≺ λ(A∗A + B∗B).

If m < n, then

s

λ(AA∗ + BB∗ ) ≺

[

] λ(A∗A + B∗B) . 0

Source: [1850, 2750]. Fact 10.22.10. Let A, B ∈ Fn, assume that A and B are positive definite.

[log λ1 (A) · · · log λn (A)] + [log λn (B) · · · log λ1 (B)] s

s

≺ [log λ1 (AB) · · · log λn (AB)] ≺ [log λ1 (A) + log λ1 (B) · · · log λn (A) + log λn (B)]. Source: [100]. Fact 10.22.11. Let f : R 7→ R be convex, define f : Hn 7→ Hn by (10.5.2), let A, B ∈ Fn×n, and

assume that A and B are Hermitian, and let α ∈ [0, 1]. Then, [ ] λ1 [ f (αA + (1 − α)B)] · · · λn [ f (αA + (1 − α)B)] ] w [ ≺ αλ1 [ f (A)] + (1 − α)λ1 [ f (B)] · · · αλn [ f (A)] + (1 − α)λn [ f (B)] . If, in addition, f is either nonincreasing or nondecreasing, then, for all i ∈ {1, . . . , n}, λi [ f (αA + (1 − α)B)] ≤ λi [α f (A) + (1 − α) f (B)].

Source: [197]. Remark: Under the assumptions of the last statement,[ the] inequality[ λi [] f (αA+(1− α)B)] ≤ αλi [ f (A)] + (1 − α)λi [ f (B)] may not hold as shown by A = 20 01 and B = 00 01 . Remark:

Convexity of f : R 7→ R does not imply convexity of f : Hn 7→ Hn. Fact 10.22.12. Let A, B ∈ Fn×n, and assume that A and B are positive semidefinite. If r ∈ [0, 1], then [ ]w[ ] λ1 [(A + B)r ] · · · λn [(A + B)r ] ≺ λ1 (Ar + Br ) · · · λn (Ar + Br ) ,

and, for all i ∈ {1, . . . , n}, If r ≥ 1, then

[

λ1 (Ar + Br ) · · ·

and, for all i ∈ {1, . . . , n},

21−rλi [(A + B)r ] ≤ λi (Ar + Br ). ]w[ λn (Ar + Br ) ≺ λ1 [(A + B)r ] · · · λi (Ar + Br ) ≤ 2r−1λi [(A + B)r ].

Source: This result follows from Fact 10.22.11. See [106, 195, 197].

] λn [(A + B)r ] ,

807

POSITIVE-SEMIDEFINITE MATRICES

Fact 10.22.13. Let A, B ∈ Fn×n, and assume that A and B are positive semidefinite. Then, for

all k ∈ {1, . . . , n},

k ∏

|λi (A − B)| ≤

k ∏

i=1

λi (A + B).

i=1

Source: [2991, p. 362]. Fact 10.22.14. Let A, B ∈ Fn×n, and assume that A and B are Hermitian. Then, s

[σ21 (A) + σ2n (B) · · · σ2n (A) + σ21 (B)] ≺ [σ21 (A + ȷB) · · · σ2n (A + ȷB)], 2 1 2 [σ1 (A +

s

ȷB) + σ2n (A + ȷB) · · · σ2n (A + ȷB) + σ21 (A + ȷB)] ≺ [σ21 (A) + σ21 (B) · · · σ2n (A) + σ2n (B)].

Now, assume that A and B are positive semidefinite. Then, s

[σ21 (A + ȷB) · · · σ2n (A + ȷB)] ≺ [σ21 (A) + σ21 (B) · · · σ2n (A) + σ2n (B)]. Furthermore, for all k ∈ {1, . . . , n}, n n ∏ ∏ |σi (A) + ȷσi (B)| ≤ |σi (A + ȷB)|. i=k

i=k

Source: [100, 714] and [2979, pp. 97, 98]. Related: Fact 10.16.5 and Fact 11.10.69. Fact 10.22.15. Let A, B ∈ Fn×n, and assume that A and B are positive semidefinite. Then, the

following statements hold: p i) If p ∈ [0, 1], then σmax (Ap − B p ) ≤ σmax (A − B). √ p p ii) If p ≥ 2, then σmax (A − B ) ≤ p[max {σmax (A), σmax (B)}] p−1 σmax (A − B). iii) If a and b are positive numbers such that aI ≤ A ≤ bI and aI ≤ B ≤ bI, then σmax (Ap − B p ) ≤ b[b p−2 + (p − 1)a p−2 ]σmax (A − B). Source: [463, 1627]. Fact 10.22.16. Let A, B ∈ Fn×n, and assume that A and B are positive semidefinite. Then, for

all i ∈ {1, . . . , n},

([ σi (A − B) ≤ σi

A 0

0 B

]) .

Source: [2587, 2975] and [2991, p. 362]. Fact 10.22.17. Let A, B ∈ Fn×n. Then, 2 2 max {σmax (A), σmax (B)} − σmax (AB) ≤ σmax (A∗A − BB∗ ), 2 2 2 2 σmax (A∗A − BB∗ ) ≤ max {σmax (A), σmax (B)} − min {σmin (A), σmin (B)}.

Furthermore, 2 2 2 2 max {σmax (A), σmax (B)} + min {σmin (A), σmin (B)} ≤ σmax (A∗A + BB∗ ), 2 2 (A), σmax (B)} + σmax (AB). σmax (A∗A + BB∗ ) ≤ max {σmax

Now, assume that A and B are positive semidefinite. Then, max {λmax (A), λmax (B)} − σmax (A1/2B1/2 ) ≤ σmax (A − B), σmax (A − B) ≤ max {λmax (A), λmax (B)} − min {λmin (A), λmin (B)}. Furthermore, max {λmax (A), λmax (B)} + min {λmin (A), λmin (B)} ≤ λmax (A + B),

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CHAPTER 10

λmax (A + B) ≤ max {λmax (A), λmax (B)} + σmax (A1/2B1/2 ). Source: [1635, 2978]. Related: Fact 10.22.19 and Fact 11.15.10. Fact 10.22.18. Let A, B ∈ Fn×n, and assume that A and B are positive semidefinite, and let

k ≥ 1. Then, for all i ∈ {1, . . . , n},

2σi [A1/2 (A + B)k−1B1/2 ] ≤ λi [(A + B)k ]. Hence,

2σmax (A1/2B1/2 ) ≤ λmax (A + B),

σmax (A1/2B1/2 ) ≤ max {λmax (A), λmax (B)}.

Source: Fact 10.22.17 and Fact 11.10.45. Fact 10.22.19. Let A, B ∈ Fn×n, and assume that A and B are positive semidefinite. Then,

max {λmax (A), λmax (B)} − σmax (A1/2B1/2 ) ≤ σmax (A − B) ≤ max {λmax (A), λmax (B)} ≤ λmax (A + B) [ ] √ 1 2 2 1/2 1/2 ≤ λmax (A) + λmax (B) + [λmax (A) − λmax (B)] + 4σmax (A B ) 2   1/2 1/2      max {λmax (A), λmax (B)} + σmax (A B )  ≤ 2 max {λmax (A), λmax (B)}. ≤      λmax (A) + λmax (B) Furthermore,

λmax (A + B) = λmax (A) + λmax (B)

if and only if

1/2 σmax (A1/2B1/2 ) = λ1/2 max (A)λmax (B).

Source: [1629, 1632, 1635] and Fact 10.22.18. Related: Fact 10.22.17, Fact 10.22.19, Fact 11.10.30, Fact 11.10.78, and Fact 11.16.18. Fact 10.22.20. Let A, B ∈ Fn×n, assume that A and B are positive semidefinite, and let z ∈ C. Then, for all k ∈ {1, . . . , n}, k ∏

σi (A − |z|B) ≤

i=1

k ∏

σi (A + zB) ≤

i=1

k ∑

σi (A − |z|B) ≤

i=1

k ∑

k ∏

σi (A + |z|B),

i=1

σi (A + zB) ≤

i=1

k ∑

σi (A + |z|B).

i=1

Source: [2975] and [2991, pp. 357, 358]. Related: Fact 11.10.24. Fact 10.22.21. Let A, B ∈ Fn×n, and assume that A and B are positive semidefinite. Then,

tr AB ≤ tr (AB2A)1/2 ≤ 14 tr (A + B)2,

tr (AB)2 ≤ tr A2B2 ≤

1 16 tr (A

+ B)4,

σmax (AB) ≤ 41 σmax [(A + B)2 ] 1  1 1 2 2 2 2     2 σmax (A + B ) ≤ 2 σmax (A ) + 2 σmax (B )   1 2 2 (B). ≤ ≤ σ (A) + 12 σmax    1 σ2 (A + B) ≤ 1 [σmax (A) + σmax (B)]2   2 max 4

max

4

Source: Fact 11.10.45. The inequalities tr AB ≤ tr (AB2A)1/2 and tr (AB)2 ≤ tr A2B2 follow from Fact 10.14.24. Related: Fact 10.22.22.

809

POSITIVE-SEMIDEFINITE MATRICES

Fact 10.22.22. Let A, B ∈ Fn×n, assume that A and B are positive semidefinite, and let k ≥ 1.

Then, for all i ∈ {1, . . . , n},

λi [(AB)k ] ≤ 41 λi [(A + B)2k ], Therefore,

tr (AB)k ≤

1 4

tr (A + B)2k ,

λi [(A2B2 )k ] ≤ tr (A2B2 )k ≤

1 λ [(A 16k i

1 16k

+ B)4k ].

tr (A + B)4k .

Source: [471, 2935]. Related: Fact 10.22.21. Fact 10.22.23. Let A, B ∈ Fn×n, and assume that A and B are positive semidefinite. Then, for

all i, j ∈ {0, 1, . . . , n − 1} such that i + j ≤ n − 1,

λn−i (A)λn− j (B) ≤ λn−i− j (AB), and, for all i, j ∈ {1, . . . , n} such that i + j ≤ n + 1, λi+ j−1 (AB) ≤ λi (A)λ j (B). Therefore, for all i ∈ {1, . . . , n}, λi (A)λn (B) ≤ λi (AB) ≤ λi (A)λ1 (B). In particular,

λmin (A)λmin (B) ≤ λmin (AB) ≤ λmin (A)λmax (B).

Source: [2403, pp. 126, 127], [2991, pp. 277, 278, 280]. Related: Fact 10.22.28 and Fact

10.22.36. Fact 10.22.24. Let A, B ∈ Fn×n, and assume that A and B are positive definite. Then, for all

i ∈ {1, . . . , n},

λ2i (AB) λ2i (AB) ≤ λi (A)λi (B) ≤ . λ1 (A)λ1 (B) λn (A)λn (B)

Source: [2403, p. 137]. Fact 10.22.25. Let A, B ∈ Fn×n, assume that A is positive semidefinite, and assume that B is

Hermitian. Then, for all k ∈ {1, . . . , n}, k ∑

λi (A)λn−i+1 (B) ≤

i=1

k ∑

λi (AB),

i=1

In particular,

n ∑

k ∑

λn−i+1 (AB) ≤

i=1

λi (A)λi (B).

i=1

λi (A)λn−i+1 (B) ≤ tr AB ≤

i=1

k ∑

n ∑

λi (A)λi (B).

i=1

Source: [1664]. Remark: The bounds on tr AB are related to Fact 2.12.8. See [453, p. 140]. Related: Fact 7.13.12, Fact 7.13.13, Fact 7.13.16, and Proposition 10.4.13. Fact 10.22.26. Let A, B ∈ Fn×n, assume that A and B are positive semidefinite, and let 1 ≤ i1
0 and β > 0 satisfy αI ≤ A ≤ βI. Then, √

σmax (AB) ≤ In particular, σmax (A) ≤

α+β α+β √ ρmax (AB) ≤ √ σmax (AB). 2 αβ 2 αβ α+β α+β √ ρmax (A) ≤ √ σmax (A). 2 αβ 2 αβ

812

CHAPTER 10

Source: [2686]. Remark: The left-hand inequality is tightest for α = λmin (A) and β = λmax (A). Credit: J.-C. Bourin. Fact 10.22.34. Let A, B ∈ Fn×n, and assume that A and B are positive semidefinite. Then, for

all i ∈ {1, . . . , n},

√ σi (AB) ≤ 12 λi (A + B).

Source: [954]. Fact 10.22.35. Let A, B ∈ Fn×n, and assume that A and B are positive semidefinite. Then, 1/2 σmax (A1/2B1/2 ) ≤ σmax (AB).

Equivalently,

2 λmax (A1/2BA1/2 ) ≤ λ1/2 max (AB A).

Furthermore, AB = 0 if and only if A1/2B1/2 = 0. Source: [1629, 1635]. Related: Fact 10.22.36. Fact 10.22.36. Let A, B ∈ Fn×n, and assume that A and B are positive semidefinite. Then, the following statements hold: q i) If q ∈ [0, 1], then σmax (AqBq ) ≤ σmax (AB). q q q q ii) If q ∈ [0, 1], then σmax (B A B ) ≤ σmax (BAB). iii) If q ∈ [0, 1], then λmax (AqBq ) ≤ λqmax (AB). q iv) If q ≥ 1, then σmax (AB) ≤ σmax (AqBq ). q v) If q ≥ 1, then λmax (AB) ≤ λmax (AqBq ). 1/q 1/p vi) If p ≥ q > 0, then σmax (AqBq ) ≤ σmax (ApB p ). 1/q 1/p vii) If p ≥ q > 0, then λmax (AqBq ) ≤ λmax (ApB p ) Source: [449, pp. 255–258] and [1117]. Remark: iii) is the Cordes inequality. Related: Fact 10.11.78, Fact 10.14.24, Fact 10.22.23, Fact 10.22.35, Fact 11.10.49, and Fact 11.10.50. Fact 10.22.37. Let A, B ∈ Fn×n, assume that A and B are positive semidefinite, and let p ≥ r ≥ 0. Then, s 1/p p p 1/p p p r r 1/r r r [λ1/r 1 (A B ) · · · λn (A B )] ≺ [λ1 (A B ) · · · λn (A B )]. Furthermore, for all q > 0,

det (AqBq )1/q = det AB.

Source: [449, p. 257], [2977, p. 20], and Fact 3.25.15. Fact 10.22.38. Let A, B ∈ Fn×n, assume that A and B are positive semidefinite, let p and r be

positive integers such that p ≤ r, let k ∈ {1, . . . , n}, and let 1 ≤ i1 < · · · < ik ≤ n. Then, k ∏

λi j (A)λn−i j +1 (B) ≤

j=1

k ∏

λri (A1/r B1/r ) ≤

i=1 k ∏

λi (AB) ≤

i=1

k ∏ i=1

p p λ1/p i (A B ) ≤

k ∏

λip (A1/pB1/p ) ≤

i=1 k ∏ i=1

r r λ1/r i (A B ) ≤

k ∏

k ∏

λi (A)λi (B).

i=1

Furthermore, if k = n, then all of the above inequalities are equalities. Finally, λn (A)λn (B) ≤ λrn (A1/r B1/r ) ≤ λnp (A1/pB1/p ) ≤ λn (AB), λ1 (AB) ≤ λ1p (A1/pB1/p ) ≤ λr1 (A1/r B1/r ) ≤ λ1 (A)λ1 (B). Source: [2991, pp. 366–368].

λi (AB),

i=1

813

POSITIVE-SEMIDEFINITE MATRICES

Fact 10.22.39. Let A, B ∈ Fn×n, and assume that A and B are positive semidefinite. Then,

σmax [(I + A)−1AB(I + B)−1 ] ≤

σmax (AB) 1/2 [1 + σmax (AB)]2

.

Source: [2756]. Fact 10.22.40. Let A, B, C ∈ Fn×n, and assume that A, B, and C are positive semidefinite. Then,

spec(CABA + CA + BA) ⊂ [0, ∞). If, in addition, A is positive definite and either B or C is positive definite, then spec(CABA + CA + BA) ⊂ (0, ∞). △

Source: Assume that A is positive definite and either B or C is positive definite, and define X = 1/2

1/2

A CA

△

and Y = A BA , at least one of which is positive definite. Then, 1/2

1/2

spec(CABA + CA + BA) = spec[A1/2 (CAB + C + B)A1/2 ] = spec(XY + X + Y) = spec(XY + X + Y + I) − {1} = spec[(X + I)(Y + I)] − {1} ⊂ R. Furthermore, it follows from Fact 10.22.23 that λmin [(X + I)(Y + I)] − 1 ≥ λmin (X + I)λmin (Y + I) − 1 > 0. The first result follows from continuity. Credit: A. Ali and M. Lin. Fact 10.22.41. Let A, B ∈ Fn×n, and assume that A and B are positive definite. Then, slog

[λ1 (log A1/2BA1/2 ) · · · λn (log A1/2BA1/2 )] ≺ [λ1 (log A + log B) · · · λn (log A + log B)]. Consequently, log det AB = tr(log A + log B) = tr log A1/2BA1/2 = log det A1/2BA1/2 . Source: [196]. Fact 10.22.42. Let A, B ∈ Fn×n, and assume that A and B are positive semidefinite. Then, the

following statements hold: 1/2 i) σmax [log(I + A)log(I + B)] ≤ (log[1 + σmax (AB)])2 . 1/3 ii) σmax [log(I + B)log(I + A) log(I + B)] ≤ (log[1 + σmax (BAB)])3 . iii) det[log(I + A)log(I + B)] ≤ det [log(I + ⟨AB⟩1/2 )]2 . iv) det[log(I + B)log(I + A) log(I + B)] ≤ det (log[I + (BAB)1/3 ])3 . Source: [2756]. Related: Fact 15.17.8. Fact 10.22.43. Let A, B, C ∈ Fn×n, and assume that A, B, and C are positive definite. Then, s

[log λ1 (AC) · · · log λn (AC)] ≺ [log λ1 (AB) + log λ1 (B−1C) · · · log λn (AB) + log λn (B−1C)]. Source: [2991, p. 371].

10.23 Facts on Alternative Partial Orderings Fact 10.23.1. Let A, B ∈ Fn×n, and assume that A and B are positive definite. Then, the follow-

ing statements are equivalent: i) log B ≤ log A. ii) For all r ∈ (0, ∞), Br ≤ (Br/2ArBr/2 )1/2. iii) For all r ∈ (0, ∞), (Ar/2BrAr/2 )1/2 ≤ Ar. 1 iv) For all p, r ∈ (0, ∞) and k ∈ N such that (k + 1)r = p + r, Br ≤ (Br/2ApBr/2 ) k+1 .

814

v) vi) vii) viii)

CHAPTER 10

For all For all For all For all

1

p, r ∈ (0, ∞) and k ∈ N such that (k + 1)r = p + r, (Ar/2B pAr/2 ) k+1 ≤ Ar. r p, r ∈ [0, ∞), Br ≤ (Br/2ApBr/2 ) r+p . r p, r ∈ [0, ∞), (Ar/2B pAr/2 ) r+p ≤ Ar. p, q, r, t ∈ R such that p ≥ 0, r ≥ 0, t ≥ 0, and q ∈ [1, 2], r+t

[Ar/2 (At/2B pAt/2 )qAr/2 ] r+qt+qp ≤ Ar+t. Source: [1088, 1828, 2939] and [1124, pp. 139, 200]. Remark: log B ≤ log A is the chaotic order.

This order is weaker than the L¨owner ordering since B ≤ A implies that log B ≤ log A, but not vice versa. Remark: Additional statements are given in [1828]. Fact 10.23.2. Let A, B ∈ Fn×n, assume that A is positive definite and B is positive semidefinite, and let α > 0. Then, the following statements are equivalent: i) Bα ≤ Aα. ii) For all p, q, r, τ ∈ R such that p ≥ α, r ≥ τ, q ≥ 1, and τ ∈ [0, α], [Ar/2 (A−τ/2B pA−τ/2 )qAr/2 ] r−qτ+qp ≤ Ar−τ. r−τ

Source: [1088]. Fact 10.23.3. Let A, B ∈ Fn×n, and assume that A is positive definite and B is positive semidef-

inite. Then, the following statements are equivalent: i) For all k ≥ 0, Bk ≤ Ak. ii) For all α > 0, Bα ≤ Aα. 2p−r

iii) For all p, r ∈ R such that p > r ≥ 0, (A−r/2B pA−r/2 ) p−r ≤ A2p−r. iv) For all p, q, r, τ ∈ R such that p ≥ τ, r ≥ τ, q ≥ 1, and τ ≥ 0, [Ar/2 (A−τ/2B pA−τ/2 )qAr/2 ] r−qτ+qp ≤ Ar−τ. r−τ

Source: [1125]. Remark: A and B are related by the spectral ordering. Fact 10.23.4. Let A, B ∈ Fn×n, and assume that A and B are positive semidefinite. Then, if two

of the following statements hold, then the remaining statement also holds: rs

i) A ≤ B. rs

ii) A2 ≤ B2. iii) AB = BA. Source: [229, 1243, 1244]. Remark: The rank subtractivity partial ordering is defined in Fact 4.30.3. Fact 10.23.5. Let A, B, C ∈ Fn×n, and assume that A, B, and C are positive semidefinite. Then, the following statements hold: i) If A2 = AB and B2 = BA, then A = B. ii) If A2 = AB and B2 = BC, then A2 = AC. Source: Fact 4.30.6 and Fact 4.30.7. ∗ Fact 10.23.6. Let A, B ∈ Fn×n, assume that A and B are Hermitian, and let A ≤ B denote ∗ A2 = AB. Then, “≤” is a partial ordering on Hn×n. Source: Use Fact 4.30.8 or Fact 10.23.5 to show ∗

∗

that “≤” is antisymmetric and transitive. Remark: “≤” is the star partial ordering. See Fact 4.30.8. Fact 10.23.7. Let A, B ∈ Fn×n, and assume that A and B are positive semidefinite. Then, the following statements are equivalent: ∗

i) A ≤ B.

815

POSITIVE-SEMIDEFINITE MATRICES ∗

ii) B+ ≤ A+. rs

rs

iii) A ≤ B and A2 ≤ B2. Remark: See [1255, 1332]. Remark: The star partial ordering is defined in Fact 10.23.6. GL

Fact 10.23.8. Let A, B ∈ Fn×m, and let A ≤ B denote the case where the following three

statements hold: i) ⟨A⟩ ≤ ⟨B⟩. ii) R(A∗ ) ⊆ R(B∗ ). iii) AB∗ = ⟨A⟩⟨B⟩. Then, the following statements hold: GL

iv) “ ≤ ” is a partial ordering on Fn×m. rs

GL

v) If A ≤ B, then A ≤ B. GL

vi) Assume that A and B are positive semidefinite. Then, A ≤ B if and only if A ≤ B. Furthermore, the following statements are equivalent: GL

vii) A ≤ B. GL

viii) A∗ ≤ B∗. ix) ρmax (B+A) ≤ 1, R(A) ⊆ R(B), R(A∗ ) ⊆ R(B∗ ), and AB∗ = ⟨A⟩⟨B⟩. GL

Source: [1347]. Remark: “ ≤ ” is the generalized L¨owner partial ordering. The polar decomposi-

tion links the L¨owner, generalized L¨owner, and star partial orderings. See [1347].

10.24 Facts on Generalized Inverses Fact 10.24.1. Let A ∈ Fn×n. Then, the following statements are equivalent:

i) A + A∗ ≥ 0. ii) A+ + A+∗ ≥ 0. If, in addition, A is group invertible, then the following statement is equivalent to i) and ii): iii) A# + A#∗ ≥ 0. Source: [2713]. Fact 10.24.2. Let A ∈ Fn×n, and assume that A is positive semidefinite. Then, the following statements hold: i) A+ = AD = A# ≥ 0. △ ii) A+1/2 = (A1/2 )+ = (A+ )1/2. iii) A1/2 = A(A+ )1/2 = (A+ )1/2A. iv) AA+ = A1/2 (A1/2 )+. [ ] + v) AA+A AA A+ is positive semidefinite. vi) A+A + AA+ ≤ A + A+. vii) A+A ⊙ AA+ ≤ A ⊙ A+. Source: [2985] and Fact 10.12.9. See Fact 10.25.60 for v)–vii). Fact 10.24.3. Let A ∈ Fn×n, and assume that A is positive semidefinite. Then, rank A ≤ (tr A) tr A+. Furthermore, equality holds if and only if rank A ≤ 1. Source: [238]. Fact 10.24.4. Let A ∈ Fn×m. Then, ⟨A∗ ⟩ = A⟨A⟩+A∗ = A⟨A⟩A+. Source: Fact 10.8.4 and Fact

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CHAPTER 10

10.8.3. Remark: This result shows that the Schur complement of ⟨A⟩ in the partitioned matrix A defined in Fact 10.12.27 is zero. △ Fact 10.24.5. Let A ∈ Fn×m, and define S ∈ Fn×n by S = ⟨A∗ ⟩ + In − AA+. Then, S is positive definite, and SAA+S = ⟨A∗ ⟩AA+⟨A∗ ⟩ = AA∗. Source: [970, p. 432]. Remark: This result provides a congruence transformation between AA+ and AA∗. Related: Fact 7.9.21. Fact 10.24.6. Let A, B ∈ Fn×n, and assume that A and B are positive semidefinite. Then, the

following statements are equivalent: i) A = B. ii) A + AA+ = B + BB+ . iii) rank A = rank B and AB+A = B. iv) rank A = rank B and 2A(A + B)+A = A. ([ ]) ([ ]) A B v) R =R . B A Source: [2625]. Fact 10.24.7. Let A, B ∈ Fn×n, and assume that A and B are positive semidefinite. Then, R(A) ⊆ R(A + B) and A = (A + B)(A + B)+A. Fact 10.24.8. Let A, B ∈ Fn×n, and assume that A and B are Hermitian. Then, the following statements are equivalent: rs

i) A ≤ B. ii) R(A) ⊆ R(B) and AB+A = A. Source: [1243, 1244]. Related: Fact 8.9.39. Fact 10.24.9. Let A, B ∈ Fn×n, assume that A and B are Hermitian, assume that ν− (A) = ν− (B), and consider the following statements: ∗

i) A ≤ B. rs

ii) A ≤ B. iii) A ≤ B. iv) R(A) ⊆ R(B) and AB+A ≤ A. Then, i) =⇒ ii) =⇒ iii) ⇐⇒ iv). If A and B are positive semidefinite, then the following statement is equivalent to iii) and iv): v) R(A) ⊆ R(B) and ρmax (B+A) ≤ 1. Source: i) =⇒ ii) is given in [1339]. See [229, 1243, 1255], [2418, p. 229], and [2532]. Related: Fact 10.24.8. Fact 10.24.10. Let A, B ∈ Fn×n, and assume that A and B are positive semidefinite. Then, the following statements are equivalent: i) A2 ≤ B2. ii) R(A) ⊆ R(B) and σmax (B+A) ≤ 1. Source: [1255]. Fact 10.24.11. Let A, B ∈ Fn×n, assume that A and B are positive semidefinite, and assume that A ≤ B. Then, the following statements are equivalent: i) B+ ≤ A+. ii) rank A = rank B.

817

POSITIVE-SEMIDEFINITE MATRICES

iii) R(A) = R(B). Furthermore, the following statements are equivalent: iv) A+ ≤ B+. v) A2 = AB. ∗

vi) A+ ≤ B+. Source: [1332, 2044]. Fact 10.24.12. Let A, B ∈ Fn×n, and assume that A and B are positive semidefinite. Then, if two of the following statements hold, then the remaining statement also holds: i) A ≤ B. ii) B+ ≤ A+. iii) rank A = rank B. Source: [230, 2044, 2868, 2914]. Fact 10.24.13. Let A, B ∈ Fn×n, and assume that A and B are Hermitian. Then, if two of the following statements hold, then the remaining statement also holds: i) A ≤ B. ii) B+ ≤ A+. iii) In A = In B. Source: [228]. Fact 10.24.14. Let A, B ∈ Fn×n, assume that A and B are positive semidefinite, and assume that A ≤ B. Then, 0 ≤ AA+ ≤ BB+. If, in addition, rank A = rank B, then

AA+ = BB+.

Fact 10.24.15. Let A, B ∈ Fn×n, assume that A and B are Hermitian, and assume that R(A) =

R(B). Then,

In A − In B = In(A − B) + In(A+ − B+ ).

Source: [2141]. Related: Fact 10.11.26. Fact 10.24.16. Let A, B ∈ Fn×n, assume that A and B are positive semidefinite, and assume that

A ≤ B. Then,

0 ≤ AB+A ≤ A ≤ A + B[(I − AA+ )B(I − AA+ )]+B ≤ B.

Source: [1332]. Fact 10.24.17. Let A, B ∈ Fn×n, and assume that A and B are positive semidefinite. Then,

spec[(A + B)+A] ⊂ [0, 1]. Source: Let C be positive definite and satisfy B ≤ C. Then, (A + C)−1/2C(A + C)−1/2 ≤ I. The result

now follows from Fact 10.24.19. Fact 10.24.18. Let A, B ∈ Fn×n, and assume that A and B are positive semidefinite. Then, the following statements are equivalent: i) For all α ∈ [0, 1], [αA + (1 − α)B]+ ≤ αA+ + (1 − α)B+ . ii) R(A) = R(B). Now, let x ∈ C. Then, the following statements are equivalent: iii) For all α ∈ [0, 1], x∗ [αA + (1 − α)B]+ x ≤ αx∗A+ x + (1 − α)x∗B+ x. iv) x ∈ [R(A) ∩ R(B)] + [N(A) ∩ N(B)].

818

CHAPTER 10

Furthermore, the following statements are equivalent: v) For all α ∈ [0, 1], x∗ [αA + (1 − α)B]+ x = αx∗A+ x + (1 − α)x∗B+ x. vi) iv) holds and (A+ − B+ )x ∈ N(A) + N(B). Source: [2142]. Fact 10.24.19. Let A, B, C ∈ Fn×n, assume that A, B, C are positive semidefinite, and assume that B ≤ C. Then, for all i ∈ {1, . . . , n}, λi [(A + B)+B] ≤ λi [(A + C)+C]. Consequently,

tr (A + B)+B ≤ tr (A + C)+C.

Source: [2815]. Related: Fact 10.24.17. Fact 10.24.20. Let A, B ∈ Fn×n, assume that A and B are positive semidefinite, and define △

A : B = A(A + B)+B. Then, the following statements hold: i) A : B is positive semidefinite. ii) A : B = limε↓0 (A + εI) : (B + εI). iii) A : A = 12 A. iv) A : B = B : A = B − B(A + B)+B = A − A(A + B)+A. v) A : B ≤ A. vi) A : B ≤ B. +     A 0 I   0  vii) A : B = −[0 0 I]  0 B I   0  .     I I I 0 viii) A : B = (A+ + B+ )+ if and only if R(A) = R(B). ix) A(A + B)+B = ACB for every (1)-inverse C of A + B. x) tr A : B ≤ (tr B) : (tr A). xi) tr A : B = (tr B) : (tr A) if and only if there exists α ∈ [0, ∞) such that either A = αB or B = αA. xii) det A : B ≤ (det B) : (det A). xiii) R(A : B) = R(A) ∩ R(B). xiv) N(A : B) = N(A) + N(B). xv) rank A : B = rank A + rank B − rank(A + B). xvi) Let S ∈ F p×n, and assume that S is right invertible. Then, S (A : B)S ∗ ≤ (SAS ∗ ) : (SBS ∗ ). xvii) Let S ∈ Fn×n, and assume that S is nonsingular. Then, S (A : B)S ∗ = (SAS ∗ ) : (SBS ∗ ). xviii) For all positive numbers α, β, (α−1A) : (β −1B) ≤ αA + βB. [ ] A xix) Let X ∈ Fn×n, and assume that X is Hermitian and A+B A A−X ≥ 0. Then, [ ] A+B A X ≤ A : B, ≥ 0. A A − A: B △

xx) ϕ: Nn × Nn 7→ −Nn defined by ϕ(A, B) = −A : B is convex. xxi) If A and B are projectors, then 2(A : B) is the projector onto R(A) ∩ R(B). xxii) If A + B is positive definite, then A : B = A(A + B)−1B.

819

POSITIVE-SEMIDEFINITE MATRICES

xxiii) A#B = [ 12 (A + B)]#[2(A : B)]. xxiv) If C, D ∈ Fn×n are positive semidefinite, then (A : B) :C = A : (B :C),

A :C + B : D ≤ (A + B) : (C + D).

xxv) If C, D ∈ Fn×n are positive semidefinite, A ≤ C, and B ≤ D, then A : B ≤ C : D. xxvi) If A and B are positive definite, then A : B = (A−1 + B−1 )−1 ≤ 21 (A#B) ≤ 41 (A + B). xxvii) If A and B are positive definite, then n n ∑ ∑ λi (A)λi (B) λi (A)λn−i+1 (B) ≤ tr A : B ≤ . λ (A) + λ (B) λ (A) + λi (B) n−i+1 i=1 i i=1 i

xxviii) Let x, y ∈ Fn. Then,

(x + y)∗ (A : B)(x + y) ≤ x∗Ax + y∗By.

xxix) Let x, y ∈ Fn. Then,

x∗ (A : B)x ≤ y∗Ay + (x − y)∗B(x − y).

xxx) Let x ∈ Fn. Then,

x∗ (A : B)x = infn [y∗Ay + (x − y)∗B(x − y)]. y∈F

xxxi) Let x ∈ F . Then, n

x∗ (A : B)x ≤ (x∗Ax) : (x∗Bx).

Source: [81, 82, 85, 1224, 1696], [2299, p. 189], [2632, 2750], and [2977, p. 9]. Remark: A : B is the parallel sum of A and B. Remark: A symmetric expression for the parallel sum of three or more positive-semidefinite matrices is given in [2632]. Remark: A#B is the geometric mean of A and B. See Fact 10.11.68. Related: Fact 8.4.8 and Fact 8.8.19. Fact 10.24.21. Let A, B ∈ Fn×n, assume that A is positive semidefinite, and assume that B is a projector. Then, △ sh(A, B) = max {X ∈ Nn : 0 ≤ X ≤ A and R(X) ⊆ R(B)}

exists. Furthermore,

sh(A, B) = A − AB⊥ (B⊥AB⊥ )+B⊥A.   A sh(A, B) = A  B⊥A

That is,

Finally,

 AB⊥   . B⊥AB⊥

sh(A, B) = lim (αB) : A. α→∞

Source: Existence of the maximum is proved in [85]. The expression for sh(A, B) is given in [1195]; a related expression involving the Schur complement is given in [81]. The last equality is shown in [85]. See also [96]. Remark: sh(A, B) is the shorted operator. Fact 10.24.22. Let B ∈ Rm×n, define △

S = {A ∈ Rn×n : A ≥ 0 and R(BTBA) ⊆ R(A)}, △

and define ϕ: S 7→ −Nm by ϕ(A) = −(BA+BT )+. Then, S is a convex cone, and ϕ is convex. Source: [1245]. Related: This result generalizes xii) of Proposition 10.6.17 in the case where r = p = 1. Fact 10.24.23. Let A, B ∈ Fn×n, and assume that A and B are positive semidefinite. If (AB)+ = + + B A , then AB is range Hermitian. Furthermore, the following statements are equivalent:

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CHAPTER 10

i) AB is range Hermitian. ii) (AB)# = B+A+. iii) (AB)+ = B+A+. Source: [2001]. Related: Fact 8.5.11. Fact 10.24.24. Let A, B ∈ Fn×n, and assume that A and B are positive semidefinite. Then, the following statements are equivalent: i) A(A + B)+B = 0. ii) B(A + B)+A = 0. iii) A(A + B)+A = A. iv) B(A + B)+B = B. v) A(A + B)+B + B(A + B)+A = 0. vi) A(A + B)+A + B(A + B)+B = A + B. vii) rank [A B] = rank A + rank B. viii) R(A) ∩ R(B) = {0}. ix) (A + B)+ = [(I − BB+ )A(I − B+B)]+ + [(I − AA+ )B(I − A+A)]+. Source: [2666]. Related: Fact 8.4.31. Fact 10.24.25. Let A ∈ Fn×n and C ∈ Fm×m, assume that A and C are positive semidefinite, let △ B ∈ Fn×m, and define X = A+1/2BC +1/2. Then, the following statements are equivalent: [ ] i) BA∗ CB is positive semidefinite. ii) AA+B = B and XX ∗ ≤ In . iii) BC +C = B and X ∗X ≤ Im . iv) B = A1/2XC 1/2 and X ∗X ≤ Im . v) There exists Y ∈ Fn×m such that B = A1/2YC 1/2 and Y ∗Y ≤ Im . Source: [2977, p. 15]. Related: Fact 10.12.15.

10.25 Facts on the Kronecker and Schur Products Fact 10.25.1. Let A ∈ Fn×n, and assume that every entry of A is nonzero. Then, A and A⊙−1 are positive semidefinite if and only if there exists x ∈ Fn such that A = xx∗ . Source: [1435, 1788]. Fact 10.25.2. Let A ∈ Fn×n, assume that A is positive semidefinite, assume that every entry of A is nonnegative, and let α ∈ [0, n − 2]. Then, A⊙α is positive semidefinite. Source: [451, 1053]. Remark: A⊙α may be positive semidefinite for all α ≥ 0. See Fact 10.9.8. Fact 10.25.3. Let A ∈ Fn×n, assume that A is positive semidefinite, and let k ≥ 1. Then, the

following statements hold: i) If r ∈ [0, 1], then (Ar )⊙k ≤ (A⊙k )r . ii) If r ∈ [1, 2], then (A⊙k )r ≤ (Ar )⊙k . iii) If A is positive definite and r ∈ [0, 1], then (A⊙k )−r ≤ (A−r )⊙k. Source: [2977, p. 8]. Fact 10.25.4. Let A ∈ Fn×n, and assume that A is positive semidefinite. Then, (I ⊙ A)2 ≤ 21 (I ⊙ A2 + A ⊙ A) ≤ I ⊙ A2, Hence,

n ∑ i=1

A2(i,i) ≤

n ∑ i=1

λ2i (A).

A ⊙ A ≤ I ⊙ A2.

821

POSITIVE-SEMIDEFINITE MATRICES

Now, assume that A is positive definite. Then, (A ⊙ A)−1 ≤ A−1 ⊙ A−1 ,

(A ⊙ A−1 )−1 ≤ I ≤ (A1/2 ⊙ A−1/2 )2 ≤ 21 (I + A ⊙ A−1 ) ≤ A ⊙ A−1.

If A is real, then

(√ 2 max

i∈{1,...,n}

Furthermore,

n ) ∑ (A ⊙ A−1 )(i,i) − 1 ≤ [(A ⊙ A−1 )(i,i) − 1]. i=1

(A ⊙ A−T )1n×1 = 1n×1 ,

△

1 = min spec(A ⊙ A−T ).

△

Next, let α = λmin (A) and β = λmax (A). Then, 2αβ αβ α2 + β2 2αβ 2 −2 1/2 2 −2 −1 I ≤ (A ⊙A ) ≤ (I+A ⊙A ) ≤ A⊙A ≤ I, 2αβ α2 + β2 α2 + β 2 α2 + β2 △

A⊙A−T ≤

△

α2 + β2 I. 2αβ △

Finally, define Φ(A) = A ⊙ A−1 , and, for all k ≥ 1, define Φ(k+1) (A) = Φ[Φ(k) (A)], where Φ(1) (A) = Φ(A). Then, for all k ≥ 1, Φ(k) (A) ≥ I, lim Φ(k)(A) = I. k→∞

Source: [1039, 1042, 1548, 2804, 2805], [1448, p. 475], [2403, pp. 304, 305], [2991, p. 251], and set B = A−1 in Fact 10.25.49. The upper bound for A ⊙ A−T is given in [1874]. Remark: A ⊙ A−T is Hermitian in the case where F = C. Remark: The convergence result holds in the case where A is an H-matrix [1548]. A ⊙ A−T is the relative gain array. Related: Fact 10.25.46 and Fact 10.25.55. Fact 10.25.5. Let A ∈ Fn×n, and assume that A is positive definite. Then, for all i ∈ {1, . . . , n},

1 ≤ A(i,i) (A−1 )(i,i) , n √ √ ∑ A(i,i) (A−1 )(i,i) − 1, 2 max A(i,i) (A−1 )(i,i) − 1 ≤ i=1,...,n

√ 2 max

i=1,...,n

i=1

A(i,i) (A−1 )(i,i) − 1 ≤

n [√ ∑

] A(i,i) (A−1 )(i,i) − 1 .

i=1

Source: [1041, p. 66-6]. △ Fact 10.25.6. Let A ∈ Fn×n, assume that A is positive definite, and define α = λn (A) and △

β = λ1 (A). Then,

2I ≤ (A + A−1 ) ⊙ I ≤

α2 + β2 I. αβ

Source: [26]. Remark: The left-hand inequality is a special case of Fact 10.25.55. △

Fact 10.25.7. Let A, B, C ∈ Fn×n , define A =

semidefinite. Then,

△

Fact 10.25.8. Let A =

A−1 =

X Y Y∗ Z

]

[

A B B∗ C

]

]

∈ F2n×2n, and assume that A is positive

∈ F(n+m)×(n+m), assume that A is positive definite, and partition

conformably with A. Then, ] [ A ⊙ A−1 0 I≤ ≤ A ⊙ A−1 , 0 Z ⊙ Z −1

Source: [282].

A B B∗ C

−A ⊙ C ≤ B ⊙ B∗ ≤ A ⊙ C.

Source: [2991, p. 236, 237].

[

[

[

X ⊙ X −1 I≤ 0

] 0 ≤ A ⊙ A−1. C ⊙ C −1

822

CHAPTER 10

Fact 10.25.9. Let A ∈ Fn×n, let p, q ∈ R, assume that A is positive semidefinite, and assume

that either p and q are nonnegative or A is positive definite. Then, A(p+q)/2 ⊙ A(p+q)/2 ≤ Ap ⊙ Aq. In particular, I ≤ A ⊙ A−1. Source: [198]. Fact 10.25.10. Let A ∈ Fn×n, assume that A is positive semidefinite, and assume that In ⊙A = In . Then, det A ≤ λmin (A ⊙ A). Source: [2842]. Fact 10.25.11. Let A ∈ Fn×n. Then, −A∗A ⊙ I ≤ A∗ ⊙ A ≤ A∗A ⊙ I. Source: Use Fact 10.25.57 with B = I. { ∗ } AA⊙I 2 Fact 10.25.12. Let A ∈ Fn×n. Then, ⟨A ⊙ A∗ ⟩ ≤ ⟨A⟩ ⊙ ⟨A∗ ⟩ ≤ σmax (A)I. Source: [2985] and Fact 10.25.37. Fact 10.25.13. Let A ∈ Fn×n and x ∈ Cn . Then, |x∗ (A ⊙ A∗ )x| ≤ x∗ (⟨A⟩ ⊙ ⟨A∗ ⟩)x. Source: [2991, p. 314]. ] ] [ [ △ B11 B12 △ A A12 (n+m)×(n+m) ∗ ∈ F(n+m)×(n+m), and assume that ∈ F and B = Fact 10.25.14. Let A = A11 A B B 22 12 22 12 A and B are positive semidefinite. Then, (A11 |A) ⊙ (B11|B) ≤ (A11 |A) ⊙ B22 ≤ (A11 ⊙ B11 )|(A ⊙ B). Source: [1800] and [2991, p. 241]. Fact 10.25.15. Let A, B ∈ Fn×n, and assume that A and B are positive semidefinite. Then,

rank(A ⊙ B) ≤ rank(A ⊗ B) = (rank A)(rank B). Now, assume that A is positive definite. Then, rank B ≤ rank(A ⊙ B) = rank(I ⊙ B). Source: Fact 9.4.20, Fact 9.6.15, Fact 10.25.21, [2263, pp. 154, 166], and [2991, p. 238]. Remark:

The first inequality is due to D. Z. Djokovic. Fact 10.25.16. Let A, B ∈ Fn×n, and assume that A and B are positive semidefinite. Then, A ⊙ B is positive semidefinite. If, in addition, A is positive definite and I ⊙ B is positive definite, then A ⊙ B is positive definite. Source: By Fact 9.4.22, A ⊗ B is positive semidefinite, and the Schur product A ⊙ B is a principal submatrix of the Kronecker product. If A is positive definite, use Fact 10.25.33 to obtain det(A ⊙ B) > 0. See [2403, p. 300] and [2991, pp. 234, 235]. Remark: The first result is Schur’s theorem. The second result is Schott’s theorem. See [1873], Fact 10.25.15, and Fact 10.25.33. Fact 10.25.17. Let A ∈ Fn×n, assume that A is positive semidefinite, and define e⊙A ∈ Fn×n by △ ⊙A (e )(i, j) = eA(i, j) . Then, e⊙A is positive semidefinite. Source: Note that e⊙A = 1n×n + 21 A ⊙ A + 3!1 A ⊙ A ⊙ A + · · · , and use Fact 10.25.16. See [922, p. 10]. Fact 10.25.18. Let A ∈ Fn×n, and assume that A is positive definite. Then, there exist positivedefinite matrices B, C ∈ Fn×n such that A = B ⊙ C. Source: [2263, pp. 154, 166]. Credit: D. Z. Djokovic. Fact 10.25.19. Let A, B ∈ Fn×n. Then, A is positive semidefinite if and only if, for every positive-semidefinite matrix B ∈ Fn×n, 11×n (A ⊙ B)1n×1 ≥ 0. Source: [1448, p. 459]. Remark: This is Fej´er’s theorem. Fact 10.25.20. Let A, B ∈ Fn×n, and assume that A and B are positive definite. Then, 11×n [(A − B) ⊙ (A−1 − B−1 )]1n×1 ≤ 0. Furthermore, equality holds if and only if A = B. Source: [304, p. 8-8].

823

POSITIVE-SEMIDEFINITE MATRICES

Fact 10.25.21. Let A, B ∈ Fn×n, and assume that A is positive definite and B is positive semidef-

inite. Then,

(11×n A−11n×1 )−1B ≤ A ⊙ B.

Source: [1044]. Remark: Setting B = 1n×n yields Fact 10.10.20. Fact 10.25.22. Let A, B ∈ Fn×n, and assume that A and B are positive definite. Then,

(11×n A−11n×1 11×n B−11n×1 )−11n×n ≤ A ⊙ B. Source: [2985]. Fact 10.25.23. Let A ∈ Fn×n, and assume that A is positive semidefinite. Then,

tr A⊙2 ≤ tr A2 ≤ (tr A)2 ,

(tr A)1/2 ≤ tr A1/2 ≤ tr (A ⊙ I)1/2 .

Now, assume that A is positive definite. Then, n2 ≤ tr (A ⊙ I)−1 ≤ tr A−1 . tr A Source: [2991, p. 213]. Related: Fact 10.13.9. Fact 10.25.24. Let A, B ∈ Fn×n. Then, A is positive semidefinite if and only if, for every positive-semidefinite matrix B ∈ Fn×n, tr(A ⊙ B) ≥ 0. Source: [2991, p. 237]. Fact 10.25.25. Let A, B ∈ Fn×n, and assume that A and B are positive semidefinite. Then, the following statements hold: i) tr AB ≤ tr(A ⊗ B). ii) tr(A ⊙ B) ≤ 12 tr(A ⊙ A + B ⊙ B). iii) tr(A ⊗ B) ≤

1 2

iv) det(A ⊗ B) ≤

tr(A ⊗ A + B ⊗ B). 1 2 [det(A ⊗ A)

+ det(B ⊗ B)].

Source: [2991, p. 238]. Fact 10.25.26. Let A ∈ Fn×m and B ∈ Fk×l. Then, ⟨A ⊗ B⟩ = ⟨A⟩ ⊗ ⟨B⟩. Source: [2991, p. 291]. Fact 10.25.27. Let A, B ∈ Fn×n, assume that A and B are Hermitian, and assume that either

A ≤ B or B ≤ A. Then,

2A ⊙ B ≤ A ⊙ A + B ⊙ B.

Source: [2991, p. 251]. Fact 10.25.28. Let A, B ∈ Fn×n, and assume that A and B are positive semidefinite. Then, the

following statements hold: i) If p ≥ 1, then tr (A ⊙ B) p ≤ tr Ap ⊙ B p. ii) If 0 ≤ p ≤ 1, then tr Ap ⊙ B p ≤ tr (A ⊙ B) p. iii) If A and B are positive definite and p ≤ 0, then tr (A ⊙ B) p ≤ tr Ap ⊙ B p. Source: [2817]. Fact 10.25.29. Let A, B ∈ Fn×n, and assume that A and B are positive semidefinite. Then, λmin (AB) ≤ λmin (A ⊙ B),

λmin (AB)I ≤ λmin (A ⊙ B)I ≤ A ⊙ B.

Source: [1541]. Related: This result interpolates an inequality in Fact 10.25.34. Fact 10.25.30. Let A, B ∈ Fn×n, and assume that A is positive semidefinite and B is positive

definite. Then,

ρmax (A ⊙ B) ≤ ρmax (B ⊙ B−1 )ρmax (AB).

Source: [1459]. Remark: Fact 10.25.4 implies that ρmax (B ⊙ B−1 ) ≥ ρmin (B ⊙ B−1 ) ≥ 1.

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CHAPTER 10

Fact 10.25.31. Let A, B ∈ Fn×n, and assume that A and B are Hermitian. Then, for all i ∈

{1, . . . , n},

min λi (A)λ j (B) ≤ λi+n2 −n (A ⊗ B) ≤ λi (A ⊙ B) ≤ λi (A ⊗ B) ≤ max λi (A)λ j (B).

i, j∈{1,...,n}

i, j∈{1,...,n}

Now, assume that A and B are positive semidefinite. Then, λn (A)λn (B) ≤ λi+n2 −n (A ⊗ B) ≤ λi (A ⊙ B) ≤ λi (A ⊗ B) ≤ λ1 (A)λ1 (B). Source: Proposition 9.1.10, Proposition 9.3.2, and Theorem 10.4.5. For A, B positive semidefinite, see [1951] and [2991, p. 278]. Fact 10.25.32. Let A, B ∈ Fn×n, and assume that A and B are positive semidefinite. Then, for all i ∈ {1, . . . , n},

dmin (A)λmin (B) ≤ di (A)λmin (B) ≤ λi (A ⊙ B) ≤ di (A)λmax (B) ≤ dmax (A)λmax (B). Source: [2403, pp. 303, 304] and [2991, pp. 274, 275]. Fact 10.25.33. Let A, B ∈ Fn×n, and assume that A and B are positive semidefinite. Then,

  n n ∏  ∏ A(i,i) B(i,i) . A(i,i)  det B ≤ det(A ⊙ B) ≤ det AB ≤  i=1

i=1

Equivalently, det AB ≤ [det(I ⊙ A)] det B ≤ det(A ⊙ B) ≤

n ∏

A(i,i) B(i,i) .

i=1

Furthermore,

  n   n  ∏  ∏ B(i,i)  det A ≤ det(A ⊙ B) + (det A) det B. A(i,i)  det B +  2 det AB ≤  i=1

i=1

Finally, the following statements hold: i) If I ⊙ A and B are positive definite, then A ⊙ B is positive definite. ii) If I ⊙ A and B are positive definite and rank A = 1, then equality holds in the right-hand equality. iii) If A and B are positive definite, then equality holds in the right-hand equality if and only if B is diagonal. Source: [1960], [2418, p. 253], [2955], and [2991, p. 242]. Remark: In the first string, the first and third inequalities follow from Hadamard’s inequality Fact 10.21.15, while the second inequality is Oppenheim’s inequality. See Fact 10.25.16. Remark: The right-hand inequality in the third string of inequalities is valid if A and B are M-matrices. See [89, 710]. Fact 10.25.34. Let A, B ∈ Fn×n, assume that A and B are positive semidefinite, let k ∈ {1, . . . , n}, and let r ∈ (0, 1]. Then, n n n n n ∏ ∏ ∏ ∏ ∏ λi (A)λi (B) ≤ σi (AB) ≤ λi (AB) ≤ λ2i (A#B) ≤ λi (A ⊙ B), i=k n ∏

i=k

λi (A)λi (B) ≤

i=k

n ∏ i=k

≤

n ∏ i=k

σi (AB) ≤

i=k n ∏

λi (AB) ≤

i=k

eλi (log A+log B) ≤

i=k n ∏

i=k

r r λ1/r i (A B )

i=k n ∏ i=k

eλi [I⊙(log A+log B)] ≤

n ∏ i=k

r r λ1/r i (A ⊙ B ) ≤

n ∏ i=k

λi (A ⊙ B).

825

POSITIVE-SEMIDEFINITE MATRICES

Consequently, λmin (AB) ≤ λmin (A ⊙ B),

det AB = det (A#B)2 ≤ det(A ⊙ B).

Source: [94, 1039, 2803], [2403, p. 305], [2977, p. 21], Fact 10.11.68, and Fact 10.22.28. Remark: Although det AB = det (A#B)2 , the matrices AB and (A#B)2 do not necessarily have the same

spectrum. Fact 10.25.35. Let A, B ∈ Fn×n, assume that A and B are positive definite, let k ∈ {1, . . . , n}, and

let r > 0. Then,

n ∏

λ−r i (A ⊙ B) ≤

n ∏

λ−r i (AB).

i=k

i=k

Source: [2802]. △ Fact 10.25.36. Let A ∈ Fn×n, assume that A is positive semidefinite, and define S = {X ∈

Nn : X ⊙ I = I}. Then,

min det(A ⊙ X) = det A. X∈S

Source: [2991, p. 243]. Remark: X is a correlation matrix. Fact 10.25.37. Let A, B ∈ Fn×n, let C, D ∈ Fm×m, assume that A, B, C, and D are Hermitian,

A ≤ B, C ≤ D, and that either A and C are positive semidefinite, A and D are positive semidefinite, or B and D are positive semidefinite. Then, A ⊗ C ≤ B ⊗ D. If, in addition, n = m, then

A ⊙ C ≤ B ⊙ D.

Source: [88, 230]. Problem: Determine conditions under which these inequalities are strict. Fact 10.25.38. Let A, B, C, D ∈ Fn×n, assume that A, B, C, D are positive semidefinite, and

assume that A ≤ B and C ≤ D. Then,

0 ≤ A ⊗ C ≤ B ⊗ D,

0 ≤ A ⊙ C ≤ B ⊙ D.

Source: Fact 10.25.37 and [2991, p. 238]. Fact 10.25.39. Let A, B ∈ Fn×n, and assume that A and B are positive semidefinite. Then, the

following statements are equivalent: i) A ≤ B ii) A ⊗ I ≤ B ⊗ I. iii) A ⊗ A ≤ B ⊗ B. Source: [1873] and [2991, p. 237]. Fact 10.25.40. Let A, B ∈ Fn×n, assume that A and B are positive semidefinite, assume that 0 ≤ A ≤ B, and let k ≥ 1. Then, A⊙k ≤ B⊙k. Source: 0 ≤ (B −A)⊙(B + A) implies that A⊙A ≤ B⊙B; that is, A⊙2 ≤ B⊙2. Fact 10.25.41. Let A1 , . . . , Ak , B1 , . . . , Bk ∈ Fn×n, and assume that A1 , . . . , Ak , B1 , . . . , Bk are positive semidefinite. Then, (A1 + B1 ) ⊗ · · · ⊗ (Ak + Bk ) ≤ A1 ⊗ · · · ⊗ Ak + B1 ⊗ · · · ⊗ Bk . Source: [2029, p. 143]. Fact 10.25.42. Let A1 , A2 , B1 , B2 ∈ Fn×n, assume that A1 , A2 , B1 , B2 are positive semidefinite,

assume that 0 ≤ A1 ≤ B1 and 0 ≤ A2 ≤ B2 , and let α ∈ [0, 1]. Then,

[αA1 + (1 − α)B1 ] ⊗ [αA2 + (1 − α)B2 ] ≤ α(A1 ⊗ A2 ) + (1 − α)(B1 ⊗ B2 ).

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CHAPTER 10

Source: [2840]. Fact 10.25.43. Let A ∈ Fn×n and B ∈ Fm×m, assume that A and B are positive semidefinite, let

r ∈ R, and assume that either A and B are positive definite or r is positive. Then, (A ⊗ B)r = Ar ⊗ Br. Source: [2074]. Fact 10.25.44. Let A, B, C ∈ Fn×n , assume A, B, C are positive semidefinite, and let k ≥ 1. Then, (A + B)⊗k + (A + C)⊗k ≤ (A + B + C)⊗k + A⊗k . Source: [2681]. Related: Fact 10.16.35. Fact 10.25.45. For all i ∈ {1, . . . , k}, let Ai ∈ Hn and Bi ∈ Hm , assume that Ak ≤ · · · ≤ A1 and

Bk ≤ · · · ≤ B1 , and let { j1 , . . . , jk } = {1, . . . , k}. Then, k ∑

Ai ⊗ Bk−i+1 ≤

k ∑

i=1

k

k ∑

Ai ⊗ B ji ≤

k ∑

i=1

Ai ⊗ Bk−i+1 ≤

i=1

k ∑

Ai ⊗ Bi ,

i=1

Ai ⊗

i=1

k ∑

k ∑

B ji ≤ k

i=1

Ai ⊗ Bi .

i=1

Now, assume that n = m, assume that Ak + Bk is positive semidefinite, and let l ≥ 1. Then, k k k ∑ ∑ ∑ (Ai + Bk−i+1 )⊗l ≤ (Ai + B ji )⊗l ≤ (Ai + Bi )⊗l . i=1

i=1

i=1

Source: [2681]. Related: Fact 10.14.62. Fact 10.25.46. For all i ∈ {1, . . . , k}, let Ai , Bi ∈ Hn , assume that Ak ≤ · · · ≤ A1 and Bk ≤ · · · ≤

B1 , and let { j1 , . . . , jk } = {1, . . . , k}. Then, k ∑

Ai ⊙ Bk−i+1 ≤

k ∑

i=1

k

k ∑

Ai ⊙ B ji ≤

k ∑

i=1

Ai ⊙ Bk−i+1 ≤

i=1

k ∑

Ai ⊙

i=1

Ai ⊙ Bi ,

i=1 k ∑

B ji ≤ k

i=1

k ∑

Ai ⊙ Bi .

i=1

Now, assume that Ak + Bk is positive semidefinite, and let l ≥ 1. Then, k k k ∑ ∑ ∑ (Ai + Bk−i+1 )⊙l ≤ (Ai + B ji )⊙l ≤ (Ai + Bi )⊙l . i=1

i=1

i=1

Now, assume that Ak and Bk are positive semidefinite. Then, k ∑

det(Ai ⊙ Bk−i+1 )l ≤

i=1

k ∑

det(Ai ⊙ B ji )l ≤

i=1

k ∑

det(Ai ⊙ Bi )l .

i=1

Finally, assume that Ak is positive definite. Then, kI ≤

k ∑ i=1

Ai ⊙ A−1 i ≤

k ∑

Ai ⊙ A−1 ji .

i=1

Source: [2681]. Related: Fact 10.14.62 and Fact 10.25.4. Fact 10.25.47. Let A ∈ Fn×m and B ∈ Fk×l. Then, ⟨A ⊗ B⟩ = ⟨A⟩ ⊗ ⟨B⟩.

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POSITIVE-SEMIDEFINITE MATRICES

Fact 10.25.48. Let A, B ∈ Fn×n , let C, D ∈ Fm×m, assume that A, B, C, D are positive semidefi-

nite, let α and β be nonnegative numbers, and let r ∈ [0, 1]. Then,

α(Ar ⊗ C 1−r ) + β(Br ⊗ D1−r ) ≤ (αA + βB)r ⊗ (αC + βD)1−r . Source: [1796]. Fact 10.25.49. Let A, B ∈ Fn×n, and assume that A and B are positive semidefinite. Then, the

following statements hold: i) If r ∈ [0, 1], then Ar ⊙ Br ≤ (A ⊙ B)r. ii) If r ∈ [1, 2], then (A ⊙ B)r ≤ Ar ⊙ Br. △ △ Now, define α = λmin (A ⊗ B) and β = λmax (A ⊗ B). Then, A ⊙B − 2

2

1 4 (β

− α) I ≤ (A ⊙ B) ≤ 2

2

1 2 2 [A

 (α+β)2 2    4αβ (A ⊙ B) ⊙ B + (AB) ] ≤ A ⊙ B ≤    (A ⊙ B)2 + 1 (β − α)2I, ⊙2

2

2

2

4

√ √ A ⊙ B − 41 ( β − α)2I ≤ (A

1/2

1/2 1/2 ⊙2

) ≤ 12 [A ⊙ B + (A B

⊙B

1/2 2

) ] ≤ A ⊙ B ≤ (A ⊙ B ) 2

2 1/2

 α+β     2 √αβ A ⊙ B ≤  (β−α)2   A ⊙ B + 4(β+α) I.

Source: [88], [1448, p. 475], [2073, 2804], [2977, p. 8], and [2991, p. 252]. Fact 10.25.50. Let A, B ∈ Fn×n, and assume that A and B are positive definite. Then, the

following statements hold: i) If r ∈ [0, 1], then (A ⊙ B)−r ≤ A−r ⊙ B−r. ii) If k, l are nonzero integers such that k ≤ l, then, (Ak ⊙ Bk )1/k ≤ (Al ⊙ Bl )1/l. In particular, for all k ≥ 1, A ⊙ B ≤ (Ak ⊙ Bk )1/k, △

A1/k ⊙ B1/k ≤ (A ⊙ B)1/k.

△

Now, define α = λmin (A ⊗ B) and β = λmax (A ⊗ B). Then,  (α+β)2 −1 −1 −1    4αβ (A ⊙ B ) −1 −1 −1 (A ⊙ B ) ≤ A ⊙ B ≤    (A−1 ⊙ B−1 )−1 + ( √β − √α)2I. Source: [26, 2073]. Fact 10.25.51. Let A, B ∈ Fn×n, assume that A and B are positive semidefinite, and define △

△

α = λmin (A ⊕ B) and β = λmax (A ⊕ B). Then, (A2 + B2 ) ⊙ I ≤

(α + β)2 [(A + B) ⊙ I]2 , 4αβ

(A2 + B2 ) ⊙ I ≤ [(A + B) ⊙ I]2 +

(β − α)2 I. 4

Now, assume that A and B are positive definite. Then, (A−1 + B−1 ) ⊙ I ≤

(α + β)2 [(A + B) ⊙ I]−1 , 4αβ

(A−1 + B−1 ) ⊙ I ≤ [(A + B) ⊙ I]−1 +

√

√ β− α I. αβ

Source: [26]. Source: A ⊙ I + I ⊙ B = (A + B) ⊙ I is the Hadamard sum of A and B. See [26]. Fact 10.25.52. Let A, B ∈ Fn×n, and assume that A and B are Hermitian. Then, there exist

unitary matrices S 1 , S 2 ∈ Fn×n such that

⟨A ⊙ B⟩ ≤ 21 [S 1 (⟨A⟩ ⊙ ⟨B⟩)S 1∗ + S 2 (⟨A⟩ ⊙ ⟨B⟩)S 2∗ ]. Source: [196]. Related: Fact 10.11.34.

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CHAPTER 10

Fact 10.25.53. Let A, B ∈ Fn×n, and assume that A is positive definite, B is positive semidefinite,

and I ⊙ B is positive definite. Then, for all i ∈ {1, . . . , n}, [(A ⊙ B)−1 ](i,i) ≤

(A−1 )(i,i) . B(i,i)

Furthermore, if rank B = 1, then equality holds. Source: [2955]. Fact 10.25.54. Let A, B ∈ Fn×n, assume that A and B are positive semidefinite, let p, q ∈ R, and assume that at least one of the following statements holds: i) p ≤ q ≤ −1, and A and B are positive definite. ii) p ≤ −1 < 1 ≤ q, and A and B are positive definite. iii) 1 ≤ p ≤ q. iv) 21 ≤ p ≤ 1 ≤ q. v) p ≤ −1 ≤ q ≤ − 12 , and A and B are positive definite. Then, (Ap ⊙ B p )1/p ≤ (Aq ⊙ Bq )1/q. Source: [2074]. Consider case iii). Since p/q ≤ 1, it follows from Fact 10.25.49 that Ap ⊙ B p = (Aq ) p/q ⊙ (Aq ) p/q ≤ (Aq ⊙ Bq ) p/q. Then, use Corollary 10.6.11 with p replaced by 1/p. Remark: See

[198] and [2977, p. 8]. Fact 10.25.55. Let A, B ∈ Fn×n, and assume that A and B are positive definite. Then, 2I ≤ A ⊙ B−1 + B ⊙ A−1. Source: [2804, 2985]. Remark: Setting B = A yields an inequality given by Fact 10.25.4. Setting

B = I yields Fact 10.25.6. Fact 10.25.56. Let A, B ∈ Fn×m, and define [ ∗ ] A A ⊙ B∗B (A ⊙ B)∗ △ A= . A⊙B I

Then, A is positive semidefinite. Furthermore, { } (A ⊙ B)∗ (A ⊙ B) ∗ ∗ −A A ⊙ B B ≤ ≤ 12 (A∗A ⊙ B∗B + A∗B ⊙ B∗A) ≤ A∗A ⊙ B∗B. A∗B ⊙ B∗A Source: [1453, 2804, 2985]. Remark: (A ⊙ B)∗ (A ⊙ B) ≤ A∗A ⊙ B∗B is Amemiya’s inequality. See

[1873]. Fact 10.25.57. Let A, B ∈ Fn×m. Then,

−A∗A ⊙ B∗B ≤ A∗B ⊙ B∗A ≤ A∗A ⊙ B∗B, | det(A∗B ⊙ B∗A)| ≤ det(A∗A ⊙ B∗B). [ ∗ ∗ ] Source: Apply Fact 10.25.60 to BA∗AA BA∗BB . Related: Fact 10.12.33 and Fact 10.25.11. Fact 10.25.58. Let A, B ∈ Fn×n, and assume that A is positive definite. Then, −A ⊙ B∗A−1B ≤ B ⊙ B∗ ≤ A ⊙ B∗A−1B,

| det(B ⊙ B∗ )| ≤ det(A ⊙ B∗A−1B).

Source: Fact 10.12.49 and Fact 10.25.60. Fact 10.25.59. Let A, B ∈ Fn×n, assume that A and B are positive definite, and let x, y ∈ Fn .

Then,

(x ⊙ y)∗ (A ⊙ B)−1 (x ⊙ y) ≤ (x∗A−1x) ⊙ (y∗B−1 y).

Source: [2991, p. 243]. Related: Fact 10.19.16.

829

POSITIVE-SEMIDEFINITE MATRICES

Fact 10.25.60. Let A, B, C ∈ Fn×n, define

[

A A= ∗ B △

] B , C

and assume that A is positive semidefinite. Then, −A ⊙ C ≤ B ⊙ B∗ ≤ A ⊙ C, | tr(B ⊙ B∗ )| ≤ tr(A ⊙ C),

| det(B ⊙ B∗ )| ≤ det(A ⊙ C), √ tr (B ⊙ B∗ )2 ≤ tr(A ⊙ C).

If, in addition, A is positive definite, then −A ⊙ C < B ⊙ B∗ < A ⊙ C,

| det(B ⊙ B∗ )| < det(A ⊙ C).

Source: [2985] and [2991, p. 244]. Related: Fact 10.12.9, Fact 10.12.10, and Fact 10.12.11. Fact 10.25.61. Let A, B ∈ Fn×n and α, β ∈ (0, ∞).

−(β−1/2I + αA∗A)⊙ (α−1/2I + βBB∗ ) ≤ (A + B) ⊙ (A + B)∗ ≤ (β−1/2I + αA∗A)⊙ (α−1/2I + βBB∗ ). Related: Fact 10.12.51. Fact 10.25.62. Let A, B ∈ Fn×m, and define

 ∗  A A ⊙ I A =  A⊙B △

 (A ⊙ B)∗   . BB∗ ⊙ I

Then, A is positive semidefinite. Now, assume that n = m. Then, −A∗A ⊙ I − BB∗ ⊙ I ≤ A ⊙ B + (A ⊙ B)∗ ≤ A∗A ⊙ I + BB∗ ⊙ I, −A∗A ⊙ BB∗ ⊙ I ≤ A ⊙ A∗ ⊙ B ⊙ B∗ ≤ A∗A ⊙ BB∗ ⊙ I. Related: Fact 10.25.60. Fact 10.25.63. Let A, B ∈ Fn×n, and assume that A and B are positive semidefinite. Then,

A ⊙ B ≤ 12 (A2 + B2 ) ⊙ I. Source: Fact 10.25.62. Fact 10.25.64. Let A, B ∈ Fn×m, assume that A is positive definite, and define

] A ⊙ (B∗A−1B) B∗ ⊙ B . B∗ ⊙ B A ⊙ (B∗A−1B) [ −1 ] Then, A is positive semidefinite. In particular, A⊙AI A⊙AI −1 is positive semidefinite. Furthermore, √ tr (B∗ ⊙ B)2 ≤ tr[A ⊙ (B∗A−1B)], det(B∗ ⊙ B) ≤ det[A ⊙ (B∗A−1B)]. △

[

A=

Finally, if I ⊙ B∗B is positive definite, then (B∗ ⊙ B)(I ⊙ B∗B)−1 (B∗ ⊙ B) ≤ I ⊙ B∗B. Source: [2991, p. 244]. Fact 10.25.65. Let A, B ∈ Fn×n, assume that A and B are positive definite, and let p, q ∈ (0, ∞)

satisfy p ≤ q. Then,

I ⊙ (log A + log B) = lim log (Ap ⊙ B p )1/p ≤ log (Ap ⊙ B p )1/p ≤ log (Aq ⊙ Bq )1/q . p↓0

In particular,

I ⊙ (log A + log B) ≤ log(A ⊙ B).

830

CHAPTER 10

Source: [88, 2803] and [2977, p. 8]. Remark: log (Ap ⊙ B p )1/p =

1 p p log(A

⊙ B p ). Related: Fact

15.15.22. Fact 10.25.66. Let A, B ∈ Fn×n, assume that A and B are positive semidefinite, and let k ≥ 0.

Then,

A(k) + B(k) ≤ (A + B)(k) .

If either α ∈ [0, ∞) or both A is positive definite and α ∈ R, then (Aα )(k) = (A(k) )α . Source: [1868]. Remark: A(k) is the kth compound of A. See Fact 9.5.18. Fact 10.25.67. Let A, B ∈ Fn×n, assume that A and B are positive definite, and let C, D ∈ Fm×n.

Then, In particular,

(C ⊙ D)(A ⊙ B)−1 (C ⊙ D)∗ ≤ (CA−1C ∗ ) ⊙ (DB−1D∗ ). (A ⊙ B)−1 ≤ A−1 ⊙ B−1 ,

(C ⊙ D)(C ⊙ D)∗ ≤ (CC ∗ ) ⊙ (DD∗ ).

Source: Consider the Schur [complement of ] [ the ∗lower ] right block of the Schur product of the ∗

positive-semidefinite matrices CA CAC−1C ∗ and DB DBD−1D∗ . See [1959, 2818], [2977, p. 13], [2983, p. 198], and [2991, p. 240]. Fact 10.25.68. Let A, B ∈ Fn×n, assume that A and B are positive semidefinite, and let p, q ∈ (1, ∞) satisfy 1/p + 1/q = 1. Then, (A ⊙ B) + (C ⊙ D) ≤ (Ap + C p )1/p ⊙ (Bq + Dq )1/q. Source: Use xxiv) of Proposition 10.6.17 with r = 1/p. See [2977, p. 10]. Fact 10.25.69. Let A1 , . . . , Am , B1 , . . . , Bm ∈ Fn×n, assume that A1 ≥ · · · ≥ Am ≥ 0 and B1 ≥

· · · ≥ Bm ≥ 0, and, for all i ∈ {1, . . . , m}, let αi ≥ 0. Then,   m  m   m  m  ∑  ∑  ∑ ∑  αi Ai  ⊙  αi Bi  ≤  αi  αi (Ai ⊙ Bi ). i=1

i=1

i=1

i=1

Source: [1982]. Remark: This is an extension of the Chebyshev inequality given by Fact 2.12.7. Fact 10.25.70. Let A1 , . . . , Am , B1 , . . . , Bm ∈ Fn×n, assume that there exist a, b ∈ (0, ∞) such

that, for all i ∈ {1, . . . , m}, aI ≤ Ai ≤ bI, and, for all i ∈ {1, . . . , m}, assume that Bi is positive semidefinite. Then,  m  m  m  m 2 2 ∑  ∑ ∑ 1/2 1/2  ∑ a + b  1/2 −1 1/2  Bi . Bi  ⊙ B Ai B ≤ B Ai B  ⊙  2ab i=1

i=1

i=1

i=1

Source: [1982]. Remark: This is an extension of the Kantorovich inequality Fact 2.11.134. Fact 10.25.71. Let A, B, C, D ∈ Fn×n, assume that A, B, C, and D are positive definite. Then,

(A#C) ⊙ (B#D) ≤ (A ⊙ B)#(C ⊙ D),

(A#B) ⊙ (A#B) ≤ (A ⊙ B).

Source: [198]. Fact 10.25.72. Let A ∈ Rn×n , assume that A is nonnegative and positive semidefinite, and let

f : [0, ∞) 7→ [0, ∞). Then, the following statements hold: i) If f is concave and f (0) = 0, then λmax [ f (A)] ≤ λmax [ f ⊙ (A)] and tr f (A) ≤ tr f ⊙ (A). ii) If f is convex and f (0) = 0, then λmax [ f ⊙ (A)] ≤ λmax [ f (A)] and tr f ⊙ (A) ≤ tr f (A). iii) If p ∈ [0, 1], then λmax (A p ) ≤ λmax (A⊙p ) and tr A p ≤ tr A⊙p . iv) If p ≥ 1, then λmax (A⊙p ) ≤ λmax (A p ) and tr A⊙p ≤ tr A p . Source: [1378]. Remark: f ⊙ (A) denotes the n × n matrix whose i, j entry is f (A(i, j) ).

POSITIVE-SEMIDEFINITE MATRICES

831

10.26 Notes The ordering A ≤ B is called the L¨owner ordering. Proposition 10.2.5 is given in [30] and [1703] with extensions in [347]. The proof of Proposition 10.2.8 is based on [583, p. 120], as suggested in [2579]. The proof given in [1139, p. 307] is incomplete. Theorem 10.3.5 is due to R. W. Newcomb [2123]. Proposition 10.4.13 is given in [1425, 2082]. Special cases such as Fact 10.14.35 appear in numerous papers. The proofs of Lemma 10.4.4 and Theorem 10.4.5 are based on [2539]. Theorem 10.4.9 can also be obtained as a corollary of the Fischer minimax theorem given in [1448, 1969, 1971], which provides a geometric characterization of the eigenvalues of a symmetric matrix. Three proofs of Theorem 10.4.5 are given in [2991, pp. 269–271]. Theorem 10.3.6 appears in [2299, p. 121]. Theorem 10.6.2 is given in [85]. Additional inequalities appear in [2051]. Functions that are nondecreasing on Pn are characterized by the theory of monotone matrix functions [449, 922]. The literature on convex maps is extensive. xiv) of Proposition 10.6.17 is given in [1819]. xxiv) is the Lieb concavity theorem. See [449, p. 271] and [1817]. xxxiv) is due to T. Ando. xlv) and xlvi) are due to K. Fan. Some extensions to strict convexity are considered in [1969, 1971]. See also [88, 2099]. Products of positive-definite matrices are studied in [263, 264, 265, 267, 2918]. Essays on the legacy of Issai Schur appear in [1558]. Schur complements are discussed in [637, 639, 1355, 1800, 1869, 2179]. Majorization and eigenvalue inequalities for sums and products of matrices are discussed in [450].

Chapter Eleven Norms Norms are used to quantify vectors and matrices. This chapter introduces vector and matrix norms and their properties.

11.1 Vector Norms Definition 11.1.1. A norm ∥ · ∥ on Fn is a function ∥ · ∥: Fn 7→ [0, ∞) that satisfies the following

statements: i) ∥x∥ = 0 if and only if x = 0. ii) For all α ∈ F and x ∈ Fn , ∥αx∥ = |α|∥x∥. iii) For all x, y ∈ Fn , ∥x + y∥ ≤ ∥x∥ + ∥y∥. iii) is the triangle inequality. The norm ∥ · ∥ on Fn is monotone if, for all x, y ∈ Fn, |x| ≤≤ |y| implies that ∥x∥ ≤ ∥y∥, while ∥ · ∥ is absolute if, for all x ∈ Fn, ∥|x|∥ = ∥x∥. Proposition 11.1.2. Let ∥ · ∥ be a norm on Fn. Then, ∥ · ∥ is monotone if and only if ∥ · ∥ is absolute. △ Proof. First, suppose that ∥ · ∥ is monotone. Let x ∈ Fn, and define y = |x|. Then, |y| = |x|, and thus |y| ≤≤ |x| and |x| ≤≤ |y|. Hence, ∥x∥ ≤ ∥y∥ and ∥y∥ ≤ ∥x∥, which implies that ∥x∥ = ∥y∥. Thus, ∥|x|∥ = ∥y∥ = ∥x∥, which proves that ∥ · ∥ is absolute. Conversely, suppose that ∥ · ∥ is absolute and, for convenience, let n = 2. Now, let x, y ∈ F2 satisfy |x| ≤≤ |y|. Then, there exist α1 , α2 ∈ [0, 1] and θ1 , θ2 ∈ R such that x(i) = αi e ȷθi y(i) for i = 1, 2. Since ∥ · ∥ is absolute, it follows that [ ] [ ] α1 eθ1 ȷ y(1) α1 |y(1) | ∥x∥ = = α2 eθ2 ȷ y(2) α2 |y(2) | [ ] [ ] [ ] 1 −|y(1) | |y(1) | |y(1) | 1 + 2 (1 − α1 ) + α1 = 2 (1 − α1 ) α2 |y(2) | α2 |y(2) | α2 |y(2) | ] [ |y(1) | ] [ |y(1) | ] [ = ≤ 21 (1 − α1 ) + 12 (1 − α1 ) + α1 α2 |y(2) | α2 |y(2) | [ ] [ ] [ ] [ ] 1 |y(1) | |y(1) | |y(1) | |y(1) | 1 ≤ = ||y|| .  = 2 (1 − α2 ) + 2 (1 − α2 ) + α2 −|y | |y | |y | |y | (2)

(2)

(2)

(2)

For x ∈ Fn , a useful class of norms consists of the H¨older norms defined by  n 1/p  ∑    p    , 1 ≤ p < ∞,  |x |   (i)   △ ∥x∥ p =  i=1      p = ∞.  max |x(i) |,

(11.1.1)

i∈{1,...,n}

Note that, for all x ∈ C and p ∈ [1, ∞], ∥x∥ p = ∥x∥ p . These norms depend on Minkowski’s inequality given by the following result. n

834

CHAPTER 11

Lemma 11.1.3. Let p ∈ [1, ∞] and x, y ∈ Fn. Then,

∥x + y∥ p ≤ ∥x∥ p + ∥y∥ p .

(11.1.2)

If p = 1, then equality holds if and only if, for all i ∈ {1, . . . , n}, there exists αi ≥ 0 such that either x(i) = αi y(i) or y(i) = αi x(i) . If p ∈ (1, ∞), then equality holds if and only if there exists α ≥ 0 such that either x = αy or y = αx. Proof. See [340, 1952] and Fact 2.12.51.  n Proposition 11.1.4. Let p ∈ [1, ∞]. Then, ∥ · ∥ p is a norm on F . For p = 1, n ∑ ∥x∥1 = |x(i) | (11.1.3) i=1

is the absolute sum norm; for p = 2,

1/2  n  ∑ √ 2 ∥x∥2 =  |x(i) |  = x∗x

(11.1.4)

i=1

is the Euclidean norm defined in (3.3.17); and, for p = ∞, ∥x∥∞ = max |x(i) |

(11.1.5)

i∈{1,...,n}

is the infinity norm. Note that, for all x ∈ Fn , lim ∥x∥ p = ∥x∥∞ .

(11.1.6)

p→∞

The H¨older norms satisfy the following monotonicity property, which is related to the powersum inequality given by Fact 2.11.90. Proposition 11.1.5. Let 1 ≤ p ≤ q ≤ ∞ and x ∈ Fn. Then, ∥x∥∞ ≤ ∥x∥q ≤ ∥x∥ p ≤ ∥x∥1 .

(11.1.7)

Assume, in addition, that 1 < p < q < ∞. Then, x has at least two nonzero components if and only if ∥x∥∞ < ∥x∥q < ∥x∥ p < ∥x∥1 . (11.1.8) Proof. If either p = q or x = 0 or x has exactly one nonzero component, then ∥x∥q = ∥x∥ p . Hence, to prove both (11.1.7) and (11.1.8), it suffices to prove (11.1.8) in the case where 1 < p < q < ∞ and x has at least two nonzero components. Thus, let n ≥ 2, let x ∈ Fn have at least two △ nonzero components, and define f : [1, ∞) → [0, ∞) by f (β) = ∥x∥β . Hence,

f ′(β) = β1 ∥x∥1−β β

n ∑

γi ,

i=1

where, for all i ∈ {1, . . . , n},

   |x |β (log |x(i) | − log ∥x∥β ), △  i γi =    0,

x(i) , 0, x(i) = 0.

If x(i) , 0, then log |x(i) | < log ∥x∥β . It thus follows that f ′(β) < 0, which implies that f is decreasing on [1, ∞). Hence, (11.1.8) holds.  The following result is H¨older’s inequality.

835

NORMS

Proposition 11.1.6. Let p, q ∈ [1, ∞] satisfy 1/p + 1/q = 1, and let x, y ∈ Fn. Then,

|x∗y| ≤ ∥x∥ p ∥y∥q .

(11.1.9)

Furthermore, equality holds if and only if |x y| = |x| |y| and   |x| ⊙ |y| = ∥y∥∞ |x|, p = 1,       1/p ⊙1/q 1/q ⊙1/p ∥y∥q |x| = ∥x∥ p |y| , 1 < p < ∞,       |x| ⊙ |y| = ∥x∥ |y|, p = ∞. T

T

(11.1.10)

∞

Proof. See [603, p. 127], [1448, p. 536], [1597, p. 71], Fact 2.12.23, and Fact 2.12.24.



The case p = q = 2 is the Cauchy-Schwarz inequality. Corollary 11.1.7. Let x, y ∈ Fn. Then, |x∗y| ≤ ∥x∥2 ∥y∥2 .

(11.1.11)

Furthermore, equality holds if and only if x and y are linearly dependent. △ Proof. Define θ = arg x∗ y ∈ (−π, π] satisfy x∗ y = eθ ȷ |x∗ y|. Then,

0 ≤

∥x∥ y − ∥y∥ eθ ȷ x

= ∥x∥ ∥y∥ (∥x∥ ∥y∥ − |x∗ y|). 2

2

2

2

2

2



2

Let x, y ∈ Fn, assume that x and y are both nonzero, let p, q ∈ [1, ∞], and assume that 1/p+ 1/q = 1. Since ∥x∥ p = ∥x∥ p , it follows that |xTy| ≤ ∥x∥ p ∥y∥q , (11.1.12) and, in particular,

|xTy| ≤ ∥x∥2 ∥y∥2 .

(11.1.13)

The angle θ ∈ [0, π] between x and y, which is defined by (3.3.20), is thus given by θ = acos

Re x∗ y . ∥x∥2 ∥y∥2

(11.1.14)

The norms ∥ · ∥ and ∥ · ∥′ on Fn are equivalent if, for all x ∈ Fn, there exist α, β > 0 such that α∥x∥ ≤ ∥x∥′ ≤ β∥x∥.

(11.1.15)

Note that these inequalities can be written as ′ 1 β ∥x∥

≤ ∥x∥ ≤ α1 ∥x∥′ .

(11.1.16)

Hence, the word “equivalent” is justified. The following result shows that every pair of norms on Fn is equivalent. Theorem 11.1.8. Let ∥ · ∥ and ∥ · ∥′ be norms on Fn. Then, ∥ · ∥ and ∥ · ∥′ are equivalent. Proof. See [1448, p. 272]. 

11.2 Matrix Norms Definition 11.2.1. A norm ∥ · ∥ on Fn×m is a function ∥ · ∥: Fn×m 7→ [0, ∞) that satisfies the

following statements: i) ∥A∥ = 0 if and only if A = 0. ii) For all α ∈ F and A ∈ Fn×m , ∥αA∥ = |α|∥A∥. iii) For all A, B ∈ Fn×m , ∥A + B∥ ≤ ∥A∥ + ∥B∥. If ∥·∥ is a norm on Fn×m, then ∥·∥′ defined by ∥x∥′ = ∥ vec−1 x∥, where x ∈ Fnm and vec−1 x ∈ Fn×m, is a norm on Fnm. Consequently, each matrix norm defines a vector norm. Conversely, if ∥ · ∥ is a

836

CHAPTER 11 △

norm on Fnm, then ∥ · ∥′ defined by ∥A∥′ = ∥ vec A∥, where A ∈ Fn×m and vec A ∈ Fnm, is a norm on Fn×m. H¨older norms are defined for matrices by choosing the vector norm ∥ · ∥ = ∥ · ∥ p . Let A ∈ Fn×m. Then, the H¨older matrix norm of A is defined by  1/p m ∑ n   ∑    p  , 1 ≤ p < ∞,    |A |  (i, j)    △  j=1 i=1 (11.2.1) ∥A∥ p =      max |A |, p = ∞.  (i, j)    i∈{1,...,n} j∈{1,...,m}

Note that the same symbol ∥·∥ p is used to denote the H¨older norm for both vectors and matrices. This notation is consistent since, if m = 1, then ∥A∥ p coincides with the vector H¨older norm. Furthermore, if 1 ≤ p ≤ ∞, then ∥A∥ p = ∥ vec A∥ p . (11.2.2) It follows from (11.1.7) that, if 1 ≤ p ≤ q ≤ ∞, then ∥A∥∞ ≤ ∥A∥q ≤ ∥A∥ p ≤ ∥A∥1 .

(11.2.3)

Finally, if 1 < p < q < ∞ and A has at least two nonzero entries, then ∥A∥∞ < ∥A∥q < ∥A∥ p < ∥A∥1 .

(11.2.4)

The H¨older norms with p = 1, 2, ∞ are the most commonly used. Let A ∈ Fn×m. For p = 2 we define the Frobenius norm ∥ · ∥F by △

∥A∥F = ∥A∥2 .

(11.2.5)

∥A∥F = ∥A∥2 = ∥ vec A∥2 = ∥ vec A∥F .

(11.2.6)

Since ∥A∥2 = ∥ vec A∥2 , it follows that

It is easy to see that

∥A∥F =

√ tr A∗A.

(11.2.7)

Let A ∈ Fn×m. The mixed H¨older norm ∥A∥ p|q of A is defined by applying ∥ · ∥ p to each column of A and then applying ∥ · ∥q to the resulting vector. Hence,

 

 ∥col1 (A)∥ p 

   △  .. 

. ∥A∥ p|q =

 (11.2.8) .

 

∥colm (A)∥ p

q

Therefore,

∥A∥ p|q

  q/p 1/q m ∑ n  ∑        p     |A(i, j) |   ,       j=1 i=1     n 1/p     ∑    p  max  |A(i, j) |     j∈{1,...,m} =   m ( i=1   )q 1/q  ∑        max |A(i, j) |      i∈{1,...,n}   j=1      max |A(i, j) |,     i∈{1,...,n} j∈{1,...,m}

p, q ∈ (1, ∞), p ∈ (1, ∞), q = ∞, (11.2.9) p = ∞, q ∈ (1, ∞), p = q = ∞.

837

NORMS

Furthermore, for all p ∈ [1, ∞], ∥A∥ p|p = ∥A∥ p .

(11.2.10)

Finally, note that ∥A ∥ p|q applies ∥ · ∥ p to each row of A and then applies ∥ · ∥q to the resulting vector. Let ∥ · ∥ be a norm on Fn×m. Then, ∥ · ∥ is unitarily invariant if, for all A ∈ Fn×m and all unitary matrices S 1 ∈ Fn×n and S 2 ∈ Fm×m, ∥S 1 AS 2 ∥ = ∥A∥. Now, let m = n. Then, ∥ · ∥ is self-adjoint if, for all A ∈ Fn×n, ∥A∥ = ∥A√∗ ∥. ∥ · ∥ is normalized if ∥In ∥ = 1. Note that the Frobenius norm is not normalized since ∥In ∥F = n. Finally, ∥ · ∥ is weakly unitarily invariant if, for all A ∈ Fn×n and all unitary matrices S ∈ Fn×n, ∥SAS ∗ ∥ = ∥A∥. Matrix norms can be defined in terms of singular values. Let σ1 (A) ≥ σ2 (A) ≥ · · · ≥ σmin {n,m} denote the singular values of A ∈ Fn×m. The following result gives a weak majorization condition for the singular values of a sum of matrices. Proposition 11.2.2. Let A, B ∈ Fn×m. Then, for all k ∈ {1, . . . , min {n, m}}, T

k k k ∑ ∑ ∑ [σi (A) − σi (B)] ≤ σi (A + B) ≤ [σi (A) + σi (B)]. i=1

In particular,

i=1

(11.2.11)

i=1

σmax (A) − σmax (B) ≤ σmax (A + B) ≤ σmax (A) + σmax (B),

(11.2.12)

tr ⟨A⟩ − tr ⟨B⟩ ≤ tr ⟨A + B⟩ ≤ tr ⟨A⟩ + tr ⟨B⟩. (11.2.13) [ ] [ ] △ △ Proof. Define A, B ∈ Hn+m by A = A0∗ A0 and B = B0∗ B0 . Then, Corollary 10.6.19 implies that, for all k ∈ {1, . . . , n + m}, k k ∑ ∑ λi (A + B) ≤ [λi (A) + λi (B)]. i=1

i=1

Now, consider k ≤ min {n, m}. Then, it follows from Proposition 7.6.5 that, for all i ∈ {1, . . . , k}, λi (A) = σi (A). Setting k = 1 yields (11.2.12), while setting k = min {n, m} and using Fact 10.21.7 yields (11.2.13).  n×m Let p ∈ [1, ∞] and A ∈ F , and define △

∥A∥σp = ∥σ(A)∥ p . In other words, ∥A∥σp Note that, for all p ∈ [1, ∞),

(11.2.14)

 1/p  {n,m}   min∑   p   , 1 ≤ p < ∞,    σ (A)    i △   = i=1       σmax (A), p = ∞. ∥A∥σp = (tr ⟨A⟩ p )1/p .

(11.2.15)

(11.2.16)

Proposition 11.2.3. Let p ∈ [1, ∞]. Then, ∥ · ∥σp is a norm on F . Proof. Let A, B ∈ Fn×m. Then, Proposition 11.2.2 and Minkowski’s inequality given by Fact n×m

2.12.51 imply that ∥A + B∥σp

min {n,m} 1/p min {n,m} 1/p  ∑   ∑  p p    =  σi (A + B) ≤  [σi (A) + σi (B)]  i=1

i=1

min {n,m} 1/p min {n,m} 1/p  ∑   ∑  p p    ≤  σi (A) +  σi (B) = ∥A∥σp + ∥B∥σp . i=1

i=1



838

CHAPTER 11

For all p ∈ [1, ∞), ∥ · ∥σp is a Schatten norm. Let A ∈ Fn×m. Special cases are ∥A∥σ1 = ∥σ(A)∥1 = σ1 (A) + · · · + σmin {n,m}(A) = tr ⟨A⟩, ∥A∥σ2 = ∥σ(A)∥2 =

[σ12 (A)

+ ··· +

2 1/2 σmin {n,m} (A)]

∗

= (tr A A)

1/2

= ∥A∥F ,

∥A∥σ∞ = ∥σ(A)∥∞ = σ1 (A) = σmax (A),

(11.2.17) (11.2.18) (11.2.19)

which are the trace norm, Frobenius norm, and spectral norm, respectively. The following result shows that the trace norm bounds the trace. Proposition 11.2.4. Let A ∈ Fn×n. Then, | Re tr A| ≤ | tr A| ≤

n ∑

ρi (A) ≤ tr ⟨A⟩ =

i=1

n ∑

σi (A) = ∥A∥σ1 .

(11.2.20)

i=1



Proof. See Fact 10.13.2.

By applying Proposition 11.1.5 to the vector σ(A), we obtain the following result. Proposition 11.2.5. Let p, q ∈ [1, ∞), where p ≤ q and A ∈ Fn×m. Then, ∥A∥σ∞ ≤ ∥A∥σq ≤ ∥A∥σp ≤ ∥A∥σ1 .

(11.2.21)

Assume, in addition, that 1 < p < q < ∞ and rank A ≥ 2. Then, ∥A∥σ∞ < ∥A∥σq < ∥A∥σp < ∥A∥σ1 .

(11.2.22)

Let x ∈ Fn = Fn×1. Then, σmax (x) = (x∗x)1/2 = ∥x∥2 , and, since rank x ≤ 1, it follows that, for all p ∈ [1, ∞], ∥x∥σp = ∥x∥2 .

(11.2.23)

Proposition 11.2.6. Let ∥ · ∥ be a norm on Fn×n, and let A ∈ Fn×n. Then,

ρmax (A) = lim ∥Ak ∥1/k. k→∞

Proof. See [1448, pp. 299, 322] and [1451, p. 349].

(11.2.24) 

11.3 Compatible Norms The norms (∥ · ∥, ∥ · ∥′ , ∥ · ∥′′ ) on Fn×l, Fn×m, and Fm×l, respectively, are compatible if, for all A ∈ Fn×m and B ∈ Fm×l, ∥AB∥ ≤ ∥A∥′ ∥B∥′′. (11.3.1) For l = 1, the norms (∥ · ∥, ∥ · ∥′, ∥ · ∥′′ ) on Fn, Fn×m, and Fm, respectively, are compatible if, for all A ∈ Fn×m and x ∈ Fm, ∥Ax∥ ≤ ∥A∥′ ∥x∥′′. (11.3.2) Furthermore, the norm ∥ · ∥ on Fn is compatible with the norm ∥ · ∥′ on Fn×n if, for all A ∈ Fn×n and x ∈ Fn, ∥Ax∥ ≤ ∥A∥′ ∥x∥. (11.3.3) Note that ∥In ∥′ ≥ 1. The norm ∥ · ∥ on Fn×n is submultiplicative if, for all A, B ∈ Fn×n, ∥AB∥ ≤ ∥A∥∥B∥.

(11.3.4)

Hence, the norm ∥ · ∥ on Fn×n is submultiplicative if and only if (∥ · ∥, ∥ · ∥, ∥ · ∥) are compatible. In this case, ∥In ∥ ≥ 1, and ∥ · ∥ is normalized if and only if ∥In ∥ = 1. Proposition 11.3.1. Let ∥ · ∥′ be a submultiplicative norm on Fn×n, and let y ∈ Fn be nonzero. △ Then, ∥x∥ = ∥xy∗ ∥′ is a norm on Fn, and ∥ · ∥ is compatible with ∥ · ∥′ .

839

NORMS

Proof. Note that ∥Ax∥ = ∥Axy∗ ∥′ ≤ ∥A∥′ ∥xy∗ ∥′ = ∥A∥′ ∥x∥. Proposition 11.3.2. Let ∥ · ∥ be a submultiplicative norm on Fn×n, and let A ∈ Fn×n. Then,

ρmax (A) ≤ ∥A∥.



(11.3.5)

Proof. Use Proposition 11.3.1 to construct a norm ∥ · ∥′ on Fn that is compatible with ∥ · ∥.

Furthermore, let A ∈ Fn×n, let λ ∈ spec(A), and let x ∈ Cn be an eigenvector of A associated with λ. Then, Ax = λx implies that |λ|∥x∥′ = ∥Ax∥′ ≤ ∥A∥∥x∥′, and thus |λ| ≤ ∥A∥, which implies (11.3.5). Alternatively, under the additional assumption that ∥ · ∥ is submultiplicative, it follows from Proposition 11.2.6 that ρmax (A) = lim ∥Ak ∥1/k ≤ lim ∥A∥k/k = ∥A∥. k→∞



k→∞

Proposition 11.3.3. Let A ∈ Fn×n and ε > 0. Then, there exists a normalized submultiplicative

norm ∥ · ∥ on Fn×n such that

ρmax (A) ≤ ∥A∥ ≤ ρmax (A) + ε.

(11.3.6)

Proof. See [1448, p. 297], [1451, pp. 347, 348], [2403, p. 167], and [2979, pp. 13, 14].



It is shown in [2979, pp. 13, 14] that ∥ · ∥ can be chosen to be an induced norm. Corollary 11.3.4. Let A ∈ Fn×n, and assume that ρmax (A) < 1. Then, there exists a submultiplicative norm ∥ · ∥ on Fn×n such that ∥A∥ < 1. We now identify several compatible norms. We begin with the H¨older norms. Proposition 11.3.5. Let A ∈ Fn×m and B ∈ Fm×l. If p ∈ [1, 2], then ∥AB∥ p ≤ ∥A∥ p ∥B∥ p .

(11.3.7)

If p ∈ [2, ∞] and q satisfies 1/p + 1/q = 1, then ∥AB∥ p ≤ ∥A∥ p ∥B∥q ,

(11.3.8)

∥AB∥ p ≤ ∥A∥q ∥B∥ p .

(11.3.9)

△

Proof. First, let 1 ≤ p ≤ 2 so that q = p/(p − 1) ≥ 2. Using H¨older’s inequality (11.1.9) along

with (11.1.7) and the fact that p ≤ q yields 1/p 1/p  n,l  n,l   ∑   ∑ ∗ ∗ p p    |coli (A ) col j (B)|  |rowi (A)col j (B)|  =  ∥AB∥ p =  i, j=1

i, j=1

1/p 1/p  n  n,l 1/p  l    ∑  ∑ ∑ p p p p ∗ ∗ ∗ ∗ ∥coli (A ) ∥ p ∥col j (B)∥q  =  ∥coli (A ) ∥ p   ∥col j (B)∥q  ≤  j=1

i=1

i, j=1

1/p  n 1/p  l  ∑  ∑ p ∗ ∗ p    ≤  ∥coli (A ) ∥ p   ∥col j (B)∥ p  = ∥A∗ ∥ p ∥B∥ p = ∥A∥ p ∥B∥ p . i=1

j=1

△

Next, let 2 ≤ p ≤ ∞ so that q = p/(p − 1) ≤ 2. Using H¨older’s inequality (11.1.9) along with (11.1.7) and the fact that q ≤ p yields 1/p  n 1/q  n 1/p  l 1/p  l   ∑  ∑ ∑  ∑ q p p p ∗ ∗ ∗ ∗    ∥AB∥ p ≤  ∥coli (A ) ∥ p   ∥col j (B)∥q  ≤  ∥coli (A ) ∥ p   ∥col j (B)∥q  i=1

j=1

i=1

j=1

∗

= ∥A ∥ p ∥B∥q = ∥A∥ p ∥B∥q . Similarly, it can be shown that (11.3.9) holds.



840

CHAPTER 11 △

Proposition 11.3.6. Let A ∈ Fn×m, B ∈ Fm×l, and p, q ∈ [1, ∞], define r = pq/(p + q), and

assume that r ≥ 1. Then,

∥AB∥σr ≤ ∥A∥σp ∥B∥σq . Furthermore, if r = 1, then n n ∑ ∑ | Re tr AB| ≤ | tr AB| ≤ ρi (AB) ≤ tr ⟨AB⟩ = σi (AB) = ∥AB∥σ1 ≤ ∥A∥σp ∥B∥σq . i=1

(11.3.10)

(11.3.11)

i=1

Proof. Proposition 11.6.2 and H¨older’s inequality with 1/(p/r) + 1/(q/r) = 1 imply

∥AB∥σr

1/r 1/r min {n,m,l} min {n,m,l}   ∑   ∑ σri (A)σri (B) σri (AB) ≤  =  i=1

i=1

 r/q 1/r r/p min {n,m,l} {n,m,l} min∑   ∑ q    p     σi (B)  = ∥A∥σp ∥B∥σq . σi (A)  ≤  i=1

i=1

Finally, the inequalities involving the trace are given in Proposition 11.2.4. Corollary 11.3.7. Let A ∈ Fn×m and B ∈ Fm×l. Then,    ∥A∥σ∞ ∥B∥σ2            ∥A∥ ∥B∥ ≤ ∥A∥σ2 ∥B∥σ2 . ∥AB∥σ∞ ≤ ∥AB∥σ2 ≤   σ2 σ∞           ∥AB∥ σ1 Equivalently,

   σmax (A)∥B∥F            ∥A∥ σ (B) ≤ ∥A∥F ∥B∥F . σmax (AB) ≤ ∥AB∥F ≤   F max          tr ⟨AB⟩ 

Furthermore, for all r ∈ [1, ∞], ∥AB∥σ2r ≤ ∥AB∥σr

   ∥A∥σr σmax (B)            σ (A)∥B∥ ≤ ∥A∥σr ∥B∥σr . ≤  max σr          ∥A∥ ∥B∥  σ2r σ2r



(11.3.12)

(11.3.13)

(11.3.14)

In particular, setting r = ∞ yields σmax (AB) ≤ σmax (A)σmax (B). Corollary 11.3.8. Let A ∈ F

and B ∈ F . Then,     σmax (A)∥B∥σ1 ∥AB∥σ1 ≤    ∥A∥σ1 σmax (B).

n×m

(11.3.15)

m×l

(11.3.16)

Note that the inequality ∥AB∥F ≤ ∥A∥F ∥B∥F in (11.3.13) is equivalent to (11.3.7) with p = 2 as well as (11.3.8) and (11.3.9) with p = q = 2. The following result is a matrix version of the Cauchy-Schwarz inequality given by Corollary 11.1.7. Corollary 11.3.9. Let A ∈ Fn×m and B ∈ Fn×m. Then, | tr A∗B| ≤ ∥A∥F ∥B∥F . Equality holds if and only if A and B are linearly dependent.

(11.3.17)

841

NORMS

The following result is an extension of the geometric series for scalars. Proposition 11.3.10. Let A ∈ Fn×n, and assume that ρmax (A) < 1. Then, there exists a submul∑ i tiplicative norm ∥ · ∥ on Fn×n such that ∥A∥ < 1. Furthermore, ∞ i=0 A converges absolutely, and (I − A)−1 =

∞ ∑

Ai.

(11.3.18)

i=0

Furthermore, 1 1 ≤ ∥(I − A)−1 ∥ ≤ + ∥I∥ − 1. 1 + ∥A∥ 1 − ∥A∥

(11.3.19)

If, in addition, ∥ · ∥ is normalized, then 1 1 ≤ ∥(I − A)−1 ∥ ≤ . 1 + ∥A∥ 1 − ∥A∥

(11.3.20)

Proof. Corollary 11.3.4 implies that there exists a submultiplicative norm ∥ · ∥ on Fn×n such that

∥A∥ < 1. It thus follows that ∞ ∑

which proves that

∑∞

i=0

∥Ai ∥ ≤ ∥I∥ − 1 +

∞ ∑

i=0

∥A∥i =

i=0

1 + ∥I∥ − 1, 1 − ∥A∥

i

A converges absolutely. Next, to verify (11.3.18), note that

(I − A)

∞ ∑ k=0

Ak =

∞ ∑

Ak −

k=0

∞ ∑

Ak = I +

∞ ∑

k=1

k=1

Ak −

∞ ∑

Ak = I,

k=1

which implies (11.3.18) and thus the right-hand inequality in (11.3.19). Furthermore, 1 ≤ ∥I∥ = ∥(I − A)(I − A)−1 ∥ ≤ ∥I − A∥∥(I − A)−1 ∥ ≤ (1 + ∥A∥)∥(I − A)−1 ∥, 

which yields the left-hand inequality in (11.3.19).

11.4 Induced Norms In this section we consider the case where there exists a nonzero vector x ∈ Fm such that (11.3.3) holds as an equality. This statement characterizes a special class of norms on Fn×n, namely, the induced norms. Definition 11.4.1. Let ∥·∥′′ and ∥·∥ be norms on Fm and Fn, respectively. Then, ∥·∥′ : Fn×m 7→ F defined by ∥Ax∥ \{0} ∥x∥′′

∥A∥′ = max m x∈F

(11.4.1)

is an induced norm on Fn×m. In this case, ∥ · ∥′ is induced by ∥ · ∥′′ and ∥ · ∥. If m = n and ∥ · ∥′′ = ∥ · ∥, then ∥ · ∥′ is induced by ∥ · ∥, and ∥ · ∥′ is an equi-induced norm. The next result confirms that ∥ · ∥′ defined by (11.4.1) is a norm. Theorem 11.4.2. Every induced norm is a norm. Furthermore, every equi-induced norm is normalized. Proof. See [1448, p. 293].  Let A ∈ Fn×m. It can be seen that (11.4.1) is equivalent to ∥A∥′ =

max

x∈{y∈Fm : ∥y∥′′ =1}

∥Ax∥.

(11.4.2)

842

CHAPTER 11

Theorem 12.4.11 implies that the maximum in (11.4.2) exists. Since, for all x , 0, ∥A∥′ = max m

x∈F \{0}

it follows that, for all x ∈ Fm,

∥Ax∥ ∥Ax∥ ≥ , ∥x∥′′ ∥x∥′′

∥Ax∥ ≤ ∥A∥′ ∥x∥′′

(11.4.3) (11.4.4)

so that (∥ · ∥, ∥ · ∥′ , ∥ · ∥′′ ) are compatible. If m = n and ∥ · ∥′′ = ∥ · ∥, then the norm ∥ · ∥ is compatible with the induced norm ∥ · ∥′ . The next result shows that compatible norms can be obtained from induced norms. Proposition 11.4.3. Let ∥·∥, ∥·∥′ , and ∥·∥′′ be norms on Fl, Fm, and Fn, respectively. Furthermore, let ∥ · ∥′′′ be the norm on Fm×l induced by ∥ · ∥ and ∥ · ∥′, let ∥ · ∥′′′′ be the norm on Fn×m induced by ∥ · ∥′ and ∥ · ∥′′, and let ∥ · ∥′′′′′ be the norm on Fn×l induced by ∥ · ∥ and ∥ · ∥′′ . If A ∈ Fn×m and B ∈ Fm×l, then ∥AB∥′′′′′ ≤ ∥A∥′′′′ ∥B∥′′′. (11.4.5) Proof. Note that, for all x ∈ Fl , ∥Bx∥′ ≤ ∥B∥′′′ ∥x∥, and, for all y ∈ Fm , ∥Ay∥′′ ≤ ∥A∥′′′′∥y∥′ .

Hence, for all x ∈ Fl, it follows that

∥ABx∥′′ ≤ ∥A∥′′′′∥Bx∥′ ≤ ∥A∥′′′′∥B∥′′′ ∥x∥, which implies that

∥ABx∥′′ ≤ ∥A∥′′′′ ∥B∥′′′ . ∥x∥ x∈Fl \{0}

∥AB∥′′′′′ = max



Corollary 11.4.4. Every equi-induced norm is submultiplicative.

The following result is a consequence of Corollary 11.4.4 and Proposition 11.3.2. Corollary 11.4.5. Let ∥ · ∥ be an equi-induced norm on Fn×n, and let A ∈ Fn×n. Then, ρmax (A) ≤ ∥A∥.

(11.4.6)

By assigning ∥ · ∥ p to Fm and ∥ · ∥q to Fn, where p ≥ 1 and q ≥ 1, the H¨older-induced norm on F is defined by ∥Ax∥q △ . (11.4.7) ∥A∥q,p = max x∈Fm \{0} ∥x∥ p n×m

Proposition 11.4.6. Let p, q, p′, q′ ∈ [1, ∞], where p ≤ p′ and q ≤ q′, and let A ∈ Fn×m. Then,

∥A∥q′,p ≤ ∥A∥q,p ≤ ∥A∥q,p′ .

(11.4.8)

 A subtlety of induced norms is that an induced norm may depend on the underlying field. In particular, the induced norm of a real matrix A computed over R may be smaller than the induced norm of A computed over C. Although the chosen field is usually not made explicit, we do so in special cases for clarity. In particular, for all A ∈ Rn×m and p, q ∈ [1, ∞], ∥A∥ p,q,F denotes the H¨older-induced norm of the real matrix A evaluated over F. Proposition 11.4.7. Let A ∈ Rn×m and p, q ∈ [1, ∞]. Then, Proof. Use Proposition 11.1.5.

∥A∥q,p,R ≤ ∥A∥q,p,C ≤ c p ∥A∥q,p,R , where

√   2, 1 ≤ p ≤ 2,   ∥x∥ p + ∥y∥ p   1−1/p △ = c p = max 2 , 2 ≤ p < ∞,   x,y∈Rn \{0} ∥x + ȷy∥ p   2, p = ∞.

(11.4.9)

(11.4.10)

843

NORMS

If either A is nonnegative or p ≤ q, then ∥A∥q,p,R = ∥A∥q,p,C .

(11.4.11)

 , p = ∞, and q = 1. Then, for all nonzero x = [x1 x2 ]T ∈ C2 ,

Proof. See [864, pp. 347, 348], [1399, p. 716], and [1439, 1933, 2040, 2358]. Example 11.4.8. Let A =

[

1 −1 1 1

]

∥Ax∥1 |x1 − x2 | + |x1 + x2 | = . ∥x∥∞ max {|x1 |, |x2 |} 2 Restricting x ∈ R√ implies that ∥A∥1,∞,R = 2. On the other hand, letting x = [1 ȷ]T yields ∥A∥1,∞,C ≥ ∥Ax∥1 /∥x∥∞ = 2 2. Hence, ∥A∥1,∞,R < ∥A∥1,∞,C , and thus the √ first inequality in (11.4.9) is strict. Furthermore,√the second inequality in (11.4.9) implies that 2 2 ≤ ∥A∥1,∞,C ≤ 2∥A∥1,∞,R = 4. In fact, ^ ∥A∥1,∞,C = 2 2. See [864, p. 176], [1399, p. 716], and [1439]. The following result gives explicit expressions for several H¨older-induced norms. Proposition 11.4.9. Let A ∈ Fn×m. Then,

∥A∥2,2 = σmax (A).

(11.4.12)

∥A∥ p,1 = ∥A∥ p|∞ = max ∥col j (A)∥ p .

(11.4.13)

If p ∈ [1, ∞], then j∈{1,...,m}

Finally, if p, q ∈ [1, ∞] satisfy 1/p + 1/q = 1, then ∥A∥∞,p = ∥AT ∥q|∞ = max ∥rowi (A)∥q . i∈{1,...,n}

(11.4.14)

Proof. Since A∗A is Hermitian, Corollary 10.4.2 implies that, for all x ∈ Fm,

x∗A∗Ax ≤ λmax (A∗A)x∗x, which implies that, for all x ∈ Fm, ∥Ax∥2 ≤ σmax (A)∥x∥2 , and thus ∥A∥2,2 ≤ σmax (A). Now, let x ∈ Fn×n be an eigenvector associated with λmax (A∗A) so that ∥Ax∥2 = σmax (A)∥x∥2 , which implies that σmax (A) ≤ ∥A∥2,2 . Hence, (11.4.12) holds. Next, note that, for all x ∈ Fm,



m m

∑

∑ ∥Ax∥ p =

x( j) col j (A)

≤ |x( j) |∥col j (A)∥ p ≤ max ∥col j (A)∥ p ∥x∥1 ,



j∈{1,...,m} j=1 j=1 p and hence ∥A∥ p,1 ≤ max j∈{1,...,m} ∥col j (A)∥ p . Next, let k ∈ {1, . . . , m} be such that ∥colk (A)∥ p = max j∈{1,...,m} ∥col j (A)∥ p . Now, since ∥ek ∥1 = 1, it follows that ∥Aek ∥ p = ∥colk (A)∥ p ∥ek ∥1, which implies that max ∥col j (A)∥ p = ∥colk (A)∥ p ≤ ∥A∥ p,1 , j∈{1,...,m}

and hence (11.4.13) holds. Next, for all x ∈ Fm, it follows from H¨older’s inequality (11.1.9) that ∥Ax∥∞ = max |rowi (A)x| ≤ max ∥rowi (A)∥q ∥x∥ p , i∈{1,...,n}

i∈{1,...,n}

which implies that ∥A∥∞,p ≤ maxi∈{1,...,n} ∥rowi (A)∥q . Next, let k ∈ {1, . . . , n} be such that ∥rowk (A)∥q = maxi∈{1,...,n} ∥rowi (A)∥q , and let nonzero x ∈ Fm satisfy |rowk (A)x| = ∥rowk (A)∥q ∥x∥ p . Hence, ∥Ax∥∞ = max |rowi (A)x| ≥ |rowk (A)x| = ∥rowk (A)∥q ∥x∥ p , i∈{1,...,n}

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CHAPTER 11

which implies that

max ∥rowi (A)∥q = ∥rowk (A)∥q ≤ ∥A∥∞,p ,

i∈{1,...,n}



and thus (11.4.14) holds. Let A ∈ Fn×m. Then, ∗ ∥A∥2|∞ = max ∥coli (A)∥2 = d1/2 max (A A),

(11.4.15)

∗ ∥AT ∥2|∞ = max ∥rowi (A)∥2 = d1/2 max (AA ).

(11.4.16)

i∈{1,...,m} i∈{1,...,n}

Therefore, Proposition 11.4.9 implies that ∥A∥1,1 = ∥A∥1|∞ = max ∥coli (A)∥1,

(11.4.17)

∗ ∥A∥2,1 = ∥A∥2|∞ = max ∥coli (A)∥2 = d1/2 max (A A),

(11.4.18)

∥A∥∞,1 = ∥A∥∞|∞ = ∥A∥∞ = max |A(i, j) |,

(11.4.19)

∗ ∥A∥∞,2 = ∥AT ∥2|∞ = max ∥rowi (A)∥2 = d1/2 max (AA ),

(11.4.20)

∥A∥∞,∞ = ∥AT ∥1|∞ = max ∥rowi (A)∥1 .

(11.4.21)

i∈{1,...,m}

i∈{1,...,m}

i∈{1,...,n} j∈{1,...,m}

i∈{1,...,n}

i∈{1,...,n}

For convenience, we define the column norm △

∥A∥col = ∥A∥1,1 = ∥A∥1|∞

(11.4.22)

and the row norm △

∥A∥row = ∥A∥∞,∞ = ∥AT ∥1|∞ .

(11.4.23)

∥AT ∥col = ∥A∥row .

(11.4.24)

Note that

The following result follows from Corollary 11.4.5. Corollary 11.4.10. Let A ∈ Fn×n. Then, ρmax (A) ≤ σmax (A), ρmax (A) ≤ ∥A∥col ,

(11.4.25) (11.4.26)

ρmax (A) ≤ ∥A∥row .

(11.4.27)

Proposition 11.4.11. Let p, q ∈ [1, ∞] satisfy 1/p + 1/q = 1, and let A ∈ F

∥A∥q,p ≤ ∥A∥q .

. Then,

n×m

(11.4.28)

Proof. For p = 1 and q = ∞, (11.4.28) follows from (11.4.19). For q < ∞ and x ∈ Fn, it follows

from H¨older’s inequality (11.1.9) that 1/q  n 1/q  n 1/q  n m ∑ ∑ ∑  ∑       q q ∥Ax∥q =  |rowi (A)x|q  ≤  ∥rowi (A)∥q ∥x∥ p  =  |A(i, j) |q  ∥x∥ p = ∥A∥q ∥x∥ p , i=1

i=1

i=1 j=1

which implies (11.4.28). Next, we specialize Proposition 11.4.3 to the H¨older-induced norms.



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NORMS

Corollary 11.4.12. Let p, q, r ∈ [1, ∞], and let A ∈ Fn×m and B ∈ Fm×l. Then,

∥AB∥r,p ≤ ∥A∥r,q ∥B∥q,p .

(11.4.29)

∥AB∥col ≤ ∥A∥col ∥B∥col ,

(11.4.30)

σmax (AB) ≤ σmax (A)σmax (B),

(11.4.31)

∥AB∥row ≤ ∥A∥row ∥B∥row ,

(11.4.32)

∥AB∥∞ ≤ ∥A∥∞ ∥B∥col ,

(11.4.33)

∥AB∥∞ ≤ ∥A∥row ∥B∥∞ ,

(11.4.34)

In particular,

∗ ∗ d1/2 max (B A AB)

≤

∗ d1/2 max (A A)∥B∥col ,

(11.4.35)

∗ ∗ 1/2 ∗ d1/2 max (B A AB) ≤ σmax (A)dmax (B B),

(11.4.36)

∗ ∗ 1/2 ∗ d1/2 max (ABB A ) ≤ dmax (AA )σmax (B),

(11.4.37)

∗ ∗ 1/2 ∗ d1/2 max (ABB A ) ≤ ∥B∥row dmax (BB ).

(11.4.38)

11.5 Induced Lower Bound We now consider a variation of the induced norm. Definition 11.5.1. Let ∥ · ∥ and ∥ · ∥′ denote norms on Fm and Fn, respectively, and let A ∈ Fn×m. Then, ℓ: Fn×m 7→ R defined by  ∥y∥′   min max , A , 0,   m △ y∈R(A)\{0} x∈{z∈F : Az=y} ∥x∥ ℓ(A) =  (11.5.1)    0, A = 0, is the lower bound induced by ∥ · ∥ and ∥ · ∥′ . Equivalently,  ∥Ax∥′   ,   min max △  x∈Fm\N(A) z∈N(A) ∥x+z∥ ℓ(A) =     0,

A , 0, (11.5.2) A = 0.

Proposition 11.5.2. Let ∥ · ∥ and ∥ · ∥′ be norms on Fm and Fn, respectively, let ∥ · ∥′′ be the

norm induced by ∥ · ∥ and ∥ · ∥′ , let ∥ · ∥′′′ be the norm induced by ∥ · ∥′ and ∥ · ∥, and let ℓ be the lower bound induced by ∥ · ∥ and ∥ · ∥′ . Then, the following statements hold: i) For all A ∈ Fn×m, ℓ(A) exists; that is, for all A ∈ Fn×m, the minimum in (11.5.1) is attained. ii) If A ∈ Fn×m, then ℓ(A) = 0 if and only if A = 0. iii) For all A ∈ Fn×m, there exists x ∈ Fm such that iv) For all A ∈ Fn×m,

ℓ(A)∥x∥ = ∥Ax∥′.

(11.5.3)

ℓ(A) ≤ ∥A∥′′.

(11.5.4)

v) If A , 0 and B is a (1)-inverse of A, then 1/∥B∥′′′ ≤ ℓ(A) ≤ ∥B∥′′′.

(11.5.5)

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vi) If A, B ∈ Fn×m and either R(A) ⊆ R(A + B) or N(A) ⊆ N(A + B), then ℓ(A) − ∥B∥′′′ ≤ ℓ(A + B). vii) If A, B ∈ F

n×m

(11.5.6)

and either R(A + B) ⊆ R(A) or N(A + B) ⊆ N(A), then ℓ(A + B) ≤ ℓ(A) + ∥B∥′′′.

viii) If n = m and A ∈ F

n×n

(11.5.7)

is nonsingular, then ℓ(A) = 1/∥A−1∥′′′.

(11.5.8)

Proof. See [1223].  ′ ′′ l m n ′′′ Proposition 11.5.3. Let ∥ · ∥, ∥ · ∥ , and ∥ · ∥ be norms on F , F , and F , respectively, let ∥ · ∥

denote the norm on Fm×l induced by ∥ · ∥ and ∥ · ∥′ , let ∥ · ∥′′′′ denote the norm on Fn×m induced by ∥ · ∥′ and ∥ · ∥′′ , and let ∥ · ∥′′′′′ denote the norm on Fn×l induced by ∥ · ∥ and ∥ · ∥′′ . If A ∈ Fn×m and B ∈ Fm×l, then ℓ(A)ℓ ′(B) ≤ ℓ ′′(AB). (11.5.9) In addition, the following statements hold: i) If either rank B = rank AB or def B = def AB, then ii) If rank A = rank AB, then iii) If rank B = m, then iv) If rank A = m, then

ℓ ′′(AB) ≤ ∥A∥′′ℓ(B).

(11.5.10)

ℓ ′′(AB) ≤ ℓ(A)∥B∥′′′′.

(11.5.11)

∥A∥′′ℓ(B) ≤ ∥AB∥′′′′′.

(11.5.12)

ℓ(A)∥B∥′′′′ ≤ ∥AB∥′′′′′.

(11.5.13)

 By assigning ∥ · ∥ p to Fm and ∥ · ∥q to Fn, where p ≥ 1 and q ≥ 1, the H¨older-induced lower bound on Fn×m is defined by  ∥y∥q   max   min ∥x∥ p , A , 0, m  △ y∈R(A)\{0} x∈{z∈F : Az=y} ℓq,p (A) =  (11.5.14)     0, A = 0. Proof. See [1223].

The following result shows that ℓ2,2 (A) is the smallest positive singular value of A. △

Proposition 11.5.4. Let A ∈ Fn×m, assume that A is nonzero, and let r = rank A. Then,

ℓ2,2 (A) = σr (A).

(11.5.15)

Proof. Use the singular value decomposition. Corollary 11.5.5. Let A ∈ Fn×m. If n ≤ m and A is right invertible, then

ℓ2,2 (A) = σmin (A) = σn (A).

 (11.5.16)

If m ≤ n and A is left invertible, then ℓ2,2 (A) = σmin (A) = σm (A).

(11.5.17)

Finally, if n = m and A is nonsingular, then ℓ2,2 (A−1 ) = σmin (A−1 ) =

1 σmax (A)

.

(11.5.18)

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NORMS

 In contrast to the submultiplicativity condition (11.4.4), which holds for the induced norm, the induced lower bound satisfies a supermultiplicativity condition. The following result is analogous to Proposition 11.4.3. Proposition 11.5.6. Let ∥ · ∥, ∥ · ∥′ , and ∥ · ∥′′ be norms on Fl, Fm, and Fn, respectively. Let ℓ(·) be the lower bound induced by ∥ · ∥ and ∥ · ∥′, let ℓ ′(·) be the lower bound induced by ∥ · ∥′ and ∥ · ∥′′, let ℓ ′′(·) be the lower bound induced by ∥ · ∥ and ∥ · ∥′′ , let A ∈ Fn×m and B ∈ Fm×l, and assume that either A or B is right invertible. Then, Proof. Use Proposition 7.6.2 and Fact 8.3.33.

ℓ ′(A)ℓ(B) ≤ ℓ ′′(AB).

(11.5.19)

ℓr,q (A)ℓq,p (B) ≤ ℓr,p (AB).

(11.5.20)

σm (A)σl (B) ≤ σl (AB).

(11.5.21)

Furthermore, if 1 ≤ p, q, r ≤ ∞, then

In particular,



Proof. See [1223] and [1738, pp. 369, 370].

11.6 Singular Value Inequalities Proposition 11.6.1. Let A ∈ Fn×m and B ∈ Fm×l. Then, for all i ∈ {1, . . . , min {n, m}} and j ∈ {1,

. . . , min {m, l}} such that i + j ≤ min {n, l} + 1,

σi+ j−1 (AB) ≤ σi (A)σj (B).

(11.6.1)

In particular, for all i ∈ {1, . . . , min {n, m, l}}, σi (AB) ≤ σmax (A)σi (B), σi (AB) ≤ σi (A)σmax (B).

(11.6.2) (11.6.3)

Proof. See [1450, p. 178] and [2979, pp. 78, 79].  Proposition 11.6.2. Let A ∈ Fn×m and B ∈ Fm×l. If r ≥ 0, then, for all k ∈ {1, . . . , min {n, m, l}}, k ∑

σri (AB) ≤

i=1

k ∑

σri (A)σri (B).

(11.6.4)

σi (A)σi (B).

(11.6.5)

i=1

In particular, for all k ∈ {1, . . . , min {n, m, l}}, k ∑

σi (AB) ≤

i=1

k ∑ i=1

If r < 0, n = m = l, and A and B are nonsingular, then n ∑

σri (AB) ≤

i=1

n ∑

σri (A)σri (B).

(11.6.6)

i=1

Proof. The first statement follows from Proposition 11.6.3 and Fact 3.25.9. For the case where r < 0, use Fact 3.25.13. See [449, p. 94] and [1450, p. 177].  n×m m×l Proposition 11.6.3. Let A ∈ F and B ∈ F . Then, for all k ∈ {1, . . . , min {n, m, l}}, k ∏ i=1

σi (AB) ≤

k ∏ i=1

σi (A)σi (B).

(11.6.7)

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CHAPTER 11

If, in addition, n = m = l, then n ∏

σi (AB) =

n ∏

i=1

σi (A)σi (B).

(11.6.8)

i=1

Proof. See [1450, p. 172].  n×m m×l Proposition 11.6.4. Let A ∈ F and B ∈ F , and let i ∈ {max {m − n, 0}, . . . , m − 1} and

j ∈ {max {m − l, 0}, . . . , m − 1} satisfy i + j ≤ m − 1. Then,

σm−i (A)σm− j (B) = σm−i− j (AB).

(11.6.9)

Consequently, if m ≤ n, then, for all i ∈ {1, . . . , min {m, l}}, σmin (A)σi (B) = σm (A)σi (B) ≤ σi (AB).

(11.6.10)

Furthermore, if m ≤ l, then, for all i ∈ {1, . . . , min {n, m}}, σi (A)σmin (B) = σi (A)σm (B) ≤ σi (AB).

(11.6.11)

Proof. Use Fact 10.22.23. To prove (11.6.10), note that Corollary 10.4.2 implies that σm2 (A)Im =

λmin (A∗A)Im ≤ A∗A, which implies that σm2 (A)B∗B ≤ B∗A∗AB. Hence, it follows from the monotonicity theorem Theorem 10.4.9 that, for all i ∈ {1, . . . , min {m, l}}, ∗ ∗ σm (A)σi (B) = λi [σm2 (A)B∗B]1/2 ≤ λ1/2 i (B A AB) = σi (AB),

which proves (11.6.10). Similarly, for all i ∈ {1, . . . , min {n, m}}, ∗ ∗ σi (A)σm (B) = λi [σm2 (B)AA∗ ]1/2 ≤ λ1/2 i (ABB A ) = σi (AB).



Specializing Proposition 11.6.1 and Proposition 11.6.4 yields the following four results. Corollary 11.6.5. Let A ∈ Fn×m and B ∈ Fm×l. Then, σm (A)σmin {n,m,l} (B) ≤ σmin {n,m,l} (AB) ≤ σmax (A)σmin {n,m,l} (B), σm (A)σmax (B) ≤ σmax (AB) ≤ σmax (A)σmax (B), σmin {n,m,l} (A)σm(B) ≤ σmin {n,m,l} (AB) ≤ σmin {n,m,l} (A)σmax (B),

(11.6.12) (11.6.13) (11.6.14)

σmax (A)σm (B) ≤ σmax (AB) ≤ σmax (A)σmax (B).

(11.6.15)

Corollary 11.6.6. Let A ∈ F

n×n

and B ∈ F . Then, for all i ∈ {1, . . . , min {n, l}}, n×l

σmin (A)σi (B) ≤ σi (AB) ≤ σmax (A)σi (B).

(11.6.16)

σmin (A)σmax (B) ≤ σmax (AB) ≤ σmax (A)σmax (B).

(11.6.17)

In particular, Corollary 11.6.7. Let A ∈ Fn×m and B ∈ Fm×m. Then, for all i ∈ {1, . . . , min {n, m}},

σi (A)σmin (B) ≤ σi (AB) ≤ σi (A)σmax (B).

(11.6.18)

σmax (A)σmin (B) ≤ σmax (AB) ≤ σmax (A)σmax (B).

(11.6.19)

In particular, Corollary 11.6.8. Let A, B ∈ Fn×n . Then, for all i ∈ {1, . . . , n},

σi (A)σmin (B) ≤ σi (AB) ≤ σi (A)σmax (B).

(11.6.20)

σmin (A)σmin (B) ≤ σmin (AB) ≤ σmin (A)σmax (B),

(11.6.21)

In particular,

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NORMS

σmax (A)σmin (B) ≤ σmax (AB) ≤ σmax (A)σmax (B).

(11.6.22)

Corollary 11.6.9. Let A ∈ Fn×m and B ∈ Fm×l. If m ≤ n, then

If m ≤ l, then

σmin (A)∥B∥F = σm (A)∥B∥F ≤ ∥AB∥F .

(11.6.23)

∥A∥F σmin (B) = ∥A∥F σm (B) ≤ ∥AB∥F .

(11.6.24)

Proposition 11.6.10. Let A, B ∈ Fn×m. Then, for all i, j ∈ {1, . . . , min {n, m}} such that i + j ≤

min {n, m} + 1,

σi+ j−1 (A + B) ≤ σi (A) + σj (B),

(11.6.25)

σi+ j−1(A) − σj (B) ≤ σi (A + B).

(11.6.26) 

Proof. See [1450, p. 178] and [2979, pp. 78, 79]. Corollary 11.6.11. Let A, B ∈ Fn×m. Then, for all i ∈ {1, . . . , min {n, m}},

σi (A) − σmax (B) ≤ σi (A + B) ≤ σi (A) + σmax (B).

(11.6.27)

If, in addition, n = m, then σmin (A) − σmax (B) ≤ σmin (A + B) ≤ σmin (A) + σmax (B).

(11.6.28)

Proof. Use Proposition 11.6.10 with j = 1. Alternatively, Lemma 10.4.3 and the CauchySchwarz inequality given by Corollary 11.1.7 imply that, for all nonzero x ∈ Fn,

λmin [(A + B)(A + B)∗ ] ≤ ≤

2x∗AB∗x x∗(AA∗ + BB∗ + AB∗ + BA∗ )x x∗AA∗x x∗BB∗x = + + Re x∗x ∥x∥22 ∥x∥22 ∥x∥22 x∗AA∗x (x∗AA∗x)1/2 2 σmax (B). + σmax (B) + 2 2 ∥x∥2 ∥x∥2

Minimizing with respect to x and using Lemma 10.4.3 yields 2 ∗ σn2 (A + B) = λmin [(A + B)(A + B)∗ ] ≤ λmin (AA∗ ) + σmax (B) + 2λ1/2 min (AA )σmax (B)

= [σn (A) + σmax (B)]2, which proves the right-hand inequality of (11.6.27). Finally, the left-hand inequality follows from the right-hand inequality with A and B replaced by A + B and −B, respectively. 

11.7 Facts on Vector Norms Fact 11.7.1. Let ∥ · ∥ be a norm on Fn, and define f : Fn 7→ [0, ∞) by f (x) = ∥x∥. Then, f is

convex. Fact 11.7.2. Let ∥ · ∥ and ∥ · ∥′ be norms on Fn, and let α, β > 0. Then, α∥ · ∥ + β∥ · ∥′ is also a norm on Fn. Furthermore, max {∥ · ∥, ∥ · ∥′ } is a norm on Fn. Remark: min {∥ · ∥, ∥ · ∥′ } is not necessarily a

norm. See [109, p. 278]. Fact 11.7.3. Let x, y ∈ Fn, and let ∥ · ∥ be a norm on Fn. Then,  ∥x∥2 + ∥y∥2    ≤ ∥x + y∥2 + ∥x − y∥2 ≤ 2∥x∥2 + 4∥x∥∥y∥ + 2∥y∥2 ≤ 4(∥x∥2 + ∥y∥2 ). 2 2  2∥x∥ − 4∥x∥∥y∥ + 2∥y∥  Source: [1124, pp. 9, 10] and [2112, p. 278]. Fact 11.7.4. Let ∥ · ∥ be a norm on Fn. Then, there exists a unique α ∈ [1, 2] such that, for all

x, y ∈ Fn, at least one of which is nonzero,

2 ∥x + y∥2 + ∥x − y∥2 ≤ ≤ 2α. α ∥x∥2 + ∥y∥2

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CHAPTER 11

Furthermore, α = 1 if and only if ∥ · ∥ = ∥ · ∥2 . Finally, if ∥ · ∥ = ∥ · ∥ p , then    2(2−p)/p , 1 ≤ p ≤ 2, α=  2(p−2)/p , p ≥ 2. Source: [605, p. 258] and [2061, p. 550]. Remark: This is the von Neumann–Jordan inequality. Remark: If p = 2, then α = 1 and this result yields v) of Fact 11.8.3. Related: Fact 11.7.5. Fact 11.7.5. Let ∥ · ∥ be a norm on Fn. Then, the following statements are equivalent:

i) ∥ · ∥ = ∥ · ∥2 . ii) For all x, y ∈ F, ∥x + y∥2 + ∥x − y∥2 = 2∥x∥2 + 2∥y∥2 . iii) For all x, y ∈ F,



1 x − 1 y

≤ 2∥x − y∥ .

∥x∥ ∥y∥ ∥x∥ + ∥y∥ iv) For all x, y, z ∈ F, ∥x + y∥2 + ∥y + z∥2 + ∥z + x∥2 = ∥x∥2 + ∥y∥2 + ∥z∥2 + ∥x + y + z∥2 . Source: [2094, 1620]. Credit: The equivalence of i) and iv) is due to M. Fr´echet. Related: Fact 11.7.4. Fact 11.7.6. Let x, y ∈ Fn, let α ∈ [0, 1], and let ∥ · ∥ be a norm on Fn. Then, ∥x + y∥ ≤ ∥αx + (1 − α)y∥ + ∥(1 − α)x + αy∥ ≤ ∥x∥ + ∥y∥. Fact 11.7.7. Let x, y ∈ Fn, and let ∥ · ∥ be a norm on Fn. Then,

  ∥x∥ + ∥y∥ − ∥x + y∥ − ∥x − y∥      { }   ∥x∥ − ∥y∥ ≤  ≤ ∥x∥ + ∥y∥. ∥x + y∥      ∥x + y∥ + ∥x − y∥ − ∥x∥ − ∥y∥ ≤ min {∥x + y∥, ∥x − y∥} ≤  ∥x − y∥  Source: [1932]. Related: Fact 2.2.44. Fact 11.7.8. Let x, y ∈ Fn, let ∥ · ∥ be a norm on Fn, assume that ∥x∥ , ∥y∥, and let p > 0. Then,

∥x∥ − ∥y∥ ≤





∥x∥ p x − ∥y∥ p y

∥x∥ − ∥y∥ ≤ ∥x − y∥. p+1 p+1 ∥x∥ − ∥y∥

Source: [2061, p. 516]. Fact 11.7.9. Let x, y ∈ Fn, and let ∥ · ∥ be a norm on Fn. Then,

√ ∥x − y∥ + ∥x∥ − ∥y∥ ≤ 2∥x − y∥2 + 2(∥x∥ − ∥y∥)2 ≤ 2∥x − y∥,

If, in addition, x and y are nonzero, then

√

2 2

1 x − 1 y

≤ 2∥x − y∥ + 2(∥x∥ − ∥y∥) .

∥x∥ ∥y∥ max {∥x∥, ∥y∥} Source: [2212]. Fact 11.7.10. Let x, y ∈ Fn be nonzero, and let ∥ · ∥ be a norm on Fn. Then,

(

) 1 1

∥x + y∥ ≤ ∥x∥ + ∥y∥ − min {∥x∥, ∥y∥} 2 −

x+ y ≤ ∥x∥ + ∥y∥, ∥x∥ ∥y∥ (

) 1 1

∥x − y∥ ≤ ∥x∥ + ∥y∥ − min {∥x∥, ∥y∥} 2 −

x+ y ≤ ∥x∥ + ∥y∥, ∥x∥ ∥y∥ (



) 1 1 ∥x∥ + ∥y∥ − max {∥x∥, ∥y∥} 2 −

x+ y ≤ ∥x + y∥ ≤ ∥x∥ + ∥y∥, ∥x∥ ∥y∥

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NORMS

(

) 1 1

∥x∥ + ∥y∥ − max {∥x∥, ∥y∥} 2 −

x+ y

≤ ∥x − y∥ ≤ ∥x∥ + ∥y∥. ∥x∥ ∥y∥ Source: [1931]. Fact 11.7.11. Let x, y ∈ Fn be nonzero, and let ∥ · ∥ be a norm on Fn. Then,

) ( (∥x∥ + ∥y∥) ∥x + y∥ − ∥x∥ − ∥y∥ 4 min {∥x∥, ∥y∥}



1 1

≤ 14 (∥x∥ + ∥y∥)

x+ y

∥x∥ ∥y∥

1 1

1 ≤ 2 max {∥x∥, ∥y∥}

x+ y ∥x∥ ∥y∥ ( ) ≤ 12 ∥x + y∥ + max {∥x∥, ∥y∥} − ∥x∥ − ∥y∥ ) ( ≤ 1 ∥x + y∥ + ∥x∥ − ∥y∥ ≤ ∥x + y∥, 2

  ∥x∥ − ∥y∥  ∥x − y∥ +      

    ∥x − y∥ − ∥x∥ − ∥y∥

1   max {∥x∥, ∥y∥} 1

   ≤

x− y

≤       min {∥x∥, ∥y∥} ∥x∥ ∥y∥   2∥x − y∥   2       ∥x∥2 + ∥y∥2   2∥x − y∥           max {∥x∥, ∥y∥}     4∥x − y∥ ≤ ≤ .      2(∥x − y∥ + ∥x∥ − ∥y∥ ) ∥x∥ + ∥y∥         ∥x∥ + ∥y∥ Source: Fact 11.7.10, [1931, 1932, 2023, 2094], and [2061, p. 516]. Remark: In the second string, the first inequality is the reverse Maligranda inequality, the second inequality is the Maligranda inequality, the second and upper fourth terms are the Massera-Schaffer inequality, and the second and fifth terms are the Dunkl-Williams inequality. Remark: The term involving the Euclidean norm is present only for ∥ · ∥ = ∥ · ∥2 . See Fact 2.21.8. Remark: Extensions to more than two vectors are given in [1588, 2211]. Remark: A matrix version is given by Fact 10.11.87. Related: Fact 10.11.87. Fact 11.7.12. Let x1 , . . . , xk ∈ Fn be nonzero, and let ∥ · ∥ be a norm on Fn. Then,













k k k k k ∑

∑



∑

∑



∑ 1

 1

  

xi  min ∥xi ∥ ≤ ∥xi ∥ ≤

xi  max ∥xi ∥, xi + k −

xi + k −

i=1



i=1 ∥xi ∥

 i∈{1,...,k}

i=1



i=1 ∥xi ∥

 i∈{1,...,k} i=1

  

k k k  1 

∑

∑ 

∑

1  

 ∥x j ∥ − ∥xi ∥  ≤

max  x j − xi

   i∈{1,...,k} ∥xi ∥

i=1 ∥xi ∥

j=1

j=1

   k k

∑  1 

∑    

∥x j ∥ − ∥xi ∥  . x j

+ ≤ min  

i∈{1,...,k} ∥xi ∥ 

j=1

j=1

If p ∈ (0, ∞), then



p

p 

( ) ∑ k k k

∑

∑  k 1

 ∑ p 1−p p−1 xi

 ≤ ∥xi ∥ − k

xi

min ∥xi ∥  ∥xi ∥ −

i∈{1,...,k}

i=1

i=1 ∥xi ∥ i=1 i=1



p  ( ) ∑ k

∑  n 1

 p−1 ≤ max ∥xi ∥  ∥xi ∥ −

xi  . i∈{1,...,k}

i=1 ∥xi ∥

 i=1

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CHAPTER 11

If α1 , . . . , αk ∈ F, then



 





k k k k k

∑  

∑

∑  ∑ 

∑

  



|α j − αi |∥x j ∥ . x j + |α j − αi |∥x j ∥ ≤

αi xi

≤ min |αi | x j − max |αi |





i∈{1,...,k}

i∈{1,...,k}

j=1

j=1

i=1

j=1

j=1

Source: [675, 939, 941, 1588, 2055, 2211]. Fact 11.7.13. Let x, y ∈ Fn be nonzero, let p ∈ R, and let ∥ · ∥ be a norm on Fn. Then, the

following statements hold: i) If p ∈ (0, 1], then



1+1/p ∥x − y∥

1 x − 1 y

≤ 2 .

∥x∥ p ∥y∥ (∥x∥ + ∥y∥ p )1/p

ii) If p ≥ 1, then



4∥x − y∥

1 x − 1 y

≤ .

∥x∥ ∥y∥ (∥x∥ p + ∥y∥ p )1/p

Source: [25, 2094]. Fact 11.7.14. Let x, y ∈ Fn, assume that x and y are nonzero, let ∥ · ∥ be a norm on Fn, and let

p, q ∈ R. Then, the following statements hold: i) If p ≤ 0, then

p−1

(2 − p) max {∥x∥ p , ∥y∥ p }

∥x∥ x − ∥y∥ p−1 y

≤ ∥x − y∥. max {∥x∥, ∥y∥} ii) If p ∈ [0, 1], then iii) If p ∈ [0, 1], then iv) If p ≥ 1, then

p−1

∥x∥ x − ∥y∥ p−1 y

≤

(2 − p)∥x − y∥ . [max {∥x∥, ∥y∥}]1−p

p−1

∥x∥ x − ∥y∥ p−1 y

≤

2∥x − y∥ . + ∥y∥1−p

∥x∥1−p

p−1

∥x∥ x − ∥y∥ p−1 y

≤ p[max {∥x∥, ∥y∥}] p−1 ∥x − y∥.

v) If p ∈ [0, 1] and q > 0, then

p−1

∥x∥ x − ∥y∥ p−1 y

≤

21+1/q ∥x − y∥ . (∥x∥(1−p)q + ∥y∥(1−p)q )1/q

Source: [811, 1931, 2094]. Fact 11.7.15. Let x, y ∈ Fn, let ∥ · ∥ be a norm on Fn, let p and q be real numbers, and assume

that 1 < p ≤ q. Then,

[ 21 (∥x + √1 y∥q + ∥x − √1 y∥q )]1/q ≤ [ 12 (∥x + √1 y∥ p + ∥x − √1 y∥ p )]1/p. q−1

q−1

p−1

p−1

Source: [1146, p. 207]. Remark: This is Bonami’s inequality. See Fact 2.2.32. Fact 11.7.16. Let A ∈ Fn×n, assume that A is nonsingular, and let ∥ · ∥ be a norm on Fn. Then, △

∥x∥′ = ∥Ax∥ is a norm on Fn. △ Fact 11.7.17. Let A ∈ Fn×n, where A is positive definite. Then, ∥x∥′ = (x∗Ax)1/2 is a norm on Fn. Fact 11.7.18. Let x ∈ Rn, and let ∥·∥ be a norm on Rn. Then, xTy > 0 for all y ∈ {z ∈ Rn : ∥z−x∥ < ∥x∥}. Fact 11.7.19. Let ∥ · ∥ be a norm on Fn , and assume that, for all x ∈ Fn and every permutation matrix A ∈ Rn×n , ∥Ax∥ = ∥x∥. Then, f : Fn 7→ R defined by f (x) = ∥x∥ is Schur-convex. Source: [2991, p. 376].

853

NORMS

Fact 11.7.20. Let k and n be positive integers, assume that k ≤ n, let p ≥ 1, and, for all x ∈ Fn ,

1/p  k  ∑ p  ↓  ∥x∥ =  (|x| )(i)  .

define

△

i=1

Then, ∥ · ∥ is a symmetric gauge function on Rn and an absolute and monotone norm on Fn . Source: [449, p. 89] and [2991, pp. 373, 376]. Remark: Setting p = 1 yields the Ky Fan k-norm. Fact 11.7.21. Let x, y ∈ Rn, assume that x and y are nonzero, assume that xTy = 0, and let ∥ · ∥ be a norm on Rn. Then, ∥x∥ ≤ ∥x + y∥. Source: If ∥x + y∥ < ∥x∥, then x + y ∈ B∥x∥(0), and thus y ∈ B∥x∥(−x). By Fact 11.7.18, xTy < 0. Remark: See [479, 1806] for related results on matrices. Fact 11.7.22. Let x, y ∈ Fn, and let ∥ · ∥ be a norm on Fn. Then, the following statements hold: i) If there exists β ≥ 0 such that either x = βy or y = βx, then ∥x + y∥ = ∥x∥ + ∥y∥. ii) If ∥x + y∥ = ∥x∥ + ∥y∥ and x and y are linearly dependent, then there exists β ≥ 0 such that either x = βy[ or ] y = βx. [ ] Remark: Let x = 10 and y = 11 , which are linearly independent. Then, ∥x+y∥∞ = ∥x∥∞ +∥y∥∞ = 2. Problem: If x and y are linearly independent and p ∈ [1, ∞), then does ∥x + y∥ p < ∥x∥ p + ∥y∥ p ?

11.8 Facts on Vector p-Norms Fact 11.8.1. For all x ∈ R2 , define

   ∥x∥2 , ∥x∥ =   ∥x∥∞ , △

x(1) ≤ 0, x(1) ≥ 0.

2 Then, Source: [1233]. Remark: Let √ [ ∥ · ]∥ is a norm [ on ] R that is neither absolute nor monotone. △ −1/2 △ 3/5 x = −1/2 and y = 3/5 so that |x| ≤≤ |y| and ∥y∥ = 53 < 22 = ∥x∥.

Fact 11.8.2. Let x, y ∈ Fn. Then, x and y are linearly dependent if and only if |x|⊙2 and |y|⊙2 are linearly dependent and |x∗y| = |x|T |y|. Remark: This clarifies the relationship between (11.1.10)

with p = 2 and Corollary 11.1.7. Fact 11.8.3. Let x, y ∈ Fn. Then, the following statements hold: i) ∥x∥22 + ∥y∥22 = ∥x + y∥22 − 2Re x∗y = ∥x − y∥22 + 2Re x∗y. √ ii) ∥x + y∥2 = ∥x∥22 + ∥y∥22 + 2Re x∗y. √ iii) ∥x − y∥2 = ∥x∥22 + ∥y∥22 − 2Re x∗y. iv) The following statements are equivalent: a) ∥x − y∥2 = ∥x + y∥2 . b) ∥x + y∥22 = ∥x∥22 + ∥y∥22 . c) ∥x − y∥22 = ∥x∥22 + ∥y∥22 . d) Re x∗y = 0. v) Let α ∈ R, and assume that α , 0. Then,

1 ∥αx + y∥22 + ∥x − y∥22 = (1 + α)∥x∥22 + (1 + α1 )∥y∥22 . α In particular,

∥x + y∥22 + ∥x − y∥22 = 2∥x∥22 + 2∥y∥22 .

vi) Re x∗y = 14 (∥x + y∥22 − ∥x − y∥22 ) = 21 (∥x + y∥22 − ∥x∥22 − ∥y∥22 ). vii) If F = R, then 4xTy = ∥x + y∥22 − ∥x − y∥22 = 2(∥x + y∥22 − ∥x∥22 − ∥y∥22 ).

854

CHAPTER 11

viii) If F = C, then Im x∗y = 41 (∥x − ȷy∥22 − ∥x + ȷy∥22 ). ix) If F = C, then 4x∗y = ∥x + y∥22 − ∥x − y∥22 + (∥x − ȷy∥22 − ∥x + ȷy∥22 ) ȷ. x) Let F = C and m ≥ 3. Then, 1 ∑ −(2iπ/m) ȷ e ∥x + e(2iπ/m) ȷ y∥22 . m i=0 m−1

x∗y =

xi) If F = C, then 2x∗y = ∥x + y∥22 − ∥x∥22 − ∥y∥22 + (∥x∥22 + ∥y∥22 − ∥x + ȷy∥22 ) ȷ. xii) ∥x + y∥2 = ∥x∥2 + ∥y∥2 if and only if there exists β ≥ 0 such that either x = βy or y = βx. If these conditions hold, then Im x∗y = 0 and Re x∗y ≥ 0. xiii) If x and y are linearly independent, then ∥x + y∥2 < ∥x∥2 + ∥y∥2 . xiv) |x∗ y|2 + 21 ∥x ⊗ y − (x ⊗ y)∗ ∥22 = ∥x∥22 ∥y∥22 . xv) If x, y ∈ Rn and x and y are nonzero, then

y

2 x

= ∥x∥2 ∥y∥2 , − xTy + 12 ∥x∥2 ∥y∥2

∥x∥2 ∥y∥2 2 2  2  1  ∥x∥2 − ∥y∥2 + ∥x − y∥2  xTy 1  ∥x − y∥2 − ∥x∥2 − ∥y∥2  1 −  ≤ 1 −   ≤  , 2 max {∥x∥2 , ∥y∥2 } ∥x∥2 ∥y∥2 2 min {∥x∥2 , ∥y∥2 }

x y

2

. (∥x∥2 + ∥y∥2 − ∥x + y∥2 )2 ≤ ∥x − y∥22 − (∥x∥2 − ∥y∥2 )2 = ∥x∥2 ∥y∥2

− ∥x∥2 ∥y∥2 2 xvi) If x and y are nonzero, then (

) 1

x y

2 ∗

, Re y x = ∥x∥2 ∥y∥2 1 −

− 2 ∥x∥2 ∥y∥2 2

(

) ȷy

2 1

x

. − Im y x = ∥x∥2 ∥y∥2 1 −

2 ∥x∥2 ∥y∥2 2 ∗

|x∗ y| ≤ |x|T |y| ≤ ∥x∥2 ∥y∥2 . ∥x + y∥2 ∥x − y∥2 ≤ 12 (∥x + y∥22 + ∥x − y∥22 ) = ∥x∥22 + ∥y∥22 . If ∥x + y∥2 ≤ 2, then (1 − ∥x∥22 )(1 − ∥y∥22 ) ≤ |1 − x∗y|2. (1 − ∥x∥22 )(1 − ∥y∥22 ) + ∥x + y∥22 + ∥x − y∥22 = (1 + ∥x∥22 )(1 + ∥y∥22 ). ∥x − y∥22 ≤ (1 + ∥x∥22 )(1 + ∥y∥22 ). ∥x + y∥2 ∥x∥2 ∥y∥2 xxii) ≤ + . 1 + ∥x + y∥2 1 + ∥x∥2 1 + ∥y∥2 xxiii) If x and y are nonzero, then





1 x − ∥x∥ y

=

1 y − ∥y∥ x

. 2 2

∥y∥2

∥x∥2 xvii) xviii) xix) xx) xxi)

2

xxiv) If x and y are nonzero, then 1 2 (∥x∥2

2



1 1

+ ∥y∥2 )

x− y

≤ ∥x − y∥2 . ∥x∥2 ∥y∥2 2

xxv) Let α be a nonzero real number. Then, ∥x∥22 ∥y∥22 − |x∗ y|2 ≤ α−2 ∥αy − x∥22 ∥x∥22 . If, in addition, Re x∗y , 0, then 2 2 −2 2 2 ∥x∥22 ∥y∥22 − |x∗y|2 ≤ α−2 0 ∥α0 y − x∥2 ∥x∥2 ≤ α ∥αy − x∥2 ∥x∥2 , △

where α0 = x∗ x/(Re x∗y).

855

NORMS

xxvi) If p ∈ [1, 2], then ∥x + y∥2p ≤ 2 p−1 (∥x∥2p + ∥y∥2p ) (∥x∥2 + ∥y∥2 ) p + |∥x∥2 − ∥y∥2 | p

} ≤ ∥x + y∥2p + ∥x − y∥2p ≤ 2(∥x∥2p + ∥y∥2p ).

xxvii) If p ≥ 2, then 2(∥x∥2p + ∥y∥2p ) ≤ ∥x + y∥2p + ∥x − y∥2p ≤ 2 p−1 (∥x∥2p + ∥y∥2p ). xxviii) If p ∈ (1, 2], q ≥ 2, and 1/p + 1/q = 1, then ∥x + y∥q2 + ∥x − y∥q2 ≤ 2(∥x∥2p + ∥y∥2p )q−1. xxix) If p ≥ 2, q ∈ (1, 2], and 1/p + 1/q = 1, then 2(∥x∥2p + ∥y∥2p )q−1 ≤ ∥x + y∥q2 + ∥x − y∥q2. xxx) If p, q > 1 and 1/p + 1/q = 1, then ∥x + y∥22 ≤ p∥x∥22 + q∥y∥22 . Furthermore, equality holds if and only if y = (p − 1)x. xxxi) If α ∈ [0, 1], then ∥x − y∥22 + ∥αx + y∥22 ≤ (1 + α)∥x∥22 + (1 + α1 )∥y∥22 . xxxii) If either α < 0 or α ≥ 1, then (1 + α)∥x∥22 + (1 + α1 )∥y∥22 ≤ ∥x − y∥22 + ∥αx + y∥22 . xxxiii) Let z ∈ F, let S ⊆ F be a subspace, and let x, y ∈ S. Then, √ √ ∥x − y∥2 ≤ ∥z − x∥22 − min ∥z − w∥22 + ∥z − y∥22 − min ∥z − w∥22 . w∈S

w∈S

Equality holds if and only if there exists α ∈ [0, 1] such that minw∈S ∥z − w∥2 = ∥z − [αx + (1 − α)y]∥2 . xxxiv) Let S ⊆ Fn be a subspace, let A ∈ Fn×n be the projector onto S, and assume that y ∈ S. Then, ∥x − y∥22 = ∥x − Ax∥22 + ∥y − Ax∥22 . Source: ii) and iii) is the cosine law (see Fact 11.10.1 for a matrix version); the equivalence of c)

and d) in iv) is the Pythagorean theorem; the first equality in v) is the generalized parallelogram law, see [1095] and Fact 10.11.84; the second equality in v) is the parallelogram law, which relates the diagonals and the sides of a parallelogram; vi) follows from ii) and iii); viii) follows from vi) by replacing y with ȷy; ix) is the polarization identity (see [2112, p. 276]); x) is a generalization of the polarization identity (see [828, p. 54]); xi) is given in [2238, p. 261]; xiv) is given in [2488]; xv) is given in [32, 2022]; xvi) is given in [33]; the first and third terms in xvii) are the CauchySchwarz inequality; xix) is given by Lemma 1 in [2933] and implies Aczel’s inequality given by Fact 2.12.37; xxiv) is the Dunkl-Williams inequality, which compares the distance between x and y with the distance between the projections of x and y into the unit sphere (see [967], [2061, p. 515], and [2983, p. 28]); xxv) is given in [1827]; xxvi)–xxix) are the Clarkson inequalities (see [1419], [2061, p. 536], and [2294, p. 253]); the lower left inequality in xxvi) is given in [940]; xxx) is Bohr’s inequality (see [1984]); xxxi) and xxxii) are given in [1095]; xxxiii), which is the Beppo Levi inequality, is given in [2333]; xxxiv) is given in [2487, p. 67] Remark: In terms of the traditional inner product notation, x∗y = ⟨y, x⟩. Remark: Many of these results are extensions of results for [ ] complex scalars given by Fact 2.21.8. Note that, if F = R and n = 2, then ∥ yx ∥2 = |x+ ȷy|. Remark: By replacing the Euclidean norm in xvi) with an arbitrary vector norm, it is possible to define an

856

CHAPTER 11

alternative For example, the nonzero vectors x and y are orthogonal with respect to

notion of

angle. √ 1 1

∥ · ∥ if ∥x∥ x − ∥y∥ y = 2. See [811, 902, 1075]. Related: Fact 14.8.5. ∑k

Fact 11.8.4. Let x1 , . . . , xk ∈ Fn, let α1 , . . . , αk be nonzero real numbers, and assume that

i=1

αi = 1. Then,

2

k k ∑ αi ∑

∑ 1 2 xi

= ∥xi ∥2 −

α αj

i=1 i=1 i 2



2

α j x − x

, i j

αi

2

where the last summation is taken over all i, j ∈ {1, . . . , k} such that i ≤ j. Source: [1095, 2989]. Related: Fact 2.21.14 and Fact 10.11.85. Fact 11.8.5. Let x, y, z ∈ Fn. Then, the following statements hold: i) ∥x − z∥22 + ∥z − y∥22 = 12 ∥x − y∥22 + 2∥z − 21 (x + y)∥22 . ii) ∥x + y∥22 + ∥y + z∥22 + ∥z + x∥22 = ∥x∥22 + ∥y∥22 + ∥z∥22 + ∥x + y + z∥22 . iii) ∥x − y∥22 + ∥y − z∥22 + ∥z − x∥22 + ∥x + y + z∥22 = 3(∥x∥22 + ∥y∥22 + ∥z∥22 ). iv) ∥x + y − z∥22 + ∥y + z − x∥22 + ∥z + x − y∥22 + ∥x + y + z∥22 = 4(∥x∥22 + ∥y∥22 + ∥z∥22 ). v) ∥x + y∥2 + ∥y + z∥2 + ∥z + x∥2 ≤ ∥x∥2 + ∥y∥2 + ∥z∥2 + ∥x + y + z∥2 . vi) ∥x∥2 + ∥y∥2 + ∥z∥2 ≤ ∥x + y − z∥2 + ∥y + z − x∥2 + ∥z + x − y∥2 . vii) Re x∗zz∗y ≤ 12 (∥x∥2 ∥y∥2 + Re x∗y)∥z∥22 , |x∗zz∗y| ≤ 21 (∥x∥2 ∥y∥2 + |x∗y|)∥z∥22 . viii) | Re(x∗zz∗y − 21 x∗y)∥z∥22 )| ≤ |x∗zz∗y − 12 x∗y∥z∥22 | ≤ 21 ∥x∥2 ∥y∥2 ∥z∥22 . √ ix) | Re(x∗zz∗y − 12 x∗y∥z∥22 )| ≤ 12 ∥z∥22 ∥x∥22 ∥y∥22 − (Im x∗y)2 ≤ 21 ∥x∥2 ∥y∥2 ∥z∥22 . √ x) | Im(x∗zz∗y − 12 x∗y∥z∥22 )| ≤ 12 ∥z∥22 ∥x∥22 ∥y∥22 − (Re x∗y)2 ≤ 21 ∥x∥2 ∥y∥2 ∥z∥22 . xi) ∥x∥22 |y∗ z|2 + ∥y∥22 |z∗ x|2 + ∥z∥22 |x∗ y|2 ≤ ∥x∥22 ∥y∥22 ∥z∥22 + 2|x∗ yy∗ zz∗ x|2 ≤ 3∥x∥22 ∥y∥22 ∥z∥22 . xii) ∥x∥22 |y∗ z|2 + ∥y∥22 |z∗ x|2 ≤ ∥x∥22 ∥y∥22 ∥z∥22 + ∥x∥2 ∥y∥2 ∥z∥22 |x∗ y| ≤ 2∥x∥22 ∥y∥22 ∥z∥22 . √ √ √ √ √ √ √ xiii) ∥x + y∥2 + ∥y + z∥2 + ∥z + x∥2 ≤ ∥x∥2 + ∥y∥2 + ∥z∥2 + ∥x + y + z∥2 . Remark: i) is the Appolonius identity (see [2238, p. 260]; ii) is Hlawka’s identity (see [2128, p. 100]); v) is Hlawka’s inequality (see [2061, p. 521], [2128, p. 100], and Fact 1.21.9); vii) is Buzano’s inequality (see [937, 1090], [2527, p. 71], and Fact 2.13.2); viii) is an extension of Buzano’s inequality (see [937]); ix) and x) are given in [2254]; xi) is given in [1897, 1908]; xii) is given in [1908]; xiii) is given in [2305]. Remark: As in Fact 11.8.3, some of these results are extensions of results for complex scalars given by Fact 2.21.8. Fact 11.8.6. Let w, x, y, z ∈ Cn . Then, |wTx|2 + |yTz|2 + 2|Re(w∗ yxT z¯ − w∗ z¯ xTy)| ≤ ∥w∥22 ∥x∥22 + ∥y∥22 ∥z∥22 . Source: [722, 1342]. Remark: This is a generalized Cauchy-Schwarz inequality. Related: Fact

11.10.2. Fact 11.8.7. Let x ∈ Fn, and let p, q ∈ [1, ∞], where p ≤ q. Then,

∥x∥q ≤ ∥x∥ p ≤ n1/p−1/q ∥x∥q . In particular, ∥x∥2 ≤ ∥x∥1 ≤

√ n∥x∥2 ,

∥x∥∞ ≤ ∥x∥1 ≤ n∥x∥∞ ,

∥x∥∞ ≤ ∥x∥2 ≤

√

n∥x∥∞ .

Source: [1388] and [1389, p. 107]. Related: Fact 2.11.90 and Fact 11.9.25. △ ∑ Fact 11.8.8. Let n ≥ 3, let x1, . . . , xn ∈ Fn , and, for all k ∈ {1, . . . , n}, define S k = ∥xi1 + · · · +

xik ∥2 , where the sum is taken over all k-tuples (i1 , . . . , ik ) such that 1 ≤ i1 < · · · < ik ≤ n. Then, for

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NORMS

all k ∈ {2, . . . , n − 1},

Sk ≤

In addition,

n−1 ∑

(n−2) k−1

S1 +

(n−2) k−2 S n .

S i ≤ (2n−2 − 1)(S 1 + S n ).

i=2

Source: [2470]. Example: If n = 3, then ∥x1 + x2 ∥2 + ∥x2 + x3 ∥2 + ∥x3 + x1 ∥2 ≤ ∥x1 ∥2 + ∥x2 ∥2 +

∥x3 ∥2 + ∥x1 + x2 + x3 ∥2 . If n = 4, then

∥x1 + x2 ∥2 + ∥x1 + x3 ∥2 + ∥x1 + x4 ∥2 + ∥x2 + x3 ∥2 + ∥x2 + x4 ∥2 + ∥x3 + x4 ∥2 ≤ 2(∥x1 ∥2 + ∥x2 ∥2 + ∥x3 ∥2 + ∥x4 ∥2 ) + ∥x1 + x2 + x3 + x4 ∥2 , ∥x1 + x2 + x3 ∥2 + ∥x1 + x2 + x4 ∥2 + ∥x1 + x3 + x4 ∥2 + ∥x2 + x3 + x4 ∥2 ≤ ∥x1 ∥2 + ∥x2 ∥2 + ∥x3 ∥2 + ∥x4 ∥2 + 2∥x1 + x2 + x3 + x4 ∥2 . Remark: These inequalities concern the diagonals of a polygon. Related: Fact 2.21.25. Fact 11.8.9. Let x, y, z ∈ Fn, assume that x, y, z are nonzero, and define △

ϕ xy = acos Then,

Re y∗ x , ∥x∥2 ∥y∥2

ϕ xz ≤ ϕ xy + ϕyz , sin ϕ xz ≤ sin ϕ xy + sin ϕyz ,

△

θ xy = acos

|y∗ x| . ∥x∥2 ∥y∥2

θ xz ≤ θ xy + θyz , sin θ xz ≤ sin θ xy + sin θyz .

Source: [1829]. Remark: The first inequality is Krein’s inequality. Related: Fact 3.15.1 and Fact

5.1.5. Fact 11.8.10. Let y, x1 , . . . , xn ∈ Fn, assume that, for all i, j ∈ {1, . . . , n}, xi∗ x j = δi, j , and let

k ∈ {1, . . . , n}. Then, the following statements hold: ∑ i) xi∗ (y − ki=1 xi∗ yxi ) = 0.

2 ∑ ∑ ii) y − ki=1 xi∗ yxi

2 = ∥y∥22 − ki=1 |xi∗ y|2 . ∑ iii) ki=1 |xi∗ y|2 ≤ ∥y∥22 . ∑ iv) y = ni=1 xi∗ yxi . ∑ v) ∥y∥2 = ni=1 |xi∗ y|2 . ∑ vi) ni=1 |xi∗ y|2 = ∥y∥22 . Source: [2238, pp. 264, 265]. Remark: iii) is Bessel’s inequality; vi) is Parseval’s identity. ∑ Fact 11.8.11. Let y, x1 , . . . , xn ∈ Fn, and assume that, for all i ∈ {1, . . . , n}, nj=1 |x∗j xi | , 0. Then, n ∑ |x∗ y|2 ∑n i ∗ ≤ ∥y∥22 . j=1 |x j xi | i=1 Source: [2527, p. 225]. Remark: This is Selberg’s inequality. Fact 11.8.12. Let x1 , . . . , xm , y1 , . . . , yn ∈ Fn. Then, m,n ∑ i, j=1

|y∗j xi |2

 m,m 1/2  n,n 1/2  ∑   ∑  ∗ 2 ∗ 2    ≤  |x j xi |   |y j yi |  . i, j=1

i, j=1

Source: [2527, p. 225]. Credit: P. Enflo. Fact 11.8.13. Let x, y ∈ R3 , and let S ⊂ R3 be the parallelogram with vertices 0, x, y, and x + y. Then, area(S) = ∥x × y∥2 . Remark: The parallelogram associated with the cross product can be

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CHAPTER 11

interpreted as a bivector. See [926, pp. 86–88] and [1268, 1759]. Related: Fact 4.12.1, Fact 5.3.1, and Fact 5.3.10. Fact 11.8.14. Let x, y ∈ Fn, and assume that x and y are nonzero. Then, Re x∗y (∥x∥2 + ∥y∥2 ) ≤ ∥x + y∥2 ≤ ∥x∥2 + ∥y∥2 . ∥x∥2 ∥y∥2 Hence, if Re x∗y = ∥x∥2 ∥y∥2 , then ∥x∥2 + ∥y∥2 = ∥x + y∥2 . Source: [2061, p. 517]. Remark: This is a reverse triangle inequality. Remark: Setting x = −y = 1 shows that the first inequality can fail with Re x∗y replaced by | Re x∗y|. △ ∑ Fact 11.8.15. Let x1 , . . . , xn ∈ Cn, let α1 , . . . , αn ∈ R, and define α = ni=1 αi . Then,

2

n n n



∑ ∑ α∑ αi

α j (x j − xi )

= αi α j ∥xi − x j ∥22 .



2 j=1 i=1 i, j=1 2

Now, let x ∈ Cn , and assume that α is nonzero. Then,

2

n n n ∑

1 ∑ 1∑ 2 αi xi

+ αi α j ∥xi − x j ∥22 , αi ∥x − xi ∥2 = α

x − α 2α

i=1 i, j=1 i=1 2

n ∑ i=1

2



2 n n n



∑ ∑ ∑

1 1 αi xi

+ α j x j

. αi ∥x − xi ∥22 = α

x − αi

xi −



α i=1 α j=1

i=1 2

2

Source: [435, 1171]. Remark: The second equality is Lagrange’s second identity. The third inequality is the Huygens-Leibniz identity. Fact 11.8.16. Let x1 , . . . , xn ∈ Fn, and let α1 , . . . , αn be nonnegative numbers. Then,





n  n  n n n



∑ ∑ ∑



∑ ∑  αi

xi − α j x j

≤ αi ∥xi ∥2 +  αi  − 2

αi xi

.



i=1

i=1 i=1 j=1 i=1 2 2





n n n n

∑

∑ ∑

∑

x j

≤ ∥xi ∥2 + (n − 2)

xi

.

i=1 i=1 j=1, j,i 2 i=1 2

In particular,

Remark: The first inequality is the generalized Hlawka inequality (also called the polygonal inequalities). The second inequality is the Djokovic inequality. See [2584] and Fact 11.8.3. Fact 11.8.17. Let x1 , . . . , xn , y1 , . . . , yn ∈ Fn. Then,

2

n n n n ∑ ∑ ∑

∑

2 2 ∥xi − x j ∥2 + ∥yi − y j ∥2 + 2

(xi − yi )

= 2 ∥xi − y j ∥22 .

i=1 i, j=1 i, j=1 i, j=1 2

Equivalently, n ∑ i, j=1,i< j

∥xi −

x j ∥22

+

n ∑ i, j=1,i< j

∥yi −

y j ∥22

2

n n ∑

∑

+

(xi − yi )

= ∥xi − y j ∥22 .

i=1

i, j=1 2

Source: [2093]. Remark: This is a generalized parallelogram law. Setting x1 = −x2 = x, x3 = −x4 = y, and y1 = y2 = y3 = y4 = 0 yields the parallelogram law ∥x + y∥22 + ∥x − y∥22 = 2∥x∥22 + 2∥y∥22 given by v) of Fact 11.8.3. Related: Fact 2.21.26 and Fact 11.10.61.

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NORMS

Fact 11.8.18. Let x, y ∈ Rn, let α and δ be positive numbers, and let p, q ∈ (0, ∞) satisfy

1/p + 1/q = 1. Then,

  

p−1 α  p p T p p−1  δ ≤ |δ − x y| + α ∥x∥2 . q α + ∥y∥2

q Equality holds if and only if x = [δ∥y∥q−2 2 /(α + ∥y∥2 )]y. In particular,

αδ2 ≤ (δ − xTy)2 + α∥x∥22 . α + ∥y∥22 Equality holds if and only if x = [δ/(α + ∥y∥22 )]y. Source: [2583]. Remark: These are generalizations of Hua’s inequality. See Fact 2.11.39 and Fact 11.8.19. Credit: The first inequality is due to J. Pecaric. The case p = q = 2 is due to S. S. Dragomir and G.-S. Yang. Fact 11.8.19. Let x1 , . . . , xn , y ∈ Rn and α ∈ (0, ∞). Then,



2 n n ∑ ∑

α 2 ∥y∥2 ≤

y − xi

+ α ∥xi ∥22 . α+n

i=1 i=1 2

Equality holds if and only if x1 = · · · = xn = [1/(α + n)]y. Source: [2583]. Related: This extends Hua’s inequality. See Fact 2.11.39 and Fact 11.8.18. √ Fact 11.8.20. Let x ∈ Fn, and let p, q ∈ [1, ∞] satisfy 1/p + 1/q = 1. Then, ∥x∥2 ≤ ∥x∥ p ∥x∥q . Fact 11.8.21. Let x, y ∈ Rn, assume that x and y are nonnegative, let p ∈ (0, 1], and define △ ∑ p 1/p ∥x∥ p = ( ni=1 x(i) ) . Then, ∥x∥ p + ∥y∥ p ≤ ∥x + y∥ p . Now, let q ∈ (0, 1], and assume that p ≤ q. Then, ∥x∥q ≤ ∥x∥ p . Remark: This notation is for convenience only since, for all p ∈ (0, 1), ∥ · ∥ p is not a norm but rather is an anti-norm on {x ∈ Rn : x ≥≥ 0}. See [1779]. Related: Fact 2.2.59, Fact 2.11.91, Fact 10.14.7,

Fact 11.10.4, Fact 11.10.54, and Fact 11.10.55. Fact 11.8.22. Let x, y ∈ Fn×n. Then, |∥x∥2 − ∥y∥2 | ≤ ∥x − y∥1 . Source: [1566, p. 12]. Fact 11.8.23. Let x, y ∈ Fn×n. If p ∈ [1, 2], then (∥x∥ p + ∥y∥ p ) p + |∥x∥ p − ∥y∥ p | p ≤ ∥x + y∥ pp + ∥x − y∥ pp , (∥x + y∥ p + ∥x − y∥ p ) p + |∥x + y∥ p − ∥x − y∥ p | p ≤ 2 p (∥x∥ pp + ∥y∥ pp ). If p ∈ [2, ∞], then ∥x + y∥ pp + ∥x − y∥ pp ≤ (∥x∥ p + ∥y∥σp ) p + |∥x∥ p − ∥y∥ p | p , 2 p (∥x∥ pp + ∥y∥ pp ) ≤ (∥x + y∥ p + ∥x − y∥ p ) p + |∥x + y∥ p − ∥x − y∥ p | p . Source: [262, 1818]. Remark: These are vector extensions of Hanner’s inequality. These follow from integral inequalities on L p by appropriate choice of measure. Remark: Equality holds for

p = 2. The case where p = 2, n = 1, and F = C is given by Fact 2.21.8. The case where p = 2 and n ≥ 1 is given by Fact 11.8.3. Remark: Matrix versions are given in Fact 11.10.65. Fact 11.8.24. Let y ∈ Fn, let ∥ · ∥ be a norm on Fn, let ∥ · ∥′ be the norm on Fn×n induced by ∥ · ∥, and define △ ∥y∥D = max |y∗x|. n x∈{z∈F : ∥z∥=1}

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Then, ∥ · ∥D is a norm on Fn. Furthermore, ∥y∥ = Hence, for all x ∈ Fn,

max

x∈{z∈Fn : ∥z∥D =1}

|x∗y| ≤ ∥x∥∥y∥D ,

|y∗x|.

∥xy∗ ∥′ = ∥x∥∥y∥D .

Finally, let p, q ∈ [1, ∞) satisfy 1/p + 1/q = 1. Then, ∥ · ∥ pD = ∥ · ∥q . Hence, for all x ∈ F , n

|x∗y| ≤ ∥x∥ p ∥y∥q ,

∥xy∗ ∥ p,p = ∥x∥ p ∥y∥q .

Source: [2539, p. 57]. Remark: ∥ · ∥D is the dual norm of ∥ · ∥. Fact 11.8.25. Let x ∈ Fn and y ∈ Fm. Then,

σmax (xy∗ ) = ∥xy∗ ∥F = ∥x∥2 ∥y∥2 ,

σmax (xx∗ ) = ∥xx∗ ∥F = ∥x∥22 .

Related: Fact 7.12.19. Fact 11.8.26. Let x ∈ Fn , y ∈ Fm, and p ∈ (0, ∞). Then,

∥x ⊗ y∥ p = ∥ vec(x ⊗ yT )∥ p = ∥ vec(xyT )∥ p = ∥xyT ∥ p = ∥x∥ p ∥y∥ p . Fact 11.8.27. Let x ∈ Cn and p, q ∈ [1, ∞]. Then, ∥x∥q,p,C = ∥x∥q . Now, assume that x ∈ Rn. Then, ∥x∥q,p,C = ∥x∥q,p,R = ∥x∥q . Related: Fact 11.9.43. ∑ Fact 11.8.28. Let x1 , . . . , xk ∈ Fn, let α1 , . . . , αk ∈ (0, ∞), and assume that ki=1 αi = 1. Then,

|11×n (x1 ⊙ · · · ⊙ xk )| ≤

k ∏

∥xi ∥1/αi .

i=1

Remark: This is the generalized H¨older inequality. See [603, p. 128]. ∑ Fact 11.8.29. Let x1 , . . . , xm ∈ Rn, assume that m i=1 xi = 0, and assume that, for all i ∈

{1, . . . , m}, ∥xi ∥2 ≤ 1. Then, there exists a permutation σ of {1, . . . , m} such that, for all k ∈ {1, . . . , m}, ∑ ∥ ki=1 xσ(i) ∥2 ≤ n. Source: [1992, pp. 71–75].

11.9 Facts on Matrix Norms for One Matrix Fact 11.9.1. Let S ⊆ Fm, assume that S is bounded, and let A ∈ Fn×m. Then, AS is bounded. Fact 11.9.2. Let A ∈ Fn×n, assume that A is a idempotent, and assume that, for all x ∈ Fn, ∥Ax∥2 ≤ ∥x∥2 . Then, A is a projector. Source: [1133, p. 42]. Fact 11.9.3. Let A ∈ Fn×n, and assume that ρmax (A) < 1. Then, there exists a submultiplicative

matrix norm ∥ · ∥ on Fn×n such that ∥A∥ < 1. Furthermore, limk→∞ Ak = 0. Fact 11.9.4. Let A ∈ Fn×n, assume that A is nonsingular, and let ∥ · ∥ be a submultiplicative norm on Fn×n. Then, ∥In ∥/∥A∥ ≤ ∥A−1∥. Fact 11.9.5. Let A ∈ Fn×n, assume that A is nonzero and idempotent, and let ∥ · ∥ be a submultiplicative norm on Fn×n. Then, ∥A∥ ≥ 1. Fact 11.9.6. Let A ∈ Fn×n, and let ∥ · ∥ be a unitarily invariant norm on Fn×n. Then, ∥A∥ = ∥A∗ ∥ = ∥⟨A⟩∥ = ∥⟨A∗ ⟩∥. If, in addition, r > 0, then ∥⟨A⟩r ∥ = ∥⟨A∗ ⟩r ∥. Remark: ∥ · ∥ is self-adjoint. △ Fact 11.9.7. Let A ∈ Fn×m, let ∥ · ∥ be a norm on Fn×m, and define ∥A∥′ = ∥A∗ ∥. Then, ∥ · ∥′ is m×n ′′ a norm on F . If, in addition, n = m and ∥ · ∥ is induced by ∥ · ∥ , then ∥ · ∥′ is induced by ∥ · ∥′′D . Source: [1448, p. 309] and Fact 11.9.21. Remark: ∥ · ∥′ is the adjoint norm of ∥ · ∥. Related: Fact 11.8.24 defines the dual norm. Problem: Extend this result to nonsquare matrices and norms that are not equi-induced.

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Fact 11.9.8. Let A ∈ Fn×n . Then,

∥A∥2F = ∥ 12 (A + A∗ )∥2F + ∥ 21 (A − A∗ )∥2F ,

∥A∗A − AA∗ ∥2F = 2[tr (A∗A)2 − tr A2∗A2 ].

Source: The second equality is given in [2476]. Fact 11.9.9. Let A ∈ F2×2 . Then,

[ ( √ )]1/2 1 2 4 2 ∥A∥F + ∥A∥F − 4| det A| σmax (A) = . 2

Source: [1485, p. 261]. Fact 11.9.10. Let A ∈ Fn×m. If p ∈ (0, 2], then ∥A∥σp ≤ ∥A∥ p . If p ≥ 2, then ∥A∥ p ≤ ∥A∥σp . △ Source: [2977, p. 50]. Remark: For p ∈ (0, 1), ∥A∥σp = ∥σ(A)∥ p is not a norm. See Fact 11.8.21. Fact 11.9.11. Let 1 ≤ p ≤ ∞. Then, ∥ · ∥σp is unitarily invariant. Fact 11.9.12. Let A ∈ Fn×m. Then, ∗

∗

∥A A∥F = ∥AA ∥F =

v u tmin {m,n} ∑ i=1

σ4i (A)

≤

min {m,n} ∑

σ2i (A)

=

∥A∥2F ,

i=1

   = ∥A∥2F , rank A ≤ 1, ∥A A∥F   < ∥A∥2 , rank A ≥ 2. F ∗

Related: Fact 11.10.31. Fact 11.9.13. Let A ∈ Fn×m, and assume that A is positive definite. Then, n3/2 ≤ ∥A−1 ∥F tr A. Source: [1476]. Fact 11.9.14. Let A ∈ Rn×n, and assume that A is positive definite. Then,

n3/2 ∥A1/2 ∥F ≤ ∥A∥F ∥A−1 ∥F , tr A1/2

2 tr A − n ≤ ∥A∥F ∥A−1 ∥F , √n det A

0≤

2 tr A3/2 − n ≤ ∥A∥F ∥A−1 ∥F . √ 2n (tr A) det A

Source: [1809, 2751]. Related: Fact 11.15.2. Fact 11.9.15. Let A ∈ Fn×m. Then, ∥AA ∥F ≤ n(2−n)/2 ∥A∥n−1 F . Furthermore, equality holds if and only if either n ≥ 2 or there exist α ∈ F and a unitary matrix B ∈ Fn×n such that A = αB. Source: [2052] and [2263, pp. 151, 165]. Related: Fact 10.21.23. Fact 11.9.16. Let A ∈ Fn×m, and assume that A is normal. Then,

1 √ σmax (A) ≤ ∥A∥∞ ≤ ρmax (A) = σmax (A). mn Source: Fact 7.15.16 and xii) of Fact 11.9.23. Fact 11.9.17. Let A ∈ Rn×n, assume that A is symmetric, and assume that every diagonal entry

of A is zero. Then, the following statements are equivalent: i) For all x ∈ Rn such that 11×n x = 0, it follows that xTAx ≤ 0. ii) There exists k ≥ 1 and x1 , . . . , xn ∈ Rk such that, for all i, j ∈ {1, . . . , n}, A(i, j) = ∥xi − x j ∥22 . Source: [36]. Remark: A is a Euclidean distance matrix. Credit: I. J. Schoenberg. Fact 11.9.18. Let A ∈ Fn×n, assume that A is normal, let B ∈ Fn×n be the strictly lower triangular part of A, and let C ∈ Fn×n be the strictly upper triangular part of A. Then, √ √ ∥B∥F ≤ n − 1∥C∥F , ∥C∥F ≤ n − 1∥B∥F . Source: [2991, p. 321]. Fact 11.9.19. Let A ∈ Fn×n , and assume that A is semicontractive. Then,

∥A∥2F ≤ | det A|2 + n − 1. Source: [2992].

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Fact 11.9.20. Let A ∈ Fn×n. If p ∈ [1, 2], then

∥A∥F ≤ ∥A∥σp ≤ n1/p−1/2 ∥A∥F . If p ∈ [2, ∞], then ∥A∥σp ≤ ∥A∥F ≤ n1/2−1/p ∥A∥σp . Source: [453, p. 174]. Fact 11.9.21. Let A ∈ Fn×m, and let p, q ∈ [1, ∞] satisfy 1/p + 1/q = 1. Then, ∥A∗ ∥ p,p = ∥A∥q,q . In particular, ∥A∗ ∥col = ∥A∥row . Source: Fact 11.9.7. Fact 11.9.22. Let A ∈ Fn×m, and let p, q ∈ [1, ∞] satisfy 1/p + 1/q = 1. Then,

[

0

A∗

[

0

A∗

In particular,

A 0

A 0

]



= max {∥A∥ p,p , ∥A∥q,q }. p,p

[ ]

0

=

A∗

col

A 0

]



= max {∥A∥col , ∥A∥row }. row

Fact 11.9.23. Let A ∈ Fn×m. Then, the following statements hold:

√ ∥A∥F ≤ ∥A∥1 ≤ mn∥A∥F . ∥A∥∞ ≤ ∥A∥1 ≤ mn∥A∥∞ . ∥A∥col ≤ ∥A∥1 ≤ m∥A∥col . ∥A∥row ≤ ∥A∥1 ≤ n∥A∥row . √ σmax (A) ≤ ∥A∥1 ≤ mn rank A σmax (A). √ ∥A∥∞ ≤ ∥A∥F ≤ mn∥A∥∞ . √ √1 ∥A∥col ≤ ∥A∥F ≤ m∥A∥col . n √ 1 viii) √m ∥A∥row ≤ ∥A∥F ≤ n∥A∥row . √ ix) σmax (A) ≤ ∥A∥F ≤ rank A σmax (A). x) n1 ∥A∥col ≤ ∥A∥∞ ≤ ∥A∥col . i) ii) iii) iv) v) vi) vii)

xi) xii) xiii) xiv) xv)

1 m ∥A∥row ≤ ∥A∥∞ ≤ ∥A∥row . √1 σmax (A) ≤ ∥A∥∞ ≤ σmax (A). mn 1 m ∥A∥row ≤ ∥A∥col ≤ n∥A∥row . √ √1 σmax (A) ≤ ∥A∥col ≤ nσmax (A). m √ 1 √ σmax (A) ≤ ∥A∥row ≤ mσmax (A). n

Source: [1448, p. 314] and [3008]. Remark: See [1389, p. 115] for equality cases. Fact 11.9.24. Let A ∈ Fn×n, let ∥ · ∥ and ∥ · ∥′ be norms on Fn, and define the induced norms △

∥A∥′′ =

max

x∈{y∈Fm : ∥y∥=1}

∥Ax∥,

△

∥A∥′′′ =

max

x∈{y∈Fm : ∥y∥′ =1}

∥Ax∥′ .

Then, ∥A∥′′ ∥A∥′′′ = max = : X,0} ∥A∥′′′ A∈{X∈Fn×n : X,0} ∥A∥′′

max n×n

A∈{X∈F

∥x∥ : y,0} ∥x∥′

max n

x∈{y∈F

∥x∥′ . : y,0} ∥x∥

max n

x∈{y∈F

Source: [1448, p. 303]. Remark: This symmetry property is evident in Fact 11.9.23.

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Fact 11.9.25. Let A ∈ Fn×n and p, q ∈ [1, ∞]. Then,

∥A∥ p,p

 1/p−1/q   ∥A∥q,q , p ≤ q,  n ≤   n1/q−1/p ∥A∥q,q , q ≤ p.

Consequently, n1/p−1∥A∥col ≤ ∥A∥ p,p ≤ n1−1/p ∥A∥col , n−|1/p−1/2| σmax (A) ≤ ∥A∥ p,p ≤ n|1/p−1/2| σmax (A), n−1/p∥A∥col ≤ ∥A∥ p,p ≤ n1/p ∥A∥row . Source: [1388] and [1389, p. 112]. Related: Fact 11.8.7. Problem: Extend these inequalities to

nonsquare matrices. △ Fact 11.9.26. Let A ∈ Fn×m, p, q ∈ [1, ∞], and α ∈ [0, 1], and define r = pq/[(1 − α)p + αq]. Then, ∥A∥r,r ≤ ∥A∥αp,p ∥A∥1−α q,q . Source: [1388] and [1389, p. 113]. Fact 11.9.27. Let A ∈ Fn×m and p ∈ [1, ∞]. Then, 1−1/p ∥A∥ p,p ≤ ∥A∥1/p col ∥A∥row .

In particular, σmax (A) ≤

√ ∥A∥col ∥A∥row ≤ 12 (∥A∥col + ∥A∥row ) ≤ max {∥A∥col , ∥A∥row }.

Source: Set α = 1/p, p = 1, and q = ∞ in Fact 11.9.26. See [1389, p. 113]. To prove the special

case p = 2, note that σ2max (A) = λmax (A∗A) ≤ ∥A∗A∥col ≤ ∥A∗ ∥col ∥A∥col = ∥A∥row ∥A∥col . Fact 11.9.28. Let A ∈ Fn×m. Then, ∥A∥2,1 ≤ σmax (A) and ∥A∥∞,2 ≤ σmax (A). Source: Proposition 11.1.5. Fact 11.9.29. Let A ∈ Fn×m and p ∈ [1, 2]. Then, 2−2/p ∥A∥ p,p ≤ ∥A∥2/p−1 col σmax (A).

Source: Let α = 2/p − 1, p = 1, and q = 2 in Fact 11.9.26. See [1389, p. 113]. Fact 11.9.30. Let A ∈ Fn×n and p ∈ [1, ∞]. Then,

∥A∥ p,p ≤ ∥|A|∥ p,p ≤ nmin {1/p,1−1/p} ∥A∥ p,p ≤

√ n∥A∥ p,p .

Remark: See [1389, p. 117]. Fact 11.9.31. Let A ∈ Fn×n and p, q, r, s ∈ [1, ∞]. Then,

∥A∥q,p ≤ α(q, s, n)α(r, p, m)∥A∥ s,r , where

   1, α(q, s, n) =   n1/q−1/s ,

q ≥ s, q < s.

Source: [1030, 1031]. Fact 11.9.32. Let A ∈ Fn×m and p, q ∈ [1, ∞]. Then, ∥A∥q,p = ∥A∥q,p . Fact 11.9.33. Let A ∈ Fn×m and p, q ∈ [1, ∞]. Then, ∥A∗ ∥q,p = ∥A∥ p/(p−1),q/(q−1) .

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Fact 11.9.34. Let A ∈ Fn×m and p, q ∈ [1, ∞]. Then,

∥A∥q,p

   ∥A∥ p/(p−1) , ≤  ∥A∥q ,

1/p + 1/q ≤ 1, 1/p + 1/q ≥ 1. △

Fact 11.9.35. Let A ∈ Fn×m, let q, r ∈ [1, ∞], assume that 1 ≤ q < r, define p = qr/(r − q), and assume that p ≥ 2. Then, ∥A∥ p ≤ ∥A∥q,r . In particular, ∥A∥∞ ≤ ∥A∥∞,∞ . Source: [1030, 1031]. Credit: G. H. Hardy and J. E. Littlewood. Fact 11.9.36. Let A ∈ Rn×m. Then,

∥AT ∥2|1 ≤

√ 2∥A∥1,∞ ,

√ 2∥A∥1,∞ ,

∥AT ∥1|2 ≤

∥A∥3/4 4/3 ≤

√ 2∥A∥1,∞ .

Source: [1146, p. 303]. Credit: The first and third results are due to J. E. Littlewood; the second

result is due to W. Orlicz. Fact 11.9.37. Let A ∈ Fn×n, and assume that A is positive semidefinite. Then, ∥A∥1,∞ =

max

x∈{z∈Fn : ∥z∥∞ =1}

x∗Ax.

Credit: P. D. Tao. See [1389, p. 116] and [2328]. Fact 11.9.38. Let A ∈ Rn×m, let p, q ∈ [1, ∞], and assume that q ≤ p. Then,

∥A∥q,p,R ≤ ∥A∥q,p,C ≤ cq,p ∥A∥q,p,R , where cq,p

 ( )1/p ( )1/q      Γ[(2 + q)/2]  √ √ 1/q−1/p Γ[(1 + p)/2]  ≤ min  2, π .      Γ[(2 + p)/2] Γ[(1 + q)/2]

Furthermore, if p ∈ [1, 2], then c p,2 = c2,p/(p−1)

√ ( )1/p 2 √ Γ[(2 + p)/2] π . = 2 Γ[(1 + p)/2]

√ In particular, c1,2 = c2,∞ = 2π/4. Source: [864, p. 377] and [2040]. Problem: Compare these constants to the constants c p given in Proposition 11.4.7. Fact 11.9.39. Let A ∈ Fn×m, let p, q ∈ [1, ∞], and assume that 1/p + 1/q = 1. Then, ∥A∥1,p ≤ ∥A∥1|q . If, in addition, A ∈ Rn×m and A is nonnegative, then ∥A∥1,p = ∥A∥1|q . Source: [2040]. Fact 11.9.40. Let A ∈ Fn×m and p, q, r, s ∈ [1, ∞]. Then,

∥A∥ p|q ≤ α(p, r, n)α(q, s, m)∥A∥r|s , where α is defined in Fact 11.9.31. Source: [1030, 1031]. Fact 11.9.41. Let A ∈ Fn×m and p, q ∈ [1, ∞]. Then, ∥A∥q,p =

max m

x∈F ,y∈F x,y,0

n

|y∗Ax| . ∥y∥q/(q−1) ∥x∥ p

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Fact 11.9.42. Let A ∈ Fn×m, and let p, q ∈ [1, ∞] satisfy 1/p + 1/q = 1. Then,

∥A∥ p,p =

|y∗Ax| = ∥y∥q ∥x∥ p

max m

x∈F ,y∈Fn x,y,0

max m

x∈F ,y∈Fn x,y,0

|y∗Ax| . ∥y∥ p/(p−1) ∥x∥ p

Related: See Fact 11.15.4 for the case p = 2. Fact 11.9.43. Let A ∈ Fn×m and p, q ∈ [1, ∞]. If m < n, then

∥[A 0n×(n−m) ]∥q,p,F = ∥A∥q,p,F . If n < m, then

[ ]



A = ∥A∥q,p,F .

0 (m−n)×m q,p,F

Related: Fact 11.8.27. Fact 11.9.44. Let A ∈ Fn×n, and let ∥·∥ be a unitarily invariant norm on Fn×n. Then, ∥⟨A⟩∥ = ∥A∥. Source: [2991, p. 376]. Fact 11.9.45. Let A, S ∈ Fn×n, assume that S is nonsingular, and let ∥ · ∥ be a unitarily invariant

norm on Fn×n. Then,

∥A∥ ≤ 12 ∥SAS −1 + S −∗AS ∗ ∥.

Source: [121, 537]. △ Fact 11.9.46. For A ∈ Fn×n , define ∥A∥ = max {| tr AX| : X ∈ Fn×n and tr X ∗X = 1}. Then, ∥ · ∥ is n×n a unitarily invariant norm on F . Source: [2991, p. 377]. △ ∑min {n,m} p Fact 11.9.47. Let A ∈ Fn×m and p, r ∈ (0, ∞). and define ∥A∥σp = ( i=1 σi (A))1/p. Then, r r ∥⟨A⟩ ∥σp = ∥A∥σrp . Source: [2436, 2438]. Remark: for p ∈ (0, 1), ∥ · ∥σp is not a norm. Related:

Fact 11.8.21. Fact 11.9.48. Let A ∈ Fn×n , let α1 ≥ · · · ≥ αn ≥ 0, assume that α1 , . . . , αn are not all zero, and

define △

∥A∥ =

n ∑

αi σi (A).

i=1

Then, ∥ · ∥ is a unitarily invariant norm on Fn×n . Source: [2991, p. 377]. Fact 11.9.49. Let A ∈ Fn×m , and let ∥ · ∥ be a unitarily invariant norm on Fn×m . Then, lim ∥⟨A⟩ p ∥1/p = σmax (A).

p→∞

Now, assume that ∥ · ∥ is a normalized unitarily invariant norm on Fn×m , and define f : (0, ∞) → R △ by f (p) = ∥⟨A⟩ p ∥1/p . Then, f is nonincreasing. Source: [1383]. Fact 11.9.50. Let A ∈ Fn×n, assume that A is positive semidefinite, and let ∥ · ∥ be a submultiplicative norm on Fn×n. Then, ∥A∥1/2 ≤ ∥A1/2 ∥. Furthermore,

1/2 σmax (A) = σmax (A1/2 ).

Fact 11.9.51. Let A ∈ Fn×n, assume that A is positive semidefinite, and let ∥ · ∥ be a unitarily invariant norm on Fn×n. If r ∈ (0,1], then ∥A∥ p ≤ ∥A p ∥. Furthermore, if r ∈ [1,∞), then ∥Ar ∥ ≤ ∥A∥r . Source: [1383, 2436, 2438]. [A A ] 12 ∗ ∈ F(n+m)×(n+m) Fact 11.9.52. Let A11 ∈ Fn×n, A12 ∈ Fn×m, and A22 ∈ Fm×m, assume that A11 12 A22 ′ n×n m×m is positive semidefinite, let ∥ · ∥ and ∥ · ∥ be unitarily invariant norms on F and F , respectively, and let p > 0. Then, p p ′ ∥⟨A12 ⟩ p ∥′2 ≤ ∥A11 ∥∥A22 ∥.

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Source: [1453]. Fact 11.9.53. Let A ∈ Fn×n, let ∥ · ∥ be a norm on Fn, let ∥ · ∥D denote the dual norm on Fn, and

let ∥ · ∥′ denote the norm induced by ∥ · ∥ on Fn×n. Then, ∥A∥′ = maxn x,y∈F x,y,0

Re y∗Ax . ∥y∥D ∥x∥

Source: [1389, p. 115]. Remark: Fact 11.8.24 defines the dual norm. Problem: Extend this

result to include Fact 11.9.41 as a special case. Fact 11.9.54. Let A ∈ Fn×n, and assume that A is positive definite. Then, √ x∗Ax 2 αβ α−β min = , min σmax (αA − I) = , α≥0 x∈Fn \{0} ∥Ax∥2 ∥x∥2 α+β α+β △

△

where α = λmax (A) and β = λmin (A). Source: [1279]. Remark: These are antieigenvalues. Fact 11.9.55. Let A ∈ Fn×n, and define △

nrad(A) = max {|x∗Ax|: x ∈ Cn and x∗x ≤ 1}. Then, the following statements hold: i) nrad(A) = max {|z|: z ∈ Θ(A)} = maxθ∈[−π,π] σmax [ 12 (eθ ȷ A + e−θ ȷ A∗ )]. ii) ρmax (A) ≤ nrad(A) ≤ nrad(|A|) = 12 ρmax (|A| + |A|T ). 2 iii) 21 σmax (A) ≤ nrad(A) ≤ 12 σmax (⟨A⟩ + ⟨A∗ ⟩) ≤ 21 [σmax (A) + σ1/2 max (A )] ≤ σmax (A) ≤ 2 nrad(A).

+ AA∗ ) ≤ [nrad(A)]2 ≤ 12 σmax (A∗A + AA∗ ). If A = 0, then nrad(A) = σmax (A). If nrad(A) = σmax (A), then σmax (A2 ) = σ2max (A). If A is normal, then nrad(A) = ρmax (A). nrad(Ak ) ≤ [nrad(A)]k for all k ∈ N. nrad(·) is a weakly unitarily invariant norm on Fn×n. nrad(·) is not a submultiplicative norm on Fn×n. △ ∥ · ∥ = α nrad(·) is a submultiplicative norm on Fn×n if and only if α ≥ 4. nrad(AB) ≤ nrad(A) nrad(B) for all A, B ∈ Fn×n such that A and B are normal. nrad(A ⊙ B) ≤ αnrad(A)nrad(B) for all A, B ∈ Fn×n if and only if α ≥ 2. nrad(A ⊕ B) = max {nrad(A), nrad(B)} for all A ∈ Fn×n and B ∈ Fm×m. Source: [8, 983], [1280, pp. 109, 115], [1448, p. 331], [1450, pp. 43, 44], [1634, 1636, 2437], and [2991, pp. 109, 110]. Remark: nrad(A) is the numerical radius of A, while Θ(A) is the numerical [ ] range of A. See Fact 10.17.8. Remark: nrad(·) is not submultiplicative. The example A = 00 10 , [ ] B = 02 20 , where B is normal, nrad(A) = 1/2, nrad(B) = 2, and nrad(AB) = 2, shows that xii) can be false if only one of the matrices A and B is normal, which corrects [1450, pp. 43, 73]. Remark: viii) is the power inequality. △ Fact 11.9.56. Let A ∈ Fn×m, let γ > σmax (A), and define β = σmax (A)/γ. Then, √ √ ∥A∥F ≤ − [γ2/(2π)] log det(I − γ−2A∗A) ≤ β −1 −log(1 − β2 )∥A∥F . iv) v) vi) vii) viii) ix) x) xi) xii) xiii) xiv)

1 ∗ 4 σmax (A A 2

Source: [557]. Fact 11.9.57. Let ∥ · ∥ be a unitarily invariant norm on Fn×n. Then, ∥A∥ = 1 for all A ∈ Fn×n such that rank A = 1 if and only if ∥E1,1 ∥ = 1. Source: ∥A∥ = ∥E1,1 ∥σmax (A). Remark: These

normalizations are used in [449] and [2539, p. 74].

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Fact 11.9.58. Let ∥ · ∥ be a unitarily invariant norm on Fn×n. Then, the following statements are

equivalent: i) σmax (A) ≤ ∥A∥ for all A ∈ Fn×n. ii) ∥ · ∥ is submultiplicative. iii) ∥A2 ∥ ≤ ∥A∥2 for all A ∈ Fn×n. iv) ∥Ak ∥ ≤ ∥A∥k for all k ≥ 1 and A ∈ Fn×n. v) ∥A ⊙ B∥ ≤ ∥A∥∥B∥ for all A, B ∈ Fn×n. vi) ρmax (A) ≤ ∥A∥ for all A ∈ Fn×n. vii) ∥Ax∥2 ≤ ∥A∥∥x∥2 for all A ∈ Fn×n and x ∈ Fn. viii) ∥A∥∞ ≤ ∥A∥ for all A ∈ Fn×n. ix) ∥E1,1 ∥ ≥ 1. x) σmax (A) ≤ ∥A∥ for all A ∈ Fn×n such that rank A = 1. xi) For all A, B, C ∈ Fn×n , ∥ABC∥ ≤ σmax (A)σmax (C)∥B∥. Source: The equivalence of i)–vii) is given in [1450, pp. 211, 336]. Since ∥A∥ = ∥E1,1 ∥σmax (A) for all A ∈ Fn×n such that rank A = 1, it follows that vii) and viii) are equivalent. To prove ix) =⇒ x), let A ∈ Fn×n satisfy rank A = 1. Then, ∥A∥ = σmax (A)∥E1,1 ∥ ≥ σmax (A). To show x) =⇒ ii), define △ ∥ · ∥′ = ∥E1,1 ∥−1 ∥ · ∥. Since ∥E1,1 ∥′ = 1, it follows from [449, p. 94] that ∥ · ∥′ is submultiplicative. Since ∥E1,1 ∥−1 ≤ 1, it follows that ∥ · ∥ is also submultiplicative. Alternatively, ∥A∥′ = σmax (A) for all A ∈ Fn×n having rank 1. Then, Corollary 3.10 of [2539, p. 80] implies that ∥ · ∥′, and thus ∥ · ∥, is submultiplicative. xi) is given in [2979, p. 101]. Related: Fact 11.10.34. Fact 11.9.59. Let Φ: Fn 7→ [0, ∞), and assume that the following statements hold: i) If x , 0, then Φ(x) > 0. ii) Φ(αx) = |α|Φ(x) for all α ∈ F. iii) Φ(x + y) ≤ Φ(x) + Φ(y) for all x, y ∈ Fn. iv) If A ∈ Rn×n is a permutation matrix, then Φ(Ax) = Φ(x) for all x ∈ Fn. v) Φ(|x|) = Φ(x) for all x ∈ Fn. Then, the following statements hold: vi) Φ is an absolute and monotone norm on Fn . vii) Φ is convex on Fn . viii) Φ is Schur-convex on [0, ∞)n . △ ix) For A ∈ Fn×m, where n ≤ m, define ∥A∥ = Φ[σ(A)]. Then, ∥ · ∥ is a unitarily invariant norm on Fn×m. x) If ∥ · ∥ is a unitarily invariant norm on Fn×m, where n ≤ m, then Φ: Fn 7→ [0, ∞) defined by

  0 0n×(m−n) 



 x(1) · · ·    △  .. .. .. 

Φ(x) =

 ... . . .

 

0 · · · x(n) 0n×(m−n)

satisfies i)–v). Finally, let x, y ∈ F1×n . Then, the following statements are equivalent: w

xi) |x| ≺ |y|. xii) For all Φ: Fn 7→ [0, ∞) satisfying i)–v), Φ(x) ≤ Φ(y). Source: [2539, pp. 75, 76] and [2991, pp. 373–376]. Remark: Φ is a symmetric gauge function. See Fact 3.25.16. Credit: J. von Neumann.

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ˆ Fn×m 7→ R Fact 11.9.60. Let ∥·∥ and ∥·∥′ denote norms on Fm and Fn, respectively, and define ℓ: by

∥Ax∥′ △ ˆ ℓ(A) = min , x∈Fm \{0} ∥x∥

or, equivalently,

△ ˆ ℓ(A) =

min

∥Ax∥′.

x∈{y∈Fm : ∥y∥=1}

Then, for all A ∈ Fn×m, the following statements hold: ˆ i) ℓ(A) ≥ 0. ˆ ii) ℓ(A) > 0 if and only if rank A = m. ˆ iii) ℓ(A) = ℓ(A) if and only if either A = 0 or rank A = m. Source: [1738, pp. 369, 370]. Remark: ℓˆ is a weaker version of ℓ. Fact 11.9.61. Let ∥ · ∥ and ∥ · ∥′ denote norms on Fm and Fn, respectively, let ∥ · ∥′′′ denote the ˆ Fn×m 7→ R by norm induced by ∥ · ∥′ and ∥ · ∥, and define ℓ: △ ˆ ℓ(A) =

∥Ax∥′ . x∈R(A )\{0} ∥x∥ min ∗

Now, let A ∈ Fn×m . If A is nonzero, then 1 ˆ ≤ ℓ(A). ∥A+ ∥′′′ If, in addition, rank A = m, then

1 ˆ = ℓ(A) = ℓ(A). ∥A+ ∥′′′

Source: [2732]. Fact 11.9.62. Let A ∈ Fn×n, let ∥·∥ be a normalized, submultiplicative norm on Fn×n, and assume that ∥I − A∥ < 1. Then, A is nonsingular. Related: Fact 11.10.88. Fact 11.9.63. Let ∥ · ∥ be a normalized, submultiplicative norm on Fn×n. Then, ∥ · ∥ is equi-

induced if and only if ∥A∥ ≤ ∥A∥′ for all A ∈ Fn×n and all normalized submultiplicative norms ∥ · ∥′ on Fn×n. Source: [2543]. Remark: Not every normalized submultiplicative norm on Fn×n is △ induced. See [684, 848]. For example, the norm ∥A∥ = max {∥A∥row , ∥A∥col } on Fn×n is normalized and submultiplicative but not induced. See [1451, p. 357].

11.10 Facts on Matrix Norms for Two or More Matrices Fact 11.10.1. Let A, B ∈ Fn×m. Then,

∥A + B∥F = Therefore, ∥A − B∥F = and thus

√

√

∥A∥2F + ∥B∥2F + 2 tr AB∗ ≤ ∥A∥F + ∥B∥F .

∥A∥2F + ∥B∥2F − 2 tr AB∗ ≤ ∥A∥F + ∥B∥F ,

∥A + B∥2F + ∥A − B∥2F = 2∥A∥2F + 2∥B∥2F .

If, in addition, A is Hermitian and B is skew Hermitian, then tr AB∗ = 0, and thus ∥A + B∥2F = ∥A − B∥2F = ∥A∥2F + ∥B∥2F .

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Finally, if A and B are nonzero, then Re tr A∗B (∥A∥F + ∥B∥F ) ≤ ∥A + B∥F ≤ ∥A∥F + ∥B∥F . ∥A∥F ∥B∥F Remark: The second equality is a matrix version of the cosine law given by Fact 11.8.3. Related:

Fact 11.8.14. Fact 11.10.2. Let A, B, C, D ∈ Cn×n . Then,

|tr A∗B|2 + |tr C ∗D|2 + 2|Re[(tr A∗C) tr B∗ D − (tr A∗D) tr B∗C]| ≤ ∥A∥2F ∥B∥2F + ∥C∥2F ∥D∥2F . Source: [2957]. Remark: This is a generalized Cauchy-Schwarz inequality. Related: Fact 11.8.6. Fact 11.10.3. Let A, B ∈ Fn×n , assume that A and B are positive semidefinite, let f : [0, ∞) 7→

[0, ∞), and assume that f is concave. Then, there exist unitary matrices U, V ∈ Fn×n such that f (A + B) ≤ U f (A)U ∗ + V f (B)V ∗ .

Now, let ∥ · ∥ be a unitarily invariant norm on Fn×n . Then, ∥ f (A + B)∥ ≤ ∥ f (A)∥ + ∥ f (B)∥. In particular,

tr f (A + B) ≤ tr f (A) + tr f (B).

Source: [544, 546]. Remark: The last result is the Rotfel’d trace inequality. Related: Fact

10.12.58. Fact 11.10.4. Let A, B ∈ Pn , let p > 0, and let ∥ · ∥ be a unitarily invariant norm on Fn×n . Then,

∥A−p ∥−1/p + ∥A−p ∥−1/p ≤ ∥(A + B)−p ∥−1/p . Source: [1779]. Remark: f (A) = ∥A−p ∥−1/p is an anti-norm on Pn . Related: Fact 2.2.59, Fact

10.14.7, Fact 11.8.21, and Fact 11.9.47. Fact 11.10.5. Let A, B ∈ Fn×n, and let ∥ · ∥ be a unitarily invariant norm on Fn×n. Then, {√ 2∥⟨A⟩ + ⟨B⟩∥ 1/2 ∗ ∗ 1/2 ∥A + B∥ ≤ ∥⟨A⟩ + ⟨B⟩∥ ∥⟨A ⟩ + ⟨B ⟩∥ ≤ 1 (∥⟨A⟩ + ⟨B⟩∥ + ∥⟨A∗ ⟩ + ⟨B∗ ⟩∥). 2 Source: [1479, 1623]. Remark: ∥⟨A∗ ⟩∥ + ∥⟨B∗ ⟩∥ ≤ 2∥⟨A⟩ + ⟨B⟩∥. See [1479] and Fact 11.10.6. Fact 11.10.6. Let A, B ∈ Fn×n, assume that A and B are positive semidefinite, and let ∥ · ∥ be a

unitarily invariant norm on Fn×n. Then, the following statements hold: i) If A ≤ B, then ∥A∥ ≤ ∥B∥. ii) ∥A∥ + ∥B∥ ≤ ∥A∥ + ∥A + B∥ ≤ 2∥A + B∥. iii) ∥A∥σ1 + ∥B∥σ1 ≤ ∥A + B∥σ1 . √ iv) ∥A∥F + ∥B∥F ≤ 2∥A + B∥F . v) If p ≥ 1, then ∥A∥σp + ∥B∥σp ≤ 2(p−1)/p ∥A + B∥σp . Source: Fact 11.10.53 and Fact 11.8.7. i) follows from the fact that, for all A ∈ Fn×n , ∥A∥ defines a symmetric gauge function, which is a monotone function of σ(A). See Fact 11.9.59. Alternatively, i) follows from the Fan dominance theorem given by Fact 11.16.23. Remark: Letting p → ∞ in v) yields ii) with ∥ · ∥ = σmax (·). Fact 11.10.7. Let A, B ∈ Fn×n . Then, √4 ∥A + B∥F ≤ 2∥⟨A⟩ + ⟨B⟩∥F .

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If, in addition, A and B are nonzero, then 1/4  ( ∥B∥ )  F  − 1 S √4  ∥A∥F  ( 2 )  ∥⟨A⟩ + ⟨B⟩∥F ≤ 2∥⟨A⟩ + ⟨B⟩∥F . ∥A + B∥F ≤ 2 − ∥B∥F   S 2 ∥A∥F

Source: [3016]. Remark: S is Specht’s ratio. See Fact 12.17.5. Related: Fact 11.10.8. Fact 11.10.8. Let A, B ∈ Fn×n, and let ∥ · ∥ be a unitarily invariant norm on Fn×n. Then,

∥A + B∥ ≤

√ 2∥⟨A⟩ + ⟨B⟩∥.

Now, assume that A and B are positive semidefinite, let S ∈ Fn×n, and assume that S is unitary. Then, √ √      2∥A + I∥  2∥A + B∥ ∥A + S ∥ ≤ ∥A + I∥ ≤ ∥A + SB∥ ≤      ∥A + 2I∥.  ∥A + B + SBS ∗ ∥, Source: [1778, 1852]. Remark: The bound ∥A + S ∥ ≤ ∥A + I∥ is due to K. Fan and A. Hoffman. Fact 11.10.9. Let A, B ∈ Fn×n, assume that A and B are normal, and let ∥ · ∥ be a unitarily

invariant norm on Fn×n. Then,

∥A + B∥ ≤ ∥⟨A⟩ + ⟨B⟩∥,

∥A ⊙ B∥ ≤ ∥⟨A⟩ ⊙ ⟨B⟩∥.

Source: [196, 543, 551], [1450, p. 213], [1457, 1637, 2167], and [2991, p. 378]. Remark: Both inequalities can fail in the case where A and B are not both normal. Furthermore, there exist A, B ∈ Fn×n such that ⟨A + B⟩ ≤ ⟨A⟩ + ⟨B⟩ does not hold. Related: Fact 10.11.34. Fact 11.10.10. Let A, B ∈ Fn×n, assume that A and B are positive semidefinite, and let ∥ · ∥ be a unitarily invariant norm on Fn×n. If r ∈ [0, 1], then

∥(A + B)r ∥ ≤ ∥Ar + Br ∥. Furthermore, if r ∈ [1, ∞), then

∥Ar + Br ∥ ≤ ∥(A + B)r ∥.

Source: [106, 469, 1383, 1479]. Fact 11.10.11. Let A, B ∈ Fn×n, and let ∥ · ∥ be a unitarily invariant norm on Fn×n. If p ∈ (0, 1],

then If p ∈ [1, ∞), then

∥⟨A + B⟩ p ∥1/p ≤ 21/p−1 (∥⟨A⟩ p ∥1/p + ∥⟨B⟩ p ∥1/p ). ∥⟨A + B⟩ p ∥1/p ≤ ∥⟨A⟩ p ∥1/p + ∥⟨B⟩ p ∥1/p

and ∥⟨·⟩ p ∥1/p is a unitarily invariant norm on Fn×n. Source: [1383, 2436]. Fact 11.10.12. Let A1 , . . . , Al ∈ Nn , let p > 0, let ∥ · ∥ be a unitarily invariant norm on Fn×n, and ∑ define f : 7→ [0, ∞) by f (t) = ∥( ni=1 Ati ) p ∥. Then, f is convex. Source: [1383]. △

Fact 11.10.13. Let A, B ∈ Fn , assume that A and B are positive semidefinite, define µ(A, B) =

limr→∞ [ 12 (Ar + Br )]1/r , and let ∥ · ∥ be a unitarily invariant norm on Fn×n. Then, the following statements hold: △ i) Define f : (0, 1] 7→ R by f (r) = ∥(Ar + Br )1/r ∥. Then, f is nonincreasing. △ ii) Assume that ∥ · ∥ is normalized, and define f : (0, ∞) 7→ R by f (r) = ∥Ar + Br ∥1/r . Then, f is nonincreasing. iii) limr→∞ ∥Ar + Br ∥1/r = σmax [µ(A, B)]. Source: [1383]. Related: Fact 10.11.57.

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Fact 11.10.14. Let A, B ∈ Fn×n. Then,

∥⟨A⟩ − ⟨B⟩∥F ≤

√ 2∥A − B∥F ,

∥⟨A⟩ − ⟨B⟩∥2F + ∥⟨A∗ ⟩ − ⟨B∗ ⟩∥2F ≤ 2∥A − B∥2F . If, in addition, A and B are normal, then ∥⟨A⟩ − ⟨B⟩∥F ≤ ∥A − B∥F . Source: [93, 164], [1391, pp. 217, 218], [1623, 1639], and [2991, pp. 319, 320]. Fact 11.10.15. Let A, B ∈ Fn×n, assume that A and B are positive semidefinite, and let ∥ · ∥ be a

unitarily invariant norm on Fn×n. If r ∈ [0, 1], then

∥Ar − Br ∥ ≤ ∥⟨A − B⟩r ∥. Furthermore, if r ∈ [1, ∞), then

∥⟨A − B⟩r ∥ ≤ ∥Ar − Br ∥.

In particular,

∥(A − B)2 ∥ ≤ ∥A2 − B2 ∥.

Source: [449, pp. 293, 294] and [1631, 2438]. Fact 11.10.16. Let A, B ∈ Fn×n, assume that A and B are positive semidefinite, let α ≥ 1, and

let p ∈ [1/α, ∞]. Then,

∥A − B∥ασαp ≤ ∥Aα − Bα ∥σp .

In particular, if p ∈ [1/2, ∞], then ∥A − B∥2σ2p ≤ ∥A2 − B2 ∥σp . Source: [1381, p. 260] and [1624]. Credit: The case where α = 2 and p = 1 is due to R. T. Powers and E. Stormer. Related: Fact 10.14.32. Fact 11.10.17. Let A, B ∈ Fn×n and p ∈ [2, ∞]. Then,

∥⟨A⟩ − ⟨B⟩∥2σp ≤ ∥A + B∥σp ∥A − B∥σp . Source: [1639, 2438]. Fact 11.10.18. Let A, B ∈ Fn×n and p ≥ 1. Then,

√ ∥⟨A⟩ − ⟨B⟩∥σp ≤ max {21/p−1/2, 1} ∥A + B∥σp ∥A − B∥σp .

Source: [93, 448]. Credit: F. Kittaneh, H. Kosaki, and R. Bhatia. Fact 11.10.19. Let A, B ∈ Fn×n, and let ∥ · ∥ be a unitarily invariant norm on Fn×n . Then,

∥⟨A⟩ − ⟨B⟩∥2 ≤ 2∥A + B∥∥A − B∥. Source: [93, 448]. Credit: The case where ∥ · ∥ = ∥ · ∥σ1 is due to H. J. Borchers and H. Kosaki.

See [1639]. Fact 11.10.20. Let A, B ∈ Fn×n, and let ∥ · ∥ be a unitarily invariant norm on Fn×n . Then,

∥(⟨A⟩ − ⟨B⟩)2 ∥ ≤ σmax (A + B)∥A − B∥. Source: [466, 2438]. Fact 11.10.21. Let A, B ∈ Fn×m, and assume that A and B are contractive, Then, for all i ∈

{1, . . . , n},

2σi [(I − A∗B)−1 ] ≤ σi [(I − A∗A)−1 + (I − B∗B)−1 ].

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Now, let ∥ · ∥ be a unitarily invariant norm on Fm×m. Then, 2∥(I − A∗B)−1 ∥ ≤ ∥(I − A∗A)−1 + (I − B∗B)−1 ∥. Source: [1840, 1852]. Fact 11.10.22. Let A, B ∈ Fn×n. Then,

σmax (⟨A⟩ − ⟨B⟩) ≤

[ ] σmax (A) + σmax (B) 2 2 + log σmax (A − B). π σmax (A − B)

Credit: T. Kato. See [1639]. Fact 11.10.23. Let A, B ∈ Fn×n, assume that A and B are Hermitian, let ∥ · ∥ be a unitarily

invariant norm on Fn×n, and let k ∈ N. Then,

∥(A − B)2k+1 ∥ ≤ 22k∥A2k+1 − B2k+1∥. Source: [449, p. 294] and [1535]. Fact 11.10.24. Let A, B ∈ Fn×n, assume that A and B are positive semidefinite, let ∥ · ∥ be a

unitarily invariant norm on Fn×n, and let z ∈ F. Then,

∥A − |z|B∥ ≤ ∥A + zB∥ ≤ ∥A + |z|B∥. In particular,

∥A − B∥ ≤ ∥A + B∥.

Source: [469, 474]. Related: Fact 10.22.20. Fact 11.10.25. Let ∥ · ∥ be a normalized unitarily invariant norm on Fn×n. Then, ∥ · ∥ is submultiplicative. Source: [449, p. 94]. Fact 11.10.26. Let A ∈ Fn×m and B ∈ Fm×l. Then, ∥AB∥∞ ≤ m∥A∥∞ ∥B∥∞ . Furthermore, if

A = 1n×m and B = 1m×l, then equality holds. △ n×n Fact 11.10.27. ∥·∥′∞ = n∥·∥∞ is a submultiplicative norm [ ] on F . Remark: It is not necessarily true that ∥AB∥∞ ≤ ∥A∥∞ ∥B∥∞ . For example, let A = B = 11 11 . Fact 11.10.28. Let A, B ∈ Fn×n, and let ∥ · ∥ be a submultiplicative norm on Fn×n. Then, ∥AB∥ ≤ ∥A∥∥B∥. If ∥A∥ ≤ 1 and ∥B∥ ≤ 1, then ∥AB∥ ≤ 1, and, if ∥A∥ < 1 and ∥B∥ < 1, then < 1. [ ∥AB∥ ] Remark: ρmax (A) < 1 and ρmax (B) < 1 do not imply that ρmax (AB) < 1. Let A = BT = 00 20 . Fact 11.10.29. Let ∥ · ∥ be a norm on Fm×m, and let { } ∥AB∥ δ > sup : A, B ∈ Fm×m , A, B , 0 . ∥A∥∥B∥ △

Then, ∥ · ∥′ = δ∥ · ∥ is a submultiplicative norm on Fm×m. Source: [1448, p. 323]. Fact 11.10.30. Let A, B ∈ Fn×n, assume that A and B are Hermitian, assume that −B ≤ A ≤ B, and let ∥ · ∥ be a unitarily invariant norm on Fn×n. Then, ∥A∥ ≤ ∥B∥. If, in addition, A and B are positive semidefinite, then ∥A − B∥ ≤ ∥A + B∥. Source: [474]. For the second statement, note that −2A ≤ 0 ≤ 2B, which implies that −(A + B) ≤ A − B ≤ A + B. Related: Fact 11.10.31. Fact 11.10.31. Let A, B ∈ Fn×m, where AA∗ ≤ BB∗ . Then, ∥AA∗ ∥F ≤ ∥BB∗ ∥F and ∥A∥F ≤ ∥B∥F . Source: Fact 11.10.30. Related: Fact 10.11.32 and Fact 11.9.12. Fact 11.10.32. Let A, B ∈ Fn×n, assume that AB is normal, and let ∥ · ∥ be a unitarily invariant norm on Fn×n. Then, ∥AB∥ ≤ ∥BA∥. Source: [449, p. 253] and [2991, p. 378]. Fact 11.10.33. Let A, B ∈ Fn×n, assume that A and B are positive semidefinite and nonzero, and let ∥ · ∥ be a submultiplicative unitarily invariant norm on Fn×n. Then, ∥A + B∥ ∥AB∥ ≤ , ∥A∥∥B∥ ∥A∥ + ∥B∥

∥A ⊙ B∥ ∥A + B∥ ≤ , ∥A∥∥B∥ ∥A∥ + ∥B∥

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NORMS

∥A ⊙ B∥ ∥(A + B) ⊙ I∥ ≤ , ∥A∥∥B∥ ∥A∥ + ∥B∥

∥A ⊗ B∥ ∥A ⊕ B∥ ≤ . ∥A∥∥B∥ ∥A∥ + ∥B∥

Source: [26, 1382]. Related: Fact 11.9.58. Fact 11.10.34. Let A, B ∈ Fn×n, and let ∥ · ∥ be a unitarily invariant norm on Fn×n. Then,

∥AB∥ ≤ σmax (A)∥B∥, Consequently, if C ∈ F

∥AB∥ ≤ ∥A∥σmax (B).

, then

n×n

∥ABC∥ ≤ σmax (A)∥B∥σmax (C). Source: [1631] and [2979, pp. 43, 101]. Related: Fact 11.9.58. Fact 11.10.35. Let A, B ∈ Fn×m, and let ∥ · ∥ be a unitarily invariant norm on Fm×m. If p > 0,

then

∥⟨A∗B⟩ p ∥2 ≤ ∥(A∗A) p ∥∥(B∗B) p ∥.

In particular, ∥(A∗BB∗A)1/4 ∥2 ≤ ∥⟨A⟩∥∥⟨B⟩∥, Furthermore,

∥⟨A∗B⟩∥ = ∥A∗B∥2 ≤ ∥A∗A∥∥B∗B∥.

[tr (A∗BB∗A)1/4 ]2 ≤ (tr ⟨A⟩) tr ⟨B⟩,  | tr A∗B|    ∗ √ √   ≤ tr ⟨A B⟩ ≤ ∥A∥F ∥B∥F . ∗ 2 ∗ ∗ | tr (A B) | ≤ tr AA BB 

Source: [1453], Fact 11.9.44, Fact 11.14.1, and Fact 11.14.2. Fact 11.10.36. Let A, B ∈ Fn×n, and assume that A and B are positive semidefinite. Then,

∥AB∥1/2 F

  1/2 2 2 1/2 2 2 1/2      (2∥A∥F∥B∥F ) ≤ (∥A∥F + ∥B∥F ) = ∥(A + B ) ∥F  ≤    1/2 2 1/2   2∥AB∥ ≤ ∥(A + B) ∥ F

≤ ∥A + B∥F ≤ ∥A∥F + ∥B∥F ≤

√

F

2(∥A∥2F

+ ∥B∥2F )1/2.

2 1/2 Source: 2∥AB∥1/2 F ≤ ∥(A + B) ∥F follows from Fact 11.10.43. Fact 11.10.37. Let A, B ∈ Fn×n, and assume that A and B are positive semidefinite. Then,

∥AB + BA∥F ≤ 2∥AB∥F ≤ ∥A2 + B2 ∥F , ∥A2B + B2A∥F ≤ ∥A2B + AB2 ∥F ≤ ∥A3 + B3 ∥F , ∥A3B + B3A∥F ≤ ∥A3B + AB3 ∥F ≤ ∥A4 + B4 ∥F . Now, let p, q ∈ (0, ∞), and assume that 1/4 ≤ p/(p + q) ≤ 3/4. Then, ∥ApBq + B pAq ∥F ≤ ∥ApBq + AqB p ∥F . In particular, if p ∈ [1/3, 3], then ∥ApB + B pA∥F ≤ ∥ApB + AB p ∥F . Source: [456, 1353] and use Fact 11.10.46. Fact 11.10.38. Let A, B ∈ Fn×m, let p, q ∈ [1, ∞], let r ∈ [1, ∞), assume that 1/p + 1/q = 1/r,

and let ∥ · ∥ be a unitarily invariant norm on Fm×m. Then,

∥⟨A∗B⟩r ∥1/r ≤ ∥⟨A⟩ p ∥∥1/p ∥⟨B⟩q ∥1/q .

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In particular,

∥⟨A∗B⟩1/2 ∥2 ≤ ∥A∥∥B∥,

∥A∗B∥ ≤ (∥A∗A∥∥B∗B∥)1/2 .

Source: [449, p. 95] and [1452]. Remark: This is H¨older’s inequality for unitarily invariant norms. Related: Fact 2.12.23 and Fact 11.8.28. Fact 11.10.39. Let A, B ∈ Fn×n, let ∥ · ∥ be a unitarily invariant norm on Fn×n, let p, q ∈ (0, ∞),

and assume that

1 p

+

1 q

= 1. Then,

∥AB∥ pq = ∥AB∥ p+q ≤ ∥⟨A⟩ p ⟨B⟩q ∥. Source: [2991, p. 378]. Fact 11.10.40. Let A ∈ Fn×n, assume that A is positive definite, and let B ∈ Fn×m. Then,

∥B∗ABB∗A−1B∥F ≤

( ) n λ1 λn + ∥B∗B∥2F . 2 λn λ1

Source: [2826]. Fact 11.10.41. Let A, B ∈ Rn×n, and assume that A and B are positive definite. Then,

√n (tr A) det B ≤ ∥A∥F ∥B∥F .

Source: [1809]. Fact 11.10.42. Let A, B ∈ Fn×n, assume that A and B are positive definite, and define △

Cos(A, B) = Then,

Cos(A, I) ≤ 1,

tr AB . ∥A∥F ∥B∥F

Cos(A, I)Cos(B, I) ≤ 12 [Cos(A, B) + 1],

Cos(A, A−1 ) ≤ Cos(A, I)Cos(A−1 , I) ≤ 12 [Cos(A, A−1 ) + 1] ≤ 1. Source: [1809, 2751]. Fact 11.10.43. Let A, B ∈ Fn×n, and let ∥ · ∥ be a unitarily invariant norm on Fn×n. Then,

∥AB∥ ≤ 14 ∥(⟨A⟩ + ⟨B∗ ⟩)2 ∥. Source: [471]. Fact 11.10.44. Let A, B ∈ Fn×n, and assume that A and B are positive semidefinite. Then, for

all i ∈ {1, . . . , n},

σi (AB) ≤ 41 σi [(A + B)2 ].

Now, let ∥ · ∥ be a unitarily invariant norm on Fn×n. Then, ∥⟨AB⟩1/2 ∥ ≤ 21 ∥A + B∥,

∥AB∥ ≤ 41 ∥(A + B)2 ∥,

∥⟨A3/4B3/4 ⟩2/3 ∥ ≤ 21 ∥A + B∥.

Source: [471, 2935] and [2977, p. 77]. Fact 11.10.45. Let A, B ∈ Fn×n, assume that A and B are positive semidefinite, let p ∈ [0, 1],

and let ∥ · ∥ be a unitarily invariant norm on Fn×n. Then,

∥A1/2B1/2 ∥ ≤ 21 ∥ApB1−p + A1−pBp ∥ ≤ 21 ∥A + B∥, ∥AB∥ ≤ 14 ∥(A + B)2 ∥ ≤ 12 ∥A2 + B2 ∥, ∥AB + BA∥ ≤ ∥AB∥ + ∥BA∥ = 2∥AB∥ ≤ ∥A2 + B2 ∥, ∥(A + B)2 ∥ ≤ ∥A2 + B2 ∥ + ∥AB + BA∥ ≤ 2∥A2 + B2 ∥.

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NORMS

Source: Let p = 1/2 and X = I in Fact 11.10.82. The second inequality follows from Fact 11.10.44. Remark: Fact 11.9.6 implies that ∥ · ∥ is self adjoint, and thus ∥BA∥ = ∥(BA)∗ ∥ = ∥AB∥. Related:

Fact 10.22.18. Fact 11.10.46. Let A, B ∈ Fn×n, assume that A and B are positive semidefinite, let p, q ∈ (0, ∞),

and let ∥ · ∥ be a unitarily invariant norm on Fn×n. Then,

∥ApBq + AqB p ∥ ≤ ∥Ap+q + B p+q ∥. Source: Use Fact 11.10.82 with X = I, A replaced by A p+q , B replaced by B p+q , and p replaced by p/(p + q). Remark: See [1353]. Fact 11.10.47. Let A, B ∈ Fn×n, assume that A and B are positive semidefinite, let α ∈ [0, 1],

and let ∥ · ∥ be a unitarily invariant norm on Fn×n. Then,

∥AαB1−α + B1−αAα ∥ ≤ 2∥αA + (1 − α)B∥, ∥A1/2B1/2 (AαB1−α + A1−αBα )∥ ≤ 12 ∥(A + B)2 ∥. ∥B1/2A1/2 (AαB1−α + A1−αBα )∥ ≤ 21 ∥(A + B)2 ∥. In particular, ∥(A1/2B1/2 )2 ∥ ≤ 41 ∥(A + B)2 ∥,

∥A1/2B1/2 (A + B)∥ ≤ 21 ∥(A + B)2 ∥,

∥B1/2AB1/2 ∥ ≤ 41 ∥(A + B)2 ∥.

Source: [3017]. Fact 11.10.48. Let A ∈ Fn×m, B ∈ Fm×l, and p, q, q′, r ∈ [1, ∞], and assume that 1/q + 1/q′ = 1.

Then, where

∥AB∥ p ≤ ε pq (n)ε pr (l)εq′r (m)∥A∥q ∥B∥r ,    p ≥ q, 1, ε pq (n) =   n1/p−1/q, q ≥ p. △

Furthermore, there exist A ∈ Fn×m and B ∈ Fm×l such that equality holds. Source: [1191]. Remark: Related results are given in [1030, 1031, 1191, 1192, 1193, 1642, 2688]. Fact 11.10.49. Let A, B ∈ Fn×n, assume that A and B are positive semidefinite, let ∥ · ∥ be a unitarily invariant norm on Fn×n, and let p ∈ [0, ∞). If p ∈ [0, 1], then ∥ApB p ∥ ≤ ∥AB∥ p . If p ∈ [1, ∞), then

∥AB∥ p ≤ ∥ApB p ∥.

Source: [458, 1117]. Related: Fact 10.22.36. Fact 11.10.50. Let A, B ∈ Fn×n, assume that A and B are positive semidefinite, and let ∥ · ∥ be a

unitarily invariant norm on Fn×n. If p ∈ [0, 1], then

∥B pApB p ∥ ≤ ∥(BAB) p ∥. Furthermore, if p ≥ 1, then

∥(BAB) p ∥ ≤ ∥B pApB p ∥.

Source: [163] and [449, p. 258]. Remark: Extensions and a reverse inequality are given in Fact 10.11.78. Related: Fact 10.14.24 and Fact 10.22.36. Fact 11.10.51. Let A, B ∈ Fn×n, assume that A and B are positive semidefinite, and let either

p = 1 or p ∈ [2, ∞]. Then,

∥⟨AB⟩1/2 ∥σp ≤ 21 ∥A + B∥σp .

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Source: [196, 471]. Remark: This inequality holds for all Q-norms. See [449]. Related: Fact

10.22.18. Fact 11.10.52. Let A ∈ Fn×m, B ∈ Fm×l, and r ∈ {1, 2}. Then,

∥AB∥σr = ∥A∥σ2r ∥B∥σ2r if and only if there exists α ≥ 0 such that AA∗ = αB∗B. Furthermore, ∥AB∥∞ = ∥A∥∞ ∥B∥∞ if and only if AA∗ and B∗B have a common eigenvector associated with λ1 (AA∗ ) and λ1 (B∗B). Source: [2896]. Related: Fact 7.16.2. Fact 11.10.53. Let A, B ∈ Fn×n , assume that A and B are positive semidefinite, let p ∈ (0, ∞), △ and define ∥A∥σp = (tr A p )1/p . If p ∈ (0, 1], then p p p p p 2 p−1 (∥A∥σp + ∥B∥σp ) ≤ (∥A∥σp + ∥B∥σp ) p ≤ ∥A + B∥σp ≤ ∥A∥σp + ∥B∥σp .

If p ∈ [1, ∞), then p p p p p ∥A∥σp + ∥B∥σp ≤ ∥A + B∥σp ≤ (∥A∥σp + ∥B∥σp ) p ≤ 2 p−1 (∥A∥σp + ∥B∥σp ).

In particular, 1 2 (∥A∥F

∥A∥σ1 + ∥B∥σ1 ≤ ∥A + B∥σ1 ≤ ∥A∥σ1 + ∥B∥σ1 , + ∥B∥F ) ≤ ∥A∥2F + ∥B∥2F ≤ ∥A + B∥2F ≤ (∥A∥F + ∥B∥F )2 ≤ 2(∥A∥2F + ∥B∥2F ). 2

Source: Fact 10.14.36 and Fact 11.10.59. Remark: For all p ∈ (0, 1), ∥ · ∥σp is an anti-norm on Nn . See [545]. Remark: The first inequality in the second string is the McCarthy inequality given by Fact 10.14.36. Related: Fact 11.8.21, Fact 11.9.47, Fact 11.10.4, and Fact 11.10.6. Fact 11.10.54. Let A, B ∈ Fn×n and p, q ∈ (0, ∞). Then, the following statements hold: i) If p ∈ (0, 2], then p p p p p p 2 p−1 (∥A∥σp + ∥B∥σp ) ≤ ∥A + B∥σp + ∥A − B∥σp ≤ 2(∥A∥σp + ∥B∥σp ).

ii) If p ∈ [2, ∞), then p p p p p p 2(∥A∥σp + ∥B∥σp ) ≤ ∥A + B∥σp + ∥A − B∥σp ≤ 2 p−1 (∥A∥σp + ∥B∥σp ).

iii) If p ∈ (1, 2] and 1/p + 1/q = 1, then p p q/p + ∥B∥σp ) . ∥A + B∥qσp + ∥A − B∥qσp ≤ 2(∥A∥σp

iv) If p ∈ [2, ∞) and 1/p + 1/q = 1, then p p q/p 2(∥A∥σp + ∥B∥σp ) ≤ ∥A + B∥qσp + ∥A − B∥qσp .

Source: [1420]. Remark: These are versions of the Clarkson inequalities. See Fact 2.21.8. Remark: See [1420] for extensions to unitarily invariant norms. See [472] for further extensions. Remark: For p ∈ (0, 1), ∥ · ∥σp is defined in Fact 11.9.10. Fact 11.10.55. Let A, B ∈ Fn×n and p ∈ (0, 1]. Then,

∥A + B∥σp ≤ 2(1−p)/p (∥A∥σp + ∥B∥σp ). Source: [2438]. Remark: For p ∈ (0, 1), ∥ · ∥σp is defined in Fact 11.9.10. ∑ Fact 11.10.56. Let A1 , . . . , Al ∈ Hn , let p1 , . . . , pl ∈ (0, 1], assume that li=1 pi = 1, let r ∈ (1, 2],

and let ∥ · ∥ be a unitarily invariant norm on Fn×n. Then,



⟨ ⟩r



∑ l

∑



l 1

Ai

≤

⟨Ai ⟩r

.

i=1

i=1 pi

877

NORMS

Source: [1984]. Related: Fact 2.21.23. Fact 11.10.57. Let A1 , . . . , Ak ∈ Fn×n , let p ∈ [2, ∞), and let ∥ · ∥ be a unitarily invariant norm

on Fn×n . Then,



 1/p

1/2

k k

∑

∑  

 

 ⟨Ai ⟩2 

≤ n1/2−1/p

 ⟨Ai ⟩ p 

.  





i=1 i=1

Source: [2436]. Fact 11.10.58. Let A1 , . . . , Ak ∈ Fn×n be positive semidefinite, and let ∥·∥ be a unitarily invariant

norm on Fn×n . Then, the following statements hold: i) If p ∈ (0, 1], then



  p

 p



k k k   ∑

∑

∑ p

1−p

 Ai 

. Ai

≤ k

 Ai 

≤

i=1

i=1

i=1 ii) If p ∈ [1, ∞), then







  p

k k k 

∑

∑



∑ p p

p−1  

Ai

. Ai 

≤ k

Ai

≤



i=1

i=1 i=1

Source: [2436]. Fact 11.10.59. Let A1 , . . . , Ak ∈ Fn×n be positive semidefinite, and let p ∈ [0, ∞). If p ∈ [0, 1],

then k p−1

n ∑

p  p

n  k n ∑  ∑

∑

p ≤  ∥Ai ∥σp  ≤

Ai

≤ ∥Ai ∥σp .

i=1 i=1 i=1

p ∥Ai ∥σp

σp

i=1

If p ∈ [1, ∞), then k ∑



p p  k k k ∑  ∑

∑

p p−1    

≤ ∥Ai ∥σp . Ai ≤  ∥Ai ∥σp  ≤ k

i=1

i=1 i=1

p ∥Ai ∥σp

σp

i=1

Source: [1423, 1629]. Remark: The first inequality in the second string extends the McCarthy inequality given by Fact 10.14.36 to more than two matrices. Related: Fact 2.2.59 and Fact 2.11.90. Fact 11.10.60. Let A, B ∈ Fn×n, and let ∥ · ∥ be a unitarily invariant norm on Fn×n . Then, the following statements hold: i) If p ∈ (0, 1], then

∥⟨A⟩ p ∥1/p + ∥⟨B⟩ p ∥1/p ≤ 21/p−1 (∥⟨A + B⟩ p ∥1/p + ∥⟨A − B⟩ p ∥1/p ). ii) If p ∈ [1, ∞), then ∥⟨A⟩ p ∥1/p + ∥⟨B⟩ p ∥1/p ≤ ∥⟨A + B⟩ p ∥1/p + ∥⟨A − B⟩ p ∥1/p . iii) If p ∈ [1, ∞), then ∥⟨A + B⟩ p + ⟨A − B⟩ p ∥1/p ≤ ∥⟨A + B⟩ p ∥1/p + ∥⟨A − B⟩ p ∥1/p . iv) If p ∈ [2, ∞), then

2∥⟨A⟩ p + ⟨B⟩ p ∥ ≤ ∥⟨A + B⟩ p + ⟨A − B⟩ p ∥.

Source: [2436]. Fact 11.10.61. Let A1 , . . . , Ak , B1 , . . . , Bk ∈ Fn×n. Then, k ∑

⟨Ai − A j ⟩2 +

i, j=1

k ∑

⟨Bi − B j ⟩2 + 2

i, j=1

⟨∑ ⟩2 k k ∑ (Ai − Bi ) = 2 ⟨Ai − B j ⟩2 , i=1

i, j=1

878

CHAPTER 11 k ∑

σmax (Ai − A j )2 +

i, j=1

k ∑ i, j=1

2  k k ∑  ∑ σmax (Ai − B j )2 , σmax (Bi − B j )2 + 2σmax  (Ai − Bi ) = 2 i, j=1

i=1

2

k k k k ∑ ∑ ∑

∑ 2 2 ∥Ai − B j ∥2F . ∥Ai − A j ∥F + ∥Bi − B j ∥F + 2

(Ai − Bi )

= 2

i, j=1 i=1 i, j=1 i, j=1 F

If p ∈ (0, 2], then 2k

p−2

k ∑

∥Ai −

i, j=1

p B j ∥σp

≤

k ∑

∥Ai −

p A j ∥σp

+

i, j=1

k ∑

∥Bi −

p B j ∥σp

σp

i, j=1

≤ 2(k2 − k + 1)1−p/2

k ∑

p

k

∑ + 2

(Ai − Bi )



i=1

p ∥Ai − B j ∥σp .

i, j=1

If p ∈ [2, ∞), then 2(k2 − k + 1)1−p/2

k ∑ i, j=1

p ∥Ai − B j ∥σp



p k ∑

p p ≤ ∥Ai − A j ∥σp + ∥Bi − B j ∥σp + 2

(Ai − Bi )

i=1

i, j=1 i, j=1 k ∑

k ∑

σp

≤ 2k p−2

k ∑

p ∥Ai − B j ∥σp .

i, j=1

Source: [2093, 2096]. Related: Fact 10.11.84. Fact 11.10.62. Let A1 , . . . , Ak , B1 , . . . , Bk ∈ Fn×n , let p ∈ [2, ∞), and let ∥ · ∥ be a unitarily

invariant norm on Fn×n . Then,

 1/p



 k 1/p



 k 1/p

k



∑

∑



∑   

       k−1+1/p

 ⟨Ai + Bi ⟩ p 

≤

 ⟨Ai ⟩ p 

+

 ⟨Bi ⟩ p 







i=1 i=1 i=1



 1/p  k 1/p

 k 

∑ 



∑ 

  ≤ k1−1/p 

 ⟨Ai + Bi ⟩ p 

+

 ⟨Ai − Bi ⟩ p 

 ,





i=1 i=1

k

−1+1/p



1/p

 1/p

1/p



 k k k

∑ 





∑

∑

 p p p

⟨Ai + Bi ⟩

≤

 ⟨Ai ⟩ 

+

 ⟨Bi ⟩ 

,







i=1

i=1 i=1







1/p

1/p 1/p



 k 1/p

 k k k 

∑



∑ ∑  



∑



⟨Ai ⟩ p

+

⟨Bi ⟩ p

≤ k1−1/p 

 ⟨Ai + Bi ⟩ p 

+

 ⟨Ai − Bi ⟩ p 

 . 





i=1

i=1

i=1 i=1 Source: [2436]. Fact 11.10.63. Let A1 , . . . , Ak , B1 , . . . , Bk ∈ Fn×n be positive semidefinite, and let ∥ · ∥ be a

unitarily invariant norm on Fn×n . Then, the following statements hold: i) If p ∈ [1, ∞), then

1/p

1/p

1/p

k k k

∑

∑

∑

p p −|1/p−1/2| p

⟨Ai + Bi ⟩

≤

⟨Ai ⟩

+

⟨Bi ⟩

k

i=1

i=1

i=1

879

NORMS



1/p 

1/p

k k  ∑ ∑





≤ k|1/p−1/2| 

⟨Ai + Bi ⟩ p

+

⟨Ai − Bi ⟩ p

 .

i=1

i=1 ii) If p ∈ (0, 1], then

1/p

1/p

1/p

k k k ∑ ∑ ∑







21−1/p k−|1/p−1/2|

⟨Ai + Bi ⟩ p

≤

⟨Ai ⟩ p

+

⟨Bi ⟩ p



i=1

i=1

i=1 

1/p 

1/p

k k  ∑ ∑



≤ 21/p−1 k|1/p−1/2| 

⟨Ai + Bi ⟩ p

+

⟨Ai − Bi ⟩ p

 .

i=1

i=1 Source: [2436]. Fact 11.10.64. Let A, B ∈ Cn×m. If p ∈ [1, 2], then p p + ∥A − B∥σp )]1/p. [∥A∥2σp + (p − 1)∥B∥2σp ]1/2 ≤ [ 12 (∥A + B∥σp

If p ∈ [2, ∞], then p p [ 12 (∥A + B∥σp + ∥A − B∥σp )]1/p ≤ [∥A∥2σp + (p − 1)∥B∥2σp ]1/2.

Source: [262, 342]. Remark: This is Beckner’s two-point inequality (also called optimal 2-

uniform convexity). Fact 11.10.65. Let A, B ∈ Fn×n. If either p ∈ [1, 4/3] or both p ∈ (4/3, 2] and A + B and A − B

are positive semidefinite, then p p (∥A∥σp + ∥B∥σp ) p + |∥A∥σp − ∥B∥σp | p ≤ ∥A + B∥σp + ∥A − B∥σp .

Furthermore, if either p ∈ [4, ∞] or both p ∈ [2, 4) and A and B are positive semidefinite, then p p ∥A + B∥σp + ∥A − B∥σp ≤ (∥A∥σp + ∥B∥σp ) p + |∥A∥σp − ∥B∥σp | p.

Source: [262, 1617]. Remark: These are matrix versions of Hanner’s inequality. Vector versions

are given in Fact 11.8.23. Fact 11.10.66. Let A, B ∈ Cn×n, and assume that A and B are Hermitian. If p ∈ [1, 2], then 21/2−1/p ∥(A2 + B2 )1/2 ∥σp ≤ ∥A + ȷB∥σp ≤ ∥(A2 + B2 )1/2 ∥σp , 21−2/p (∥A∥2σp + ∥B∥2σp ) ≤ ∥A + ȷB∥2σp ≤ 22/p−1 (∥A∥2σp + ∥B∥2σp ). Furthermore, if p ∈ [2, ∞), then ∥(A2 + B2 )1/2 ∥σp ≤ ∥A + ȷB∥σp ≤ 21/2−1/p ∥(A2 + B2 )1/2 ∥σp , 22/p−1 (∥A∥2σp + ∥B∥2σp ) ≤ ∥A + ȷB∥2σp ≤ 21−2/p (∥A∥2σp + ∥B∥2σp ). Source: [470]. Fact 11.10.67. Let A, B ∈ Cn×n, and assume that A and B are Hermitian. If p ∈ [1, 2], then p p p 21−2/p (∥A∥σp + ∥B∥σp ) ≤ ∥A + ȷB∥σp .

If p ∈ [2, ∞], then In particular, Source: [470, 480].

p p p ∥A + ȷB∥σp ≤ 21−2/p (∥A∥σp + ∥B∥σp ).

∥A + ȷB∥2F = ∥A∥2F + ∥B∥2F = ∥(A2 + B2 )1/2 ∥2F .

880

CHAPTER 11

Fact 11.10.68. Let A, B ∈ Cn×n, and assume that A is positive semidefinite and B is Hermitian.

If p ∈ [1, 2], then

If p ∈ [2, ∞], then In particular,

∥A∥2σp + 21−2/p ∥B∥2σp ≤ ∥A + ȷB∥2σp . ∥A + ȷB∥2σp ≤ ∥A∥2σp + 21−2/p ∥B∥2σp . (tr ⟨A⟩)2 + (tr ⟨B⟩)2 ≤ (tr ⟨A + ȷB⟩)2, ∥A + ȷB∥2F = ∥A∥2F + ∥B∥2F , 2 2 2 σmax (A + ȷB) ≤ σmax (A) + 2σmax (B).

In addition,

∥A∥2σ1 + ∥B∥2σ1 ≤ ∥A + ȷB∥2σ1.

Source: [480]. Fact 11.10.69. Let A, B ∈ Cn×n, and assume that A and B are positive semidefinite. If p ∈ [1, 2],

then

∥A∥2σp + ∥B∥2σp ≤ ∥A + ȷB∥2σp .

If p ∈ [2, ∞], then Hence,

∥A + ȷB∥2σp ≤ ∥A∥2σp + ∥B∥2σp . ∥A∥2σ2 + ∥B∥2σ2 = ∥A + ȷB∥2σ2 .

In particular,

(tr A)2 + (tr B)2 ≤ (tr ⟨A + ȷB⟩)2, ∥A + ȷB∥2F = ∥A∥2F + ∥B∥2F , 2 2 2 σmax (A + ȷB) ≤ σmax (A) + σmax (A).

Source: [480]. Related: Fact 10.22.14. Fact 11.10.70. Let A, B ∈ Fn×n, assume that A and B are Hermitian, and let ∥ · ∥ be a unitarily

invariant norm on Fn×n. Then,

∥A + B∥ ≤

√ 2∥A + ȷB∥.

Source: [475]. Fact 11.10.71. Let A, B ∈ Fn×n, and let ∥ · ∥ be a unitarily invariant norm on Fn×n. Then,

∥(A + B)(A + B)∗ ∥ ≤ ∥AA∗ + BB∗ + 2AB∗ ∥ ≤

{

∥(A − B)(A − B)∗ + 4AB∗ ∥ (∥A∥ + ∥B∥)2 .

Source: [1983]. Fact 11.10.72. Let A ∈ Fn×n, B ∈ Fm×m, and X ∈ Fn×m , assume that A and B are nonsingular

and either both Hermitian or both skew Hermitian, and let ∥ · ∥ be a unitarily invariant norm on Fn×m. Then, ∥X∥ ≤ 12 ∥AXB−1 + A−1XB∥. Source: Replace X by A−1XB−1 in Fact 11.10.79. See [1449]. Fact 11.10.73. Let A, B ∈ Fn×n, and let ∥ · ∥ be a unitarily invariant norm on Fn×n. Then, the

following statements hold: i) If B is Hermitian, then ∥A − 21 (A + A∗ )∥ ≤ ∥A − B∥. ii) If B is skew Hermitian, then ∥A − 21 (A − A∗ )∥ ≤ ∥A − B∥.

881

NORMS

iii) Let A = MS , where M, S ∈ Fn×n , M is positive semidefinite, and S is unitary. If B is unitary, then ∥A − S ∥ ≤ ∥A − B∥ ≤ ∥A + S ∥. iv) Let A = S 1 DS 2 , where S 1 , D, S 2 ∈ Fn×n , S 1 , S 2 are unitary, and D is nonnegative and diagonal. If B is unitary, then ∥A − S 1 S 2 ∥ ≤ ∥A − B∥ ≤ ∥A + S 1 S 2 ∥. v) If A is normal and B is nonsingular, then ∥A∥ ≤ ∥B−1AB∥. vi) If A is positive semidefinite and B is unitary, then ∥A − I∥ ≤ ∥A − B∥ ≤ ∥A + I∥. Source: [449, p. 275], [2263, p. 150], and [2991, p. 378]. Related: Fact 11.16.26 and Fact 11.16.27 Fact 11.10.74. Let A, B ∈ Fn×m, let X ∈ Fm×m , and let ∥ · ∥ be a unitarily invariant norm on Fn×m. Then, ∥AA+B − B∥ ≤ ∥AX − B∥. Source: [2991, p. 377]. Fact 11.10.75. Let A, B ∈ Fn×m , and let ∥ · ∥ be a unitarily invariant norm on Fn×m . Then,

∥A+ − B+ ∥ ≤ [σmax (A+ )σmax (B+ ) + max {σ2max (A+ ), σ2max (B+ )}]∥A − B∥ ≤ 2 max {σ2max (A+ ), σ2max (B+ )}∥A − B∥. If, in addition, rank A = rank B, then ∥A+ − B+ ∥ ≤ [σmax (A+ )σmax (B+ ) + (σmax (A+ ) + σmax (B+ )) max {σmax (A+ ), σmax (B+ )}]∥A − B∥ ≤ 3σmax (A+ )σmax (B+ )∥A − B∥. Source: [612]. Fact 11.10.76. Let A, B ∈ Fn×n, assume that A is nonsingular and B is Hermitian, and let ∥ · ∥

be a unitarily invariant norm on Fn×n. Then,

∥B∥ ≤ 12 ∥ABA−1 + A−1BA∥. Source: [780, 1093]. Fact 11.10.77. Let A, M, S, B ∈ Fn×n, assume that A = MS, M is positive semidefinite, and S

and B are unitary, and let ∥ · ∥ be a unitarily invariant norm on Fn×n. Then, ∥A − S ∥ ≤ ∥A − B∥. Source: [449, p. 276] and [2263, p. 150]. Remark: A = MS is the polar decomposition of A. See Corollary 7.6.4. Fact 11.10.78. Let A, B ∈ Fn×n, assume that A and B are positive semidefinite, and let ∥ · ∥ be a unitarily invariant norm on F2n×2n. Then,

[ ] [ ] [ ]

A + B 0



A 0



A1/2B1/2

0 ≤ + .

0 0



0 B



0 A1/2B1/2

In particular,

σmax (A + B) ≤ max {σmax (A), σmax (B)} + σmax (A1/2B1/2 ),

and, for all p ∈ [1, ∞), p 1/p p ∥A + B∥σp ≤ (∥A∥σp + ∥B∥σp ) + 21/p ∥A1/2B1/2 ∥σp .

Source: [1629, 1632, 1637]. Remark: Fact 11.16.18 gives a refined upper bound for σmax (A + B). Fact 11.10.79. Let A ∈ Fn×n, B ∈ Fm×m, and X ∈ Fn×m , and let ∥ · ∥ be a unitarily invariant norm

on Fn×m. Then,

In particular, if n = m, then

∥A∗XB∥ ≤ 21 ∥AA∗X + XBB∗ ∥. ∥A∗B∥ ≤ 21 ∥AA∗ + BB∗ ∥.

882

CHAPTER 11

Source: [121, 457, 468, 1119, 1449, 1626]. Remark: The first result is McIntosh’s inequality. Related: Fact 11.16.31. Fact 11.10.80. Let A, X, B ∈ Fn×n, assume that X is positive semidefinite, and let ∥ · ∥ be a

unitarily invariant norm on Fn×n. Then,

∥A∗XB + B∗XA∥ ≤ ∥A∗XA + B∗XB∥. In particular,

∥A∗B + B∗A∥ ≤ ∥A∗A + B∗B∥.

Source: [468, 1630]. Remark: See [1630] for the case of indefinite X. △ Fact 11.10.81. Let A1 , . . . , Am ∈ Fn×n , define Am+1 = A1 , and let ∥ · ∥ be a unitarily invariant

norm on Fn×n. Then,





m m



∑

∑ ∗ ∗

Ai Ai

. Ai Ai+1

≤

i=1

i=1

Source: [2167]. Related: Fact 11.10.80. Fact 11.10.82. Let A, X, B ∈ Fn×n, assume that A and B are positive semidefinite, let p ∈ [0, 1],

and let ∥ · ∥ be a unitarily invariant norm on Fn×n. Then,

∥ApXB1−p + A1−pXB p ∥ ≤ ∥AX + XB∥, ∥ApXB1−p − A1−pXB p ∥ ≤ |2p − 1|∥AX − XB∥. Furthermore, for all i ∈ {1, . . . , n}, σi (ApB1−p + A1−pB p ) ≤ σi (A + B). Source: [121, 458, 477, 1086] and [2979, pp. 87, 88]. See [190, 1640] for the last inequality. Remark: These are the Heinz inequalities. Related: Fact 11.10.46. Fact 11.10.83. Let A, B ∈ Pn , and define △

V = (A−1/2BA−1/2 )1/2A1/2B−1/2 ,

△

U = A1/2B1/2 ⟨A1/2B1/2 ⟩−1 .

Then, U and V are unitary. If, in addition, ∥ · ∥ is a unitarily invariant norm, then ∥A + VBV ∗ ∥ ≤ ∥A + B∥ ≤ ∥A + UBU ∗ ∥. Furthermore,

det(A + UBU ∗ ) ≤ det(A + B) ≤ det(A + VBV ∗ ).

Source: [1779]. Related: Fact 10.11.68. Fact 11.10.84. Let A, B, C, D ∈ Fn×n , and let ∥ · ∥ be a unitarily invariant norm on Fn×n. If

p ∈ [1, ∞), then

∥⟨A + B⟩ p + ⟨C + D⟩ p ∥1/p ≤ 2|1/p−1/2| ∥⟨A⟩ p + ⟨C⟩ p ∥1/p + ∥⟨B⟩ p + ⟨D⟩ p ∥1/p . If p, r ∈ [1, ∞], then (1/p)(1−1/r) p p 1/p ∥⟨A + B⟩ p + ⟨C + D⟩ p ∥1/p ∥⟨A⟩ p + ⟨C⟩ p ∥1/p σr ≤ 2 σr + ∥⟨B⟩ + ⟨D⟩ ∥σr .

If p, r ∈ [1, ∞] and p ≤ r, then ∥(⟨A + B⟩ p + ⟨C + D⟩ p )1/p ∥σr ≤ 2|1/p−1/2| ∥(⟨A⟩ p + ⟨C⟩ p )1/p ∥σr + ∥(⟨B⟩ p + ⟨D⟩ p )1/p ∥σr . If p, r ∈ [1, ∞], then ∥(⟨A + B⟩ p + ⟨C + D⟩ p )1/p ∥σr ≤ 2|1/p−1/r| ∥(⟨A⟩ p + ⟨C⟩ p )1/p ∥σr + ∥(⟨B⟩ p + ⟨D⟩ p )1/p ∥σr . Source: [1383].

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NORMS

Fact 11.10.85. Let A, B, C, D ∈ Fn×n , let p, q ∈ [1, ∞], assume that 1/p + 1/q = 1, and let ∥ · ∥

be a unitarily invariant norm on Fn×n. Then,

∥C ∗A + D∗B∥ ≤ 2|1/p−1/2| ∥⟨A⟩ p + ⟨B⟩ p ∥1/p ∥⟨C⟩q + ⟨D⟩q ∥1/q . If, in addition, r ∈ [1, ∞], then q q 1/q ∥C ∗A + D∗B∥σr ≤ 21−1/r ∥⟨A⟩ p + ⟨B⟩ p ∥1/p σr ∥⟨C⟩ + ⟨D⟩ ∥σr .

Source: [1383]. Fact 11.10.86. Let A, B ∈ Fn×n, assume that A and B are positive semidefinite, and let ∥ · ∥ be a

unitarily invariant norm on Fn×n. Then,

∥ log(I + A) − log(I + B)∥ ≤ ∥ log(I + ⟨A − B⟩)∥, ∥ log(I + A + B)∥ ≤ ∥ log(I + A) + log(I + B)∥. Source: [106] and [449, p. 293]. Related: Fact 15.17.18. Fact 11.10.87. Let A, X, B ∈ Fn×n, assume that A and B are positive definite, and let ∥ · ∥ be a

unitarily invariant norm on Fn×n. Then,

∥(log A)X − X(log B)∥ ≤ ∥A1/2XB−1/2 − A−1/2XB1/2∥. Source: [477]. Related: Fact 15.17.19. Fact 11.10.88. Let A, B ∈ Fn×n, assume that A is nonsingular, let ∥ · ∥ be a normalized submultiplicative norm on Fn×n, and assume that ∥A − B∥ < 1/∥A−1 ∥. Then, B is nonsingular. Related: Fact

11.9.62. Fact 11.10.89. Let A, B ∈ Fn×n, assume that A is nonsingular, let ∥ · ∥ be a normalized sub-

multiplicative norm on Fn×n, let γ > 0, and assume that ∥A−1 ∥ < γ and ∥A − B∥ < 1/γ. Then, B is nonsingular and γ ∥B−1 ∥ ≤ , ∥A−1 − B−1 ∥ ≤ γ2 ∥A − B∥. 1 − γ∥B − A∥ Source: [970, p. 148]. Related: Fact 11.9.62. Fact 11.10.90. Let A, B ∈ Fn×n, let λ ∈ C, assume that λI − A is nonsingular, let ∥ · ∥ be

a normalized submultiplicative norm on Fn×n, let γ > 0, and assume that ∥(λI − A)−1 ∥ < γ and ∥A − B∥ < 1/γ. Then, λI − B is nonsingular and ∥(λI − B)−1 ∥ ≤

γ , 1 − γ∥B − A∥

∥(λI − A)−1 − (λI − B)−1 ∥ ≤

γ2 ∥A − B∥ . 1 − γ∥A − B∥

Source: [970, pp. 149, 150]. Related: Fact 11.10.89. Fact 11.10.91. Let A, B ∈ Fn×n, assume that A and A + B are nonsingular, and let ∥ · ∥ be a

normalized submultiplicative norm on Fn×n. Then,

∥A−1 − (A + B)−1 ∥ ≤ ∥A−1 ∥ ∥(A + B)−1 ∥ ∥B∥. If, in addition, ∥A−1B∥ < 1, then ∥A−1 + (A + B)−1 ∥ ≤

∥A−1 ∥∥A−1B∥ . 1 − ∥A−1B∥

Furthermore, if ∥B∥ < 1/∥A−1 ∥, then ∥A−1 − (A + B)−1 ∥ ≤

∥A−1 ∥2 ∥B∥ . 1 − ∥A−1 ∥∥B∥

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Fact 11.10.92. Let A ∈ Fn×n, assume that A is nonsingular, let E ∈ Fn×n, and let ∥ · ∥ be a

normalized norm on Fn×n. Then, as ∥E∥ → 0,

(A + E)−1 = A−1 (I + EA−1 )−1 = A−1 − A−1EA−1 + O(∥E∥2 ). Fact 11.10.93. Let A ∈ Fn×m and B ∈ Fl×k. Then,

∥A ⊗ B∥col = ∥A∥col∥B∥col , Furthermore, if p ∈ [1, ∞], then Fact 11.10.94. Let A ∈ F

∥A ⊗ B∥∞ = ∥A∥∞ ∥B∥∞ ,

∥A ⊗ B∥row = ∥A∥row∥B∥row .

∥A ⊗ B∥ p = ∥A∥ p ∥B∥ p .

n×m

nonnegative, then

and B ∈ Fl×k. If either p, q ∈ [1, ∞] and p ≤ q or A and B are ∥A ⊗ B∥q,p ≤ ∥A∥q,p ∥B∥q,p .

If p ∈ [1, 2], then

∥A ⊗ B∥ p,2,C ≤ Γ

( p+2 )−1/p 2

∥A∥ p,2,C ∥B∥ p,2,C .

Source: [2040]. Fact 11.10.95. Let A ∈ Fn×m, B ∈ Fl×k, and p ∈ [1, ∞]. Then,

∥A ⊗ B∥σp = ∥A∥σp ∥B∥σp . In particular,

σmax (A ⊗ B) = σmax (A)σmax (B),

∥A ⊗ B∥F = ∥A∥F ∥B∥F .

Source: [1399, p. 722]. Fact 11.10.96. Let A, B ∈ Fn×n, and let ∥ · ∥ be a unitarily invariant norm on Fn×n. Then,

∥A ⊙ B∥2 ≤ ∥A∗A∥∥B∗B∥. If, in addition, k ≥ 1, then

∥(A ⊙ B)k ∥2F ≤ ∥(A∗A)k ∥F ∥(B∗B)k ∥F .

Source: [1452, 2762].

11.11 Facts on Matrix Norms for Commutators △

Fact 11.11.1. Let A, B ∈ Fn×n, and let ∥·∥ be a submultiplicative norm on Fn×n. Then, ∥·∥′ = 2∥·∥ ′

′

′

is a submultiplicative norm on F and satisfies ∥[A, B]∥ ≤ ∥A∥ ∥B∥ . Fact 11.11.2. Let A ∈ Fn×n. Then, the following statements are equivalent: i) There exist projectors Q, P ∈ Rn×n such that A = [P, Q]. ii) σmax (A) ≤ 1/2, A and −A are unitarily similar, and A is skew Hermitian. Source: [1813]. Remark: Extensions are discussed in [1995]. Remark: For F = R, if A is skew symmetric, then A and −A are orthogonally similar. See Fact 7.10.11. Related: Fact 4.22.10 considers idempotent matrices. √ Fact 11.11.3. Let A, B ∈ Rn×n. Then, ∥AB − BA∥F ≤ 2∥A∥F ∥B∥F . Source: [532, 2807]. √ Remark: 2 is valid for all n. Remark: Extensions to complex matrices are given in [533]. Fact 11.11.4. Let A, B ∈ Fn×n, and assume that A and B are positive semidefinite. Then, n×n

∥AB − BA∥2F + ∥(A − B)2 ∥2F ≤ ∥A2 − B2 ∥2F . Source: [1631]. Fact 11.11.5. Let A, B ∈ Fn×n, let p be a positive number, and assume that either A is normal

and p ∈ [2, ∞], or A is Hermitian and p ≥ 1. Then,

∥⟨A⟩B − B⟨A⟩∥σp ≤ ∥AB − BA∥σp .

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NORMS

Source: [1]. Fact 11.11.6. Let ∥ · ∥ be a unitarily invariant norm on Fn×n, and let A, X, B ∈ Fn×n. Then,

∥AX − XB∥ ≤ [σmax (A) + σmax (B)]∥X∥. In particular,

σmax (AX − XA) ≤ 2σmax (A)σmax (X).

Now, assume that A and B are positive semidefinite. Then, ∥AX − XB∥ ≤ max {σmax (A), σmax (B)}∥X∥. In particular,

σmax (AX − XA) ≤ σmax (A)σmax (X).

Finally, assume that A and X are positive semidefinite. Then,

[ ]

X 0

1 . ∥AX − XA∥ ≤ 2 σmax (A)

0 X

In particular,

σmax (AX − XA) ≤ 12 σmax (A)σmax (X).

[

]

[

]

0 and X = 0 1 . Source: [473]. Remark: Equality holds in the first inequality with A = B = 10 −1 −1 0 Remark: ∥ · ∥ can be extended to F2n×2n by considering the n largest singular values of matrices in

F2n×2n. See [449, pp. 90, 98]. Fact 11.11.7. Let ∥ · ∥ be a unitarily invariant norm on Fn×n, let A, X ∈ Fn×n, and assume that A is Hermitian. Then, ∥AX − XA∥ ≤ δ(A)∥X∥.

Source: [473]. Remark: δ(A) = λmax (A) − λmin (A) is the spread of A. Fact 11.11.8. Let ∥ · ∥ be a unitarily invariant norm on Fn×n, let A, X ∈ Fn×n, and assume that A

is normal. Then,

∥AX − XA∥ ≤

√ 2δ(A)∥X∥.

Furthermore, let p ∈ [1, ∞]. Then, ∥AX − XA∥σp ≤ 2|2−p|/(2p) δ(A)∥X∥σp . In particular,

∥AX − XA∥F ≤ δ(A)∥X∥F ,

σmax (AX − XA) ≤

√

2δ(A)σmax (X).

Source: [473]. Remark: δ(A) is the spread of A.

11.12 Facts on Matrix Norms for Partitioned Matrices

[ ∗] and AB BC are positive semidefinite, and let ∥ · ∥ be a unitarily invariant norm on Fn×n. Then, 2∥B∥ ≤ ∥A + C∥. Source: [1840]. Fact 11.12.2. Let A, B ∈ Fn×m, and let ∥ · ∥ be a unitarily invariant norm on F2n×2n. Then,

[

[ ] ]

AA∗ + BB∗ 0



AA∗ + BB∗ AB∗ + BA∗

. ≤ 2



AB∗ + BA∗ AA∗ + BB∗

0 0

Fact 11.12.1. Let A, B, C ∈ Fn×n, assume that

Now, assume that n = m. Then,

[ ] [ 1

A + B 0



A ≤

A + B



0 2 0

0 B

[

A B B∗ C

]

]



[



⟨A⟩

≤

0

0 ⟨B⟩

]



[ ]



⟨A⟩ + ⟨B⟩ 0

,

≤

0 0

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CHAPTER 11

[

AB

0

0 AB

]



[ ∗



A A

≤

0

0 B∗B

]



,

[

A ⊙ B

0

If, in addition, A and B are Hermitian, then

[ ] [

A 0



A + B

0 A

≤

0 Source: [449, p. 97], [2750], and [2991, p. 377]. Fact 11.12.3. Let A, B, C ∈ Fn×n , and assume that

0 A⊙B 0 A−B [

A B B∗ C

]



[ ∗



A A

≤

0

0 B∗B

]



.

]



.

]

is positive semidefinite. Then, the following statements hold: i) Let p > 0, and let ∥ · ∥ be a unitarily invariant norm on F2n×2n . Then,

[ ]p

A B

|p−1| (∥(A + C) p ∥ + ∥⟨B − B∗ ⟩ p ∥).

B∗ C

≤ 2 ii) Assume that B is Hermitian, and let ∥ · ∥ be a unitarily invariant norm on F2n×2n . Then,

[ ] [ ]

A B



A + C 0

.

B∗ C

≤

0 0

iii) Let p ≥ 1, and let ∥ · ∥ be a unitarily invariant norm on F2n×2n . Then,

[ ]

A B

p p 1/p 1−1/p (∥A∥σp + ∥C∥σp ) .

B∗ C

≤ 2 σp

iv) Let nrad(B) denote the numerical radius of B. Then, ([ ]) A B σmax ≤ σmax (A + C) + 2 nrad(B) B∗ C ≤ σmax (A + C) + 2σmax (B) √ ≤ σmax (A) + σmax (C) + 2 σmax (A)σmax (C) ≤ 4 max {σmax (A), σmax (C)}. ([

v) σmax

A B∗

B C

]) ≤ 2 max {σmax (A), σmax (C)}.

Source: [548]. Related: Fact 10.12.58 and Fact 10.12.59. Fact 11.12.4. Let A ∈ Fn×m be the partitioned matrix

  A11  A  21 A =  ..  .  Ak1

A12 A22 .. . Ak2

··· ··· . · .· · ···

 A1k   A2k  ..  , .  Akk

where Ai j ∈ Fni ×n j for all i, j ∈ {1, . . . , k}. Furthermore, define µ(A) ∈ Rk×k by    σmax (A11 ) σmax (A12 ) · · · σmax (A1k )   σmax (A21 ) σmax (A22 ) · · · σmax (A2k )  △   . µ(A) =  .. .. . ..   · .· · . . .   σmax (Ak1 ) σmax (Ak2 ) · · · σmax (Akk ) Finally, let B ∈ Fn×m be partitioned conformally with A. Then, the following statements hold:

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NORMS

For all α ∈ F, µ(αA) ≤ |α|µ(A). µ(A + B) ≤ µ(A) + µ(B). µ(AB) ≤ µ(A)µ(B). ρmax (A) ≤ ρmax [µ(A)]. σmax (A) ≤ σmax [µ(A)]. If k = 2, then nrad(A) ≤ nrad[µ(A)]. Source: [881, 1633, 2177, 2473]. Remark: µ is a matricial norm. Remark: This is a normcompression inequality. Fact 11.12.5. Let A ∈ Fn×m be the partitioned matrix    A11 A12 · · · A1k    A  21 A22 · · · A2k  A =  .. .. . ..  ,  . · .· · . .   Ak1 Ak2 · · · Akk i) ii) iii) iv) v) vi)

where Ai j ∈ Fni ×n j for all i, j ∈ {1, . . . , k}. Then, the following statements hold: i) If p ∈ [1, 2], then k k ∑ ∑ ∥Ai j ∥2σp ≤ ∥A∥2σp ≤ k4/p−2 ∥Ai j ∥2σp . i, j=1

i, j=1

ii) If p ∈ [2, ∞], then k4/p−2

k ∑

∥Ai j ∥2σp ≤ ∥A∥2σp ≤

i, j=1

iii) If p ∈ [1, 2], then p ∥A∥σp ≤

k ∑

∥Ai j ∥2σp .

i, j=1 k ∑

p p ∥Ai j ∥σp ≤ k2−p ∥A∥σp .

i, j=1

iv) If p ∈ [2, ∞), then p k2−p ∥A∥σp ≤

k ∑

p p ∥Ai j ∥σp ≤ ∥A∥σp .

i, j=1

∑k

v) ∥A∥2σ2 = i, j=1 ∥Ai j ∥2σ2 . vi) For all p ∈ [1, ∞],

1/p  k  ∑ p  ∥Aii ∥σp  ≤ ∥A∥σp . i=1

vii) For all i ∈ {1, . . . , k}, σmax (Aii ) ≤ σmax (A). Source: [277, 467]. Fact 11.12.6. For all i, j ∈ {1, . . . , k}, let Ai j ∈ Fn×n be Hermitian, define the partitioned matrix

A ∈ Fkn×kn by

  A11  A  21 △  A =  ..  .  Ak1

A12 A22 .. . Ak2

··· ··· . · .· · ···

 A1k   A2k  ..  , .  Akk

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CHAPTER 11

assume that A is positive semidefinite, and let p ∈ [1, ∞]. Then,



k

∑ Aii

. ∥A∥σp ≤

i=1 σp

Source: [549]. Remark: This is Hiroshima’s theorem. Related: Fact 10.12.58. Fact 11.12.7. For all i, j ∈ {1, 2, 3}, let Ai j ∈ Fn×n , define A ∈ F3n×3n by

  A11  △  A =  A∗12  ∗ A13

A12 A22 A∗23

 A13   A23  ,  A33

assume that A is positive semidefinite, and define B ∈ F3×3 and C ∈ R3×3 by      ∥A11 ∥σ1 ∥A12 ∥σ1 ∥A13 ∥σ1   tr A11 tr A12 tr A13    △  △  B =  tr A∗12 tr A22 tr A23  , C =  ∥A∗12 ∥σ1 ∥A22 ∥σ1 ∥A23 ∥σ1  .   ∗   ∗ ∗ ∗ ∥A13 ∥σ1 ∥A23 ∥σ1 ∥A33 ∥σ1 tr A13 tr A23 tr A33 Then, the following statements hold: i) B, |B|, and C are positive semidefinite. ii) If p ∈ [1, 2], then ∥ |B| ∥σp ≤ ∥B∥σp . iii) If p ∈ [2, ∞], then ∥B∥σp ≤ ∥ |B| ∥σp . Source: [955, 1808, 1849]. Remark: Fact 10.10.2 implies that |B| is positive semidefinite. Related: Fact 10.12.64. Fact 11.12.8. Let A, B ∈ Fn×n, and define A ∈ Fkn×kn by    A B B · · · B     B A B · · · B    .. △  . B  . A =  B B A   ..  ..  .. .. . . . . .   . .   B B B ··· A Then,

σmax (A) = max {σmax (A + (k − 1)B), σmax (A − B)}.

Now, let p ∈ [1, ∞). Then, p p 1/p ∥A∥σp = (∥A + (k − 1)B∥σp + (k − 1)∥A − B∥σp ) .

Source: [277]. Fact 11.12.9. Let A ∈ Fn×n, and define A ∈ Fkn×kn by

 A A ···  A   −A A A ···  .. △   . A =  −A −A A  .. .. ..  .. . . .  . −A −A −A · · ·

 A   A    A  .  ..  .   A

√

Then, σmax (A) =

2 σmax (A). 1 − cos(π/k)

889

NORMS

Furthermore, define A0 ∈ Fkn×kn by

 A A ···  0   −A 0 A ···  ..  △  . A0 =  −A −A 0  .. .. ..  .. . . .  .  −A −A −A · · · √

Then, σmax (A0 ) =

 A   A    A  .  ..  .   0

1 + cos(π/k) σmax (A). 1 − cos(π/k)

Source: [277]. Remark: Extensions to Schatten norms are given in [277]. Fact 11.12.10. Let A, B, C, D ∈ Fn×n. Then,

([

1 2

max {σmax (A + B + C + D), σmax (A − B − C + D)} ≤ σmax

Now, let p ∈ [1, ∞). Then, 1 2 (∥A

+ B+C +

p D∥σp

+ ∥A − B − C +

p 1/p D∥σp )

[

A ≤

C

A C

B D

B D

]) .

]



. σp

Source: [277]. ] △ [ Fact 11.12.11. Let A = CA DB ∈ F(n+m)×(n+m) , and assume that A is normal. Then, ∥B∥F = ∥C∥F . Source: [2991, p. 323]. △

Fact 11.12.12. Let A, B, C ∈ Fn×n, define A =

let p ∈ [1, ∞], and define

  ∥A∥σp △  A0 =  ∥B∥σp

If p ∈ [1, 2], then

[

]

, assume that A is positive semidefinite,  ∥B∥σp   . ∥C∥σp A B B∗ C

∥A0 ∥σp ≤ ∥A∥σp .

Furthermore, if p ∈ [2, ∞], then Hence, if p = 2, then

∥A∥σp ≤ ∥A0 ∥σp . ∥A0 ∥σp = ∥A∥σp .

Finally, if A = C, B is Hermitian, and p is an integer, then p p p ∥A∥σp = ∥A + B∥σp + ∥A − B∥σp , p ∥A0 ∥σp

= (∥A∥σp + ∥B∥σp ) p + |∥A∥σp − ∥B∥σp | p.

Source: [1616]. Remark: This is a norm-compression inequality. Related: Fact 11.16.15. △

Fact 11.12.13. Let A ∈ Fn×n, B ∈ Fn×m, and C ∈ Fm×m, define A =

positive semidefinite, and let p ≥ 1. Then, the following statements hold: i) If p ∈ [1, 2], then p p p p ∥A∥σp ≤ ∥A∥σp + (2 p − 2)∥B∥σp + ∥C∥σp . ii) If p ∈ [0, 1] ∪ [2, ∞), then p p p p ∥A∥σp + (2 p − 2)∥B∥σp + ∥C∥σp ≤ ∥A∥σp .

[

A B B∗ C

]

, assume that A is

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CHAPTER 11

iii) If p = 2, then

p p p p ∥A∥σp = ∥A∥σp + (2 p − 2)∥B∥σp + ∥C∥σp .

iv) Assume that C is positive definite. If p ∈ (−∞, −2] ∪ [1, 2], then [ ∗ −1 ] [ ∗ −1 ] B C B B∗ BC B 0 p tr ≤ tr + (2 p − 2)∥B∥σp . B C 0 C v) Assume that C is positive definite. If p ∈ [0, 1] ∪ [2, 3], then [ ∗ −1 ] [ ∗ −1 BC B 0 BC B p p tr + (2 − 2)∥B∥σp ≤ tr 0 C B

] B∗ . C

Source: [189, 194]. ii) for p > 3 is a conjecture. Fact 11.12.14. Let A ∈ Fn×m be the partitioned matrix

[

A=

A11 A21

··· ···

] A1k , A2k

where Ai j ∈ Fni ×n j for all i, j ∈ {1, . . . , k}. Then, the following statements are conjectured to hold: i) If p ∈ [1, 2], then

 

 ∥A11 ∥σp · · · ∥A1k ∥σp 

 ≤ ∥A∥σp .



∥A21 ∥σp · · · ∥A2k ∥σp

σp ii) If p ≥ 2, then ∥A∥σp



 ∥A11 ∥σp ≤



∥A21 ∥σp

 ∥A1k ∥σp 

 . ∥A ∥

··· ···

2k σp

σp

Source: [191]. This result is true if either p ≥ 4 or all blocks have rank 1. Remark: This is a

norm-compression inequality.

11.13 Facts on Matrix Norms and Eigenvalues for One Matrix Fact 11.13.1. Let A ∈ Fn×n. Then,

| det A| ≤

n ∏

∥rowi (A)∥2 ,

| det A| ≤

i=1

n ∏

∥coli (A)∥2 .

i=1

Source: Use Hadamard’s inequality. See Fact 10.21.15. Fact 11.13.2. Let A ∈ Fn×n. Then,

|Re tr A2 | ≤ | tr A2 | ≤ ∥ρ(A)∥22 ≤ ∥A2 ∥σ1 = tr ⟨A2 ⟩ = ∥σ(A2 )∥1 ≤ ∥σ(A)∥22 = tr A∗A = tr ⟨A⟩2 = ∥A∥2σ2 = ∥A∥2F , ∥A∥2F −

√

∗ n3 −n 12 ∥[A, A ]∥F

≤ ∥ρ(A)∥22 ≤

√

∥A∥4F − 21 ∥[A, A∗ ]∥2F ≤ ∥A∥2F .

Consequently, A is normal if and only if ∥A∥2F = ∥ρ(A)∥22 . Furthermore, √ ∥ρ(A)∥22 ≤ ∥A∥4F − 14 (tr |[A, A∗ ]|)2 ≤ ∥A∥2F , √ 2 ∥ρ(A)∥22 ≤ ∥A∥4F − n4 | det [A, A∗ ]|2/n ≤ ∥A∥2F . Finally, A is Hermitian if and only if tr A2 = ∥A∥2F . Source: Fact 10.13.2 and Fact 10.21.10. The lower bound involving the commutator is due to P. Henrici; the corresponding upper bound is given in [1704]. The bounds in the penultimate statement are given in [1704]. The last statement follows

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from Fact 4.10.13. Remark: tr (A + A∗ )2 ≥ 0 and tr (A − A∗ )2 ≤ 0 yield | tr A2 | ≤ ∥A∥2F . Remark: ∑n 2 2 i=1 ρi (A) ≤ ∥A∥F is Schur’s inequality. See Fact 10.21.10. Related: Fact 7.12.13, Fact 10.13.5, Fact 11.13.4, and Fact 11.15.19. Fact 11.13.3. Let A ∈ Fn×n. Then, √ | tr A2 | ≤ (rank A) ∥A∥4F − 12 ∥[A, A∗ ]∥2F . Source: [706]. Fact 11.13.4. Let A ∈ Fn×n, and define △

δ=

√

(∥A∥2F − 1n | tr A|2 )2 − 21 ∥[A, A∗ ]∥2F + n1 | tr A|2.

Then, ∥ρ(A)∥22 ≤ δ ≤

√

∥A∥4F − 12 ∥[A, A∗ ]∥2F ≤ ∥A∥2F ,

∥α(A)∥22 ≤ 12 (δ + Re tr A2 ),

∥β(A)∥22 ≤ 21 (δ − Re tr A2 ). ∑ Source: [1482]. Remark: The first string interpolates the upper bound for ni=1 ρ2i (A) in the second string in Fact 11.13.2. Fact 11.13.5. Let A ∈ Fn×n and p ∈ (0, 2]. Then, p ∥ρ(A)∥ pp ≤ ∥σ(A)∥ pp = ∥A∥σp ≤ ∥A∥ pp .

Source: The left-hand inequality, which holds for all p > 0, follows from Weyl’s inequality in Fact 10.21.10. The right-hand inequality is given by Fact 11.9.10. Remark: This is the generalized Schur inequality. Remark: Equality is discussed in [1499] for p ∈ [1, 2). Fact 11.13.6. Let A ∈ Fn×n. Then,

0 ≤ ∥A∥2F − ∥ρ(A)∥22 = 12 ∥A + A∗ ∥2F − 2∥α(A)∥22 = 12 ∥A − A∗ ∥2F − 2∥β(A)∥22 . Furthermore, the following statements are equivalent: i) ∥A∥2F = ∥ρ(A)∥22 . ii) ∥A + A∗ ∥2F = 4∥α(A)∥22 . iii) ∥A − A∗ ∥2F = 4∥β(A)∥22 . iv) A is normal. Source: Fact 7.12.29, [1172, pp. 11–17], and [2295, pp. 57, 58]. Remark: This is an extension of Browne’s theorem. See Fact 7.12.24. Related: Fact 7.12.30. Fact 11.13.7. Let A ∈ Fn×n. Then, the following statements hold: i) ρmax (A) ≤ n∥A∥∞ . ii) max {|αmin (A)|, |αmax (A)|} ≤ 2n ∥A + A∗ ∥∞ . √

iii) max {|βmin (A)|, |βmax (A)|} ≤ 2n√−n ∥A − A∗ ∥∞ ≤ n∥A − A∗ ∥∞ . 2 √ iv) | det A| ≤ ( n∥A∥∞ )n . Now, assume that A is normal. Then, the following statements hold: v) ∥A + A∥ ≤ 2 max {|αmin (A)|, |αmax (A)|}. vi) ∥A − A∥ ≤ 2 max {|βmin (A)|, |βmax (A)|}. Source: [1952, p. 140] and [2991, p. 323]. Remark: i) and ii) are Hirsch’s theorems, while iii) is Bendixson’s theorem. See Fact 7.12.24. Fact 11.13.8. Let A ∈ Fn×n. Then, the following statements hold: 2

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i) Let p ∈ [1, 2] and q ∈ [2, ∞), assume that p < q, and define r = pq/(q − p). Then, ∥ρ(A)∥r ≤ 4∥A∥ p,q . ii) Let p ∈ [1, 2]. Then, ∥ρ(A)∥ p ≤ 4∥A∥ p,∞ . √ △ iii) Let p ∈ [1, 2), and define r = 2p/(2 − p). Then, ∥σ(A)∥r ≤ 2∥A∥ p,2 . iv) Let p ∈ [2, ∞) and q ∈ (1, 2], and assume that 1/p + 1/q = 1. Then, ∥λ(A)∥ p ≤ ∥A∥q|p . Source: [1667, Chapter 2] and [2040]. Fact 11.13.9. Let A ∈ Rn×n, and assume that A is nonnegative. Then, min ∥col j (A)∥1 ≤ ρmax (A) ≤ ∥A∥col ,

j∈{1,...,n}

min ∥rowi (A)∥1 ≤ ρmax (A) ≤ ∥A∥row .

i∈{1,...,n}

Source: [2403, pp. 318, 319], [2979, p. 126], and [2991, pp. 166, 167]. Remark: The upper bounds are given by (11.4.26) and (11.4.27). Fact 11.13.10. Let A ∈ Fn×n. Then, the following statements hold: √ √ i) δ(A) ≤ 2∥A∥2F − 2n | tr A|2 ≤ 2∥A∥F . ( ) √ √ n−1 1 2 2 ≤ ii) Let λ ∈ spec(A). Then, λ − trnA ≤ n−1 ∥A∥ − | tr A| F n n n ∥A∥F .

iii) δ(A) ≤ maxi, j∈{1,...,n} (|A(i,i) − A( j, j) | − A(i,i) − A( j, j) + ∥rowi (A)∥1 + ∥row j (A)∥1 ). √ iv) If A is normal, then 3∥A − I ⊙ A∥∞ ≤ δ(A). v) If A is Hermitian, then 2∥A − I ⊙ A∥∞ ≤ δ(A). Source: [2916] and [2991, p. 324]. Remark: i) is due to L. Mirsky. Fact 11.13.11. Let A ∈ Fn×n, let β > ρmax (A), and let ∥ · ∥ be a submultiplicative norm on n×n F . Then, there exists γ > 1 such that, for all k ≥ 0, ∥Ak ∥ ≤ γβk. Remark: If A is discrete-time asymptotically stable, then β can be chosen to satisfy ρmax (A) < β < 1 so that the bound converges to zero. Related: Fact 15.22.18. Fact 11.13.12. Let A ∈ Fn×n, assume that A is Hermitian, let λ ∈ spec(A), let x, z ∈ Fn , assume that x is an eigenvector of A associated with λ, assume that ∥x∥2 = 1, and assume that Az , λz. Then, ∥z∥22 ∥Az∥22 − (z∗Az)2 . |z∗ x|2 ≤ ∥Az − λz∥22 Source: [438].

11.14 Facts on Matrix Norms and Eigenvalues for Two or More Matrices Fact 11.14.1. Let A, B ∈ Fn×m, and let p, q ∈ [1, ∞] satisfy 1/p + 1/q = 1. Then, ∗

∗

∗

∗

| tr A B| ≤ ∥ρ(A B)∥1 ≤ ∥A B∥σ1 = ∥σ(A B∥1 ≤

min {m,n} ∑

σi (A)σi (B) ≤ ∥A∥σp ∥B∥σq .

i=1

In particular,

| tr A∗B| ≤ ∥A∥F ∥B∥F .

Source: Fact 10.13.2 and Proposition 11.6.2. The last inequality in the first string is H¨older’s inequality. Related: Fact 7.13.14, Fact 11.10.35, and Fact 11.16.2. Fact 11.14.2. Let A, B ∈ Fn×m. Then,

| tr (A∗B)2 | ≤ ∥ρ(A∗B)∥22 ≤ ∥σ(A∗B)∥22 = tr AA∗BB∗ = ∥A∗B∥2F ≤ ∥A∥2F ∥B∥2F . Source: Fact 10.21.10.

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Fact 11.14.3. Let A, B ∈ Fn×n, and assume that A and B are Hermitian. Then,

∥α(A + ȷB)∥22 ≤ ∥A∥2F ,

∥β(A + ȷB)∥22 ≤ ∥B∥2F .

Source: [2263, p. 146]. Fact 11.14.4. Let A, B ∈ Fn×n, assume that A and B are Hermitian, and let ∥ · ∥ be a weakly

unitarily invariant norm on Fn×n. Then,

∥ diag[λ(A) − λ(B)]∥ ≤ ∥A − B∥ ≤ ∥ diag[λ(A) − λ↑(B)]∥. In particular, max |λi (A) − λi (B)| ≤ σmax (A − B) ≤ max |λi (A) − λn−i+1 (B)|,

i∈{1,...,n}

i∈{1,...,n}

i=1

i=1

 n 1/2  n 1/2 ∑  ∑  2 2  [λi (A) − λi (B)]  ≤ ∥A − B∥F ≤  [λi (A) − λn−i+1 (B)]  . Source: [93], [447, p. 38], [449, pp. 63, 69], [453, p. 38], [1590, p. 126], [1768, p. 134], [1799], and [2539, p. 202]. Remark: The first inequality is the Lidskii-Mirsky-Wielandt theorem. This result can be stated without norms using Fact 11.9.59. See [1799]. Remark: The case where A and B are normal is considered in Fact 11.14.6. Related: Fact 11.14.5 and Fact 11.16.40. Fact 11.14.5. Let A, B ∈ Fn×n, assume that A and B are positive semidefinite, and let ∥ · ∥ be a

weakly unitarily invariant norm on Fn×n. Then,

∥ diag[λ(A) + λ↑(B)]∥ ≤ ∥A + B∥ ≤ ∥ diag[λ(A) + λ(B)]∥. In particular, max [λi (A) + λn−i+1 (B)] ≤ σmax (A + B) ≤ max [λi (A) + λi (B)],

i∈{1,...,n}

i∈{1,...,n}

i=1

i=1

1/2  n 1/2  n  ∑  ∑ 2 2  [λi (A) + λn−i+1 (B)]  ≤ ∥A + B∥F ≤  [λi (A) + λi (B)]  . Source: [1779]. Remark: This is the Fan-Lidskii theorem. Related: Fact 11.14.4. Fact 11.14.6. Let A, B ∈ Fn×n, assume that A and B are normal, and let spec(A) = {λ1 , . . . , λq }

and spec(B) = {µ1 , . . . , µr }. Then, σmax (A − B) ≤

√ 2 max {|λi − µ j | : i ∈ {1, . . . , q}, j ∈ {1, . . . , r}}.

Source: [449, p. 164] and [453, p. 154]. Related: Fact 11.14.4. Fact 11.14.7. Let A, B ∈ Fn×n, mspec(A) = {λ1 , . . . , λn }ms , and mspec(B) = {µ1 , . . . , µn }ms .

Then, there exists a permutation σ of (1, . . . , n) such that

4 max |λi − µσ(i) | ≤ √n [σmax (A) + σmax (B)](n−1)/n σ1/n max (A − B). 2

i∈{1,...,n}

Now, let ∥ · ∥ be an equi-induced norm on Fn×n . Then, there exists a permutation σ of (1, . . . , n) such that √ n max |λi − µσ(i) | ≤ 8 n (max {∥A∥, ∥B∥})(n−1)/n ∥A − B∥1/n . i∈{1,...,n} 4 Source: [461, 2230] and [2979, pp. 104, 105]. Related: Fact 12.13.3. Fact 11.14.8. Let A, B ∈ Fn×n, let mspec(A) = {λ1 , . . . , λn }ms and mspec(B) = {µ1 , . . . , µn }ms ,

and assume that at least one of the following statements holds: i) A and B are Hermitian.

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CHAPTER 11

ii) A is Hermitian, and B is skew Hermitian. iii) A is skew Hermitian, and B is Hermitian. iv) A and B are unitary. v) There exist nonzero α, β ∈ C such that αA and βB are unitary. vi) A, B, and A − B are normal. vii) n = 2 and A and B are normal. Then, there exists a permutation σ of (1, . . . , n) such that max |λi − µσ(i) | ≤ σmax (A − B).

i∈{1,...,n}

Source: [453, pp. 52, 152]. Fact 11.14.9. Let A, B ∈ Fn×n, assume that A is normal, and let mspec(A) = {λ1 , . . . , λn }ms and

mspec(B) = {µ1 , . . . , µn }ms . Then, there exists a permutation σ of (1, . . . , n) such that 1/2  n  ∑ √ 2  |λi − µσ(i) |  ≤ n∥A − B∥F . i=1

Source: [453, p. 173] and [2979, pp. 108, 109]. Credit: J. G. Sun. Fact 11.14.10. Let A, B ∈ Fn×n, assume that A is Hermitian, and let mspec(B) = {µ1 , . . . , µn }ms .

Then, there exists a permutation σ of (1, . . . , n) such that 1/2  n √  ∑  |λi (A) − µσ(i) |2  ≤ 2∥A − B∥F . i=1

Source: [453, p. 174]. Fact 11.14.11. Let A, B ∈ Fn×n, assume that A is normal, and let mspec(A) = {λ1 , . . . , λn }ms and

mspec(B) = {µ1 , . . . , µn }ms . Then, there exists a permutation σ of (1, . . . , n) such that max |λi − µσ(i) | ≤ n∥A − B∥F .

i∈{1,...,n}

Source: [2979, p. 109]. Fact 11.14.12. Let A, B ∈ Fn×n, assume that A and B are normal, and let mspec(A) = {λ1 , . . . ,

λn }ms and mspec(B) = {µ1 , . . . , µn }ms . Then, there exists a permutation σ of (1, . . . , n) such that max |λi − µσ(i) | ≤ 3σmax (A − B).

i∈{1,...,n}

Source: [453, pp. 153, 159]. Remark: Analogous inequalities for Schatten norms are given in [453, p. 159]. Remark: If A − B is normal, then Fact 11.14.8 yields a stronger inequality. Fact 11.14.13. Let A, B ∈ Fn×n, assume that A and B are normal, and let mspec(A) = {λ1 , . . . ,

λn }ms and mspec(B) = {µ1 , . . . , µn }ms . Then, there exists a permutation σ of (1, . . . , n) such that  n 1/2 ∑   |λi − µσ(i) |2  ≤ ∥A − B∥F . i=1

Source: [1448, p. 368], [1451, p. 407], [2263, pp. 160, 161], [2979, pp. 106, 107], and [2991, pp. 320, 321]. Remark: This is the Hoffman-Wielandt theorem. Related: Fact 11.14.4. The case

where B is not necessarily normal is considered in Fact 11.14.9. Fact 11.14.14. Let A, B ∈ Fn×n, assume that A and B are normal, and let mspec(A) = {λ1 , . . . , λn }ms and mspec(B) = {µ1 , . . . , µn }ms . Then, there exists a permutation σ of (1, . . . , n) such that, for all i ∈ {1, . . . , n}, |λi − µσ(i) | ≤ nσmax (A − B).

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NORMS

Source: [2991, p. 324]. Fact 11.14.15. Let A, B ∈ Fn×n, and assume that A is Hermitian and B is normal, and let

mspec(B) = {µ1 , . . . , µn }ms , where Re µn ≤ · · · ≤ Re µ1 . Then,  n 1/2 ∑   |λi (A) − µi |2  ≤ ∥A − B∥F . i=1

Source: [1448, p. 370] and [1451, pp. 407, 408]. Related: Fact 11.14.13. Fact 11.14.16. Let A, B ∈ Fn×n, assume that A is Hermitian and B is skew Hermitian, let

mspec(A) = {λ1 , . . . , λn }ms and mspec(B) = {µ1 , . . . , µn }ms , assume that |λ1 | ≥ · · · ≥ |λn | and |µ1 | ≥ · · · ≥ |µn |, and let ∥ · ∥ be a unitarily invariant norm. Then, ∥ diag(λ1 − µ1 , . . . , λn − µn )∥ ≤ ∥A − B∥.

Source: [2979, p. 109]. Fact 11.14.17. Let A, B ∈ Fn×n, assume that A and B are Hermitian, let mspec(A) = {λ1 , . . . ,

λn }ms and mspec(B) = {µ1 , . . . , µn }ms , assume that |λ1 | ≥ · · · ≥ |λn | and |µ1 | ≥ · · · ≥ |µn |, and let ∥ · ∥ be a unitarily invariant norm. Then, √ ∥ diag(λ1 + µ1 ȷ, . . . , λn + µn ȷ)∥ ≤ 2∥A + ȷB∥. Source: [2979, pp. 110, 111]. Fact 11.14.18. Let A, B ∈ Fn×n, and let ∥ · ∥ be an induced norm on Fn×n. Then,

 ∥A − B∥(∥A∥n − ∥B∥n )    , ∥A∥ , ∥B∥,   ∥A∥ − ∥B∥ | det A − det B| ≤     n∥A − B∥∥A∥n−1, ∥A∥ = ∥B∥.

Source: [1078]. Related: Fact 2.21.8.

11.15 Facts on Matrix Norms and Singular Values for One Matrix Fact 11.15.1. Let A ∈ Fn×m. Then,

(

σmax (A) = max m

x∈F \{0}

and thus

(

∗ λ1/2 min (A A) = min n

x∈F \{0}

,

x∗A∗Ax x∗x

)1/2 ,

∗ λ1/2 min (A A)∥x∥2 ≤ ∥Ax∥2 .

If, in addition, m ≤ n, then

( σm (A) = min n

x∈F \{0}

and thus

)1/2

∥Ax∥2 ≤ σmax (A)∥x∥2 .

Furthermore,

and thus

x∗A∗Ax x∗x

x∗A∗Ax x∗x

)1/2 ,

σm (A)∥x∥2 ≤ ∥Ax∥2 .

Finally, if m = n, then

( σmin (A) = min n

x∈F \{0}

x∗A∗Ax x∗x

)1/2 ,

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CHAPTER 11

and thus

σmin (A)∥x∥2 ≤ ∥Ax∥2 .

Source: Lemma 10.4.3. Related: Fact 7.12.8. Fact 11.15.2. Let A ∈ Fn×m, and assume that A is nonsingular. Then,

} )n ( ∥A∥nF σmax (A) ∥A∥F 2 ∥A∥F ≤ ≤ max 1, ≤ , √ σmin (A) | det A| | det A| n| det A|1/n n  n−1 √   n − 1 (n−1)/2  n − 1  △  β = |det A|  ≤ σmin (A).  ≤ |det A|  ∥A∥F ∥A∥2F − β2 {

Now, assume that n ≥ 3. Then, for all k ∈ {2, . . . , n − 1}, )n+k−1 ( 2k ∥A∥F σmax (A) ≤ . √ ∏ σmin (A) | det A| ki=2 σi (A) n+k−1 Source: [1264, 1811, 2231, 3015]. Related: Fact 11.9.14. Fact 11.15.3. Let A ∈ Fn×n. Then, the following statements are equivalent:

There exists α ≥ 0 such that, for all x ∈ Fn , ∥Ax∥2 = α∥x∥2 . There exists α ≥ 0 such that A∗A = αI. For all x ∈ Fn , ∥Ax∥2 = σmin (A)∥x∥2 . A∗A = σmin (A)I. For all x ∈ Fn , ∥Ax∥2 = σmax (A)∥x∥2 . A∗A = σmax (A)I. σmin (A) = σmax (A). Source: [2991, p. 111]. Fact 11.15.4. Let A ∈ Fn×m. Then, i) ii) iii) iv) v) vi) vii)

σmax (A) = max {|y∗Ax|: x ∈ Fm, y ∈ Fn, ∥x∥2 = ∥y∥2 = 1} = max {|y∗Ax|: x ∈ Fm, y ∈ Fn, ∥x∥2 ≤ 1, ∥y∥2 ≤ 1}. Related: Fact 11.9.42. △ Fact 11.15.5. Let x ∈ Fn and y ∈ Fm, and define S = {A ∈ Fn×m : σmax (A) ≤ 1}. Then,

max x∗Ay = A∈S

√

x∗xy∗y.

Fact 11.15.6. Let ∥ · ∥ be an equi-induced unitarily invariant norm on Fn×n. Then, ∥ · ∥ = σmax (·). Fact 11.15.7. Let ∥ · ∥ be an equi-induced self-adjoint norm on Fn×n. Then, ∥ · ∥ = σmax (·). Fact 11.15.8. Let A ∈ Fn×n. Then, σmin (A)−1 ≤ σmin (A+I) ≤ σmin (A)+1. Source: Proposition

11.6.10. k Fact 11.15.9. Let A ∈ Fn×n, assume that A is normal, and let k ≥ 0. Then, σmax (A[k ) = ]σmax (A).

Remark: Matrices that are not normal might also satisfy these statements. Consider Fact 11.15.10. Let A ∈ Fn×n. Then, 2 2 2 σmax (A) − σmax (A2 ) ≤ σmax (A∗A − AA∗ ) ≤ σmax (A) − σmin (A), 2 2 2 σmax (A) + σmin (A) ≤ σmax (A∗A + AA∗ ) ≤ σmax (A) + σmax (A2 ).

If A2 = 0, then

2 σmax (A∗A − AA∗ ) = σmax (A).

100 000 010

.

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NORMS

2 Source: [1064, 1631, 1635]. Remark: If A is normal, then σmax (A) ≤ σmax (A2 ). Fact 11.15.9 implies that equality holds. Related: Fact 10.22.17. Fact 11.15.11. Let A ∈ Fn×n. Then, the following statements are equivalent:

i) ρmax (A) = σmax (A). i ii) σmax (Ai ) = σmax (A) for all i ≥ 1. n iii) σmax (An ) = σmax (A). Source: [1056] and [1450, p. 44]. Remark: Additional results are given in [1194]. Credit: iii) =⇒ i) is due to V. Ptak. √ Fact 11.15.12. Let A ∈ Fn×n. Then, σmax (A) ≤ σmax (|A|) ≤ rank Aσmax (A). Source: [1389, p. 111]. Fact 11.15.13. Let A ∈ Fn×n, and let p ∈ [2, ∞) be an even integer. Then, ∥A∥σp ≤ ∥|A|∥σp . In particular,

∥A∥F ≤ ∥|A|∥F ,

σmax (A) ≤ σmax (|A|).

Finally, let ∥ · ∥ be a unitarily invariant norm on Cn×m. Then, ∥A∥ = ∥|A|∥ for all A ∈ Cn×m if and only if there exists c > 0 such that ∥ · ∥ = c∥ · ∥F . Source: [1452] and [1480]. △ Fact 11.15.14. Let A ∈ Rn×n, and assume that r = rank A ≥ 2. If r tr A2 ≤ (tr A)2, then √ (tr A)2 − tr A2 ≤ ρmax (A). r(r − 1) If (tr A)2 ≤ r tr A2, then | tr A| + r

√

r tr A2 − (tr A)2 ≤ ρmax (A). r2 (r − 1)

If rank A = 2, then equality holds in both cases. Finally, if A is skew symmetric, then √ 3 ∥A∥F ≤ ρmax (A). r(r − 1) Source: [1461]. Fact 11.15.15. Let A ∈ Rn×n. Then,

√

1 (∥A∥2F 2(n2 −n)

Furthermore, if ∥A∥F ≤ tr A, then σmax (A) ≤ tr A + 1 n

+ tr A2 ) ≤ σmax (A). √

n−1 2 n [∥A∥F

− 1n (tr A)2 ].

Source: [2025], which considers the complex case. Fact 11.15.16. Let n ≥ 2, A ∈ Cn×n, and z ∈ C. Then,

σmax [(zI − A)A ] ≤

(

1 n−1 [n|z|

) n−1 + ∥A∥2F − 2 Re(z tr A)] 2 .

Source: [1172, pp. 28, 29]. Related: Fact 10.21.23 and Fact 11.15.17. Fact 11.15.17. Let n ≥ 2, A ∈ Cn×n, and mspec(A) = {λ1 , . . . , λn }ms , and assume that |λ1 | =

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CHAPTER 11

min {|λ1 |, . . . , |λn |}. Then,

√

i/2 ( ) n 1 n − 1  2 ∑ 2  ∥A∥ − |λ j |  , σmax (A ) ≤ (n − 1)i i  F j=1 i=0 i/2  n−1 √  n ( ) n ∏  ∑  1 n − 1  2 ∑ 2    A   ∥A∥ − max {1, |λi |} σmax (A ) ≤  |λ j |  . (n − 1)i i  F n−1 ∑

A

n−i−1 ρmax (A)

i=0

i=2

j=1

Source: [1172, pp. 30–32]. Related: Fact 10.21.23 and Fact 11.15.16. Fact 11.15.18. Let A ∈ Fn×n. Then, the polynomial p ∈ R[s] defined by △

p(s) = sn − ∥A∥2F s + (n − 1)| det A|2/(n−1) has either exactly one or exactly two positive roots 0 < α ≤ β. Furthermore, α and β satisfy α(n−1)/2 ≤ σmin (A) ≤ σmax (A) ≤ β(n−1)/2. Source: [2329]. w

w

Fact 11.15.19. Let A ∈ Fn×n and p ≥ 0. Then, ρ(A)⊙p ≺ σ(A)⊙p . In particular, ρ(A) ≺ σ(A),

and thus

| tr A| ≤ ∥ρ(A)∥1 ≤ tr ⟨A⟩ = ∥A∥1 .

Source: [449, p. 42]. Remark: This is Weyl’s majorant theorem. Related: Fact 11.13.2. Fact 11.15.20. Let A ∈ Fn×n. Then, slog

slog

ρ(A)⊙2 ≺ σ(A2 ) ≺ σ(A)⊙2 . That is, for all k ∈ {1, . . . , n}, k ∏

ρ2i (A) ≤

i=1 n ∏

k ∏

σi (A2 ) ≤

i=1

ρ2i (A) =

n ∏

i=1

k ∏

σ2i (A),

i=1

σi (A2 ) =

i=1

n ∏

σ2i (A) = | det A|2 .

i=1

Source: [1450, p. 172], Fact 3.25.15, and Fact 7.12.32. Credit: H. Weyl. Related: Fact 10.13.2 and Fact 10.22.28. slog

Fact 11.15.21. Let A ∈ Fn×n and k ≥ 1. Then, σ(Ak ) ≺ σ(A)⊙k . Source: [1810] and [2991, p.

371]. Fact 11.15.22. Let A ∈ Fn×n, and assume that A is normal. Then, ∥A∥∞ ≤ ρmax (A). Source:

[2991, p. 322]. △

Fact 11.15.23. Let A ∈ Fn×n, assume that A is normal, let S ⊆ {1, . . . , n}, and define k = card(S). Then, |11×n A(S) 1n×1 | ≤ kρmax (A). Source: [2991, p. 323]. Fact 11.15.24. Let A ∈ Fn×n, and define △

ri =

n ∑

△

|A(i, j) |,

ci =

j=1 △

rˆi =

n ∑ j=1 j,i

|A(i, j) |,

n ∑

|A( j,i) |,

j=1 △

cˆ i =

n ∑

|A( j,i) |,

j=1 j,i

Then, the following statements hold:

△

rmin = min ri ,

△

α = min

i∈{1,...,n}

i∈{1,...,n}

( ) |A(i,i) | − rˆi ,

△

cmin = min ci , i∈{1,...,n}

△

β = min

i∈{1,...,n}

(

) |A(i,i) | − cˆ i .

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NORMS

i) ii) iii) iv)

If α > 0, then A is nonsingular and ∥A−1 ∥row < 1/α. If β > 0, then A is nonsingular and ∥A−1 ∥col < 1/β. √ If α > 0 and β > 0, then A is nonsingular, and αβ ≤ σmin (A). mini=1,...,n 12 [2|A(i,i) | − rˆi − cˆ i ] ≤ σmin (A).

v) mini=1,...,n 12 [(4|A(i,i) |2 + [ˆri − cˆ i ]2 )1/2 − rˆi − cˆ i ] ≤ σmin (A). } { ( )(n−1)/2 ∏cnmin , ∏rnmin vi) n−1 | det A| max ci ri ≤ σmin (A). n i=1

i=1

Source: Fact 11.9.27, [1442], [1450, pp. 227, 231], and [1540, 1550, 2782]. Fact 11.15.25. Let A ∈ Fn×n. Then, for all i ∈ {1, . . . , n}, k lim σ1/k i (A ) = ρi (A).

k→∞

In particular,

1/k lim σmax (Ak ) = ρmax (A).

k→∞

Source: [1450, p. 180] and [2979, p. 85]. Credit: T. Yamamoto. Related: The expression for

ρmax (A) is a special case of Proposition 11.2.6. Fact 11.15.26. Let A ∈ Fn×n, and assume that A is nonzero. Then, 1 = σmax (A)

min

B∈{X∈Fn×n : det(I−AX)=0}

σmax (B).

Furthermore, there exists B0 ∈ Fn×n such that rank B0 = 1, det(I − AB0 ) = 0, and 1 σmax (A)

= σmax (B0 ).

Source: If σmax (B) < 1/σmax (A), then ρmax (AB) ≤ σmax (AB) < 1, and thus I − AB is nonsingular.

Hence, 1 σmax (A)

= = ≤

min

B∈{X∈Fn×n : σmax (X)≥1/σmax (A)}

min

σmax (B)

B∈{X∈Fn×n : σmax (X) 0. Then, 2 2 σmin (A + B) ≤ [(1 + α)σmin (A) + (1 + α−1 )σmax (B)]1/2 , 2 2 σmax (A + B) ≤ [(1 + α)σmax (A) + (1 + α−1 )σmax (B)]1/2 .

Fact 11.16.21. Let A, B ∈ Fn×n. Then,

σmin (A) − σmax (B) ≤ | det(A + B)|1/n ≤

n ∏

|σi (A) + σn−i+1 (B)|1/n ≤ σmax (A) + σmax (B).

i=1

Source: [1467, p. 63] and [1798]. Fact 11.16.22. Let A, B ∈ Fn×n, and assume that σmax (B) ≤ σmin (A). Then,

0 ≤ [σmin (A) − σmax (B)]n ≤ ≤ | det(A + B)| ≤

n ∏

n ∏

|σi (A) − σn−i+1 (B)|

i=1

|σi (A) + σn−i+1 (B)| ≤ [σmax (A) + σmax (B)]n.

i=1

Hence, if σmax (B) < σmin (A), then A is nonsingular and, for all α ∈ [−1, 1], A + αB is nonsingular. Source: [1798]. Related: Fact 7.13.24 and Fact 15.19.16. △ Fact 11.16.23. Let A, B ∈ Fn×m, and define r = min {n, m}. Then, the following statements are

equivalent: w

i) σ(A) ≺ σ(B). ii) For all unitarily invariant norms ∥ · ∥ on Fn×m , ∥A∥ ≤ ∥B∥. Source: [1450, pp. 205, 206] and [2991, p. 375]. Remark: This is the Fan dominance theorem. Fact 11.16.24. Let A, B ∈ Fn×n. Then, { σ(A) + σ(B) w w σ(A) + σ(B)↑ ≺ σ(A + B) ≺ σ(⟨A⟩ + ⟨B⟩). If, in addition, A and B are Hermitian, then w

w

λ(A) + λ(B)↑ ≺ λ(A + B) ≺ λ(A) + λ(B). Source: [1798, 2750] and [2991, pp. 357, 362]. Related: Fact 11.16.25. Fact 11.16.25. Let A, B ∈ Fn×n. Then, w

|σ(A) − σ(B)| ≺ σ(A − B).

904

CHAPTER 11

Furthermore, if either σmax (A) < σmin (B) or σmax (B) < σmin (A), then w

σ(A + B) ≺ |σ(A) − σ↑(B)|. Source: Proposition 11.2.2, [1450, pp. 196, 197], [1798], [1969, p. 243], [1971, p. 330], and [2991, pp. 357, 361, 362]. Related: Fact 11.16.24. Fact 11.16.26. Let M, S, U ∈ Fn×n, assume that M is positive semidefinite, and assume that S

and U are unitary. Then, w

w

σ(SM − S ) ≺ σ(SM − U) ≺ σ(SM + S ). In particular,

w

w

σ(M − I) ≺ σ(M − U) ≺ σ(M + I). Source: [2991, p. 359]. Remark: SM is a polar decomposition. Related: Fact 11.10.73. Fact 11.16.27. Let U, V, W, D ∈ Fn×n, assume that D is nonnegative and diagonal, and assume

that U, V, and W are unitary. Then, w

w

σ(UDV − UV) ≺ σ(UDV − W) ≺ σ(UDV + UV). Source: [2991, p. 361]. Remark: UDV is a singular value decomposition. Related: Fact

11.10.73. Fact 11.16.28. Let A, B, C ∈ Fn×n, assume that B is Hermitian, and assume that C is skew

Hermitian. Then, w

σ[A − 21 (A + A∗ )] ≺ σ(A − B),

w

σ[A − 21 (A − A∗ )] ≺ σ(A − C).

Source: [2991, p. 359]. Fact 11.16.29. Let A, B ∈ Fn×m and α ∈ [0, 1]. Then, for all i ∈ {1, . . . , min {n, m}},

])  ([  A 0    σ  i  0 B   ([ √ ]) σi [αA + (1 − α)B] ≤    2αA √ 0    σ ,   i 0 2(1 − α)B 2σi (AB∗ ) ≤ σi (⟨A⟩2 + ⟨B⟩2 ).

Furthermore,

⟨αA + (1 − α)B⟩2 ≤ α⟨A⟩2 + (1 − α)⟨B⟩2.

If, in addition, n = m, then, for all i ∈ {1, . . . , n}, 1 2 σi (A

([

∗

+ A ) ≤ σi

A 0

0 A

]) .

Source: [1422]. Related: Fact 11.16.31. Fact 11.16.30. Let A ∈ Fn×m and B ∈ Fl×m, and let p, q ∈ (0, ∞) satisfy 1/p + 1/q = 1. Then,

for all i ∈ {1, . . . , min {n, m, l}},

σi (AB∗ ) ≤ σi

(

p 1 p ⟨A⟩

) + q1 ⟨B⟩q .

Source: [93, 95, 1418] and [2977, p. 28]. Related: Fact 2.2.50, Fact 2.2.53, Fact 10.11.73, Fact

10.14.8, Fact 10.14.33, and Fact 10.14.34. Fact 11.16.31. Let A ∈ Fn×m and B ∈ Fl×m. Then, for all i ∈ {1, . . . , min {n, m, l}}, σi (AB∗ ) ≤ 21 σi (A∗A + B∗B).

905

NORMS

Source: Set p = q = 2 in Fact 10.11.38. See [468], [2979, p. 88], and [2991, p. 354]. Related:

Fact 11.10.79 and Fact 11.16.29. Fact 11.16.32. Let A, B ∈ Fn×n, and assume that A and B are semicontractive. Then, for all i ∈ {1, . . . , n}, σi (2I − A∗A − B∗B) ≤ 2σi (I − A∗B). Now, assume that A and B are contractive. Then, for all i ∈ {1, . . . , n}, 2σi [(I − A∗B)−1 ] ≤ σi [(I − A∗A)−1 + (I − B∗B)−1 ]. Now, let ∥ · ∥ be a unitarily invariant norm on Fn×n. Then, 2∥(I − A∗B)−1 ∥ ≤ ∥(I − A∗A)−1 + (I − B∗B)−1 ∥. Source: [1840]. Fact 11.16.33. Let A, B, C, D ∈ Fn×m. Then, for all i ∈ {1, . . . , min {n, m}},

√ 2σi (⟨AB∗ + CD∗ ⟩) ≤ σi

([

A C

B D

])

.

Source: [1417]. Fact 11.16.34. Let A, B, C, D, X ∈ Fn×n, assume that A, B, C, D are positive semidefinite, and

assume that 0 ≤ A ≤ C and 0 ≤ B ≤ D. Then, for all i ∈ {1, . . . , n}, σi (A1/2XB1/2 ) ≤ σi (C 1/2XD1/2 ).

Source: [1422, 1627]. Fact 11.16.35. Let A1 , . . . , Ak ∈ Fn×n. Then, for all l ∈ {1, . . . , n}, l ∑ i=1

  k l ∏ k ∏  ∑  σi (Aj ). Aj  ≤ σi  i=1 j=1

j=1

Source: [709]. Fact 11.16.36. Let A ∈ Fn×m, B ∈ Fm×n, 1 ≤ k ≤ min {n, m}, and 1 ≤ i1 < · · · < ik ≤ min {n, m}.

Then,

k ∑

σi j (A)σn− j+1 (B) ≤

j=1 k ∑

k ∑

σi j (AB) ≤

j=1

σi j (A)σn−i j +1 (B) ≤

j=1

k ∑

σ j (AB),

j=1

k ∏

σi j (A)σn−i j +1 (B) ≤

j=1

k ∑

σi j (A)σ j (B),

j=1 k ∑

σ2i j (A)σ2n− j+1 (B) ≤

j=1

k ∏

σ j (AB),

j=1

k ∑

σ2i j (AB),

j=1

k ∏ j=1

σi j (AB) ≤

k ∏

σi j (A)σ j (B).

j=1

If, in addition, r > 0, then k ∑

σrij (A)σrn−i j +1 (B) ≤

j=1

k ∑

σrj (AB),

j=1

k ∑

σrij (A)σrn− j+1 (B) ≤

j=1

k ∑

σrij (AB).

j=1

Source: [2812, 2814] and [2991, p. 364] Remark: Extensions to products of three or more matrices are given in [2814]. Credit: The second inequality in the last string is due to A. Horn. Related:

Fact 10.22.26 and Fact 10.22.29. Fact 11.16.37. Let A, B ∈ Fn×n . Then, the following statements hold: w

w

i) σ(A) ⊙ σ(B)↑ ≺ σ(AB) ≺ σ(A) ⊙ σ(B).

906

CHAPTER 11 w

ii) [σ(A) ⊙ σ(B)↑ ]⊙2 ≺ σ(AB)⊙2 . slog

slog

iii) σ(A) ⊙ σ(B)↑ ≺ σ(AB) ≺ σ(A) ⊙ σ(B). w

iv) σ(A ⊙ B) ≺ σ(A) ⊙ σ(B). slog

v) If A and B are positive semidefinite, then λ(AB) ≺ λ(A) ⊙ λ(B). slog

w

vi) If A and B are Hermitian, then λ(eA ) + λ(eB )↑ ≺ λ(eA+B ) ≺ λ(eA ) ⊙ λ(eB ). Source: [2750] and [2991, pp. 353, 364]. Fact 11.16.38. Let A, B ∈ Fn×n . Then,    1  [ ] ]  σ[e 12 (A+B) ]  slog [  σ[ 2 (A + B)]  w σ(A) σ(eA )   ,  ≺ ,  1  ≺  1 σ(B) σ(eB )  σ[ 2 (A + B)] σ[e 2 (A+B) ]  [    ] [ ]  σ(A ⊙ B)  w σ⊙2 (A)  σ(AB)  slog σ⊙2 (A)    ≺  ,   ≺ σ⊙2 (B) . σ⊙2 (B) σ(A ⊙ B) σ(AB) Now, let p ≥ 1. Then,  [  ⊙p ]  σ (AB)  w σ⊙2p (A)  ≺ ⊙2p  , σ (B) σ⊙p (AB) If A and B are Hermitian, then

 [  ⊙p ]  σ (A ⊙ B)  w σ⊙2p (A)  ≺ ⊙2p  . σ (B) σ⊙p (A ⊙ B)

 1  [ ]  λ[ 2 (A + B)]  w λ(A)  ≺ .  1 λ(B) λ[ 2 (A + B)]

Finally, if A and B are positive semidefinite, then  1  [ ] [ ] [ ]  λ[ 2 (A + B)]  w λ(A) w λ(A + B) w λ(A) + λ(B)   ≺ ≺ ≺ . λ(B) 0 0 λ[ 1 (A + B)] 2

Source: [2750]. Fact 11.16.39. Let A ∈ Fn×m, let k ≥ 1 satisfy k < rank A, and let ∥ · ∥ be a unitarily invariant

norm on Fn×m. Then,

min

B∈{X∈Fn×m : rank X≤k}

∥A − B∥ = ∥A − B0 ∥,

where B0 is formed by replacing the (rank A) − k smallest positive singular values in the singular value decomposition of A by 0’s. Furthermore, v t r ∑ σmax (A − B0 ) = σk+1 (A), ∥A − B0 ∥F = σ2i (A). i=k+1

Finally, B0 is the unique solution if and only if σk+1 (A) < σk (A). Source: Use Fact 11.16.40 with △ Bσ = diag[σ1 (A), . . . , σk (A), 0(n−k)×(m−k) ], S 1 = In , and S 2 = Im . See [1196, p. 79] and [2539, p. 208]. Remark: This is the Schmidt-Mirsky theorem. For the Frobenius norm, this is the EckartYoung theorem. See [1081] and [2539, p. 210]. Related: Fact 11.17.14. Fact 11.16.40. Let A, B ∈ Fn×m, define Aσ, Bσ ∈ Fn×m by △

Aσ = diag(σ1 (A), . . . , σr (A), 0(n−r)×(m−r) ), △

△

△

Bσ = diag(σ1 (B), . . . , σl (B), 0(n−l)×(m−l) ),

where r = rank A and l = rank B, let S 1 ∈ Fn×n and S 2 ∈ Fm×m be unitary matrices, and let ∥ · ∥ be a

907

NORMS

unitarily invariant norm on Fn×m. Then, ∥Aσ − Bσ ∥ ≤ ∥A − S 1BS 2 ∥ ≤ ∥Aσ + Bσ ∥. In particular, max

i∈{1,...,max {r,l}}

|σi (A) − σi (B)| ≤ σmax (A − B) ≤ σmax (A) + σmax (B).

Source: [2815] and [2991, p. 377]. Remark: In the case where S 1 = In and S 2 = Im , the left-hand inequality is Mirsky’s theorem. See [2539, p. 204]. Related: Fact 11.14.4. Fact 11.16.41. Let A, B ∈ Fn×m, and assume that rank A = rank B. Then,

σmax [AA+(I − BB+ )] = σmax [BB+(I − AA+ )] ≤ min {σmax (A+ ), σmax (B+ )}σmax (A − B). Source: [1389, p. 400] and [2539, p. 141]. Fact 11.16.42. Let A, B ∈ Fn×m. Then, for all k ∈ {1, . . . , min {n, m}}, k ∑

σi (A ⊙ B) ≤

k ∑

i=1

i=1

k ∑

k ∑

σi (A ⊙ B) ≤

i=1

1/2 ∗ ∗ d1/2 i (A A)di (BB )

∗ 1/2 ∗ d1/2 i (AA )di (B B)

i=1

 ∑k 1/2 ∗  k     i=1 di (A A)σi (B)   ∑ ≤ ≤ σi (A)σi (B),  ∑   k σi (A)d1/2 (BB∗ )   i=1 i=1 i   ∑k 1/2 k ∗     ∑  i=1 di (AA )σi (B)  ≤ σi (A)σi (B). ≤  ∑    k σi (A)d1/2 (B∗B)  i=1

In particular,

i

i=1

σmax (A ⊙ B) ≤ ∥A∥2,1 ∥B∥∞,2

       ∥A∥2,1 σmax (B)  ≤ σmax (A)σmax (B), ≤     σmax (A)∥B∥∞,2 

σmax (A ⊙ B) ≤ ∥A∥∞,2 ∥B∥2,1

      ∥A∥∞,2 σmax (B)   ≤ ≤ σmax (A)σmax (B).    σmax (A)∥B∥2,1  

Source: [104], [1450, pp. 332, 334], [1985, 2973], Fact 3.25.2, Fact 10.21.11, and Fact 11.9.28. 1/2 ∗ ∗ Remark: d1/2 i (A A) and di (AA ) are the ith largest Euclidean norms of the columns and rows of A, respectively. Remark: Related results are given in [2749]. Equality is discussed in [713]. Fact 11.16.43. Let A, B ∈ Fn×m. Then,

σmax (A ⊙ B) ≤

√

n∥A∥∞ σmax (B).

Now, assume that n = m and that either A is positive semidefinite and B is Hermitian or A and B are nonnegative and symmetric. Then, σmax (A ⊙ B) ≤ ∥A∥∞ σmax (B). Source: [1157]. Fact 11.16.44. Let A, B ∈ Rn×m, assume that B is positive semidefinite, and let β denote the

smallest positive entry of |B|. Then, ρmax (A ⊙ B) ≤

∥A∥∞ ∥B∥∞ σmax (B), β

ρmax (B) ≤ ρmax (|B|) ≤

∥B∥∞ ρmax (B). β

Source: [1157]. Fact 11.16.45. Let A, B ∈ Fn×m, and let p ∈ [2, ∞) be an even integer. Then,

∥A ⊙ B∥2σp ≤ ∥A ⊙ A∥σp ∥B ⊙ B∥σp .

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In particular,

∥A ⊙ B∥2F ≤ ∥A ⊙ A∥F ∥B ⊙ B∥F , σ2max (A ⊙ B) ≤ σmax (A ⊙ A)σmax (B ⊙ B) ≤ σ2max (A)σ2max (B).

If B = A, then equality holds. Furthermore, ∥A ⊙ A∥σp ≤ ∥A ⊙ A∥σp . In particular,

∥A ⊙ A∥F ≤ ∥A ⊙ A∥F ,

Now, assume that n = m. Then, In particular,

In particular,

∥A ⊙ AT ∥σp ≤ ∥A ⊙ A∥σp .

∥A ⊙ AT ∥F ≤ ∥A ⊙ A∥F ,

Finally,

σmax (A ⊙ A) ≤ σmax (A ⊙ A).

σmax (A ⊙ AT ) ≤ σmax (A ⊙ A).

∥A ⊙ A∗ ∥σp ≤ ∥A ⊙ A∥σp . ∥A ⊙ A∗ ∥F ≤ ∥A ⊙ A∥F ,

σmax (A ⊙ A∗ ) ≤ σmax (A ⊙ A).

Source: [1450, p. 340] and [1452, 2441]. Related: Fact 9.6.23 and Fact 11.16.42. Fact 11.16.46. Let A, B ∈ Cn×m. Then,

∥A ⊙ B∥2F =

n ∑

σ2i (A ⊙ B) = tr (A ⊙ B)(A ⊙ B)T = tr (A ⊙ A)(B ⊙ B)T

i=1

≤

n ∑

σi [(A ⊙ A)(B ⊙ B)T ] ≤

n ∑

σi (A ⊙ A)σi (B ⊙ B).

i=1

i=1

Source: [1480]. Fact 11.16.47. Let A, B ∈ Fn×n. Then,

∥(A ⊙ B)(A ⊙ B)T ∥σ1 ≤ ∥(A ⊙ A)(B ⊙ B)T ∥σ1 , ∥(A ⊙ B)(A ⊙ B)T ∥F ≤ ∥(A ⊙ A)(B ⊙ B)T ∥F , σmax [(A ⊙ B)(A ⊙ B)T ] ≤ σmax [(A ⊙ A)(B ⊙ B)T ]. Source: [958]. Fact 11.16.48. Let A, B ∈ Rn×n, and assume that A and B are nonnegative. Then,

σmax (A ⊙ B) ≤ ρmax (ATB). Source: [2991, p. 170]. Fact 11.16.49. Let A, B ∈ Rn×n, assume that A and B are nonnegative, and let α ∈ [0, 1]. Then, α 1−α σmax (A⊙α ⊙ B⊙(1−α) ) ≤ σmax (A)σmax (B), √ σmax (A⊙1/2 ⊙ B⊙1/2 ) ≤ σmax (A)σmax (B),

σmax (A⊙1/2 ⊙ A⊙1/2T ) ≤ σmax (A⊙α ⊙ A⊙(1−α)T ) ≤ σmax (A). Source: [2441]. Related: Fact 9.6.24.

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Fact 11.16.50. Let ∥ · ∥ be a unitarily invariant norm on Cn×n, and let A, X, B ∈ Cn×n. Then,

∥A ⊙ X ⊙ B∥ ≤

1 2

√ n∥A ⊙ X ⊙ A + B ⊙ X ⊙ B∥,

∥A ⊙ X ⊙ B∥2 ≤ n∥A ⊙ X ⊙ A∥∥B ⊙ X ⊙ B∥, ∥A ⊙ X ⊙ B∥F ≤ 21 ∥A ⊙ X ⊙ A + B ⊙ X ⊙ B∥F . Source: [1480].

11.17 Facts on Linear Equations and Least Squares Fact 11.17.1. Let A ∈ Rn×n, assume that A is nonsingular, let b ∈ Rn, and let xˆ ∈ Rn. Then,

1 ∥A xˆ − b∥ ∥ xˆ − A−1b∥ ∥A xˆ − b∥ ≤ , ≤ κ(A) −1 κ(A) ∥b∥ ∥b∥ ∥A b∥ △

where κ(A) = ∥A∥∥A−1 ∥ and the vector and matrix norms are compatible. Equivalently, letting △ △ bˆ = A xˆ and x = A−1b, it follows that 1 ∥bˆ − b∥ ∥ xˆ − x∥ ∥bˆ − b∥ ≤ ≤ κ(A) . κ(A) ∥b∥ ∥x∥ ∥b∥ Remark: This result estimates the accuracy of an approximate solution xˆ to Ax = b. κ(A) is the condition number of A. Remark: For ∥A∥ = σmax (A), κ(A) = κ(A−1 ). Remark: See [3008]. ˜ < 1, Fact 11.17.2. Let A ∈ Rn×n, assume that A is nonsingular, let A˜ ∈ Rn×n, assume that ∥A−1A∥

˜ ˜ xˆ = b + b. and let b, b˜ ∈ Rn. Furthermore, let x ∈ Rn satisfy Ax = b, and let xˆ ∈ Rn satisfy (A + A) Then, ) ( ˜ ˜ ∥ xˆ − x∥ κ(A) ∥A∥ ∥b∥ ≤ + , ˜ ∥b∥ ∥A∥ ∥x∥ 1 − ∥A−1A∥

△ ˜ < where κ(A) = ∥A∥∥A−1 ∥ and the vector and matrix norms are compatible. If, in addition, ∥A−1 ∥∥A∥ 1, then ˜ ˜ ∥b˜ − Ax∥ ∥ xˆ − x∥ κ(A) ∥b˜ − Ax∥ 1 ≤ ≤ . ˜ κ(A) + 1 ∥b∥ ∥x∥ ∥b∥ 1 − ∥A−1A∥

Source: [905, 906]. ˆ < 1, let b ∈ R(A), let bˆ ∈ Rn, and assume that Fact 11.17.3. Let A, Aˆ ∈ Rn×n satisfy ∥A+A∥

△ ˆ Then, x = ˆ Furthermore, let xˆ ∈ Rn satisfy (A + A) ˆ xˆ = b + b. b + bˆ ∈ R(A + A). A+ b + (I − A+A) xˆ satisfies Ax = b and ( ) ˆ ˆ ∥b∥ κ(A) ∥A∥ ∥ xˆ − x∥ ≤ + , ˆ ∥b∥ ∥A∥ ∥x∥ 1 − ∥A+A∥ △

where κ(A) = ∥A∥∥A−1∥ and the vector and matrix norms are compatible. Source: [905]. Remark: See [906] for a lower bound. Fact 11.17.4. Let A ∈ Fn×m, assume that A is left invertible, and let AL ∈ Fm×n be a left inverse of A. Then, A+A+∗ ≤ ALAL∗ . Therefore,

σmax (A+ ) ≤ σmax (AL ),

∥A+ ∥F ≤ ∥AL ∥F .

Furthermore, A+A+∗ = ALAL∗ if and only if AL = A+. Source: Fact 8.3.14, Fact 10.11.32, and Fact 11.10.31. Fact 11.17.5. Let A ∈ Fn×m, assume that A is right invertible, and let AR ∈ Fm×n denote a right inverse of A. Then, A+∗A+ ≤ AR∗AR.

910

Therefore,

CHAPTER 11

σmax (A+ ) ≤ σmax (AR ),

∥A+ ∥F ≤ ∥AR ∥F .

Furthermore, A+∗A+ = AR∗AR if and only if AR = A+. Source: Fact 8.3.15, Fact 10.11.32, and Fact 11.10.31. Fact 11.17.6. Let A ∈ Fn×m , let b ∈ Fn, assume that b < R(A), and let B ∈ Fm×n . Then, the following statements are equivalent: i) B is a (1,3)-inverse of A. ii) For all x ∈ Fm , ∥ABb − b∥2 ≤ ∥Ax − b∥2 . Source: [1275, p. 10] and [2238, p. 285]. Remark: Bb is a least-squares solution of Ax = b. See Fact 8.3.6. Fact 11.17.7. Let A ∈ Fn×m , let b ∈ Fn, assume that b ∈ R(A), and let B ∈ Fm×n . Then, the following statements are equivalent: i) B is a (1,4)-inverse of A. ii) For all x ∈ {x : Ax = b} such that x , Bb, it follows that ∥Bb∥2 < ∥x∥2 . iii) If C ∈ Fm×n , ACb = b, and, for all x ∈ Fm such that Ax = b and x , Cb, ∥CB∥2 < ∥x∥2 , then C is a (1,4)-inverse of A. Source: [281, pp. 8, 114], [1275, p. 9], and [2238, p. 284]. Remark: Bb is a minimum-norm solution of Ax = b. See Fact 8.3.7. △ ∑ Fact 11.17.8. Let x, x1 , . . . , xn ∈ Fn, let α1 , . . . , αn be real numbers, define α = ni=1 αi , and assume that α > 0. Then, n n ∑ 1 ∑ αi ∥x − xi ∥22 . αi α j ∥xi − x j ∥22 ≤ 2α i, j=1 i=1 ∑ Equality holds if and only if x = α1 ni=1 αi xi . Source: [1171]. Remark: This is a weighted least squares problem. Fact 11.17.9. Let A ∈ Fn×m and b ∈ Fn, define f : Fm 7→ R by △

f (x) = (Ax − b)∗ (Ax − b) = ∥Ax − b∥22 , and let B ∈ Fm×n be a (1,3)-inverse of A. Then, Bb minimizes f. Now, let x0 ∈ Fm. Then, the following statements are equivalent: i) x0 minimizes f. ii) f (x0 ) = b∗ (I − AB)b. iii) Ax0 = ABb. iv) There exists y ∈ Rm such that x0 = Bb + (I − BA)y. Source: [2403, pp. 233–236]. Remark: Existence of solutions of Ax = b is not assumed. Fact 11.17.10. Let A ∈ Fn×m and b ∈ Fn, and define f : Fm 7→ R by △

f (x) = (Ax − b)∗ (Ax − b) = ∥Ax − b∥22 . Then, f has a minimizer. Now, let x0 ∈ Fm . Then, the following statements are equivalent: i) x0 minimizes f. ii) A∗Ax0 = A∗ b. iii) There exists y ∈ Fm such that x0 = A+b + (I − A+A)y. If i)–iii) are satisfied, then, for all x ∈ Fn , f (x) = (x − x0 )∗A∗A(x − x0 ) + b∗ b − b∗AA+b.

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NORMS

Therefore, if x0 minimizes f, then f (x0 ) = b∗b − x0∗ A∗Ax0 = b∗(I − AA+ )b. Furthermore, if y ∈ Fm and (I − A+A)y is nonzero, then √ ∥A+b∥2 < ∥A+b + (I − A+A)y∥2 = ∥A+b∥22 + ∥(I − A+A)y∥22 . Finally, A+b is the unique minimizer of f if and only if A is left invertible. Remark: This is a least squares problem. See [31, 489, 2535]. The expression for x is identical to the expression given by vii) of Proposition 8.1.9 for solutions of Ax = b. Therefore, x satisfies Ax = b if and only if x is optimal in the least-squares sense. However, unlike Proposition 8.1.9, consistency is not assumed; that is, there need not exist a solution to Ax = b. Related: Fact 10.17.18. Fact 11.17.11. Let A ∈ Fn×m and B ∈ Fn×l, and define f : Fm×l → R by [ ] △ f (X) = tr (AX − B)∗(AX − B) = ∥AX − B∥2F . Then, X = A+B minimizes f. Remark: This is the orthogonal Procrustes problem. See [1196, pp. 327, 328]. Related: Fact 11.17.12. Problem: Determine all minimizers. Fact 11.17.12. Let A ∈ Fn×m and B ∈ Fl×m, and define f : Fl×n → R by [ ] △ f (X) = tr (XA − B)∗(XA − B) = ∥XA − B∥2F . Then, X = BA+ minimizes f. Related: Fact 11.17.11. Fact 11.17.13. Let A, B ∈ Fn×m, and define f : U(m) → R by △

f (X) = tr [(AX − B)∗(AX − B)] = ∥AX − B∥2F . [ ] Then, X = S 1 S 2 minimizes f, where S 1 B0ˆ 00 S 2 is the singular value decomposition of A∗B. Source: [300, p. 224], [1969, pp. 269, 270], and [1971, p. 375]. Fact 11.17.14. Let A ∈ Fn×m, B ∈ Fn×p, and C ∈ Fq×m, and let k ≥ 1 satisfy k < rank A. Then, min

X∈{Y∈F p×q: rank Y≤k}

∥A − BXC∥F = ∥A − BX0C∥F ,

where X0 = B+S C + and S is formed by replacing all but the k largest singular values in the singular value decomposition of BB+AC +C by 0’s. Furthermore, X0 is a solution that minimizes ∥X∥F . Finally, X0 is the unique solution if and only if either rank BB+AC +C ≤ k or both k ≤ BB+AC +C and σk+1 (BB+AC +C) < σk (BB+AC +C). Source: [1081]. Related: This result generalizes Fact 11.16.39. Fact 11.17.15. Let A, B ∈ Fn×n , and assume that A and B are Hermitian. Then, v t n ∑ ∗ min ∥A − S BS ∥F = [λi (A) − λi (B)]2 . S ∈U(n)

i=1

Source: [775]. Related: Fact 7.13.20. Fact 11.17.16. Let A ∈ Fn×m, assume that m ≤ n, let k ≤ rank A, let S, S 1 , S 2 ∈ Fm×k , as-

sume that S , S 1 , and S 2 are left inner, and assume that A∗AS 1 = S 1 diag[σ2m−k+1 (A), . . . , σ2m (A)] and A∗AS 2 = S 2 diag[σ21 (A), . . . , σ2k (A)]. Then, ∥AS 1 ∥F ≤ ∥AS ∥F ≤ ∥AS 2 ∥F . Source: [775]. Related: Fact 7.13.19. Fact 11.17.17. Let A, B ∈ Rn×n, and define △

f (X1 , X2 ) = tr [(X1 AX2 − B)T (X1 AX2 − B)] = ∥X1 AX2 − B∥2F ,

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CHAPTER 11

[ ] where X1 , X2 ∈ Rn×n are orthogonal. Then, (X1 , X2 ) = (V2TU1T, V1T U2T ) minimizes f, where U1 A0ˆ 00 V1 [ ] is the singular value decomposition of A and U2 B0ˆ 00 V2 is the singular value decomposition of B. Source: [1969, p. 270] and [1971, p. 375]. Credit: W. Kristof. Related: Fact 4.11.5. Problem: Extend this result to C and nonsquare matrices. Fact 11.17.18. Let A ∈ Rn×m, let b ∈ Rn, and assume that rank [A b] = m + 1. Furthermore, consider the singular value decomposition of [A b] given by [ ] Σ [A b] = U V, 0(n−m−1)×(m+1) △

where U ∈ Rn×n and V ∈ R(m+1)×(m+1) are orthogonal and Σ = diag[σ1 (A), . . . , σm+1 (A)]. Furthermore, define Aˆ ∈ Rn×m and bˆ ∈ Rn by [ ] Σ0 △ ˆ ˆ [A b] = U V, 0(n−m−1)×(m+1) △

where Σ0 = diag[σ1 (A), . . . , σm (A), 0]. Finally, assume that V(m+1,m+1) , 0, and define    V(m+1,1)  1   △ .. xˆ = −  .  . V(m+1,m+1)   V(m+1,m) ˆ Remark: xˆ is the total least squares solution. See [2773]. Related: The construction Then, Aˆ xˆ = b. ˆ is based on Fact 11.16.39. of [Aˆ b]

11.18 Notes The equivalence of absolute and monotone norms given by Proposition 11.1.2 is given in [329]. More general monotonicity conditions are considered in [1544]. Induced lower bounds are treated in [1738, pp. 369, 370]. See also [2539, pp. 33, 80]. The induced norms (11.4.13) and (11.4.14) are given in [686] and [1389, p. 116]. Alternative norms for the convolution operator are given in [686, 2888]. Proposition 11.3.6 is given in [2311, p. 97]. Norm-related topics are discussed in [351]. Spectral perturbation theory in finite and infinite dimensions is treated in [1590], where the emphasis is on the regularity of the spectrum as a function of the perturbation rather than on bounds for finite perturbations. The trace norm is also called the nuclear norm.

Chapter Twelve Functions, Limits, Sequences, Series, Infinite Products, and Derivatives The norms discussed in Chapter 11 provide the foundation for the development in this chapter of some basic results in topology and analysis.

12.1 Open Sets and Closed Sets Definition 12.1.1. Let ∥ · ∥ be a norm on Fn, let x ∈ Fn, and let ε > 0. Then, the open ball of

radius ε centered at x is defined by

△

Bε (x) = {y ∈ Fn : ∥x − y∥ < ε},

(12.1.1)

and the sphere of radius ε centered at x is defined by △

Sε (x) = {y ∈ Fn : ∥x − y∥ = ε}.

(12.1.2)

It follows from Theorem 11.1.8 on the equivalence of norms on Fn that the following definitions are independent of the norm assigned to Fn. Definition 12.1.2. Let x ∈ S ⊆ Fn. Then, x is an interior point of S if there exists ε > 0 such that Bε (x) ⊆ S. The interior of S is the set △

int S = {x ∈ S: x is an interior point of S}.

(12.1.3)

Finally, S is open if S = int S. Note that int S ⊆ S. Hence, S is open if and only if every element of S is an interior point of S. Definition 12.1.3. Let x ∈ S ⊆ S′ ⊆ Fn. Then, x is an interior point of S relative to S′ if there exists ε > 0 such that Bε (x) ∩ S′ ⊆ S; equivalently, Bε (x) ∩ S = Bε (x) ∩ S′. The interior of S relative to S′ is the set △ intS′ S = {x ∈ S: x is an interior point of S relative to S′ }. (12.1.4) In particular, the relative interior of S is △

relint S = intaffin S S.

(12.1.5)

′

Finally, S is open relative to S if S = intS′ S, and S is relatively open if S = relint S. As an example, the interval [0, 1) is open relative to the interval [0, 2]. Proposition 12.1.4. Let S ⊆ Fn , let S′ ⊆ Fn, and assume that S ⊆ int S′ ⊆ Fn. Then, intS′ S = int S. Furthermore, S ⊆ Fn is open if and only if S is open relative to S′ . In particular, S ⊆ Fn is open if and only if S is open relative to Fn. Proposition 12.1.5. Let S ⊆ S′ ⊆ S′′ ⊆ Fn. Then, intS′′ S ⊆ intS′ S.

(12.1.6)

int S ⊆ relint S.

(12.1.7)

In particular,

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Definition 12.1.6. Let S ⊆ Fn and x ∈ S. Then, x ∈ Fn is a closure point of S if, for all ε > 0,

Bε (x) ∩ S is nonempty. The closure of S is the set △

cl S = {x ∈ Fn : x is a closure point of S}.

(12.1.8)

The vector x ∈ F is an essential closure point of S if, for all ε > 0, the set [Bε (x)\{x}] ∩ S is nonempty. The essential closure of S is the set n

△

ecl S = {x ∈ Fn : x is an essential closure point of S}.

(12.1.9)

Finally, S is closed if S = cl S. Note that S ⊆ cl S. Hence, S is closed if and only if every closure point of S is an element of S. △ △ For example, let S = (0, 1). Then, ecl S = cl S = [0, 1]. As another example, let S = (0, 1) ∪ {2}. Then, [0, 1] = ecl S ⊂ cl S = [0, 1] ∪ {2}. As another example, let S = {1, 1/2, 1/3, . . .}. Then, {0} = ecl S ⊂ cl S = S ∪ {0}. Finally, every affine subspace is closed. Definition 12.1.7. Let S ⊆ Fn and x ∈ S. Then, x ∈ S is an isolated point of S if there exists ε > 0 such that Bε (x) ∩ S = {x}. The isolated subset of S is the set △

iso S = {x ∈ S: x is an isolated point of S}.

(12.1.10)

Proposition 12.1.8. Let S ⊆ F . Then, n

 iso S = S\(ecl S) ⊆ S    ⊆ cl S = S ∪ ecl S = (iso S) ∪ (ecl S).   cl relint S ⊆ ecl S = cl ecl S = (cl S)\(iso S) 

(12.1.11)

Furthermore, the following statements hold: i) ecl S and iso S are disjoint. ii) ecl S is closed. iii) ecl S ⊆ S if and only if S is closed. iv) ecl S = cl S if and only if iso S is empty. v) ecl S = S if and only if S is closed and iso S is empty. vi) If S0 ⊆ S, then ecl S0 ⊆ ecl S. vii) If S0 ⊆ S and S\S0 is finite, then ecl S0 = ecl S. viii) If S is finite, then ecl S is empty. Definition 12.1.9. Let S ⊆ S′ ⊆ Fn. Then, the closure of S relative to S′ is the set △

clS′ S = {x ∈ S′ : x is a closure point of S}.

(12.1.12)

The essential closure of S relative to S′ is the set △

eclS′ S = {x ∈ S′ : x is an essential closure point of S}.

(12.1.13)

Finally, S is closed relative to S′ if S = clS′ S, and S is essentially closed relative to S′ if S = eclS′ S. As an example, the interval (0, 1] is closed relative to the interval (0, 2] and essentially closed relative to the interval (0, 2]. As another example, the set (0, 1] ∪ {2} is closed relative to (0, ∞) but is not essentially closed relative to (0, ∞). Note that S ⊆ Fn is closed if and only if S is closed relative to Fn. Furthermore, S ⊆ Fn is essentially closed if and only if S is essentially closed relative to Fn. The empty set is both open and closed, and int ∅ = cl ∅ = int∅ ∅ = cl∅ ∅ = relint ∅ = ∅. Proposition 12.1.10. Let S ⊆ S′ ⊆ S′′ ⊆ Fn. Then, clS′ S ⊆ clS′′ S.

(12.1.14)

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FUNCTIONS, LIMITS, SEQUENCES, SERIES, INFINITE PRODUCTS, AND DERIVATIVES

If, in addition, S′ is closed, then clS′ S = cl S.

(12.1.15)

claffin S S = cl S.

(12.1.16)

clS′ S = (cl S) ∩ S′,

(12.1.17)

In particular,

Let S ⊆ S′ ⊆ Fn. Then,

′

′

intS′ S = S \cl(S \S),

(12.1.18)

int S ⊆ intS′ S ⊆ S ⊆ clS′ S ⊆ cl S.

(12.1.19)

cl S = (cl S) ∩ affin S,

(12.1.20)

relint S = (affin S)\cl[(affin S)\S],

(12.1.21)

In particular,

∼

∼

int S = [cl(S )] ,

(12.1.22)

int S ⊆ relint S ⊆ S ⊆ cl S. ′

(12.1.23) ′

Definition 12.1.11. Let S ⊆ S ⊆ F . Then, the boundary of S relative to S is the set n

△

bdS′ S = (clS′ S)\intS′ S.

(12.1.24)

In particular, the boundary of S is the set △

bd S = (cl S)\int S.

(12.1.25)

and the relative boundary of S is △

relbd S = (cl S)\relint S.

(12.1.26)

cl S = (relint S) ∪ relbd S.

(12.1.27)

Note that, if S ⊆ Fn , then Definition 12.1.12. Let S ⊆ Fn. Then, S is solid if int S is nonempty. Furthermore, S is com-

pletely solid if S is solid and cl int S = cl S. The empty set is neither solid nor completely solid, and every nonempty, open set is both solid and completely solid. Definition 12.1.13. Let S ⊆ Fn. Then, S is bounded if there exists δ > 0 such that, for all x, y ∈ S, ∥x − y∥ < δ. (12.1.28) The set S is compact if it is both closed and bounded. Definition 12.1.14. Let S ⊂ Fn. Then, S is disconnected if there exist nonempty, disjoint subsets S1 , S2 of S that are open relative to S and whose union is S. Furthermore, S is connected if S is not disconnected. The empty set is compact and connected.

12.2 Limits of Sequences Proposition 12.2.1. Let (xi )∞ i=1 ⊂ F. Then, there exists at most one x ∈ F such that, for all

ε > 0, there exists n ≥ 1 such that, for all i > n, |xi − x| < ε.

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CHAPTER 12 △

Definition 12.2.2. Let X = (xi )∞ i=1 ⊂ F. Then, X converges to the limit x ∈ F if, for all ε > 0,

there exists n ≥ 1 such that, for all i > n, |xi −x| < ε. In this case, we write either lim X = limi→∞ xi = x or xi → x as i → ∞. Finally, X converges and limi→∞ xi exists if there exists x ∈ F such that (xi )∞ i=1 converges to x. △ Definition 12.2.3. Let X = (xi )∞ i=1 ⊂ R. Then, X has the limit ∞ if, for all M > 0, there exists n ≥ 1 such that, for all i > n, xi > M. In this case, we write either lim X = limi→∞ xi = ∞ or xi → ∞ as i → ∞. Likewise, X has the limit −∞ if, for all M > 0, there exists n ≥ 1 such that, for all i > n, xi < −M. In this case, we write either lim X = limi→∞ xi = −∞ or xi → −∞ as i → ∞. Note that lim X ∈ R ∪ {−∞, ∞}. However, the terminology “converges” and “limi→∞ xi exists” are used only in the case where lim X is finite. △ n n Definition 12.2.4. Let X = (xi )∞ i=1 ⊂ F . Then, X converges to the limit x ∈ F if limi→∞ ∥x − n xi ∥ = 0, where ∥ · ∥ is a norm on F . In this case, we write either x = limi→∞ xi or xi → x as i → ∞, where i ∈ P. X converges if there exists x ∈ Fn such that (xi )∞ i=1 converges to x. Now, let △ ∞ n×m n×m A = (Ai )i=1 ⊂ F . Then, A converges to A ∈ F if limi→∞ ∥A − Ai ∥ = 0, where ∥ · ∥ is a norm on Fn×m. In this case, we write either A = limi→∞ Ai or Ai → A as i → ∞, where i ∈ P. Finally, A converges and limi→∞ Ai exists if there exists A ∈ Fn×m such that (Ai )∞ i=1 converges to A. Theorem 11.1.8 implies that convergence of a sequence is independent of the choice of norm. Proposition 12.2.5. Let S ⊆ Fn, and let x ∈ Fn. Then, x ∈ cl S if and only if there exists a sequence (xi )∞ i=1 ⊆ S that converges to x. Furthermore, x ∈ ecl S if and only if there exists a sequence (xi )∞ i=1 ⊆ S\{x} that converges to x. Proof. Suppose that x ∈ ecl S. Then, for all i ∈ P, there exists xi ∈ S\{x} such that ∥x − xi ∥ < 1/i. ∞ Hence, x − xi → 0 as i → ∞ and thus (xi )∞ i=1 converges to x. Conversely, suppose that (xi )i=1 ⊆ S\{x} is such that xi → x as i → ∞, and let ε > 0. Then, there exists a positive integer p such that ∥x − xi ∥ < ε for all i > p. Therefore, x p+1 ∈ [Bε (x)\{x}] ∩ S, and thus [Bε (x)\{x}] ∩ S is nonempty. Hence, x is an essential closure point of S.  △ n Let X = (xi )∞ , where, for all i ∈ P, x ∈ F . Recall from Chapter 1 that X can be viewed as i i=1 the subset {x1 , x2 , . . .} of Fn , where the multiplicity of the components of the sequence X is ignored. The following result gives necessary conditions for X to converge to x ∈ Fn . △ n n Proposition 12.2.6. Let X = (xi )∞ i=1 ⊂ F , let x ∈ F , and assume that X converges to x. Then, the following statements hold: i) ecl X ⊆ {x} ⊆ cl X = X ∪ {x} = X ∪ ecl X. ii) X = cl X if and only if x ∈ X. iii) The following statements are equivalent: a) ecl X = ∅. b) X is finite. c) There exists a positive integer l such that, for all i ≥ l, xi = x. iv) The following statements are equivalent: a) ecl X = {x}. b) X is not finite. c) The subsequence Xˆ of X obtained by deleting all components of X that are equal to x converges to x. To illustrate iii), let X = (1, 1/2, 0, 0, 0, . . .). Then, limi→∞ xi = 0, ecl X = ∅, card(X) = 3, and 0 ∈ cl X = X = {0, 1/2, 1}. To illustrate iv), let X = (0, 1, 1/2, 1/3, 1/4, . . .). Then, limi→∞ xi = 0, ecl X = {0}, card(X) is not finite, and {i ∈ P : xi = 0} is finite. Also to illustrate iv), let X = (1, 0, 1/2, 0, 1/3, 0, 1/4, 0, . . .). Then, limi→∞ xi = 0, ecl X = {0}, card(X) is not finite, {i ∈ P : xi = 0}

FUNCTIONS, LIMITS, SEQUENCES, SERIES, INFINITE PRODUCTS, AND DERIVATIVES

917

is not finite, and the subsequence Xˆ of X obtained by deleting all components of X that are equal to 0 converges to 0. △ n The following result gives a necessary and sufficient condition for X = (xi )∞ i=1 ⊂ F to converge n to x ∈ F . △ n n Proposition 12.2.7. Let X = (xi )∞ i=1 ⊂ F , and let x ∈ F . Then, X converges to x if and only if, for all ε > 0, {i ∈ P : ∥x − xi ∥ > ε} is finite. △ △ n ∞ n Let X = (xi )∞ i=1 ⊆ R and Y = (yi )i=1 ⊆ R . Then, Y is a rearrangement of X if there exists σ : P 7→ P such that σ is one-to-one and onto and such that, for all i ∈ P, yi = xσ(i) . Note that Y is a rearrangement of X if and only if X is a rearrangement of Y. △ n Proposition 12.2.8. Let X = (xi )∞ i=1 ⊂ F . Then, the following statements are equivalent: i) X converges. ii) Every subsequence of X converges to lim X. iii) Every rearrangement of X converges to lim X. iv) Every rearrangement of every subsequence of X converges to lim X. Proof. i) =⇒ iii) follows from Proposition 12.2.7.  Theorem 12.2.9. Let S ⊂ Fn be compact, and let (xi )∞ ⊆ S. Then, there exists a subsequence i=1 ∞ (xi j )∞j=1 of (xi )∞ i=1 such that (xi j ) j=1 converges and lim j→∞ xi j ∈ S. Proof. See [2112, p. 145].  ∞ The sequence X = (xi )i=1 ⊂ R is (decreasing, nonincreasing, increasing, nondecreasing) if, for all i ≥ 1, (xi+1 < xi , xi+1 ≤ xi , xi+1 > xi , xi+1 ≥ xi ). Furthermore, (xi )∞ i=1 ⊂ R is bounded if the set {x1 , x2 , . . .} is bounded. The same terminology is used for sequences of vectors and matrices with a specified partial ordering. Proposition 12.2.10. Let X = (xi )∞ i=1 ⊂ R. Then, the following statements hold: i) X is bounded if and only if inf X ∈ R and sup X ∈ R. ii) Assume that X is nonincreasing. Then, lim X ∈ R ∪ {−∞}. Furthermore, lim X ∈ R if and only if inf X ∈ R. △ iii) Assume that sup X ∈ R. Then, Y = (sup (x j )∞j=i )∞ i=1 ⊂ R, and Y is nonincreasing. Furthermore, lim Y ∈ R if and only if inf X ∈ R. iv) Assume that X is nondecreasing. Then, lim X ∈ R ∪ {∞}. Furthermore, lim X ∈ R if and only if sup X ∈ R. △ v) Assume that inf X ∈ R. Then, Y = (inf (x j )∞j=i )∞ i=1 ⊂ R, and Y is nondecreasing. Furthermore, lim Y ∈ R if and only if sup X ∈ R. ∞ Definition 12.2.11. Let X = (xi )∞ i=1 ⊂ R. Then, the limit inferior of (xi )i=1 is defined by    inf X = −∞, △ △ −∞, lim inf X = lim inf xi =  (12.2.1)  limi→∞ inf (x j )∞j=i , inf X ∈ R. i→∞ Furthermore, the limit superior of (xi )∞ i=1 is defined by  ∞   △ △ limi→∞ sup (x j ) j=i , lim sup X = lim sup xi =   ∞, i→∞

sup X ∈ R, sup X = ∞.

Proposition 12.2.12. Let X = (xi )∞ i=1 ⊂ R. Then, the following statements hold:

i) inf X ≤ lim inf X ≤ lim sup X ≤ sup X. ii) inf X , ∞, and sup X , −∞. iii) lim inf X ∈ R ∪ {−∞, ∞}, and lim sup X ∈ R ∪ {−∞, ∞}.

(12.2.2)

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CHAPTER 12

iv) v) vi) vii) viii) ix) x) xi) xii)

inf X = −∞ if and only if lim inf X = −∞. If lim inf X ∈ R, then inf X ∈ R. If inf X ∈ R, then lim inf X ∈ R ∪ {∞}. sup X = ∞ if and only if lim sup X = ∞. If lim sup X ∈ R, then sup X ∈ R. If sup X ∈ R, then lim sup X ∈ R ∪ {−∞}. lim inf X ∈ R and lim sup X ∈ R if and only if X is bounded. X converges if and only if lim inf X = lim sup X ∈ R. If X converges, then X is bounded and −∞ < inf X ≤ lim inf X = lim X = lim sup X ≤ sup X < ∞.

(12.2.3)

xiii) There exists a subsequence (xi j )∞j=1 of (xi )∞ i=1 such that lim j→∞ xi j = lim inf i→∞ xi . ∞ xiv) There exists a subsequence (xi j ) j=1 of (xi )∞ i=1 such that lim j→∞ xi j = lim supi→∞ xi . Definition 12.2.13. Let X = (xi )∞ ⊂ R, and let α ≥ 0. Then, as n → ∞, i=1 xn = O(n−α ) if (i

α

xi )∞ i=1

(12.2.4)

is bounded. Furthermore, as n → ∞, xn = o(n−α )

(12.2.5)

α

if limn→∞ n xn = 0. Note that, as n → ∞, xn = O(1) means that X is bounded, whereas xn = o(1) means that X converges to zero. Furthermore, if α > β ≥ 0 and, as n → ∞, xn = O(n−α ), then, as n → ∞, xn = o(n−β ). ∞ Definition 12.2.14. Let X = (xi )∞ i=1 ⊂ R and Y = (yi )i=1 ⊂ R, and assume that Y has a finite number of components that are zero. Then, as n → ∞, xn ∼ yn

(12.2.6)

if limn→∞ xn /yn = 1. ∞ Proposition 12.2.15. Let X = (xi )∞ i=1 ⊂ R and Y = (yi )i=1 ⊂ R, let α and β be nonnegative

numbers, assume that β ≤ α, assume that Y has a finite number of components that are zero, assume that yn = O(n−β ), and assume that, as n → ∞, xn = yn + o(n−α ). Then, as n → ∞, xn ∼ yn . △ ∑∞ p 1/p , and Definition 12.2.16. Let X = (xi )∞ i=1 ⊂ F and p ∈ [1, ∞). Then, ∥X∥ p = ( i=1 |xi | ) △

ℓ p = {Y = (yi )∞ i=1 ⊂ F : ∥Y∥ p ∈ [0, ∞)}.

(12.2.7)

△

Furthermore, ∥X∥∞ = supi≥1 |xi |, and △

ℓ∞ = {Y = (yi )∞ i=1 ⊂ F : ∥Y∥∞ ∈ [0, ∞)}.

(12.2.8)

Finally, △

c0 = {Y = (yi )∞ i=1 ⊂ F : lim yi = 0}. i→∞

(12.2.9)

∞ Proposition 12.2.17. Let X = (xi )∞ i=1 ⊂ F, Y = (yi )i=1 ⊂ F, and p, q ∈ [1, ∞]. Then, the following statements hold: i) ∥X∥ p = 0 if and only if, for all i ≥ 1, xi = 0. ii) If α ∈ F and X ∈ ℓ p , then ∥αX∥ p = |α|∥X∥ p . iii) If p ≤ q, then ∥X∥∞ ≤ ∥X∥q ≤ ∥X∥ p ≤ ∥X∥1 .

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FUNCTIONS, LIMITS, SEQUENCES, SERIES, INFINITE PRODUCTS, AND DERIVATIVES

If 1 < p < q < ∞, then ℓ1 ⊂ ℓ p ⊂ ℓq ⊂ c0 ⊂ ℓ∞ . If p ∈ (1, ∞) and X ∈ ℓ p , then limr→∞ ∥X∥r = ∥X∥∞ . If X, Y ∈ ℓ p , then ∥X + Y∥ p ≤ ∥X∥ p + ∥Y∥ p and thus X + Y ∈ ℓ p . If X, Y ∈ c0 , then X + Y ∈ c0 . For all i ≥ 1, let Xi ∈ ℓ p and assume that, for all ε > 0, there exists n ≥ 1 such that, for all k, l > n, ∥Xk − Xl ∥ p < ε. Then, there exists X ∈ ℓ p such that limi→∞ ∥Xi − X∥ p = 0. ix) For all i ≥ 1, let Xi ∈ c0 and assume that, for all ε > 0, there exists n ≥ 1 such that, for all k, l > n, ∥Xk − Xl ∥∞ < ε. Then, there exists X ∈ c0 such that limi→∞ ∥Xi − X∥∞ = 0. ∑ x) For all i ≥ 1, let Xi ∈ ℓ p and assume that ∞ i=1 ∥Xi ∥ p < ∞. Then, there exists X ∈ ℓ p such ∑i that limi→∞ ∥ j=1 X j − X∥ p = 0. ∑ xi) For all i ≥ 1, let Xi ∈ c0 and assume that ∞ i=1 ∥Xi ∥∞ < ∞. Then, there exists X ∈ c0 such ∑i that limi→∞ ∥ j=1 X j − X∥∞ = 0.

iv) v) vi) vii) viii)

△

xii) For all i ≥ 1, let Xi = (xi, j )∞j=1 ∈ ℓ p , and assume that, for all j ≥ 1, x∞, j = limi→∞ xi, j exists, △

define X = (x∞, j )∞j=1 , and assume that (∥Xi ∥ p )∞ i=1 is bounded. Then, X ∈ ℓ p . △

i j−1 ) , and let p ∈ [1, ∞). Then, for all i ≥ 1, For all i ≥ 1, define Xi = (xi, j )∞j=1 , where xi, j = ( i+1

∥Xi ∥ p = √p

i+1 . (i + 1) p − i p

Hence, Xi ∈ ℓ p . Note that, for all j ≥ 1, limi→∞ xi, j = 1. However, the limiting sequence (1, 1, 1, . . .) is not an element of ℓ p . Furthermore, for all i ≥ 1, Xi ∈ c0 , but (1, 1, 1, . . .) is not an element of c0 . Alternatively, define Xi = (xi, j )∞j=1 by xi, j = min {i, j}. Then, for all i ≥ 1, ∥Xi ∥∞ = i, and, for all j ≥ 1, limi→∞ xi, j = j. However, the limiting sequence (1, 2, 3, . . .) is not an element of ℓ∞ . These examples show that, for all p ∈ [1, ∞], the component-wise limit of a sequence of elements of ℓ p is not necessarily an element of ℓ p , and likewise for c0 . Finally, for all i ≥ 1, define Xi = (xi, j )∞j=1 ∈ c0 , where, for all j ∈ {1, . . . , i}, xi, j = 1 and, for all j ≥ i + 1, xi, j = 0. Then, for all i ≥ 1, ∥Xi ∥∞ = 1 and, for all j ≥ 1, limi→∞ xi, j = 1. Since the limiting sequence (1, 1, 1, . . .) is not an element of c0 , it follows that xii) does not hold in the case where ℓ p is replaced by c0 and ∥ · ∥ p is replaced by ∥ · ∥∞ .

12.3 Series, Power Series, and Bi-power Series

∑ n is a series. Consider the sequence (xi )∞ . Then, the sequence of partial sums ( ki=1 xi )∞ i=1 ⊂ F∑ k=1 ∑k ∞ The limit of this series is denoted by i=1 xi . For convenience, we denote the series ( i=1 xi )∞ k=1 by ∑k ∑∞ ∞ i=1 xi whether or not ( i=1 xi )k=1 converges. ∑∞ n n Definition 12.3.1. Let (xi )∞ i=1 xi i=1 ⊂ F , and let∑∥ · ∥ be a norm on F . Then, the series ∑ ∑k ∞ ∞ ∞ converges if ( i=1 xi )k=1 converges. Furthermore, i=1 xi converges absolutely if the series i=1 ∥xi ∥ converges. ∑∞ n Proposition 12.3.2. Let (xi )∞ i=1 xi converges absolutely. i=1 ⊂ F , and assume that the series ∑∞ Then, the series i=1 xi converges. ∑ n×m Definition 12.3.3. Let (Ai )∞ , and let ∥ · ∥ be a norm on Fn×m. Then, the series ∞ i=1 Ai i=1 ⊂ F ∑k ∑ ∑ ∞ ∞ ∞ converges if ( i=1 Ai )k=1 converges. Furthermore, i=1 Ai converges absolutely if the series i=1 ∥Ai ∥ converges. ∑ n×m Proposition 12.3.4. Let (Ai )∞ , and assume that the series ∞ i=1 Ai converges absoi=1 ⊂ F ∑∞ lutely. Then, the series i=1 Ai converges. Definition 12.3.5. Let (βi )∞ i=0 ⊂ C, let z0 ∈ C, let D ⊆ C denote the set of all z ∈ C such that

920

∑∞

i=0

CHAPTER 12

βi (z − z0 )i converges, and define f : D 7→ C by △

f (z) =

∞ ∑

βi (z − z0 )i .

(12.3.1)

i=0

Then, f is a power series, and D is the domain of convergence of f . For all z ∈ D, the power series ∑ f converges at z. The power series f is absolutely convergent at z if ni=0 |βi ||z − z0 |i converges. ∞ Finally, f is a generating function for the sequence (βi )i=0 . Proposition 12.3.6. Let (βi )∞ i=0 ⊂ C, let z0 ∈ C, and define the power series f with domain of convergence D by (12.3.1). Then, exactly one of the following statements holds: i) D = {z0 }. ii) There exist ρ > 0 and Dρ ⊆ Sρ (z0 ) such that D = {z ∈ C : |z − z0 | < ρ} ∪ Dρ .

(12.3.2)

iii) D = C. In cases ii) and iii), (12.3.1) converges absolutely for all z ∈ int D. Proof. See [2113, pp. 67–69].  △ ∑ 1 i z . The domain of convergence D of f As an example, consider the power series f (z) = ∞ i=1 i is CIUD \{1}, which is case ii) of Proposition 12.3.6. Furthermore, f converges absolutely for all z ∈ OIUD . However, for all z ∈ UC, f does not converge absolutely. Consider the power series f defined by (12.3.1). Then, the domain of convergence D of f is either the point z0 , the union of Bρ (z0 ) and a subset of Sρ (z0 ), or the entire complex plane. ρ is the radius of convergence of f . In case iii), we set ρ = ∞. The following result gives the Cauchy-Hadamard formula, d’Alembert’s ratio test, and Cauchy’s root test, respectively, for the radius of convergence of a power series. Proposition 12.3.7. Let (βi )∞ i=1 ⊂ C, let z0 ∈ C, and define the power series f by (12.3.1) with domain of convergence D and radius of convergence ρ. Then, ρ=

1 lim supi→∞ |βi |1/i

.

(12.3.3)

Furthermore, if either limi→∞ |βi |1/i exists or is ∞, then ρ=

1 limi→∞ |βi |1/i

.

(12.3.4)

Finally, if either limi→∞ ββi+1i exists or is ∞, then

βi . ρ = lim i→∞ βi+1

(12.3.5)

Definition 12.3.8. A bi-sequence (xi )∞ i=−∞ = (. . . , x−2 , x−1 , x0 , x1 , x2 , . . .) is a tuple with a count-

ably infinite number of components. Now, let · · · < i−2 < i−1 < i0 < i1 < i2 < · · · . Then, (xi j )∞j=−∞ is a bisubsequence of (xi )∞ i=−∞ . Definition 12.3.9. Let (βi )∞ i=1−∞ ⊂ C, let z0 ∈ C, let D ⊆ C denote the set of all z ∈ C such that ∑∞ ∑∞ −i β (z − z ) and β (z − z0 )i converge, and define f : D 7→ C by −i 0 i i=1 i=0 △

f (z) =

∞ ∑

βi (z − z0 )i .

(12.3.6)

i=−∞

Then, f is a bi-power series; for all z ∈ D, f converges at z; and D is the domain of convergence of ∑ i f . The bi-power series f is absolutely convergent at z if ∞ i=−∞ |βi ||(z − z0 ) | converges. Finally, f is

FUNCTIONS, LIMITS, SEQUENCES, SERIES, INFINITE PRODUCTS, AND DERIVATIVES

921

a generating function for the bi-sequence (βi )∞ i=−∞ . ∞ Proposition 12.3.10. Let (βi )i=−∞ ⊂ C, and define the bi-power series f with domain of convergence D by (12.3.6). Then, exactly one of the following statements holds: i) D = ∅. ii) D = {z0 }. iii) D = C\{z0 }. iv) D = C. v) There exist ρ > 0 and Dρ ⊆ Sρ (z0 ) such that D = Bρ (z0 ) ∪ Dρ .

(12.3.7)

vi) There exist ρ > 0 and Dρ ⊆ Sρ (z0 ) such that D = {z ∈ C : 0 < |z − z0 | < ρ} ∪ Dρ .

(12.3.8)

vii) There exist r > 0, ρ > r, Dr ⊆ Sr (z0 ), and Dρ ⊆ Sρ (z0 ) such that D = {z ∈ C : r < |z − z0 | < ρ} ∪ Dr ∪ Dρ .

(12.3.9)

viii) There exist r > 0 and Dr ⊆ Sr (z0 ) such that D = {z ∈ C : r < |z − z0 |} ∪ Dr .

(12.3.10)

ix) D is nonempty, and there exists r > 0 such that D ⊆ Sr (z0 ). In cases iii)–viii), (12.3.1) converges absolutely for all z ∈ int D. Consider the bi-power series f defined by (12.3.6). Then, the domain of convergence D of f is either empty, the point z0 , the entire complex plane except for z0 , the entire complex plane, the union of Bρ (z0 ) and a subset of its boundary, the union of the punctured open disk Bρ (z0 )\{z0 } and a subset of its boundary, the union of the open annulus {z ∈ C : r < |z − z0 | < ρ} and a subset of its boundary, the union of the open outer disk {z ∈ C : r < |z − z0 |} and a subset of its boundary, or a nonempty subset of the circle Sr (z0 ). r is the inner radius of convergence of f , and ρ is the outer radius of convergence of f . In cases i), ii), v), and vi), we set r = 0; in cases iii), iv), and viii), we set ρ = ∞; in case ix), we set ρ = r. Proposition 12.3.11. Let (βi )∞ i=−∞ ⊂ C, and define the bi-power series f by (12.3.6) with domain of convergence D, assume that one of the statements ii)–ix) of Proposition 12.3.6 holds with inner radius of convergence r and outer radius of convergence ρ, where r ≤ ρ. Then, r = lim sup |β−i |1/i , i→∞

ρ=

1 . lim supi→∞ |βi |1/i

(12.3.11)

12.4 Continuity Definition 12.4.1. Let D ⊆ Fm, f : D 7→ Fn, and x ∈ D. Then, f is continuous at x if, for every

sequence (xi )∞ i=1 ⊆ D such that limi→∞ xi = x, it follows that limi→∞ f (xi ) = f (x). Furthermore, let D0 ⊆ D. Then, f is continuous on D0 if, for all x ∈ D0 , f is continuous at x. Finally, f is continuous if it is continuous on D. Definition 12.4.2. Let D ⊆ R, f : D 7→ Fn, and x0 ∈ cl D. Then, lim x↓x0 f (x) exists if there exists α ∈ Fn such that, for every sequence (xi )∞ i=1 ⊆ D ∩ [x0 , ∞) such that limi→∞ xi = x0 , △ limi→∞ f (xi ) = α. In this case, lim x↓x0 f (x) = α. Definition 12.4.3. Let D ⊆ R, f : D 7→ Fn, and x0 ∈ cl D. Then, lim x↑x0 f (x) exists if there exists α ∈ Fn such that, for every sequence (xi )∞ i=1 ⊆ D ∩ (−∞, x0 ] such that limi→∞ xi = x0 , △ limi→∞ f (xi ) = α. In this case, lim x↑x0 f (x) = α.

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Definition 12.4.4. Let D ⊆ R, f : D 7→ Fn, and x0 ∈ cl D. Then, lim x→x0 f (x) exists if there

exists α ∈ Fn such that, for every sequence (xi )∞ i=1 ⊆ D such that limi→∞ xi = x0 , limi→∞ f (xi ) = α. △ In this case, lim x→x0 f (x) = α. Proposition 12.4.5. Let D ⊆ Fm, f : D 7→ Fn, and x0 ∈ D. Then, f is continuous at x if and only if lim x→x0 f (x) = f (x0 ). Theorem 12.4.6. Let D ⊆ Fn be convex, and let f : D → F be convex. Then, f is continuous on relint D. Proof. See [333, p. 81] and [2319, p. 82].  △ Corollary 12.4.7. Let A ∈ Fn×m, and define f : Fm → Fn by f (x) = Ax. Then, f is continuous. m Proof. This is a consequence of Theorem 12.4.6. Alternatively, let x ∈ Fm, and let (xi )∞ i=1 ⊂ F ′ ′′ n n×m satisfy xi → x as i → ∞. Furthermore, let ∥ · ∥, ∥ · ∥ , and ∥ · ∥ be compatible norms on F , F , and Fm, respectively. Since ∥Ax − Axi ∥ ≤ ∥A∥′ ∥x − xi ∥′′, it follows that Axi → Ax as i → ∞.  The following result is a consequence of Corollary 12.4.7. △ △ n ∞ m Proposition 12.4.8. Let X = (xi )∞ i=1 ⊂ F and Y = (yi )i=1 ⊂ F , assume that X and Y converge, △ △ △ p×n p×m p define x = limi→∞ xi and y = limi→∞ yi , let A ∈ F and B ∈ F , and let Z = (zi )∞ i=1 ⊂ F , where, △ for all i ≥ 0, zi = Axi + Byi . Then, Z converges, and lim zi = Ax + By.

i→∞

(12.4.1)

The following result characterizes continuity of a function f in terms of properties of the inverse mapping f inv . Theorem 12.4.9. Let D ⊆ Fm and f : D 7→ Fn. Then, the following statements are equivalent: i) f is continuous. ii) For all open S ⊆ Fn, f inv (S) is open relative to D. iii) For all closed S ⊆ Fn, f inv (S) is closed relative to D. iv) For all S ⊆ Fn, f inv (int S) ⊆ intD f inv (S). v) For all S ⊆ Fn, cl f inv (S) ⊆ f inv (cl S). vi) For all D0 ⊆ D, f (cl D0 ) ⊆ cl f (D0 ). Proof. See [874, pp. 9, 10] and [2112, pp. 87, 110].  Corollary 12.4.10. Let A ∈ Fn×m and S ⊆ Fn. If S is open, then Ainv (S) is open. If S is closed, then Ainv (S) is closed. Theorem 12.4.11. Let D ⊂ Fm be compact, and let f: D 7→ Fn be continuous. Then, f (D) is compact. Proof. See [837, p. 125] and [2112, p. 146].  The following corollary of Theorem 12.4.11 shows that a continuous real-valued function defined on a compact set has a minimizer and a maximizer. Corollary 12.4.12. Let D ⊂ Fm be compact, and let f: D 7→ R be continuous. Then, there exist x0 , x1 ∈ D such that, for all x ∈ D, f (x0 ) ≤ f (x) ≤ f (x1 ). Corollary 12.4.13. Let A ∈ Fn×m, and let S ⊆ Fm be compact. Then, AS is compact. If S ⊆ Fm is closed but not bounded, then AS is not necessarily closed. For example, consider [ ] A = [1 0] and S = { yx ∈ R2 : x > 0, y = 1x }. Then, AS = (0, ∞). Corollary 12.4.14. Let D ⊆ Fm and f: D 7→ Fn , assume that f is one-to-one and f Inv : f (D) 7→ D is continuous, and let S ⊆ f (D) be compact. Then, f Inv (S) is compact. Proof. The result follows from Theorem 12.4.11.  Corollary 12.4.15. Let A ∈ Fn×m be left invertible, and let S ⊂ Fn be compact. Then, Ainv (S) is

923

FUNCTIONS, LIMITS, SEQUENCES, SERIES, INFINITE PRODUCTS, AND DERIVATIVES

compact. Furthermore, Ainv (S) = A+ [S ∩ R(A)].

(12.4.2) △

Proof. Since A is left invertible, it follows that f : Fm 7→ R(A) defined by f (x) = Ax is invertible.

Let A ∈ F be a left inverse of A. Then, the inverse f : R(A) 7→ F of f is given by f Inv (y) = L Inv A y. Since f is linear, it follows from Corollary 12.4.7 that f Inv is continuous. Corollary 12.4.14 thus implies that Ainv (S) is compact. The expression for Ainv (S) follows from Proposition 8.1.10.  Definition 12.4.16. Let D ⊆ R and f : D 7→ Fn. Then, f is an open mapping if, for all D0 ⊆ D such that D0 is open relative to D, f (D0 ) is open. Theorem 12.4.17. Let D ⊆ Fm and f : D 7→ Fn. Then, the following statements are equivalent: i) f is an open mapping. ii) For all D0 ⊆ D, f (intD D0 ) ⊆ int f (D0 ). Proof. See [874, pp. 11, 12].  m n Theorem 12.4.18. Let D ⊆ F , S ⊆ F , and f : D 7→ S, and assume that f is one-to-one and onto. Then, the following statements are equivalent: i) f and f Inv are continuous. ii) f is continuous, and f is an open mapping. iii) For all D0 ⊆ D, f (clD D0 ) = cl f (D0 ). If these conditions hold and D and S have nonempty interior, then m = n. Proof. See [874, p. 12].  The following result, called invariance of domain, implies that a function that is continuous and one-to-one is an open mapping. This result assumes that the domain and range of f have the same dimension. Theorem 12.4.19. Let D ⊆ Fn be open, let S ⊆ Fn , and let f: D 7→ S be one-to-one, onto, and continuous. Then, S is open, and f Inv : S 7→ Fn is continuous. Proof. See [2505, p. 3].  The following result is a variation of Theorem 12.4.19. Theorem 12.4.20. Let D ⊆ Fn be compact, and let f: D 7→ Fn be continuous and one-to-one. Then, f (D) is compact, and f Inv : f (D) 7→ Rn is continuous. Proof. See [837, p. 371].  The following result specializes Theorem 12.4.19 to the case f (x) = Ax, where A ∈ Fn×n . In this case, it follows from Corollary 3.7.7 that f is one-to-one if and only if A is nonsingular. Corollary 12.4.21. Let S ⊆ Fn, assume that S is open, let A ∈ Fn×n, and assume that A is nonsingular. Then, AS is open. The following result is the open mapping theorem. This is an extension of Corollary 12.4.21, where the domain and range of A need not have the same dimension. The nonsingularity condition of Corollary 12.4.21 is replaced by the assumption that A is right invertible; that is, rank A = n ≤ m. Theorem 12.4.22. Let S ⊆ Fm, let A ∈ Fn×m, assume that S is open, and assume that A is right invertible. Then, AS is open. Definition 12.4.23. Let S ⊆ Fn. Then, S is pathwise connected if, for all x1 , x2 ∈ S, there exists a continuous function f : [0, 1] 7→ S such that f (0) = x1 and f (1) = x2 . Proposition 12.4.24. Let S ⊆ Fn be pathwise connected. Then, S is connected. Proof. See [837, p. 259].  The converse of Proposition 12.4.24 is false. Let S = S1 ∪ S2 , where S1 = {0} × [0, 1] and [ ] S2 = { yx ∈ R2 : x ∈ (0, 1], y = sin 1x }. Then, S1 is not open relative to S. In fact, intS S1 = ∅. Thus, S L

m×n

Inv

n

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is connected. However, S is not pathwise connected. See [837, p. 260]. Theorem 12.4.25. Let D ⊂ Fm be connected, and let f: D 7→ Fn be continuous. Then, f (D) is connected. Proof. See [837, p. 259].  The following result is the Schauder fixed-point theorem. Theorem 12.4.26. Let D ⊆ Fm be nonempty, closed, and convex, let f : D → D be continuous, and assume that f (D) is bounded. Then, there exists x ∈ D such that f (x) = x. Proof. See [2836, p. 167].  The following corollary for the case of a bounded domain is the Brouwer fixed-point theorem. Corollary 12.4.27. Let D ⊆ Fm be nonempty, compact, and convex, and let f : D → D be continuous. Then, there exists x ∈ D such that f (x) = x. Proof. See [2836, p. 163].  m n Definition 12.4.28. Let D ⊆ F , and let f : D 7→ F . Then, f is locally Lipschitz if, for all ξ ∈ D and ε > 0, there exists µ > 0 such that, for all x, y ∈ Bε (ξ) ∩ D, ∥ f (x) − f (y)∥2 ≤ µ∥x − y∥2 . Furthermore, f is globally Lipschitz if there exists µ > 0 such that, for all x, y ∈ D, ∥ f (x) − f (y)∥2 ≤ µ∥x − y∥2 . The function f : R 7→ R defined by f (x) = x2 is locally Lipschitz but not globally Lipschitz.

12.5 Derivatives Let D ⊆ Fm, and let x0 ∈ D. Then, the feasible cone of D with respect to x0 is the set △

fcone(D, x0 ) = {ξ ∈ Fm : there exists α0 > 0 such that, for all α ∈ [0, α0 ), x0 + αξ ∈ D}. (12.5.1) Note that fcone(D, x0 ) is a pointed cone, although it may consist of only ξ = 0 as can be seen from the example x0 = 0 and { } 3 2 D = x ∈ R2 : 0 ≤ x(1) ≤ 1, x(1) ≤ x(2) ≤ x(1) . Now, let D ⊆ Fm and f : D → Fn. If ξ ∈ fcone(D, x0 ), then the one-sided directional differential of f at x0 in the direction ξ is defined by [ ] △ (12.5.2) D+ f (x0 ; ξ) = lim α1 f (x0 + αξ) − f (x0 ) α↓0

in the case where the limit exists. Similarly, if ξ ∈ fcone(D, x0 ) and −ξ ∈ fcone(D, x0 ), then the two-sided directional differential Df (x0 ; ξ) of f at x0 in the direction ξ is defined by [ ] △ D f (x0 ; ξ) = lim α1 f (x0 + αξ) − f (x0 ) (12.5.3) α→0

in the case where the limit exists. If ξ = ei so that the direction ξ is one of the coordinate axes, then the partial derivative of f with respect to x(i) at x0 , denoted by ∂f∂x(x(i)0 ) , is given by

Equivalently,

∂f (x0 ) △ = lim α1 [ f (x0 + αei ) − f (x0 )]. α→0 ∂x(i)

(12.5.4)

∂f (x0 ) = Df (x0 ; ei ), ∂x(i)

(12.5.5)

in the case where the two-sided directional differential Df (x0 ; ei ) exists. Note that

∂f (x0 ) ∂x(i)

∈ Fn×1.

Proposition 12.5.1. Let D ⊆ Fm be convex, let f : D 7→ Fn be convex, and let x0 ∈ int D.

Then, D+ f (x0 ; ξ) exists for all ξ ∈ fcone(D, x0 ). Proof. See [333, p. 83].



FUNCTIONS, LIMITS, SEQUENCES, SERIES, INFINITE PRODUCTS, AND DERIVATIVES

925

Note that D+ f (x0 ; ξ) = ±∞ is possible if x0 is an element of the boundary of√D. For example, consider the convex, continuous function f : [0, ∞) 7→ R given by f (x) = 1 − x. In this case, D+ f (0; 1) = −∞ and thus does not exist. Next, we consider a stronger form of differentiation. Note that this result does not assume that x0 is contained in the interior of the domain of f. Proposition 12.5.2. Let D ⊆ Fm and f : D 7→ Fn, assume that D is a solid, convex set, and let x0 ∈ D. Then, there exists at most one matrix F ∈ Fn×m satisfying lim ∥x − x0 ∥−1 [ f (x) − f (x0 ) − F(x − x0 )] = 0.

x→x0 x∈D\{x0 }

(12.5.6)

 In (12.5.6) the limit is taken over all sequences that are contained in D, do not include x0 , and converge to x0 . The restriction to sequences that do not include x0 is necessitated by the fact that ∥x − x0 ∥−1 [ f (x) − f (x0 ) − F(x − x0 )] is not defined for x = x0 . The restriction to sequences that are confined to D is not explicitly mentioned henceforth. Finally, note that D is not necessarily open, and x0 may be an element of D ∩ bd D. However, x0 must be an element of D since f (x0 ) plays a role in the limit. Definition 12.5.3. Let D ⊆ Fm , let f : D 7→ Fn, assume that D is a solid, convex set, let x0 ∈ D, and assume that there exists F ∈ Fn×m satisfying (12.5.6). Then, f is differentiable at x0 , and the matrix F is the derivative of f at x0 . In this case, we write f ′(x0 ) = F and Proof. See [2836, p. 170].

lim ∥x − x0 ∥−1 [ f (x) − f (x0 ) − f ′(x0 )(x − x0 )] = 0.

x→x0

(12.5.7)

f is differentiable if, for all x ∈ D, f is differentiable at x. (x0 ) We alternatively write d fdx for f ′(x0 ). Proposition 12.5.4. Let D ⊆ Fm be a solid, convex set, let f : D 7→ Fn, let x ∈ D, and assume that f is differentiable at x0 . Then, f is continuous at x0 . Let D ⊆ Fm be a solid, convex set, and let f : D 7→ Fn. In terms of its scalar components, f can be written as f = [ f1 · · · fn ]T, where fi : D 7→ F for all i ∈ {1, . . . , n} and f (x) = [ f1 (x) · · · fn (x)]T for all x ∈ D. With this notation, if f ′(x0 ) exists, then it can be written as   ′  f1 (x0 )     (12.5.8) f ′(x0 ) =  ...  ,   ′ fn (x0 ) where fi′(x0 ) ∈ F1×m is the gradient of fi at x0 and f ′(x0 ) is the Jacobian of f at x0 . Furthermore, if x ∈ int D, then f ′(x0 ) is related to the partial derivatives of f by [ ] ∂f (x0 ) ∂f (x0 ) ··· f ′(x0 ) = , (12.5.9) ∂x(1) ∂x(m) ∂f (x0 ) n×1 for all i ∈ {1, . . . , m}. Finally, note ∂x(i) ∈ F ∂fi(x0 ) n n×n , then ∂x( j) . For example, if x ∈ F and A ∈ F

where is

that the (i, j) entry of the n × m matrix f ′(x0 )

d Ax = A. dx

(12.5.10)

Note that the existence of the partial derivatives of f at x0 does not imply that f ′(x0 ) exists. That is, f may not be differentiable at x0 since f ′(x0 ) given by (12.5.9) may not satisfy (12.5.7). Let D ⊆ Fm and f : D 7→ Fn, and assume that f ′(x) exists for all x ∈ D and f ′ : D 7→ Fn×m is continuous. Then, f is continuously differentiable, which is also called C1. For all x0 ∈ D,

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f ′ (x0 ) ∈ Fn×m, and thus f ′ (x0 ) : Fm 7→ Fn. The second derivative of f at x0 ∈ D, denoted by f ′′(x0 ), is the derivative of f ′ : D 7→ Fn×m at x0 ∈ D. By analogy with the first derivative, it follows that f ′′(x0 ): Fm 7→ Fn×m is linear. Therefore, for all η ∈ Fm, f ′′ (x0 )η ∈ Fn×m, and, thus, for all η, ηˆ ∈ Fm, △ [ f ′′ (x0 )η]ˆη ∈ Fn. Defining f ′′(x0 )(η, η) ˆ = [ f ′′ (x0 )η]ηˆ , it follows that f ′′(x0 ): Fm × Fm 7→ Fn is m bilinear; that is, for all ηˆ ∈ F , the mapping η 7→ f ′′(x0 )(η, η) ˆ is linear, and, for all η ∈ Fm, the ′′ T mapping ηˆ 7→ f (x0 )(η, ηˆ ) is linear. Letting f = [ f1 · · · fn ] , it follows that   T ′′   T ′′  η f1 (x0 )ηˆ   η f1 (x0 )         .. ..  , (12.5.11) , f ′′(x0 )(η, η) ˆ =  f ′′(x0 )η =   . .    T ′′   T ′′ η fn (x0 )ηˆ η fn (x0 ) where, for all i ∈ {1, . . . , n}, the matrix fi′′(x0 ) is the m × m Hessian of fi at x0 . We write f (2) (x0 ) for f ′′(x0 ) and f (k) (x0 ) for the kth derivative of f at x0 . f is Ck if f (k) (x) exists for all x ∈ D and f (k) is continuous on D. The following result is the inverse function theorem [974, p. 185]. Theorem 12.5.5. Let D ⊆ Fn be open, let f : D 7→ Fn, and assume that f is Ck. Furthermore, let x0 ∈ D satisfy det f ′(x0 ) , 0. Then, there exist open sets N ⊂ Fn containing x0 and M ⊂ Fn containing f (x0 ) and a Ck function g: M 7→ N such that, for all x ∈ N, g[ f (x)] = x and, for all y ∈ M, f [g(y)] = y. Let S : [t0 , t1 ] 7→ Fn×m, and assume that every entry of S (t) is differentiable. Then, define △ ˙ S (t) = dSdt(t) ∈ Fn×m for all t ∈ [t0 , t1 ] entrywise; that is, for all i ∈ {1, . . . , n} and j ∈ {1, . . . , m}, d S (i, j) (t). (12.5.12) dt If either t = t0 or t = t1, then either d+/dt or∫ d−/dt (or just d/dt) denotes the right and left onet1 sided derivatives, respectively. Finally, define t S (t) dt entrywise; that is, for all i ∈ {1, . . . , n} and 0 j ∈ {1, . . . , m}, [ ∫ t1 ] ∫ t1 △ S (t) dt = S (i, j) (t) dt. (12.5.13) △

[S˙ (t)](i, j) =

t0

(i, j)

t0

12.6 Complex-Valued Functions Complex-valued functions of a complex variable possess properties that warrant separate consideration. The following definition specializes Definition 12.5.3 to the case F = C and m = n = 1. Definition 12.6.1. Let D ⊆ C be a solid, convex set, let f : D 7→ C, let z0 ∈ D, and assume that △

f ′ (z0 ) = lim

z→z0

f (z) − f (z0 ) z − z0

(12.6.1)

exists. Then, f is differentiable at z0 , and f ′(z0 ) is the derivative of f at z0 . Proposition 12.6.2. Let D ⊆ C be a solid, convex set, let f : D 7→ C, let z0 ∈ D, and assume △ △ ˆ = that f ′ (z0 ) exists. Furthermore, let D {(x, y) ∈ R2 : x + y ȷ ∈ D}, define u(x, y) = Re f (x + y ȷ), △ 2 ˆ 7→ R by v(x, y) = Im f (x + y ȷ), and fˆ : D [ ] △ u(x, y) fˆ(x, y) = , (12.6.2) v(x, y) and let z0 = x0 + y0 ȷ, where x0 , y0 ∈ R. Then, fˆ is differentiable at (x0 , y0 ), and  ∂u(x , y ) ∂u(x , y )  0 0  0 0     ∂x ∂y   . ′ fˆ (x0 , y0 ) =   ∂v(x0 , y0 ) ∂v(x0 , y0 )    ∂x ∂y

(12.6.3)

FUNCTIONS, LIMITS, SEQUENCES, SERIES, INFINITE PRODUCTS, AND DERIVATIVES

For convenience, we rewrite (12.6.3) as fˆ′ (x0 , y0 ) =

[

] u x (x0 , y0 ) uy (x0 , y0 ) . v x (x0 , y0 ) vy (x0 , y0 )

927

(12.6.4)

As an example, define f : C 7→ C by f (z) = z2 . Then, f ′ (z) = 2z and [ ] ˆf ′ (x, y) = 2x −2y . 2y 2x Likewise, define f : C 7→ C by f (z) = z3 . Then, f ′ (z) = 3z2 and [ 2 ] 3x − 3y2 −6xy fˆ′ (x, y) = . 6xy 3x2 − 3y2 The structure of fˆ′ (x, y) is explained by the following result. Proposition 12.6.3. Let D ⊆ C, let f : D 7→ C, and, for all z = x + y ȷ ∈ D, where x, y ∈ R, △ △ define u(x, y) = Re f (x + y ȷ) and v(x, y) = Im f (x + y ȷ). Then, the following statements hold: i) Let z0 = x0 + y0 ȷ ∈ int D, and assume that f ′ (z0 ) exists. Then, u x (x0 , y0 ) = vy (x0 , y0 ),

uy (x0 , y0 ) = −v x (x0 , y0 ).

(12.6.5)

ii) Let z0 = x0 + y0 ȷ ∈ int D, assume that there exists an open subset D0 ⊆ D such that z0 ∈ D0 and such that, for all z = x + y ȷ ∈ D0 , the partial derivatives u x (x, y), uy (x, y), v x (x, y), and vy (x, y) exist and are continuous at (x0 , y0 ), and assume that (12.6.5) holds. Then, f ′ (z0 ) exists. In addition, f ′ (z0 ) = u x (x0 , y0 ) + v x (x0 , y0 ) ȷ. iii) Assume that there exists an open subset D0 ⊆ D such that, for all z ∈ D0 , f ′ (z) exists, assume that u xx (x0 , y0 ), u xy (x0 , y0 ), uyx (x0 , y0 ), uyy (x0 , y0 ), v xx (x0 , y0 ), v xy (x0 , y0 ), vyx (x0 , y0 ), and vyy (x0 , y0 ) exist, and assume that u xy (x0 , y0 ) = uyx (x0 , y0 ) and v xy (x0 , y0 ) = vyx (x0 , y0 ). Then, u xx (x0 , y0 ) + uyy (x0 , y0 ) = 0,

v xx (x0 , y0 ) + vyy (x0 , y0 ) = 0.

(12.6.6)

 (12.6.5) is the Cauchy-Riemann equations. (12.6.6) shows that u and y are harmonic functions. Define f : C 7→ C by f (z) = z. Then, for all (x, y) ∈ R2 , u(x, y) = x and v(x, y) = −y. Consequently, for all (x, y) ∈ R2 , u x (x, y) = 1 , −1 = vy (x, y). It thus follows from Proposition 12.6.3 that, ˆ′ for all z ∈ C, f ′ (z) does not exist. However, [ in the ] notation of Proposition 12.6.2, f (x, y) exists for 1 0 all (x, y) ∈ R2 and is given by fˆ′ (x, y) = . 0 −1 As another example, define f : C 7→ C by f (z) = |z|2 . Then, for all (x, y) ∈ R2 , u(x, y) = x2 + y2 and v(x, y) = 0. Consequently, u x (x, y) = 2x = 0 = vy (x, y) and uy (x, y) = 2y = 0 = −v x (x, y) if and only if x = y = 0. It thus follows from Proposition 12.6.3 that f ′ (z) exists if and only if z = 0. However, [in the notation of Proposition 12.6.2, fˆ′ (x, y) exists for all (x, y) ∈ R2 and is given by ] 2x 2y fˆ′ (x, y) = . 0 0 Definition 12.6.4. Let D ⊆ C, let f : D 7→ C, let D0 ⊆ D, assume that D0 is open, assume that f ′ (z) exists for all z ∈ D0 , and let z0 ∈ D0 . Then, f is analytic in D0 , and f is analytic at z0 . The function f : C 7→ C defined by f (z) = z2 is analytic in C. The function f : C\{0} 7→ C defined by f (z) = 1/z is analytic in C\{0}. If D ⊆ C is an open set, then the function f : D 7→ C defined by f (z) = |z|2 is not analytic in D since f ′ (z) exists only if z = 0. Finally, if D ⊆ C is an Proof. See [582, pp. 63–68, 78, 79].

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open set, then the function f : D 7→ C defined by f (z) = z is not analytic since, for all z ∈ C, f ′ (z) does not exist. Proposition 12.6.5. Let D ⊆ C, assume that D is open and connected, let f : D 7→ C, assume that f is analytic in D, let D0 ⊆ D, assume that D0 is either a line segment of positive length or is open and nonempty, and assume that, for all z ∈ D0 , f (z) = 0. Then, for all z ∈ D, f (z) = 0. Proof. See [582, pp. 83, 84].  Definition 12.6.6. Let S1 , S2 ⊆ C, assume that S1 and S2 are open, let f : S1 7→ S2 , and assume that f is analytic in S1 , one-to-one, and onto. Then, f is a conformal mapping. Proposition 12.6.7. Let D ⊆ C, assume that D is open, let f : D 7→ C, and assume that f is analytic in D. Then, for all n ≥ 1 and z ∈ D, f (n) (z) exists. Furthermore, for all n ≥ 1, f (n) is analytic in D. Proof. See [582, p. 168].  The following shows that an analytic function has a power series expansion in each open disk. Proposition 12.6.8. Let D ⊆ C, assume that D is open, let f : D 7→ C, assume that f is △ analytic in D, let z0 ∈ D, let ρ0 > 0 satisfy D0 = {z ∈ C : |z − z0 | < ρ} ⊆ D, and, for all i ≥ 0, define △

βi =

f (i) (z0 ) . i!

(12.6.7)

Then, for all z ∈ D0 , f (z) =

∞ ∑

βi (z − z0 )i .

(12.6.8)

i=0

Furthermore, (12.6.8) converges absolutely in D0 . Proof. See [582, p. 189].  The power series (12.6.8) is the Taylor series of f at z0 . The radius of convergence of the Taylor series (12.6.8) is characterized by Proposition 12.3.6. Proposition 12.6.9. Let D ⊆ C, let f : D 7→ C, let z0 ∈ C, let r, R ∈ [0, ∞] satisfy r < R and △ D0 = {z ∈ C : r < |z − z0 | < R} ⊆ D, and assume that f is analytic in D0 . Then, there exists a unique bi-sequence (βi )∞ i=−∞ ⊂ C such that, for all z ∈ D0 , f (z) =

∞ ∑

βi (z − z0 )i .

(12.6.9)

i=−∞

Furthermore, (12.6.9) converges absolutely in D0 . Finally, for all i ∈ Z, I 1 f (z) βi = dz, 2π ȷ C (z − z0 )i+1

(12.6.10)

where C is a counterclockwise circle centered at z0 and lying in D0 . Proof. See [1136, pp. 165–168].  The series in (12.6.9) is the Laurent series of f at z0 . If r = 0 and R is finite, then D0 is a punctured open disk. If r = 0 and R is infinite, then D0 is a punctured plane. If r > 0 and R is finite, then D0 is an open annulus. If r > 0 and R is infinite, then D0 is an open outer disk. If r = 0 and f is analytic at z0 , then the Laurent series is a power series. Define f , D, z0 , r, R, D0 , and (βi )∞ i=−∞ as in Proposition 12.6.9, and assume that z0 < D, and thus f is not defined at z0 . Then, (12.6.9) can be written as f (z) = fouter (z) + finner (z), △

where fouter (z) =

∑−1

i=−∞

△

(12.6.11) △

βi (z − z0 )i is analytic on Douter = {z ∈ C : r < |z − z0 |} and finner (z) =

FUNCTIONS, LIMITS, SEQUENCES, SERIES, INFINITE PRODUCTS, AND DERIVATIVES

∑∞

929

△

βi (z − z0 )i is analytic on Dinner = {z ∈ C : |z − z0 | < R}. If r = 0, then z0 is an isolated singularity of f . If z0 is an isolated singularity of f and, for all i < 0, βi = 0, then f˜ : D0 7→ C defined by △ △ f˜(z) = f (z) for all z ∈ D0 and f˜(z0 ) = β0 is analytic on D0 . In this case, z0 is a removable singularity of f , and the radius of convergence ρ of (12.6.11) satisfies R ≤ ρ. If z0 is an isolated singularity of f and {i < 0 : βi , 0} is finite, then z0 is a pole of f. If z0 is a pole of f and max {i < 0 : βi , 0} = −1, then z0 is a simple pole. If z0 is an isolated singularity of f and is neither a removable singularity nor a pole, then z0 is an essential singularity. As an example, define f : C\{1} 7→ C by f (z) = z/(z − 1). Then, the Laurent series of f at 0 in OOUD = {z ∈ C : |z| > 1} is given by ∞ ∑ 1 f (z) = . zi i=1 i=0

Note that 0 is not an isolated singularity of f since r = 1. On the other hand, the Laurent series of f centered at 1 in the punctured plane {z ∈ C : |z − 1| > 0} is given by 1 + 1, z−1 which shows that 1 is a pole of f (z). As another example, the function f : C\{0} 7→ C defined by f (z) = e1/z has the Laurent series ∞ ∑ 11 f (z) = , i! zi i=0 f (z) =

which shows that 0 is an essential singularity of f. Finally, the function f : C\{0} 7→ C defined by △ f (z) = (sin z)/z has a removable singularity at 0 since f˜ : C 7→ C defined by f˜(z) = f (z) for all △ z ∈ C\{0} and f˜(0) = 1 is analytic in C. For n ≥ 2, consider the function f : C 7→ C defined by f (z) = z1/n , which is the principal nth root. Note that, for all z ∈ (−∞, 0), f is not continuous at z. For example, it follows from (1.6.32) that limθ↑π (eθ ȷ )1/2 = ȷ and limθ↓−π (eθ ȷ )1/2 = − ȷ. Therefore, there does not exist an open set D ⊂ C such that 0 ∈ D and f is analytic on D\{0}. Consequently, f does not have a Laurent series at z = 0. As a final example, consider the function f : C\{1, 2} 7→ C defined by f (z) = 1/[(z − 1)(z − 2)]. Then, f has a power series in the region {z ∈ C : |z| < 1} and Laurent series in the regions {z ∈ C : 1 < |z| < 2} and {z ∈ C : |z| > 2} given by ) ∑ ∞ (  1    1 − zi , |z| < 1,   i+1  2   i=0     ∞ ∞     ∑ 1 i ∑1 − z − , 1 < |z| < 2, (12.6.12) f (z) =    zi 2i+1   i=0 i=1     ∞  ∑   1  i−1   |z| > 2.   (2 − 1) zi , i=1 For details, see [582, pp. 203–205].

12.7 Infinite Products Definition 12.7.1. Let (xi )∞ i=1 ⊂ C. Then, the infinite product

k ≥ 1 such that limn→∞

∏n

∏∞

i=k xi exists and is nonzero. In this case,  ∞  ∏  0 ∈ (xi )∞ △ 0, i=1 , xi =  ∏n  limn→∞ i=1 xi , 0 < (xi )∞ . i=1 i=1

i=1 xi

converges if there exists

(12.7.1)

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∏n ∏∞ △ If (xi )∞ i=1 xi = ±∞, respectively. i=1 xi = ±∞, then i=1 ⊂ R and limn→∞ ∏ ∏ i For all k ≥ 1, limn→∞ ni=k (−1)i does not exist, and thus ∞ i=1 (−1) does not converge. For ∏n ∏∞ all k ≥ 1, limn→∞ i=k i = ∞, and thus i=1 i does not converge. If (xi )∞ i=1 = (0, 1, 2, 3, . . .), ∏ ∏ ∏∞ then limn→∞ ni=1 xi = 0 and, for all k ≥ 2, ∞ xi does not converge. If i = ∞, and thus i=1 i=k x ∏n ∏ (xi )∞ 0, and thus ∞ not converge. i=1 xi does i=k xi = ∏ i=1 = (0, 1, 0, 1, . . .), then, for all k ≥ 1, limn→∞ ∏ n If (xi )∞ xi = 0, and thus ∞ xi does not i=1 i=k i=1 = (1, 1/2, 1/3, 1/4, . . .), then, for all k ≥ 1, limn→∞ ∏ ∏ converge. However, if (xi )∞ = (0, 1, 1, 1, . . .), then limn→∞ ni=2 xi = 1, and thus ∞ xi converges; i=1 i=1 ∏ in fact, ∞ i=1 xi = 0. Proposition 12.7.2. Let (xi )∞ i=1 ⊂ C. Then, the following statements are equivalent: ∏∞ i) i=1 xi converges. ∑ ii) limi→∞ xi = 1 and {i∈P : xi ,0} log xi converges. ∑ iii) card({i ∈ P : xi = 0}) is finite and {i∈P : xi ,0} log xi converges. If (xi )∞ i=1 ⊂ (0, ∞), then the following statements are equivalent: ∏ iv) ∞ xi converges. ∑ i=1 log xi converges. v) ∞ ∏∞ i=1 If i=1 xi converges, then the following statements hold: vi) card {i ∈ P : xi = 0} is finite. ∑∞ ∏∞ i=1 log xi , 0. vii) If 0 < (xi )∞ i=1 xi = e i=1 , then The following statements are equivalent: ∏ (1 + |xi |) converges. viii) ∞ ∑∞i=1 ix) i=1 |xi | converges. Now, assume that (xi )∞ i=1 ⊂ R. Then, the following statements are equivalent: ∏∞ x) i=1 xi = ∞. ∑∞ ∑ 1 xi) Re ∞ i=1 log xi = ∞ and π Im i=1 log xi is an even integer. Furthermore, the following statements are equivalent: ∏ xii) ∞ i=1 xi = −∞. ∑ ∑∞ 1 xiii) Re ∞ i=1 log xi = ∞ and π Im i=1 log xi is an odd integer. Proof. See [116, pp. 595–597] and [1136, pp. 352, 353]. 

12.8 Functions of a Matrix Let (βi )∞ i=0 ⊂ C, let f : D 7→ C denote the power series △

f (z) =

∞ ∑

β i zi ,

(12.8.1)

i=0

where D is the domain of convergence of f , and let ρ be the radius of convergence of f. Next, let A ∈ Cn×n satisfy ρmax (A) < ρ, and define f (A) by the infinite series

FUNCTIONS, LIMITS, SEQUENCES, SERIES, INFINITE PRODUCTS, AND DERIVATIVES

△

f (A) =

∞ ∑

βi Ai .

931

(12.8.2)

i=0

To show that the infinite series (12.8.2) converges, we express A as A = SBS −1, where S ∈ Cn×n is nonsingular and B ∈ Cn×n. It thus follows that f (A) = S f (B)S −1.

(12.8.3)

Now, let B = diag(J1, . . . , Jr ) be a Jordan form of A. Then, f (A) = S diag[ f (J1 ), . . . , f (Jr )]S −1.

(12.8.4)

Letting λ ∈ D be an eigenvalue of A and J = λIk + Nk denote a k × k Jordan block, expanding the ∑ i infinite series ∞ i=1 βi J shows that f (J) is the k × k upper triangular, Toeplitz matrix f (J) = f (λ)Ik + f ′(λ)Nk + 21 f ′′(λ)Nk2 + · · · +   f (λ)   0   =  0   ..  .  0

f ′(λ)

1 ′′ 2 f (λ)

···

f (λ)

f ′(λ)

···

0

f(λ)

···

.. .

..

..

0

0

.

.

···

1 f (k−1)(λ)Nkk−1 (k − 1)!

   1 (k−2) (λ)  (k−2)! f   1 (k−3) f (λ)  . (k−3)!   ..  .   f (λ) 1 (k−1) (λ) (k−1)! f

(12.8.5)

Note that every entry of f (J) exists since f converges in D and all of its derivatives exist in D. Therefore, the infinite series (12.8.2) converges. Alternatively, since ρmax (A) < ρ, Proposition 11.3.3 implies that there exists a normalized submultiplicative norm ∥ · ∥ on Fn×n such that ∥A∥ < ρ. Therefore, since ρ is the radius of convergence ∑∞ ∑∞ ∑ i i i of f, it follows that ∞ i=0 βi A is absolutely convergent. i=0 |βi |∥A∥ exists. Hence, i=0 ∥βi A ∥ ≤ Next, we extend the definition f (A) to functions f : D ⊆ C 7→ C that are not necessarily of the form (12.8.1). Definition 12.8.1. Let f : D ⊆ C 7→ C, let A ∈ Cn×n, where spec(A) ⊂ D, and assume that, for △ all λi ∈ spec(A), f is ki − 1 times differentiable at λi , where ki = indA (λi ) is the largest size of all of the Jordan blocks associated with λi as given by Theorem 7.4.2. Then, f is defined at A, and f (A) is given by (12.8.3) and (12.8.4), where f (Ji ) is defined by (12.8.5) with k = ki and λ = λi . The following result shows that f (A) in Definition 12.8.1 is well-defined in the sense that f (A) is independent of the decomposition A = SBS −1 used to define f (A) in (12.8.3). △ Theorem 12.8.2. Let A ∈ Fn×n, let spec(A) = {λ1 , . . . , λr }, and, for all i ∈ {1, . . . , r}, let ki = indA (λi ). Furthermore, suppose that f : D ⊆ C 7→ C is defined at A. Then, there exists a polynomial p ∈ F[s] such that f (A) = p(A). Furthermore, there exists a unique polynomial p of degree less than ∑r i=1 ki satisfying f (A) = p(A) and such that, for all i ∈ {1, . . . , r} and j ∈ {0, 1, . . . , ki − 1}, f ( j)(λi ) = p( j)(λi ).

(12.8.6)

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This polynomial is given by

        ki −1  ∑ r r  k ∑ ∏ f (s) 1 d   (s − λ j )n j p(s) = ∏r  k k l  l=1 (s − λl )  k=0 k! ds j=1 i=1 l,i j,i

   (s − λi )k  . 

(12.8.7)

s=λi

If, in addition, A is simple, then p is given by p(s) =

r ∑

f (λi )

i=1

r ∏ s − λj . λ − λj j=1 i

(12.8.8)

j,i



Proof. See [799, pp. 263, 264].

The polynomial (12.8.7) is the Lagrange-Hermite interpolation polynomial for f. The following result, called the identity theorem, is a special case of Theorem 12.8.2. △ Theorem 12.8.3. Let A ∈ Fn×n, let spec(A) = {λ1 , . . . , λr }, and, for all i ∈ {1, . . . , r}, let ki = indA (λi ). Furthermore, let f : D ⊆ C 7→ C and g: D ⊆ C 7→ C be analytic on a neighborhood of spec(A). Then, f (A) = g(A) if and only if, for all i ∈ {1, . . . , r} and j ∈ {0, 1, . . . , ki − 1}, f ( j)(λi ) = g( j)(λi ).

(12.8.9)

Corollary 12.8.4. Let A ∈ Fn×n, and let f : D ⊂ C 7→ C be analytic on a neighborhood of

mspec(A). Then,

mspec[ f (A)] = f [mspec(A)].

(12.8.10)

12.9 Matrix Square Root and Matrix Sign Functions Theorem 12.9.1. Let A ∈ Cn×n, and assume that A is group invertible and has no eigenvalues

in (−∞, 0). Then, there exists a unique matrix B ∈ Cn×n such that spec(B) ⊂ ORHP ∪ {0} and such that B2 = A. If, in addition, A is real, then B is real. Proof. See [1391, pp. 20, 31].  The matrix B given by Theorem 12.9.1 is the principal square root of A. This matrix is denoted by A1/2. The existence of a square root that is not necessarily the principal square root is discussed in Fact 7.17.21. The following result defines the matrix sign function. Definition 12.9.2. Let A ∈ Cn×n, assume that A has no eigenvalues on the imaginary axis, and [ ] let J 0 −1 A=S 1 S , 0 J2 where S ∈ Cn×n is nonsingular, J1 ∈ C p×p and J2 ∈ Cq×q are Jordan matrices, and spec(J1 ) ⊂ OLHP and spec(J2 ) ⊂ ORHP. Then, the matrix sign of A is defined by [ ] −Ip 0 −1 △ Sign(A) = S S . 0 Iq

12.10 Vector and Matrix Derivatives In this section we consider derivatives of scalar-valued functions with matrix arguments. Consider the linear function f : Fm×n 7→ F given by f (X) = tr AX, where A ∈ Fn×m and X ∈ Fm×n. In △ terms of vectors x = vec X ∈ Fmn, we can define the linear function fˆ(x) = f (X) = (vec AT )T x. Cond m×n m×n 7→ F can be represented for all Y ∈ Fm×n sequently, for all X0 ∈ F , the function dX f (X0 ) : F

933

FUNCTIONS, LIMITS, SEQUENCES, SERIES, INFINITE PRODUCTS, AND DERIVATIVES

by d f (X0 )Y = fˆ′(vec X0 ) vec Y = (vec AT )T vec Y = tr AY. dX Noting that fˆ′(vec X0 ) = (vec AT )T and identifying derivative of f : D ⊆ Fm×n 7→ F by

d dX

(12.10.1)

f (X0 ) with the matrix A, we define the matrix

d △ f (X) = (vec−1 [ fˆ′(vec X)]T )T , dX

(12.10.2)

(X) which is the n × m matrix A whose (i, j) entry is ∂f ∂X( j,i) . Note the ordering of the indices. The matrix derivative is a representation of the derivative in the sense that

lim

X→X0 X∈D\{X0 }

f (X) − f (X0 ) − tr[F(X − X0 )] = 0, ∥X − X0 ∥

f (X0 ) and ∥ · ∥ is a norm on Fm×n. Proposition 12.10.1. Let x ∈ Fn. Then, the following statements hold: i) If A ∈ Fn×n, then d T x Ax = xT (A + AT ). dx ii) If A ∈ Fn×n is symmetric, then d T x Ax = 2xTA. dx iii) If A ∈ Fn×n is Hermitian, then d ∗ x Ax = 2x∗A. dx

where F denotes

(12.10.3)

d dX

(12.10.4)

(12.10.5)

(12.10.6)

Proposition 12.10.2. Let A ∈ Fn×m and B ∈ Fl×n. Then, the following statements hold:

i) For all X ∈ Fm×n, ii) For all X ∈ Fm×l, iii) For all X ∈ Fl×m,

d tr AX = A. dX

(12.10.7)

d tr AXB = BA. dX

(12.10.8)

d tr AXTB = ATBT. dX

(12.10.9)

d tr (AXB)k = kB(AXB)k−1A. dX

(12.10.10)

d det AXB = B(AXB)AA. dX

(12.10.11)

iv) For all X ∈ Fm×l and k ≥ 1,

v) For all X ∈ Fm×l,

vi) For all X ∈ Fm×l such that AXB is nonsingular, d log det AXB = B(AXB)−1A. (12.10.12) dX Proposition 12.10.3. Let A ∈ Fn×m and B ∈ Fm×n. Then, the following statements hold:

934

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i) For all X ∈ Fm×m and k ≥ 1, ∑ d tr AXkB = Xk−1−iBAX i. dX i=0

(12.10.13)

d tr AX −1B = −X −1BAX −1. dX

(12.10.14)

k−1

ii) For all nonsingular X ∈ Fm×m,

iii) For all nonsingular X ∈ Fm×m, d det AX −1B = −X −1B(AX −1B)AAX −1. dX iv) For all nonsingular X ∈ Fm×m, d log det AX −1B = −X −1B(AX −1B)−1AX −1. dX Proposition 12.10.4. The following statements hold: i) Let A, B ∈ Fn×m. Then, for all X ∈ Fm×n,

ii) Let A ∈ Fn×n

d tr AXBX = AXB + BXA. dX and B ∈ Fm×m. Then, for all X ∈ Fn×m,

d tr AXBXT = BXTA + BTXTAT. dX iii) Let A ∈ Fn×n. Then, for all X ∈ Fn×m, d tr XTAX = XT(A + AT ). dX

(12.10.15)

(12.10.16)

(12.10.17)

(12.10.18)

(12.10.19)

iv) Let A ∈ Fk×l, B ∈ Fl×m, C ∈ Fn×l, D ∈ Fl×l, and E ∈ Fl×k, and let p be a positive integer. Then, for all X ∈ Fm×n , ∑ d tr A(D + BXC) pE = C(D + BXC) p−iEA(D + BXC)i−1B. dX i=1 p

(12.10.20)

v) Let A ∈ Fk×l, B ∈ Fl×m, C ∈ Fn×l, D ∈ Fl×l, and E ∈ Fl×k. Then, for all X ∈ Fm×n such that D + BXC is nonsingular, d tr A(D + BXC)−1E = −C(D + BXC)−1EA(D + BXC)−1B. dX

(12.10.21)

vi) Let A ∈ Fk×l, B ∈ Fl×m, C ∈ Fn×l, D ∈ Fl×l, and E ∈ Fl×k. Then, for all X ∈ Fn×m such that D + BXTC is nonsingular, d tr A(D + BXTC)−1E = −BT (D + BXTC)−TATE T (D + BXTC)−TCT. dX

12.11 Facts on One Set Fact 12.11.1. Let x ∈ R, and assume that x is an irrational number. Then,

cl {nx + m : n, m ∈ Z} = R. Source: [2294, pp. 44, 45]. Remark: This is Kronecker’s theorem.

(12.10.22)

FUNCTIONS, LIMITS, SEQUENCES, SERIES, INFINITE PRODUCTS, AND DERIVATIVES

935

Fact 12.11.2. Let S ⊂ Fn, let ∥ · ∥ be a norm on Fn, assume that there exists δ > 0 such that, for

all x, y ∈ S, ∥x − y∥ < δ, and let x0 ∈ S. Then, S ⊆ Bδ (x0 ). △ Fact 12.11.3. Let S ⊂ Rn, assume that S is bounded, and let δ = sup {∥x − y∥2 : x, y ∈ S}. Then, there exists x0 ∈ Rn such that S ⊆ B √ n δ (x0 ). Source: [1260, p. 49]. Remark: This is Jung’s 2n+2

theorem. Remark: For n = 2, the bound is achieved in the case where S is an equilateral triangle. Fact 12.11.4. Let S ⊆ Fn. Then, cl S is the smallest closed set containing S, and int S is the largest open set contained in S. Fact 12.11.5. Let S ⊆ Fn. If S is (open, closed), then S∼ is (closed, open). Fact 12.11.6. Let S ⊆ S′ ⊆ Fn. If S is (open relative to S′, closed relative to S′ ), then S′ \S is (closed relative to S′, open relative to S′ ). Fact 12.11.7. Let S ⊆ Fn. Then, (int S)∼ = cl(S∼ ),

bd S = cl bd S = bd(S∼ ) = (cl S) ∩ cl(S∼ ) = [(int S) ∪ int(S∼ )]∼ .

Fact 12.11.8. Let S ⊆ Fn, and assume that int S = ∅. Then, bd S = cl S. Hence, if S is closed,

then S = bd S, whereas, if S is not closed, then S ⊂ bd S. Fact 12.11.9. Let S ⊆ Fn, and assume that S is either open or closed. Then, int bd S is empty. Source: [154, p. 68]. Remark: Let S = {x ∈ R : x is a rational number}. Then, bd S = R. Fact 12.11.10. Let S ⊆ Fn, and assume that int S is nonempty. Then, dim S = n. Fact 12.11.11. Let S ⊆ Fn, and assume that S is an affine subspace. Then, S is closed. Furthermore, S is open if and only if either S = ∅ or S = Fn. Fact 12.11.12. Let S ⊆ Fn. Then, cl S ⊆ affin cl S = cl affin S = affin S. Source: [523, p. 7]. Fact 12.11.13. Let n ≥ 2, let 1 ≤ k ≤ n − 1, and let x1 , . . . , xk ∈ Fn. Then, int affin {x1 , . . . , xk } = ∅. Related: Fact 3.11.14. Fact 12.11.14. Let x ∈ Fn, and let ε > 0. Then, Bε (x) is completely solid and convex. Fact 12.11.15. Let S ⊆ Fn, and assume that S is a (cone, convex set, convex cone, affine subspace, subspace). Then, so are cl S, int S, and relint S. Source: [1260, p. 44], [2319, p. 45], and [2320, p. 64]. Fact 12.11.16. Let S ⊆ Fn, and assume that S is convex. Then, relint cl S = relint S, int cl S = int S, and S ⊆ cl relint S = cl S. Source: [1260, p. 44] and [2487, p. 95], and use Fact 12.12.4. Fact 12.11.17. Let S ⊆ Fn, and assume that S is solid. Then, conv S is completely solid. Fact 12.11.18. Let S ⊆ Fn. Then, conv relint S ⊆ relint conv S, and conv int S ⊆ int conv S. If, in addition, S is completely solid, then conv int S = int conv S. Fact 12.11.19. Let S ⊆ Fn, and assume that S is open. Then, conv S is open. Source: Assume that S is nonempty, so that S is completely solid. Fact 12.11.18 implies that conv S = conv int S = int conv S. Fact 12.11.20. Let S ⊆ Fn, and assume that S is convex. Then, the following statements are equivalent: i) S is solid. ii) S is completely solid. iii) dim S = n. iv) affin S = Fn. Fact 12.11.21. Let S ⊆ Fn. Then, the following statements hold: i) conv cl S ⊆ cl conv S. ii) If S is bounded, then conv S is bounded. iii) If S is either bounded or convex, then conv cl S = cl conv S.

936

CHAPTER 12

iv) If S is convex, then cl S is convex. v) If S is compact, then conv S is compact. vi) conv relint S ⊆ relint conv S. 2 2 Source: [1260, p. 45] and [2487, pp. 128–130]. Remark: S = {x ∈ R2 : x(1) x(2) = 1 for all x(1) > 0} is closed, but conv S is not closed. Hence, conv cl S ⊂ cl conv S. Likewise, S = {[0 0]T } ∪ {[x 1]T ∈ R2 : x ∈ R} satisfies conv cl S ⊂ cl conv S. Fact 12.11.22. Let A ∈ Fn×m , S1 ⊆ Fm , and S2 ⊆ Fn . Then, relint conv AS1 = A relint conv S1 , relint conv A (S2 ) = Ainv (relint conv[S2 ∩ R(A)]) ⊆ Ainv (conv S2 ). inv

Source: [2487, p. 135]. Related: Fact 3.11.17. Fact 12.11.23. Let S ⊆ Fn. Then, relbd S is closed. Now, assume that S is convex. Then,

relbd S = relbd cl S = relbd relint S. Furthermore, relbd S is empty if and only if S is an affine subspace. Source: [2487, p. 104]. Fact 12.11.24. Let S1 ⊆ Fm, S2 ⊆ Fn, and A ∈ Fn×m. Then, relint AS1 = A relint S1 , relint A (S2 ) = Ainv (relint[S2 ∩ R(A)]) ⊆ Ainv (relint S2 ). inv

If, in addition, (relint S2 ) ∩ R(A) , ∅, then relint Ainv (S2 ) = Ainv (relint S2 ). Source: [2487, pp. 92, 93]. Related: Fact 12.11.25 and Fact 12.11.26. Fact 12.11.25. Let S1 ⊆ Fm, S2 ⊆ Fn, and A ∈ Fn×m. Then,

relint conv AS1 = A relint conv S1 , relint conv A (S2 ) = Ainv (relint conv[S2 ∩ R(A)]) ⊆ Ainv (relint conv S2 ). inv

Source: [2487, p. 135]. Related: Fact 12.11.24. Fact 12.11.26. Let S1 ⊆ Fm, S2 ⊆ Fn, and A ∈ Fn×m. Then,

A cl S1 ⊆ cl AS1 , cl A (S2 ) = A (cl[S2 ∩ R(A)]) ⊆ Ainv (cl S2 ). inv

inv

Furthermore, the following statements hold: i) If either A is left invertible or S1 is bounded, then A cl S1 = cl AS1 . ii) A cl S1 = cl AS1 if and only if (cl S1 ) + N(A) is closed. iii) If relint S2 ∩ R(A) , ∅, then cl Ainv (S2 ) = Ainv (cl S2 ). Source: [2487, pp. 99, 100, 103]. Related: Fact 12.11.24. Fact 12.11.27. Let S ⊆ Fn, and assume that S is solid. Then, dim S = n. Fact 12.11.28. Let S ⊆ Fm, assume that S is solid, let A ∈ Fn×m, and assume that A is right invertible. Then, AS is solid. Source: Theorem 12.4.22. Related: Fact 3.13.4. Fact 12.11.29. Nn is a closed and completely solid subset of Fn(n+1)/2. Furthermore, int Nn = Pn. Fact 12.11.30. Let D ⊆ Fn, and let x0 belong to a solid, convex subset of D. Then, dim fcone(D, x0 ) = n.

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Fact 12.11.31. Let S ⊂ Fn, assume that S is equilibrated, solid, compact, and convex, and, for

all x ∈ Fn, define

△

∥x∥ = min {α ≥ 0: x ∈ αS} = max {α ≥ 0: αx ∈ S}.

Then, ∥ · ∥ is a norm on Fn, and cl B1 (0) = S. Conversely, let ∥ · ∥ be a norm on Fn. Then, cl B1 (0) is equilibrated, solid, compact, and convex, and, in addition, ∥x∥ = min {α ≥ 0: x ∈ α cl B1 (0)} = max {α ≥ 0: αx ∈ cl B1 (0)}. Source: [1467, pp. 38, 39]. Remark: In all cases, B1(0) is defined with respect to ∥ · ∥. Remark: S

is equilibrated if, for all λ ∈ F such that |λ| = 1, λS = S. If F = R, then S is equilibrated if and only if it is symmetric; that is, S = −S. In the case where F = C, let z = x + y ȷ ∈ C, where x, y ∈ R. Then, √ △ f (z) = 2x2 + y2 does not satisfy f ( ȷz) = | ȷ| f (z), and thus f is not a norm on C. In fact, if ∥ · ∥ is a norm on C, then, for all z ∈ C, ∥z∥ = ∥z · 1∥ = |z|∥1∥. Hence, ∥ · ∥ is proportional to | · |, whose unit ball is circular. Credit: H. Minkowski. Related: Fact 11.7.1. Fact 12.11.32. Let S ⊆ Rn, assume that S is a nonempty, closed, convex set, and let E ⊆ S denote the set of elements of S that cannot be represented as nontrivial convex combinations of two distinct elements of S. Then, E is nonempty and S = conv E. If, in addition, n = 2, then E is closed. Source: [970, pp. 482–484]. Remark: E is the set of extreme points of S. Remark: The last result is the Krein-Milman theorem. E is not necessarily closed for n ≥ 3. See [970, p. 483].

12.12 Facts on Two or More Sets Fact 12.12.1. Let S1 ⊆ S2 ⊆ Fn. Then, cl S1 ⊆ cl S2 and int S1 ⊆ int S2 . Fact 12.12.2. Let S1 , S2 ⊆ Fn. Then, the following statements hold:

(int S1 ) ∩ (int S2 ) = int(S1 ∩ S2 ). (int S1 ) ∪ (int S2 ) ⊆ int(S1 ∪ S2 ). (cl S1 ) ∪ (cl S2 ) = cl(S1 ∪ S2 ). bd(S1 ∪ S2 ) ⊆ (bd S1 ) ∪ (bd S2 ). If (cl S1 ) ∩ (cl S2 ) = ∅, then bd(S1 ∪ S2 ) = (bd S1 ) ∪ (bd S2 ). Assume that S1 and S2 are convex. Then, cl(S1 ∩ S2 ) ⊆ (cl S1 ) ∩ cl S2 . If, in addition, (relint S1 ) ∩ relint S2 , ∅, then cl(S1 ∩ S2 ) = (cl S1 ) ∩ cl S2 . Source: [154, p. 65]. The last statement is given in [2487, p. 97]. Fact 12.12.3. Let S1 , S2 ⊆ Fn, assume that either S1 is closed or S2 is closed, and assume that int S1 = int S2 = ∅. Then, int(S1 ∪ S2 ) is empty. Source: [154, p. 69]. Remark: The set int(S1 ∪ S2 ) is not necessarily empty if neither S1 nor S2 is closed. Consider the sets of rational and irrational numbers. Fact 12.12.4. Let S1 , S2 ⊆ Fn, and assume that S1 is open, S2 is convex, and S1 ⊆ cl S2 . Then, S1 ⊆ S2 . Source: [2403, pp. 72, 73]. Remark: The statement is false without the assumption that S2 is convex. Let S1 = (0, 3/2) and S2 = (0, 1) ∪ (1, 2). Related: Fact 12.11.16. Fact 12.12.5. Let S1 , S2 ⊆ Fn, and assume that S1 is open. Then, S1 + S2 is open. Source: Note that S1 + S2 = ∪ x∈S2 (S1 + {x}) is the union of a collection of open sets. The result now follows from Fact 12.12.16. Remark: See [1267, p. 107]. △ △ Fact 12.12.6. Define S1 , S2 ⊆ R by S1 = Z and S2 = {i + 1/i + π : i ∈ P}. Then, S1 and S2 are closed, but S1 + S2 is not closed. Remark: The sum of two closed, convex sets in R2 is not necessarily closed. See [2487, p. 98]. The sum of two closed, convex cones in R3 is not necessarily closed. See [309, p. 65]. Fact 12.12.7. Let S1 , S2 ⊆ Fn. Then, cl S1 + cl S2 ⊆ cl(S1 + S2 ). Now, assume that at least one of the following statements holds: i) ii) iii) iv) v) vi)

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i) Either S1 or S2 is bounded. ii) There exist affine subspaces S3 ⊆ Fn and S4 ⊆ Fn such that S1 ⊆ S3 , S2 ⊆ S4 , and the subspaces S′3 and S′4 that are parallel to S3 and S4 , respectively, satisfy S′3 ∩ S′4 = {0}. Then, cl S1 + cl S2 = cl(S1 + S2 ). Source: [2487, pp. 10, 98]. Related: Fact 12.12.8. Fact 12.12.8. Let S1 , S2 ⊆ Fn, and assume that S1 is closed and S2 is compact. Then, S1 + S2 is closed. If, in addition, S1 is compact, then S1 + S2 is compact. Source: [309, p. 34], [962, p. 209], and [1260, p. 81]. Remark: This result follows from Fact 12.12.7. Fact 12.12.9. Let S1 , S2 ⊆ Fn, and assume that S1 and S2 are convex. Then, cl S1 = cl S2 if and only if relint S1 = relint S2 . Source: [2487, p. 95]. Fact 12.12.10. Let S1 , S2 , S3 ⊆ Fn, assume that S1 , S2 , and S3 are closed, convex sets, assume that S1 ∩ S2 , ∅, S2 ∩ S3 , ∅, and S3 ∩ S1 , ∅, and assume that S1 ∪ S2 ∪ S3 is convex. Then, S1 ∩ S2 ∩ S3 , ∅. Source: [309, p. 32]. Fact 12.12.11. Let S1 , S2 , S3 ⊆ Fn, assume that S1 and S2 are convex sets, S2 is a closed set, and S3 is a bounded set, and assume that S1 + S3 ⊆ S2 + S3 . Then, S1 ⊆ S2 . Source: [523, p. 5]. Credit: H. Radstrom. Fact 12.12.12. Let S1 , . . . , Sk ⊆ Fn , and let α1 , . . . , αk ∈ F. Then, relint conv

k ∑

αi Si =

i=1

k ∑

αi relint conv Si .

i=1

If, in addition, S1 , . . . , Sk ⊆ Fn are convex, then relint

k ∑

αi Si =

i=1

k ∑

αi relint Si .

i=1

Source: [2487, pp. 89, 90, 132]. Related: Fact 3.12.5. Fact 12.12.13. Let S1 , . . . , Sk ⊆ Fn . Then,

relint conv

k ∪

Si ⊆ conv

i=1

k ∪

relint Si .

i=1

If, in addition, ∩ki=1 relint Si , ∅, then relint conv

k ∪ i=1

Si = conv

k ∪

relint Si .

i=1

Source: [2487, pp. 134, 135]. Fact 12.12.14. Let S, S1 , S2 ⊆ Fn. Then, the following statements hold:

i) ii) iii) iv) v) vi) vii) viii) ix) x)

polar S is a closed, convex set containing the origin. polar Fn = {0}, and polar {0} = Fn. If α > 0, then polar αS = α1 polar S. S ⊆ polar polar S. If S is nonempty, then polar polar polar S = polar S. If S is nonempty, then polar polar S = cl conv(S ∪ {0}). If 0 ∈ S and S is a closed, convex set, then polar polar S = S. If S1 ⊆ S2 , then polar S2 ⊆ polar S1 . polar(S1 ∪ S2 ) = (polar S1 ) ∩ (polar S2 ). If S is a convex cone, then polar S = dcone S.

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xi) dcone dcone S = cl coco S. Source: [309, pp. 143–147]. For xi), see [523, p. 54]. Fact 12.12.15. Let S1 , S2 ⊆ Fn, and assume that S1 and S2 are cones. Then, dcone(S1 + S2 ) = (dcone S1 ) ∩ (dcone S1 ). If, in addition, S1 and S2 are closed, convex sets, then dcone(S1 ∩ S2 ) = cl[(dcone S1 ) + (dcone S2 )]. Source: [309, p. 147] and [523, pp. 58, 59]. Fact 12.12.16. Let A be a collection of open subsets of Fn. Then, the union of all elements of A is open. If, in addition, A is finite, then the intersection of all elements of A is open. Source:

[154, p. 50]. Fact 12.12.17. Let A be a collection of closed subsets of Fn. Then, the intersection of all

elements of A is closed. If, in addition, A is finite, then the union of all elements of A is closed. Source: [154, p. 50]. Fact 12.12.18. Let S ⊆ Fn . Then, clS is the intersection of all closed subsets of Fn that contain S. Fact 12.12.19. Let S ⊆ Fn , and define the following terminology: i) S is Gδ if it is the intersection of a countable number of open subsets of Fn . ii) S is Fσ if it is the union of a countable number of closed subsets of Fn . Then, the following statements hold: iii) The intersection of a countable number of Gδ sets is Gδ . iv) The union of a countable number of Fσ sets is Fσ . v) If S is either open or closed, then it is both Gδ and Fσ . vi) S is Gδ if and only if S∼ is Fσ . vii) The intersection of all open sets that contain S is S. viii) The intersection of a countable collection of open sets that contain S is a Gδ set that contains S. ix) The union of all closed sets that are contained in S is S. x) The union of a countable collection of closed sets that are contained in S is an Fσ set that is contained in S. xi) The set Q of rational numbers is Fσ , and the set R\Q of irrational numbers is Gδ . △ xii) Let a, b ∈ R, assume that a < b, and let f : [a, b] 7→ R. Then, S = {x ∈ [a, b] : f is continuous at x} is Gδ . Fact 12.12.20. Let A = (Ai )∞ i=1 be a nonincreasing sequence of nonempty, compact subsets of n R . Then, glb(A) is compact and nonempty. Source: [154, p. 56]. Remark: This is the Cantor intersection theorem, where glb(A) = ∩∞ i=1 Ai . Fact 12.12.21. Let ∥ · ∥ be a norm on Fn, let S ⊂ Fn, assume that S is a subspace, let y ∈ Fn, and define △ µ = max |y∗x|. x∈{z∈S: ∥z∥=1}

Then, there exists w ∈ S⊥ such that max

x∈{z∈Fn : ∥z∥=1}

|(y + w)∗x| = µ.

Source: [2539, p. 57]. Remark: This is a version of the Hahn-Banach theorem. Problem: Find a

simple interpretation in R2.

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Fact 12.12.22. Let S ⊂ Rn, assume that S is a convex cone, let x ∈ Rn, and assume that x < int S. Then, there exists a nonzero vector λ ∈ Rn such that λTx ≤ 0 and λTz ≥ 0 for all z ∈ S. Source: [1769, p. 37], [2260, p. 443], [2319, pp. 95–101], and [2544, pp. 96–100]. Remark: This is a separation theorem. Remark: Every convex cone that is a proper subset of Rn is contained in a

closed half space. Fact 12.12.23. Let S1 , S2 ⊂ Rn, and assume that S1 and S2 are nonempty, convex sets. Then,

the following statements are equivalent: i) There exist a nonzero vector λ ∈ Rn and α ∈ R such that λTx ≤ α for all x ∈ S1 , λTy ≥ α for all y ∈ S2 , and either S1 or S2 is not contained in the affine hyperplane {x ∈ Rn: λTx = α}. ii) intaffin S1 S1 and intaffin S2 S2 are disjoint. Source: [421, p. 82] and [1267, p. 148]. Remark: This is a proper separation theorem. Fact 12.12.24. Let ∥ · ∥ be a norm on Fn, let y ∈ Fn, let S ⊆ Fn, and assume that S is nonempty and closed. Then, there exists x0 ∈ S such that ∥y − x0 ∥ = min ∥y − x∥. x∈S

Now, assume that S is convex. Then, there exists a unique vector x0 ∈ S such that ∥y − x0 ∥ = min ∥y − x∥. x∈S

Consequently, there exists x0 ∈ S such that, for all x ∈ S\{x0 }, ∥y − x0 ∥ < ∥y − x∥. Source: [970, pp. 470, 471]. Related: Fact 12.12.26. Fact 12.12.25. Let ∥ · ∥ be a norm on Fn, let y1 , y2 ∈ Fn, let S ⊆ Fn, assume that S is a nonempty, closed, convex set, and let x1 and x2 denote the unique elements of S that are closest to y1 and y2 , respectively. Then, ∥x1 − x2 ∥ ≤ ∥y1 − y2 ∥. Source: [970, pp. 474, 475]. Fact 12.12.26. Let S ⊆ Rn, assume that S is a subspace, let A ∈ Fn×n be the projector onto S,

and let x ∈ Fn. Then,

min ∥x − y∥2 = ∥A⊥ x∥2 . y∈S

Source: [1133, p. 41] and [2539, p. 91]. Related: Fact 12.12.24. Fact 12.12.27. Let S, S1 , S2 , S3 ⊆ Fn, let ∥ · ∥ be a norm on Fn , and define △

dist(x, S) = inf ∥x − y∥, y∈S                 △ H(S1 , S2 ) = max  sup dist(x, S ), sup dist(y, S ) = max sup inf ∥x − y∥, sup inf ∥x − y∥ .    2 1      x∈S1    y∈S2 x∈S1 y∈S2 y∈S2 x∈S1 Then, the following statements hold: i) H(S1 , S2 ) = H(S2 , S1 ). ii) H(S1 , S2 ) ≥ 0. iii) If S1 and S2 are bounded, then H(S1 , S2 ) is finite. iv) If H(S1 , S2 ) is finite, then S1 and S2 are either both bounded or both unbounded. v) H(S1 , S2 ) = 0 if and only if cl S1 = cl S2 . vi) H(S1 , S3 ) ≤ H(S1 , S2 ) + H(S2 , S3 ). Remark: The function “H” is the Hausdorff distance, which is a metric on the set of nonempty, compact subsets of Fn . See [344, pp. 85, 86].

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Fact 12.12.28. Let S1 , S2 , S3 ⊆ Rn, assume that S1 , S2 , and S3 are nonzero subspaces, let A1

and A2 be the projectors onto S1 and S2 , respectively, define “dist” and “H” as in Fact 12.12.27 with ∥ · ∥ = ∥ · ∥2 , and define { } △ G(S1 , S2 ) = max max dist(x, S2 ), max dist(y, S1 ) x∈S1 ∩S1 (0) y∈S2 ∩S1 (0)           . ∥x − y∥ ∥x − y∥ , max min max min = max  2 2   y∈S x∈S   2 x∈S1   ∥x∥ =11 y∈S2 ∥y∥ =1 2

2

Then, the following statements hold: i) G(S1 , S2 ) = G(S2 , S1 ). ii) 0 ≤ G(S1 , S2 ) ≤ 1. iii) G(S1 , S2 ) = 0 if and only if S1 = S2 . iv) G(S1 , S2 ) ≤ H(S1 ∩ S1 (0), S2 ∩ S1 (0)) ≤ 2G(S1 , S2 ). v) If dim S1 = dim S2 , then G(S1 , S2 ) = sin θ, where θ is the minimal principal angle defined in Fact 7.12.42. vi) G(S1 , S2 ) = σmax (A1 − A2 ). vii) G(S1 , S3 ) ≤ G(S1 , S2 ) + G(S2 , S3 ). Source: [1187, Chapter 13], [1590, pp. 199, 200], and [2539, pp. 91–93]. Remark: The function “G” is the gap, which is a metric on the set of subspaces in Fn . See [1590, pp. 199, 200]. Remark: △ If ∥ · ∥ is a norm on Fn×n, then d(S1 , S2 ) = ∥A1 − A2 ∥ is a metric on the set of subspaces of Fn, yielding the gap topology. See [2539, p. 93]. Related: Fact 7.13.27.

12.13 Facts on Functions △

Fact 12.13.1. Let p ∈ R[s], let α ∈ R, assume that p(α) , 0, and define m = card[mroots(p) ∩

(α, ∞)]. Then, sign[p(α)] = (−1) . Fact 12.13.2. Let p ∈ C[s], where p(s) = sn +an−1 sn−1 +· · ·+a0 , let roots(p) = {λ1 , . . . , λr }, and, for all i ∈ {1, . . . , r}, let αi ∈ R satisfy 0 < αi < min j,i |λi − λ j |. Furthermore, for all ε0 , . . . , εn−1 ∈ R, define △ pε0 ,...,εn−1 (s) = sn + (an−1 + εn−1 )sn−1 + · · · + (a1 + ε1 )s + a0 + ε0 . m

Then, there exists ε > 0 such that, for all ε0 , . . . , εn−1 satisfying |εi | < ε for all i ∈ {1, . . . , n − 1}, it follows that, for all i ∈ {1, . . . , r}, the polynomial pε0 ,...,εn−1 has exactly mult p (λi ) roots in the open disk {s ∈ C: |s − λi | < αi }. Source: [2047]. Remark: This result shows that the roots of a polynomial are continuous functions of the coefficients. Remark: λ1 , . . . , λr are the distinct roots of p. Fact 12.13.3. Let p, q ∈ C[s], where p(s) = sn + an−1 sn−1 + · · · + a1 s + a0 and q(s) = sn + △ bn−1 sn−1 +· · ·+b1 s+b0 , define γ = 2 maxi∈{1,...,n} {|an−i |1/i , |bn−i |1/i }, and let mroots(p) = {λ1 , . . . , λn }ms and mroots(q) = {µ1 , . . . , µn }ms . Then, there exists a permutation σ of (1, . . . , n) such that  n 1/n  4 ∑ i max |λi − µσ(i) | ≤ √n  |ai − bi |γ  . i∈{1,...,n} 2 i=1 Source: [461]. Related: Fact 11.14.7. Fact 12.13.4. Let f : [0, 1] 7→ C, assume that f is continuous, and let z1 , . . . , zn ∈ C. Then,

there exists α ∈ [0, 1] such that

| f (1) − f (0)|n ∏ ≤ | f (α) − zi |. 22n−1 i=1 n

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Source: [461]. Related: Fact 12.13.5. △ Fact 12.13.5. Let p ∈ C[s], assume that p is monic, and define n = deg p. Then,

1 ≤ max |p(x)|. 22n−1 x∈[0,1] Source: [461]. Credit: P. L. Chebyshev. Related: Fact 12.13.4 and Fact 12.13.6. ∑ Fact 12.13.6. Let p ∈ C[s], where p(s) = ni=0 βi si . Then,

|βn | + |β0 | ≤

max

s∈{z∈C : |z|=1}

|p(s)|.

Furthermore, equality holds if and only if p(s) = βn sn + β0 . Source: [1512]. Credit: P. L. Visser. Related: Fact 12.13.5. △ Fact 12.13.7. Let p ∈ C[s], define n = deg p, and let r > 1. Then, max |p(z)| ≤ rn max |p(z)|.

z∈Sr (0)

z∈UC

If, in addition, ρmin (p) ≥ 1, then   n rn − 1 r +1      max |p(z)| − min |p(z)|        2 z∈UC  rn + 1  2 z∈UC ≤ max |p(z)|. max |p(z)| ≤       rn + ρmin (p) z∈Sr (0) 2 z∈UC       max |p(z)|   1 + ρ (p) z∈UC min

Source: [3011]. Use Fact 2.4.2 for the last inequality. Related: Fact 12.16.3. Fact 12.13.8. Let S1 ⊆ Fn, assume that S1 is compact, let S2 ⊂ Fm, let f : S1 × S2 → R, and

assume that f is continuous. Furthermore, for each x ∈ S1 , define h x : S2 7→ R by h x (y) = f (x, y), △ and define g: S2 → R by g(y) = max x∈S1 h x (y). Then, the following statements hold: i) g is continuous. ii) Assume that S2 is convex and, for all x ∈ S1 , h x is convex. Then, g is convex. iii) Assume that S2 is open and, for all (x, y) ∈ S1 × S2 , ∂ f /∂x exists and is continuous on S1 × S2 . Then, for all z ∈ Fn, Dg(y; z) = max x∈S1

∂f (x, y)z. ∂x

Remark: See [1267, p. 20]. A related result is given in [962, p. 208]. Remark: This is Danskin’s

theorem. Fact 12.13.9. Define f : R2 7→ R by

   min {x/y, y/x}, x > 0 and y > 0, f (x, y) =   0, otherwise. △

△

Furthermore, for all x ∈ R, define g x : R 7→ R by g x (y) = f (x, y), and, for all y ∈ R, define hy : R 7→ R △ by hy (x) = f (x, y). Then, the following statements hold: i) For all x ∈ R, g x is continuous. ii) For all y ∈ R, hy is continuous. iii) f is not continuous at (0, 0).

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Fact 12.13.10. Let l1 , . . . , ln and m1 , . . . , mn be positive integers, and define f : Rn 7→ R by

 ∏ n li   i=1 x(i)     ∑n 2mi , f (x) =   i=1 x(i)    0,

x , 0, x = 0.

Then, the following statements are equivalent: i) f is continuous. ∑ ii) ni=1 mlii > 2. Now, assume that n ≥ 2 and m1 ≤ · · · ≤ mn . Then, the following conditions are equivalent: iii) For every hyperplane S ⊂ Rn , f is continuous on S. ∑ k < ni=1 mlii . iv) For all k ∈ {2, . . . , n}, 2 + mlkk − mlk−1 Example: f : R2 7→ R defined by

  xy2    ,  2 f (x, y) =  x + y4    0,

x2 + y2 > 0, x = y = 0,

is not continuous at x = y = 0 but is continuous on every line that contains zero. Likewise, f : R3 7→ R defined by   xyz2    , x2 + y2 + z2 > 0,  2 f (x, y, z) =  x + y4 + z8    0, x = y = z = 0, is not continuous at x = y = z = 0 but is continuous on every plane that contains zero. Source: [746]. Fact 12.13.11. Let S ⊆ Fn, assume that S is pathwise connected, let f : S 7→ Fn, and assume that f is continuous. Then, f (S) is pathwise connected. Source: [2588, p. 65]. Fact 12.13.12. Let f : [0, ∞) → R, assume that f is continuous, and define ∫ t ∫ t △ △ 1 f (τ) dτ g(t) = f (τ) dτ, h(t) = t 0 0 Then, the following statements hold: i) Let I be a bounded subset of [0, ∞). Then, f, g, and h are bounded on I. ii) If f is bounded, then h is bounded. iii) If limt→∞ f (t) = 0, then either limt→∞ g(t) exists, limt→∞ g(t) = ∞, or limt→∞ g(t) = −∞. iv) Assume that limt→∞ f (t) exists. If limt→∞ f (t) > 0, then limt→∞ g(t) = ∞. If limt→∞ f (t) < 0, then limt→∞ g(t) = −∞. v) If limt→∞ f (t) = ∞, then limt→∞ g(t) = ∞. If limt→∞ f (t) = −∞, then limt→∞ g(t) = −∞. vi) If limt→∞ f (t) = ∞, then limt→∞ h(t) = ∞. If limt→∞ f (t) = −∞, then limt→∞ h(t) = −∞. vii) If limt→∞ f (t) exists, then limt→∞ h(t) = limt→∞ f (t). Fact 12.13.13. Let f : [0, 1] → R, and assume that f is continuous. Then, ∫ 1 lim αtα−1 f (t) dt = f (0). α↓0

Source: [116, p. 439].

0

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Fact 12.13.14. Let f : R → R, assume that f is continuous, assume that, for all x ∈ R, f (x + 1) = f (x), and let a be an irrational real number. Then, ∫ 1 n 1∑ f (ia) = f (x) dx. lim n→∞ n 0 i=1 Source: [1158, pp. 166, 167]. Fact 12.13.15. Let a and b be real numbers such that a < b, let f : [a, b] → R, assume that f

is continuous, let g: [a, b] → R, and assume that g is integrable. If, for all x ∈ [a, b], g(x) ≥ 0, then there exists c ∈ (a, b) such that ∫ b ∫ b f (x)g(x) dx = f (c) g(x) dx. a

a

In particular, there exists c ∈ (a, b) such that ∫ b f (x) dx = f (c)(b − a). a

If f is nondecreasing on [a, b], then there exists c ∈ (a, b) such that ∫ b ∫ c ∫ b f (x)g(x) dx = f (a) g(x) dx + f (b) g(x) dx. a

a

c

Source: [2687]. Remark: The second statement is the mean-value theorem. Fact 12.13.16. Let a be a positive number, let f : [0, a] → R, assume that f (0) = 0, assume

that f is continuous and increasing, and let b ∈ [0, f (a)]. Then, ∫ a ∫ b ab ≤ f (x) dx + f Inv (x) dx. 0

0

Equality holds if and only if b = f (a). Source: [3002]. Remark: This is Young’s inequality. Related: Fact 2.2.39. Fact 12.13.17. Let a > 0, let p > 1, let f : [0, a] → R, and assume that f p is integrable. Then, )p ( )p ∫ a ∫ a( ∫ x 1 p f (t) dt dx ≤ f p (x) dx. x 0 p−1 0 0 In particular, if f 2 is integrable, then )2 ∫ a ∫ a( ∫ x 1 f (t) dt dx ≤ 4 f 2 (x) dx. x 0 0 0 Source: [2527, p. 166]. Remark: This is the Hardy inequality. Remark: 4 cannot be replaced by a smaller value. Related: Fact 2.11.132. Fact 12.13.18. Let a, b ∈ R, where a < b, let f : [a, b] → R, and assume that f is convex. Then, ∫ b 1 1 f [ 2 (a + b)] ≤ f (x) dx ≤ 21 [ f (a) + f (b)]. b−a a Source: [943]. Remark: This is the Hermite-Hadamard inequality. Fact 12.13.19. Let I ⊆ R be either a finite or infinite interval, let f : I → R, assume that f

is continuous, and assume that, for all x, y ∈ I, it follows that f [ 12 (x + y)] ≤ convex. Source: [2128, p. 10]. Credit: J. Jensen. Related: Fact 1.21.7.

1 2

f (x + y). Then, f is

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FUNCTIONS, LIMITS, SEQUENCES, SERIES, INFINITE PRODUCTS, AND DERIVATIVES

Fact 12.13.20. Let I ⊆ R be either a finite or infinite interval, let f, g: I → R, assume that f and g are nonincreasing. Then, f + g is nonincreasing. Now, assume that f, g: I → [0, ∞). Then, f g is nonincreasing. Fact 12.13.21. Let A0 ∈ Fn×n, let ∥ · ∥ be a norm on Fn×n, and let ε > 0. Then, there exists δ > 0 such that, if A ∈ Fn×n and ∥A − A0 ∥ < δ, then

dist[mspec(A) − mspec(A0 )] < ε, where

△

dist[mspec(A) − mspec(A0 )] = min max |λσ(i)(A) − λi (A0 )| σ

i∈{1,...,n}

and the minimum is taken over all permutations σ of (1, . . . , n). Source: [1399, p. 399]. Fact 12.13.22. Let I ⊆ R be an interval, let A: I 7→ Fn×n, and assume that A is continuous. Then, for i ∈ {1, . . . , n}, there exist continuous functions λi : I 7→ C such that, for all t ∈ I, mspec(A(t)) = {λ1 (t), . . . , λn (t)}ms . Source: [1399, p. 399]. Remark: If A is continuously parameterized by either two or more variables, then the spectrum of A is not necessarily continuously parameterizable. See [1399, p. 399]. Fact 12.13.23. Let a be a positive number, let f : [0, a) 7→ R, assume that f (0) ≥ 0, assume that, for all x ∈ (0, a), f is differentiable at x, assume that f ′ is convex on (0, a), and define g : (0, a) 7→ R △ by g(x) = f (x)/x. Then, g is convex. Source: [1378]. Fact 12.13.24. Let S ⊆ Fn be convex, let f : S 7→ R, assume that f is convex, and let α ∈ R. Then, f inv [(−∞, α)] and f inv [(−∞, α]] are convex sets. Remark: f inv [(−∞, α)] and f inv [(−∞, α]] are sublevel sets. Remark: The converse is false. Let f (x) = x3 . Fact 12.13.25. Let S ⊆ Rn , assume that S is convex, let f : S 7→ R, and define △ [ ] epi( f ) = { yx : x ∈ S, y ≥ f (x)}. Then, f is convex if and only if epi( f ) is convex. Remark: epi( f ) is the epigraph of f. Fact 12.13.26. Let A ∈ Fn×m , and let ∥ · ∥ be a norm on Fn×m . Then, the following statements are equivalent: i) rank A = min {n, m}. ii) There exists ε > 0 such that, for all B ∈ Fn×m such that ∥B∥ < ε, rank(A + B) = rank A. Furthermore, the following statements are equivalent: iii) rank A < min {n, m}. iv) For all ε > 0, there exists B ∈ Fn×m such that ∥B∥ < ε and rank(A + B) > rank A. △ n×m Now, let (Ai )∞ , and assume that A∞ = limi→∞ Ai exists. Then, i=1 ⊂ F rank A∞ ≤ lim inf rank Ai . i→∞

△

Source: To prove iii) =⇒iv), let r = rank A, and let A0 ∈ Fr×r be a nonsingular submatrix of A. By continuity of the determinant, there exists ε > 0 such that, for all C ∈ Fr×r such that ∥C∥F < ε, det(A0 +C) , 0. Now, let B ∈ Fn×m satisfy ∥B∥F < ε, and let B0 be the submatrix of B whose rows and columns correspond to those of A0 as a submatrix of A. Then, ∥B0 ∥F < ε, and thus det(A0 + B0 ) , 0, and therefore rank(A + B) ≥ r. By norm equivalence, ∥ · ∥F can be replaced by ∥ · ∥. Remark: This result shows that the rank function is lower semicontinuous. See [850, 857].

12.14 Facts on Functions of a Complex Variable △

△

Fact 12.14.1. Define f : C\{0} 7→ C by f (z) = e−1/z , and define g: C 7→ C by g(z) = f (z) for △

z , 0 and g(0) = 0. Then, the following statements hold: i) f is analytic.

2

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ii) 0 is an essential singularity of f. iii) g is not continuous at 0. iv) The restriction of g to R is C∞ . ∑ i Fact 12.14.2. Let f ∈ F(s), assume that f has no pole at zero, and let ∞ i=0 βi z be the Taylor series of f at 0. Then, the following statements are equivalent: i) f is analytic in the CIUD. ii) CIUD is a subset of the domain of convergence of the Taylor series of f at 0. iii) (βi )∞ i=1 ∈ ℓ1 . iv) For all p ∈ [1, ∞), (βi )∞ i=1 ∈ ℓ p . v) There exists p ∈ [1, ∞) such that (βi )∞ i=1 ∈ ℓ p . Fact 12.14.3. Let G be a proper rational function, let z0 ∈ C, consider the Laurent series of G △ given by (12.6.9), and define R = max {|λ| : λ is a pole of G}. Then, the following statements hold: i) R = lim supi→∞ |βi |1/i . ii) R ∈ [0, 1) if and only if limi→∞ βi = 0. iii) R = 1 and G has no repeated poles in S1 (z0 ) if and only if 0 < lim supi→∞ |βi | < ∞. iv) Either R ∈ (1, ∞) or both R = 1 and G has at least one repeated pole in S1 (z0 ) if and only if lim supi→∞ |βi | = ∞. Credit: S. Dai. Fact 12.14.4. Let f : OIUD 7→ OIUD, and assume that f is analytic in OIUD, and f (0) = 0. Then, | f ′ (0)| ≤ 1, and, for all z ∈ OIUD, | f (z)| ≤ |z|. Furthermore, if either | f ′ (0)| = 1 or there exists nonzero z ∈ OIUD such that | f (z)| = |z|, then there exists θ ∈ R such that, for all z ∈ OIUD, f (z) = eθ ȷ z. Remark: f is a rotation. Remark: This is the Schwarz lemma. See [1697, p. 78]. Fact 12.14.5. Let f : OIUD 7→ OIUD, assume that f is analytic in OIUD, and assume that 2 there exist a, b ∈ OIUD such that f (a) = b. Then, | f ′ (a)| ≤ 1−|b| . Furthermore, let a1 , a2 ∈ OIUD. 1−|a|2 Then, f (a2 ) − f (a1 ) a2 − a1 . ≤ 1 − a1 a2 1 − f (a1 ) f (a2 ) Finally, if equality holds in either of these equalities with a1 , a2 in the second inequality, then f is one-to-one and f (OIUD) = OIUD. Remark: f is a conformal mapping. Remark: This is the Schwarz-Pick lemma. See [1697, p. 78]. Fact 12.14.6. Let f : OIUD 7→ OIUD and assume that f is analytic in OIUD . Then, f is a conformal mapping if and only if there exist θ ∈ R and a ∈ OIUD such that, for all z ∈ OIUD, z−a . f (z) = eθ ȷ 1 − az Remark: Every conformal mapping on the unit disk is the product of a rotation and a M¨obius transformation Fact 12.14.7. Let D ⊆ C, assume that D is open and connected, let f : D 7→ C, and assume that f is analytic in D. Then, f (D) is either a single point or an open set. Remark: This is the open mapping theorem. See [1697, p. 73]. Fact 12.14.8. Let D ⊆ C, assume that D is open and connected, let f : D 7→ C, and assume that f is analytic in D. Then, the following statements hold: i) If there exists z0 ∈ D such that, for all z ∈ D, | f (z)| ≤ | f (z0 )|, then f is a constant function. ii) If D = C and f is bounded, then f is a constant function. iii) If there exists ε > 0 and z0 ∈ D such that, for all z ∈ Bε (z0 ), | f (z)| ≤ | f (z0 )|, then f is a

FUNCTIONS, LIMITS, SEQUENCES, SERIES, INFINITE PRODUCTS, AND DERIVATIVES

947

constant function. iv) If there exists a connected compact set D0 ⊂ D and z0 ∈ D0 such that, for all z ∈ D0 , | f (z)| ≤ | f (z0 )|, then z0 ∈ bd D0 . v) If, for all z ∈ D, f (z) , 0 and there exists z0 ∈ D such that, for all z ∈ D, | f (z0 )| ≤ | f (z)|, then f is a constant function. vi) If there exists a connected compact set D0 ⊂ D and z0 ∈ D0 such that, for all z ∈ D0 , | f (z0 )| ≤ | f (z)|, then z0 ∈ bd D0 . vii) Assume that D is bounded, let g : cl D 7→ C, assume that g is analytic on D and continuous on cl D, and assume that, for all z ∈ cl D, g(z) , 0. Then, there exists z0 ∈ bd D such that, for all z ∈ cl D, |g(z0 )| ≤ |g(z)|. Remark: See [1697, pp. 31, 32, 76, 77]. Remark: These are versions of the maximum modulus principle. ii) is Liouville’s theorem. Note that i) does not imply ii). Remark: The absolute value of f : OIUD 7→ C defined by f (z) = z2 is minimized by z = 0. Therefore, v) is false if the assumption that f is nonzero on D is removed. Fact 12.14.9. Let f : D 7→ C, and assume that f is analytic and not constant in C. Then, card(C\ f (C)) ≤ 1. Source: [2964]. Remark: Let f (z) = ez . Then, f (C) = C\{0}. Remark: This is Picard’s theorem. Fact 12.14.10. Let D ⊂ C, assume that D is open and simply connected. Then, there exists a conformal mapping f : D 7→ OIUD . Remark: This is the Riemann mapping theorem. See [1697, p. 86]. Remark: D and OIUD are homeomorphic, and ϕ is a homeomorphism. Remark: Note that D is assumed to be a proper subset of C. Fact 12.14.11. Let D ⊆ C, assume that D is open and connected, let f : D 7→ C, let g : D 7→ C, assume that f and g are analytic in D, and assume that there exists a nonempty, open set D0 ⊆ D such that, for all z ∈ D0 , f (z) = g(z). Then, for all z ∈ D, f (z) = g(z). Remark: This result shows that every analytic function has at most one analytic extension. See [1697, p. 123]. The process of constructing the analytic extension is called analytic continuation. See [1697, pp. 123–135] and [2249, Chapter 13]. Fact 12.14.12. Let f : OIUD 7→ C, and assume that f is one-to-one, f (0) = 0, f is analytic in OIUD, and f ′ (0) = 1. Then, the following statements hold: ∑ i i) There exist β2 , β3 , . . . ∈ C such that, for all z ∈ OIUD, f (z) = z + ∞ i=2 βi z . ii) For all i ≥ 2, |βi | ≤ i. iii) For all i ≥ 2, |βi | = i if and only if there exists θ ∈ [0, 2π) such that, for all z ∈ OIUD, z f (z) = . (1 + eθ ȷ z)2 Remark: f is a Schlicht function. f given by iii) is a K¨obe function. See [1697, pp. 149, 150]. Remark: ii) is de Branges’s theorem. See [1671]. Fact 12.14.13. Let D ⊆ C, assume that D is open and contains OIUD, let f : D 7→ C, assume △

that f is analytic in D, for all i ≥ 0, define βi = f (i) (z0 )/i!, and assume that the radius of convergence ∑ i of ∞ i=0 βi z is at least 1. Then, for all z ∈ OIUD,  i   i  ∞ ∞  f (z) ∑ ∑  i f (z2 ) ∑ ∑ j  β j  z ,  β j z  zi . = = 1 − z i=0 j=0 1−z i=0 j=0 Source: [2880, p. 39]. Fact 12.14.14. Let (βi )∞ i=0 ⊂ C, let z0 ∈ C, define the power series f with domain of convergence

D by (12.3.1), let ρ be the radius of convergence of (12.3.1), assume that ρ ≥ 1, and assume that,

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∑ i for all z ∈ OIUD, | f (z)| < 1. Then, for all z ∈ C such that |z| < 13 , ∞ i=0 |βi z | < 1. Remark: 1/3 is the Bohr radius. See [22]. Fact 12.14.15. Let f ∈ F(s), assume that f is proper, let ρ > 0 be such that f has no poles ∑ −i in ρ COUD, and, for all z ∈ ρ COUD, let g(z) = ∞ i=0 αi z . Then, the following statements are equivalent: i) f has no poles in COUD . ∑ p 1/p ii) For all p ∈ [1, ∞), ( ∞ exists. i=0 |αi | ) ∑ p 1/p iii) There exists p ∈ [1, ∞) such that ( ∞ exists. i=0 |αi | ) iv) limi→∞ αi = 0. Source: [430].

12.15 Facts on Functions of a Matrix Fact 12.15.1. Let A ∈ Cn×n, and assume that A is group invertible and has no eigenvalues in

(−∞, 0). Then, 1/2

A

2 = A π

∫

∞

(t2I + A)−1 dt.

0

Source: [1391, p. 133]. Fact 12.15.2. Let A ∈ Cn×n, and assume that A has no eigenvalues on the imaginary axis. Then,

the following statements hold: i) Sign(A) is involutory. ii) A = Sign(A) if and only if A is involutory. iii) [A, Sign(A)] = 0. iv) Sign(A) = Sign(A−1 ). v) If A is real, then Sign(A) is real. vi) Sign(A) = A(A2 )−1/2 = A−1 (A2 )1/2. vii) Sign(A) is given by ∫ ∞ 2 Sign(A) = A (t2I + A2 )−1 dt. π 0 Source: [1391, pp. 39, 40 and Chapter 5] and [1602]. Remark: The square root in vi) is the

principal square root. Fact 12.15.3. Let A, B ∈ Cn×n, assume that AB has no eigenvalues in (−∞, 0], and define △ C = A(BA)−1/2 . Then, ([ ]) [ ] 0 A 0 C Sign = −1 . B 0 C 0 If, in addition, A has no eigenvalues in (−∞, 0], then ]) [ ([ 0 A 0 Sign = −1/2 A I 0

] A1/2 . 0

Source: vi) of Fact 12.15.2 and [1391, p. 108]. Remark: The square root is the principal square

root. Fact 12.15.4. Let A, B ∈ Cn×n, and assume that A and B are positive definite. Then,

([

Sign

0 A−1

B 0

])

[

=

0 (A#B)−1

] A#B . 0

Source: [1391, p. 131]. Remark: The geometric mean is defined in Fact 10.11.68.

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Fact 12.15.5. Let A ∈ Fn×n , B ∈ Fm×m , and C ∈ Fn×m , and assume that ρmax (B) < ρmin (A). Then,

the unique solution X ∈ Fn×m of AX + XB + C = 0 is given by

∞ ∑ X= (−1)i+1A−i−1CBi . i=0

42]. Fact 9.4.18 implies that ρmax (BT ⊗ A−1 ) = ρmax (B)ρmax (A−1 ) = ρmax (B)/ρmin (A) < 1. Using Proposition 11.3.10, it follows that

Source: [2979, p.

X = − vec−1 (I + BT ⊗ A−1 )−1 vec(A−1C) = − vec−1

∞ ∑ (−BT ⊗ A−1 )i vec(A−1C) i=0

∞ ∞ ∑ ∑ = − vec−1 (−1)i (BiT ⊗ A−i ) vec(A−1C) = (−1)i+1 A−i−1CBi . i=0

i=0

12.16 Facts on Derivatives Fact 12.16.1. Let (βi )∞ i=0 ⊂ C, let z0 ∈ C, define the power series f with domain of convergence

D by (12.3.1), let ρ be the radius of convergence of (12.3.1), and assume that ρ > 0. Then, f is analytic in D, f ′ is given by ∞ ∑ △ f ′ (z) = βi i(z − z0 )i−1 , i=1 ′

and the radius of convergence of f is ρ. Source: [152]. △ Fact 12.16.2. Let p ∈ R[s], assume that roots(p) ⊂ R, and define n = deg p. Then, for all x ∈ R, p(x)p′′ (x) ≤

n−1 ′ [p (x)]2 ≤ [p′ (x)]2 . n

Source: [108, pp. 420, 421]. △ Fact 12.16.3. Let p ∈ C[s], define n = deg p. Then,

max |p′ (z)| ≤ n max |p(z)|. z∈UC

z∈UC

If, in addition, ρmin (p) ≥ 1, then

) n(     max |p(z)| − min |p(z)|       z∈UC  2 z∈UC  n ′ ≤ max |p(z)|. max |p (z)| ≤     n   z∈UC 2 z∈UC    max |p(z)|    1 + ρmin (p) z∈UC

Source: [2295, pp. 508–510] and [3011]. Remark: The first inequality is Bernstein’s theorem. Related: Fact 12.13.7. △ Fact 12.16.4. Let p ∈ C[s], define n = deg p, assume that roots(p) ⊂ UC, and define q ∈ C[s] △ ′ by q(z) = 2zp (z) − np(z). Then, roots(q) ⊂ UC. Source: [108, p. 426]. Fact 12.16.5. Let p ∈ C[s]. Then, roots(p′ ) ⊆ conv roots(p). Furthermore, let mroots(p) =

∑ 1 ∑n−1 {λ1 , . . . , λn }ms and mroots(p′ ) = {µ1 , . . . , µn−1 }ms . Then, n1 ni=1 λi = n−1 i=1 µi . If, in addition, the roots of p are distinct and not contained in a single line, then roots(p′ ) ∩ bd conv roots(p) = ∅. Source: [970, p. 488], [1574], and [2295, pp. 71–74]. Remark: This is the Gauss-Lucas theorem. Related: Fact 5.5.3 and Fact 15.18.1. △ Fact 12.16.6. Let p, q ∈ C[s], assume that n = deg q ≥ 3, assume that q has n distinct roots

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λ1 , . . . , λn ∈ C, and assume that deg p ≤ n − 2. Then, n ∑ p(λi ) = 0. ′ (λ ) q i i=1

Source: [2979, pp. 225, 226]. Remark: This is Abel’s theorem. Related: Fact 2.11.4. Fact 12.16.7. Let f : R2 7→ R, g: R 7→ R, and h: R 7→ R, and assume that g and h are

differentiable. Then, assuming each of the following integrals exists, ∫ ∫ h(α) d h(α) ∂ ′ ′ f (t, α) dt. f (t, α) dt = f (h(α), α)h (α) − f (g(α), α)g (α) + dα g(α) ∂α g(α) Remark: This is Leibniz’s rule. Fact 12.16.8. Let D ⊂ R, let f : D 7→ R and g : D 7→ R, let x ∈ D, assume that 1/x ∈ D and

x , 0, let n ≥ 1, and assume that f and g are n times differentiable at x. Then, ( ) n ∑ dn−i g(x) dn i n 1 (i) [g(x) f (1/x)] = f (1/x) n−i i . (−1) n i i x dx dx x i=0

Therefore,

(−1)n dn n−1 [x f (1/x)] = n+1 f (n) (1/x). n dx x

In particular, dn n−1 (n − 1)! dn n dn n−1 1/x (−1)n (x log x) = , (x log x) = n!(Hn + log x), (x e ) = n+1 e1/x . n n n dx x dx dx x Source: [771, p. 161]. ( ) △ Fact 12.16.9. Let n ≥ 1, let 0 ≤ m ≤ n, and define f : [0, ∞) 7→ R by f (x) = x+n m . Then, ( ) ( ) n n f ′ (0) = (Hn − Hn−m ), f ′′ (0) = [(Hn − Hn−m )2 − Hn,2 + Hn−m,2 ], m m ( ) n f ′′′ (0) = [(Hn − Hn−m )3 − 3(Hn − Hn−m )(Hn,2 − Hn−m,2 ) + 2(Hn,3 − Hn−m,3 )]. m Source: [2830]. Fact 12.16.10. Let D ⊆ Rm, assume that D is an open, convex set, let f : D 7→ R, and assume

that f is C1 on D. Then, the following statements hold: i) f is convex if and only if, for all x, y ∈ D, f (x) + (y − x)T f ′ (x) ≤ f (y). ii) f is strictly convex if and only if, for all distinct x, y ∈ D, f (x) + (y − x)T f ′ (x) < f (y). Remark: If f is not differentiable, then these inequalities can be stated in terms of either directional differentials of f or the subdifferential of f . See [2128, pp. 29–31, 128–145]. Fact 12.16.11. Let f : D ⊆ Fm 7→ Fn, and assume that D+ f (0; ξ) exists. Then, for all β > 0, D+ f (0; βξ) = βD+ f (0; ξ). △

△

Fact 12.16.12. Let f : R 7→ R, assume that f is C2 , and assume that a = sup x∈R | f (x)| and

b = sup x∈R | f ′′ (x)| are finite. Then,

sup | f ′ (x)| ≤

√

2ab.

x∈R

Source: [2294, p. 239]. Remark: This is Landau’s inequality. Extensions to higher order derivatives are given in [2294, pp. 240–242].

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FUNCTIONS, LIMITS, SEQUENCES, SERIES, INFINITE PRODUCTS, AND DERIVATIVES △

Fact 12.16.13. Let f : [0, ∞) 7→ R, assume that f is C2 , and assume that a = △

b=

∫∞ 0

| f ′′ (x)|2 dx are finite. Then,

∫

∞

∫∞ 0

| f (x)|2 dx and

√ | f ′ (x)|2 dx ≤ 2 ab.

0

Source: [2294, p. 239]. Remark: This is Hardy and Littlewood’s inequality. △ Fact 12.16.14. Define f : R 7→ R by f (x) = |x|. Then, for all ξ ∈ R, △

Now, define f : Rn 7→ Rn by f (x) =

√

D+ f (0; ξ) = |ξ|. xTx. Then, for all ξ ∈ Rn, √ D+ f (0; ξ) = ξTξ.

Fact 12.16.15. For all k ≥ 0, let fk : [0, T ] 7→ Fn be continuous, let ck ≥ 0, and assume

∑ ∑∞ that maxt∈[0,T ] ∥ fk (t)∥2 ≤ ck and ∞ k=0 ck exists. Then, k=0 fk converges absolutely and uniformly. Source: [2271, p. 217]. Remark: This is the Weierstrass M-test. Fact 12.16.16. For all k ≥ 0, let fk : [0, T ] 7→ Fn be continuous, and assume that ( fk )∞ k=1 △ converges uniformly. Then, f = limk→∞ fk is continuous. Now, assume, in addition, that, for all k ≥ 0, fk is differentiable on [0, T ] and that ( f˙k )∞ k=1 converges uniformly. Then, f is differentiable on [0, T ], and, for all t ∈ [0, T ], f˙(t) = limk→∞ f˙k (t). Source: [2271, pp. 213, 219, 220]. Fact 12.16.17. Let A, B ∈ Fn×n. Then, for all s ∈ F, d (A + sB)2 = AB + BA + 2sB2. ds d (A + sB)2 = AB + BA. ds s=0

Hence, Furthermore, for all k ≥ 1,

k−1 ∑ d (A + sB)k = AiBAk−1−i. ds s=0 i=0 △

Fact 12.16.18. Let A, B ∈ Fn×n, and let D = {s ∈ F: det(A + sB) , 0}. Then, for all s ∈ D,

d (A + sB)−1 = −(A + sB)−1B(A + sB)−1. ds Hence, if A is nonsingular, then d (A + sB)−1 = −A−1BA−1. ds s=0 Fact 12.16.19. Let A, B ∈ Fn×n. Then, for all s ∈ F,

d det(A + sB) = tr B(A + sB)A . ds Hence,

n ∑ d i det(A + sB) = tr BAA = det[A ← coli (B)]. ds s=0 i=1

Source: Fact 3.19.8 and Fact 12.16.22. Related: This result generalizes Lemma 6.4.8. △ Fact 12.16.20. Let A, B ∈ Fn×n, and let D = {s ∈ F: det(A + sB) , 0}. Then, for all s ∈ D,

d d (A + sB)A = [det(A + sB)](A + sB)−1 ds ds

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CHAPTER 12

= [tr B(A + sB)A ](A + sB)−1 − [det(A + sB)](A + sB)−1B(A + sB)−1 Hence, if A is nonsingular, then

d (A + sB)A = (tr BAA )A−1 − (det A)A−1BA−1 . ds s=0 △

Fact 12.16.21. Let A, B ∈ Fn×n, and let D = {s ∈ F: det(A + sB) , 0}. Then, for all s ∈ D,

d2 d tr B(A + sB)A det(A + sB) = ds ds2 d = ([det(A + sB)] tr B(A + sB)−1 ) ds = [tr B(A + sB)A ] tr B(A + sB)−1 − [det(A + sB)] tr [B(A + sB)−1 ]2 . Hence, if A is nonsingular, then

d2 = (tr BAA ) tr BA−1 − (det A) tr (BA−1 )2 . det(A + sB) s=0 ds2

Fact 12.16.22. Let D ⊆ F, let A: D 7→ Fn×n, and assume that A is differentiable. Then,

( ) ∑ n 1 d d det A(s) = tr AA(s)A′(s) = tr A(s) AA (s) = det Ai (s), ds n −1 ds i=1

where Ai(s) is obtained by differentiating the entries of the ith row of A(s). If A(s) is nonsingular for all s ∈ D, then d log det A(s) = tr A−1(s)A′(s). ds If A(s) is positive definite for all s ∈ D, then 1 d det A1/n (s) = [det A1/n (s)] tr A−1 (s)A′(s). ds n Finally, A(s) is nonsingular and has no negative eigenvalues for all s ∈ D, then ∫ 1 d log A(s) = [(A(s) − I)t + I]−1 A′(s)[(A(s) − I)t + I]−1 dt, ds 0 d [log A(s)]2 = 2[tr log A(s)]A−1 (s)A′(s). ds Source: [799, p. 267], [1190, 2065], [2263, pp. 199, 212], [2314, p. 430], and [2417]. Related: Fact 16.20.6. Fact 12.16.23. Let D ⊆ F, let A: D 7→ Fn×n, let k ≥ 1, and assume that A is k-times differentiable. Then,   di1  dsi1 row1 [A(s)]  ( ) k ∑   k d  ..  , det A(s) = det  . k  i1 , . . . , in ds  in d rown [A(s)] dsin where the sum is taken over all n-tuples (i1 , . . . , in ) of nonnegative integers whose components sum to k. Source: [578]. Remark: An alternative expression is given in [465]. Fact 12.16.24. Let D ⊆ F, let A: D 7→ Fn×n, assume that A is differentiable, and assume that A(s) is nonsingular for all s ∈ D. Then, ( ) d −1 d A (s) = −A−1(s)A′(s)A−1 (s), tr A−1(s)A′(s) = − tr A(s) A−1 (s) . ds ds

FUNCTIONS, LIMITS, SEQUENCES, SERIES, INFINITE PRODUCTS, AND DERIVATIVES

953

Source: [1450, p. 491] and [2263, pp. 198, 212]. Fact 12.16.25. Let A ∈ Rn×n, assume that A is symmetric, let X ∈ Rm×n, and assume that XAXT

is nonsingular. Then,

d det XAXT = 2(det XAXT )ATXT (XAXT )−1. dX

Source: [786]. Fact 12.16.26. Let D ⊆ R be an open interval, and, for all i ∈ {1, . . . , n}, let fi : D 7→ R and

assume that fi is n − 1 times differentiable. Furthermore, for all x ∈ D, define   ··· fn (x)   f1 (x)   ′  f1 (x) ··· fn′ (x)  △   , A(x) =   .. . ..  · .· ·  . .    (n−1) (n−1) f1 (x) · · · fn (x)

and assume that, for all x ∈ D, det A(x) = 0. Then, there exist an open interval D′ ⊆ D and ∑ α1 , . . . , αn ∈ R such that α1 , . . . , αn are not all zero and such that, for all x ∈ D′ , ni=1 αi fi (x) = 0. Source: [1706]. Remark: det A(x) is the Wronskian of f1 , . . . , fn . Example: The open interval D′ may need to be a proper subset of D. For example, let D = R, f1 (x) = x2 , and f2 (x) = x|x|, so that det A(x) = 0 for all x ∈ R. Then, f1 and f2 are linearly dependent on every open interval contained in (−∞, 0] and every open interval contained in [0, ∞). However, on every open interval that includes zero, f1 and f2 are linearly independent. See [1706]. Remark: Extensions to analytic functions are considered in [530]. Fact 12.16.27. Let ϕ : R3 7→ R, let ψ : R3 7→ R, and define  ∂     ∂x     ∂  ∂2 ∂2 ∂2 △  △ ∇ =   , ∇2 = ∇T ∇ = 2 + 2 + 2 .  ∂y  ∂x ∂y ∂z    ∂  ∂z Then,  ∂ϕ     ∂x    2 2 2  ∂ϕ  △   , ∇T ϕ = ∂ϕ + ∂ϕ + ∂ϕ , ∇2 ϕ = ∂ ϕ + ∂ ϕ + ∂ ϕ , ∇ × ∇ϕ = 0, ∇ϕ =   2 2  ∂y  ∂x ∂y ∂z ∂x ∂y ∂z2    ∂ϕ  ∂z ∇(ϕ + ψ) = ∇ϕ + ∇ψ, ∇(ϕψ) = ψ∇ϕ + ϕ∇ψ, ∇2 (ϕψ) = ϕ∇2 ψ + ψ∇2 ϕ + 2(∇ϕ)T ∇ψ. Now, let f = [ f1 f2 f3 ]T : R3 7→ R3 . Then,   0    ∂ ∇ × f = K(∇) f =   ∂z   ∂ − ∂y

−

∂ ∂z 0

∂ ∂x

 ∂f ∂   3 −   ∂y ∂y     ∂ f1 ∂  −  f =  −  ∂z ∂x      ∂ f2 0  − ∂x

∂ f2   ∂z   ∂ f3   , ∂x   ∂ f1   ∂y

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∇T (∇ × f ) = 0,

K(∇ × f ) = 12 [(∇ f T )T − ∇ f T ],

∇T (ϕ f ) = ϕ∇T f + f T ∇ϕ,

∇ × (ϕ f ) = ϕ∇ × f + (∇ϕ) × f,

∇( f f ) = 2( f ∇) f − 2(∇ × f ) × f, T

∇2 f = ∇(∇T f ) − ∇ × (∇ × f ),

T

∇2 (∇T f ) = ∇T (∇2 f ), tr ∇ f T = ∇T f,

∇ × (∇ × f ) = ∇(∇T f ) − ∇2 f,

∇2 ( f T f ) = 2∇T ( f T ∇) f − ∇T [(∇ × f ) × f ], tr (∇ f T )2 = ∇T ( f T ∇) f − ( f T ∇)(∇T f ).

Now, let g = [g1 g2 g3 ]T : R3 7→ R3 . Then, ∇( f T g) = f × (∇ × g) + g × (∇ × f ) + ( f T ∇)g + (gT ∇) f, ∇T ( f × g) = gT (∇ × f ) − f T (∇ × g), ∇ × ( f × g) = (gT ∇) f − ( f T ∇)g + f (∇T g) − g(∇T f ). Remark: ∇, ∇T , and K(∇) denote gradient, divergence, and curl, respectively. Remark: For  

 [∇ f1 (x)]T    x ∈ R , K(x) is defined in Fact 4.12.1. Remark: ϕ (x) = [∇ϕ(x)] , and f (x) =  [∇ f2 (x)]T  .   [∇ f3 (x)]T ′

3

T

′

12.17 Facts on Limits of Functions Fact 12.17.1. Let α be a positive number. Then,

lim α x = lim α x = lim α x = lim α1/x = lim α1/x = 1. x↓0

x→∞

x→0

x↑0

Furthermore,

x→−∞

lim x x = lim x1/x = 1. x→∞

x↓0

Remark: lim x↓0 x = 1 motivates the convention 00 = 1. Fact 12.17.2. Let α be a real number. Then, x

   ∞, 0 < α < 1, lim α =   0, α > 1, x→−∞ x

   0, 0 < α < 1, lim α =   ∞, α > 1, x→∞ x

   0, α < 0, lim x =   ∞, α > 0. x→∞

   ∞, α < 0, lim x =   0, α > 0, x↓0

α

α

If α < 0, then

lim |xα | = 0,

x→−∞

If α > 0, then

lim |xα | = ∞. x↑0

lim |xα | = 0. x↑0

Source: [2294, p. 116]. Remark: In the last three limits, xα may be complex. Fact 12.17.3. Let α > 0, and assume that α , 1. Then,

[

αx − 1 lim x→∞ (α − 1)x

]1/x

   1, 0 < α < 1, =  α, α > 1.

Source: [2294, p. 134]. Fact 12.17.4. Let a be a real number. Then,

lim x tan

x→∞

a a = lim x tan = a. x→−∞ x x

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FUNCTIONS, LIMITS, SEQUENCES, SERIES, INFINITE PRODUCTS, AND DERIVATIVES

Fact 12.17.5. Define S : (0, ∞) 7→ R by

  x1/(x−1)    , △  S (x) =  e log x1/(x−1)    1,

x , 1, x = 1.

Then, the following statements hold: i) S is continuous and strictly convex. ii) For all x ∈ (0, ∞), S (x) ≥ 1. iii) S (x) = 1 if and only if x = 1. iv) If x > 0, then S (x) = S (1/x). v) lim x↓0 S (x) = lim x→∞ S (x) = ∞. Remark: S is Specht’s ratio. Related: Fact 2.2.54, Fact 2.2.66, Fact 2.11.95, Fact 11.10.7, and Fact 15.15.23. Fact 12.17.6.

)x ( )x ( )x/2 ( 1 x+1 1 = lim 1 + = lim = e, lim 1 + x→−∞ x→∞ x − 1 x→∞ x x (

x−2 lim x→∞ x

) x−1

1 = 2, e

( lim

x→∞

x−1 x

) x−1 =

1 , e

] xx (x + 1) x+1 − lim = e. x→∞ xx (x − 1) x−1 [

Now, let z ∈ C. Then,

( z )x lim 1 + = ez . x→∞ x Source: The first two limits are given in [2294, p. 123], the third limit is given in [2363], and the sixth limit is given in [1648]. Fact 12.17.7. Let a > 0. As x → 0, a x = 1 + (log a)x + 21 (log2 a)x2 + 61 (log3 a)x3 + O(x4 ). Therefore,

ax − 1 a x − 1 − (log a)x = = log a, lim x→0 x→0 x x2 Fact 12.17.8. Let α > 0 and p > 1. Then, lim

lim

x→∞

1 2

log2 a.

xα = 0. px

Fact 12.17.9. Let α > 0. Then,

lim

x→∞

log x = lim xα log x = 0. x↓0 xα

Fact 12.17.10. Let x, y, p, q be positive numbers, and assume that 1 ≤ p ≤ q. Then,

[ ] [ ] xy ≤ 12 (x + y) ≤ max {x, y} = lim ∥ yx ∥r = ∥ yx ∥∞ r→∞ [ ] [ ] [ ] [ ] ≤ ∥ yx ∥q ≤ ∥ yx ∥ p ≤ ∥ yx ∥1 = lim ∥ yx ∥r = x + y.

lim [ 21 (xr + yr )]1/r = r→0

√

r→1

Source: [1566, p. 49]. Related: Proposition 11.1.5 and Fact 12.17.11.

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Fact 12.17.11. Let x ∈ Rn , assume that x ≥≥ 0, let p, q be positive numbers, and assume that

1 ≤ p ≤ q. Then,

v t   n n n ∏  1 ∑ r 1/r 1∑ x(i) ≤ x(i)  = n x(i) ≤ max {x(1) , . . . x(n) } = lim ∥x∥r lim  r→∞ r→0 n n i=1 i=1 i=1 = ∥x∥∞ ≤ ∥x∥q ≤ ∥x∥ p ≤ ∥x∥1 = lim ∥x∥1 = r→∞

n ∑

x(i) .

i=1

Related: Proposition 11.1.5 and Fact 12.17.10. Fact 12.17.12. Let f : [0, ∞) → R. Then, the following statements hold:

√ i) Let f (x) = x. Then, lim x→∞ f (x) = ∞ and lim x→∞ f ′ (x) = 0. 2 ii) Let f (x) = e−x sin e x . Then, lim x→∞ f (x) = 0, lim x→∞ f ′ (x) does not exist, lim inf x→∞ f ′ (x) = −∞, and lim sup x→∞ f ′ (x) = ∞. √ iii) Let f (x) = sin x. Then, lim x→∞ f (x) does not exist, lim inf x→∞ f (x) = −1, lim sup x→∞ f (x) = 1, and lim x→∞ f ′ (x) = 0. Fact 12.17.13. Let S ⊂ R, let f : S 7→ R and g : S 7→ R, assume that f and g are differentiable, let a ∈ R, and let ε > 0. Then, the following statements hold: △ i) Assume that I = (a − ε, a) ∪ (a, a + ε) ⊆ S, 0 < g(I) ∪ g′ (I), either lim x→a f (x) ′ = (x) lim x→a g(x) = 0 or both lim x→a f (x) = ±∞ and lim x→a g(x) = ±∞, and either lim x→a gf ′ (x) exists or lim x→a

f ′ (x) g′ (x)

= ±∞. Then, lim

x→a

f (x) f ′ (x) = lim ′ . g(x) x→a g (x)

△

ii) Assume that I = (a, a + ε) ⊆ S, 0 < g(I) ∪ g′ (I), either lim x↓a f (x) = lim x↓a g(x) = 0 or both ′ ′ (x) (x) exists or lim x↓a gf ′ (x) = ±∞. lim x↓a f (x) = ±∞ and lim x↓a g(x) = ±∞, and either lim x↓a gf ′ (x) Then, f (x) f ′ (x) lim = lim ′ . x↓a g(x) x↓a g (x) △

iii) Assume that I = (a − ε, a) ⊆ S, 0 < g(I) ∪ g′ (I), either lim x↑a f (x) = lim x↑a g(x) = 0 or both ′ ′ (x) (x) lim x↑a f (x) = ±∞ and lim x↑a g(x) = ±∞, and either lim x↑a gf ′ (x) exists or lim x↑a gf ′ (x) = ±∞. Then, f (x) f ′ (x) lim = lim ′ . x↑a g(x) x↑a g (x) △

iv) Assume that I = (−∞, a) ⊆ S, 0 < g(I) ∪ g′ (I), either lim x→−∞ f (x) = lim x→−∞ g(x) = 0 ′ (x) exists or or both lim x→−∞ f (x) = ±∞ and lim x→−∞ g(x) = ±∞, and either lim x→−∞ gf ′ (x) lim x→−∞

f ′ (x) g′ (x)

= ±∞. Then, lim

x→−∞

f (x) f ′ (x) = lim ′ . g(x) x→−∞ g (x)

△

v) Assume that I = (a, ∞) ⊆ S, 0 < g(I) ∪ g′ (I), either lim x→∞ f (x) ′ = lim x→∞ g(x) = 0 or′ both (x) (x) lim x→∞ f (x) = ±∞ and lim x→∞ g(x) = ±∞, and either lim x→∞ gf ′ (x) exists or lim x→∞ gf ′ (x) = ±∞. Then, f (x) f ′ (x) lim = lim ′ . x→∞ g(x) x→∞ g (x) Source: [2406, p. 200]. Remark: This is l’Hˆopital’s rule. Example: lim x→0

= 1 and

x lim x→∞ √e x

√

= lim x→∞ 2 xe x = ∞.

sin x x

= lim x→0 cos x

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957

Fact∫ 12.17.14. Let f : [0, ∞) 7→ R, assume that f is uniformly continuous, and assume that t limt→∞ 0 f (x) dx exists. Then, limt→∞ f (t) = 0. Source: [781, 1021]. Remark: This is Barbalat’s lemma. An equivalent form is given by Fact 12.17.17. Fact 12.17.15. Let f : [a, b] 7→ R, and assume that f is integrable. Then, ∫ b ∫ b lim f (x) sin nx dx = lim f (x) cos nx dx = 0. n→∞

a

n→∞

a

Source: [1103, p. 261]. Fact 12.17.16. Let f : [0, ∞) 7→ R, and assume that f is C1 , f ′ is uniformly continuous, and lim x→∞ f (x) exists. Then, lim x→∞ f ′ (x) = 0. Remark: This result follows from Fact 12.17.14. Fact 12.17.17. Let f : [0, ∞) 7→ R, and assume that f is C2 , f ′′ is bounded, and lim x→∞ f (x) exists. Then, lim x→∞ f ′ (x) = 0. Source: This result follows from Fact 12.17.14. See [2527, p.

118]. Fact 12.17.18. Let D ⊆ R2 , assume that D is open and convex, let z ∈ D, let I1 and I2 be

open intervals in R such that z ∈ I1 × I2 ⊆ D, let f : D 7→ R, and assume that, for all convex, C1 functions g1 : I1 7→ R and g2 : I2 7→ R such that g1 (z(2) ) = z(1) and g2 (z(1) ) = z(2) , it follows that lim x→z(2) f (g1 (x), x) = lim x→z(1) f (x, g2 (x)) = f (z). Then, limξ→z f (ξ) = f (z). Source: [2111]. Credit: A. Rosenthal.

12.18 Facts on Limits of Sequences and Series Fact 12.18.1. The following statements hold:

∑ that ∞ and let Xˆ = ( xˆi )∞ i) Let X = (xi )∞ i=1 xi converges absolutely, i=1 be a i=1 ⊂ F, assume ∑ ∑∞ ∑∞ x . rearrangement of X. Then, i=1 xˆi converges absolutely, and i=1 xˆi = ∞ i i=1 ∑ i+1 Hi = log 2. ii) ∞ i=1 (−1) ∑∞ 1 iii) Let (xi )∞ i=1 xi = 2 log 2. i=1 = (1, −H2 , −H4 , H3 , −H6 , −H8 , H5 , −H10 , −H12 , . . .). Then, ∑ ∞ 3 iv) Let (xi )∞ i=1 xi = 2 log 2. i=1 = (1, H3 , −H2 , H5 , H7 , −H4 , H9 , H11 , −H6 , . . .). Then, 12.2.8 implies Source: [1350, p. 102]. Remark: If X = (xi )∞ i=1 ⊂ F converges, then Proposition ∑ xi converges but that every rearrangement of X converges to the same limit as X. Assuming that ∞ i=1 ∑ not absolutely, this example shows that a rearrangement of X may yield a series ( ki=1 xˆi )∞ k=1 with a limit that is different from the limit of the series based on X; the series is conditionally convergent. ∑ is conditionally convergent, then, for all α ∈ R, the sequence Let F = R. If the series ( ki=1 xi )∞ ∑ k=1 x ˆ X can be rearranged so that ∞ i=1 i = α. For a power series with a finite radius of convergence, conditional convergence can occur only on the boundary of the domain of convergence. Credit: B. Riemann. Fact 12.18.2. The following statements hold: i) If X = (−1, −2, −3, −4, . . .), then −∞ = inf X = lim inf X = lim sup X < sup X = −1. ii) If X = (−1, 0, −2, 0, −3, 0, . . .), then −∞ = inf X = lim inf X < lim sup X = sup X = 0. iii) If X = (−1, 1, −2, 0, −3, 0, . . .), then −∞ = inf X = lim inf X < lim sup X = 0 < sup X = 1. iv) If X = (−1, 1, −2, 2, −3, 3, . . .), then −∞ = inf X = lim inf X < lim sup X = sup X = ∞. v) If X = (1, 1, 1, 1, . . .), then inf X = lim inf X = lim sup X = sup X = 1. vi) If X = (0, 1, 1, 1, 1, . . .), then 0 = inf X < lim inf X = lim sup X = sup X = 1. vii) If X = (1, 2, 1, 2, 1, 2, . . .), then 1 = inf X = lim inf X < lim sup X = sup X = 2. viii) If X = (2, 1, 1, 1, 1, . . .), then 1 = inf X = lim inf X = lim sup X = 1 < sup X = 2. ix) If X = (0, 1, 2, 1, 2, 1, 2, . . .), then 0 = inf X < lim inf X = 1 < lim sup X = sup X = 2. x) If X = (0, 2, 1, 1, 1, 1, . . .), then 0 = inf X < lim inf X = lim sup X = 1 < sup X = 2.

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= (2, 4, 1, 3, 1, 3, 1, 3, . . .), then 1 = inf X = lim inf X < lim sup X = 3 < sup X = 4. = (0, 3, 1, 2, 1, 2, 1, 2, . . .), then 0 = inf X < lim inf X < lim sup X = 2 < sup X = 3. = (1, 2, 1, 3, 1, 4, . . .), then 1 = inf X = lim inf X < lim sup X = sup X = ∞. = (0, 1, 2, 1, 3, 1, 4, . . .), then 0 = inf X < lim inf X = 1 < lim sup X = sup X = ∞. = (1, 2, 3, 4, . . .), then 1 = inf X < lim inf X = lim sup X = sup X = ∞. Remark: These 15 cases exhaust all possible forms of i) in Proposition 12.2.12. △ n Fact 12.18.3. Let S = (S i )∞ i=1 be a sequence of subsets of F , and define xi) xii) xiii) xiv) xv)

If X If X If X If X If X

△

liminf(S) = {x ∈ Fn : there exists (xi )∞ i=1 such that, for all i ≥ 1, xi ∈ S i , and x = lim xi }, i→∞

limsup(S) = {x ∈ Fn : there exists a subsequence (S i j )∞j=1 of S and a sequence (xi j )∞j=1 such that, for all j ≥ 1, xi ∈ S i j , such that x = lim xi j }. j→∞

△

Furthermore, if liminf(S) = limsup(S), then define lim(S) = liminf(S) = limsup(S). Then, {   lub(S)    esslub(S) ⊆  limsup(S) glb(S) ⊆ essglb(S) ⊆      liminf(S) ⊆ limsup(S). Furthermore, the following statements hold: i) If lim(S) and esslim(S) exist, then esslim(S) ⊆ lim(S). ii) liminf(S) and limsup(S) are closed. iii) If lim(S) exists, then lim(S) is closed. iv) Let A ∈ Fm×n , and assume that there exists a bounded set S 0 ⊂ Fn such that, for all i ≥ 1, S i ⊂ S 0 . If lim(S) exists, then lim(AS) = A lim(S). Source: [1185, pp. 97–101]. n Fact 12.18.4. Let (xi )∞ i=1 ⊂ F . Then, limi→∞ xi = x if and only if, for all j ∈ {1, . . . , n}, limi→∞ xi( j) = x( j) . Fact 12.18.5. Let (xi )∞ i=1 ⊂ R. Then, the following statements are equivalent: i) limi→∞ xi = 0. ii) limi→∞ |xi | = 0. iii) lim inf i→∞ −|xi | = 0. iv) lim supi→∞ |xi | = 0. Fact 12.18.6. Let (xi )∞ i=1 ⊂ (0, ∞). Then, √ √ xi+1 xi+1 ≤ lim inf i xi ≤ lim sup i xi ≤ lim sup . lim inf i→∞ i→∞ xi xi i→∞ i→∞ Source: [2294, p. 6]. n ∞ n Fact 12.18.7. Let X = (xi )∞ i=1 ⊂ F and Y = (yi )i=1 ⊂ F , and assume that X and Y converge. Then,

X + Y and X ⊙ Y converge, and

lim(X + Y) = lim X + lim Y,

lim(X ⊙ Y) = (lim X) ⊙ (lim Y).

∞ Fact 12.18.8. Let X = (xi )∞ i=1 ⊆ R and Y = (yi )i=1 ⊆ R. Then, the following statements hold:

i) If X is a rearrangement of Y, then lim inf X = lim inf Y and lim sup X = lim sup Y. ii) If X is either nonincreasing or nondecreasing, then lim inf X = lim sup X = lim X.

FUNCTIONS, LIMITS, SEQUENCES, SERIES, INFINITE PRODUCTS, AND DERIVATIVES

959

iii) lim inf X +lim inf Y ≤ lim inf(X +Y) ≤ lim inf X +lim sup Y ≤ lim sup(X +Y) ≤ lim sup X + lim sup Y. iv) If X ⊂ [0, ∞) and Y ⊂ [0, ∞) are bounded, then (lim inf X) lim inf Y ≤ lim inf(XY) and lim sup(XY) ≤ (lim sup X) lim sup Y. △ Source: [1566, p. 44]. Remark: XY = (xi yi )∞ i=1 . ∞ Fact 12.18.9. Let (xi )i=1 ⊂ C, and assume that at least one of the following statements holds: i) lim supi→∞ |xi |1/i < 1. ii) 0 < (xi )∞ i=1 and limi→∞ |xi+1 /xi | < 1. ∑ Then, ∞ x i=1 i converges. Source: Under i) and ii), the Cauchy-Hadamard formula and ratio test of Proposition 12.3.7 imply that the unit circle in C is contained in the domain of convergence of ∑∞ i i=1 xi z . ∞ Fact 12.18.10. Let (xi )∞ i=1 ⊂ (0, ∞), assume that (xi )i=1 is nonincreasing, and assume that limi→∞ x2i /xi exists. Then, the following statements hold: ∑ i) If limi→∞ x2i /xi < 1/2, then ∞ xi converges. ∑i=1 ∞ ii) If limi→∞ x2i /xi > 1/2, then i=1 xi = ∞. Source: [703]. ∞ Fact 12.18.11. Let (xi )∞ i=1 ⊂ (0, ∞), and assume that (xi )i=1 is nonincreasing and limi→∞ xi = 0. ∑∞ i Then, i=1 (−1) xi converges. Remark: This is Leibniz’s test. ∑n ∞ let (yi )∞ Fact 12.18.12. Let (xi )∞ i=1 ⊂ [0, ∞), assume i=1 ⊂ R, assume that ( i=1 xi )n=1 is bounded, ∑∞ ∞ that (yi )i=1 is nonincreasing, and assume that limi→∞ yi = 0. Then, i=1 xi yi converges. Source: [2294, p. 63]. Remark: This is Dirichlet’s test. Example: 1 − 21 + 31 − 14 + · · · converges. See [1024]. ∑∞ (y )∞ ⊂ R, and assume that Fact 12.18.13. Let (xi )∞ i=1 xi converges, let i=1 ⊂ R, assume that ∑∞ i i=1 ∞ (yi )i=1 is bounded and either nonincreasing or nondecreasing. Then, i=1 xi yi converges. Source: [2294, p. 63]. Remark: This is Abel’s test. Fact 12.18.14. Let (xi )∞ i=1 ⊂ (0, ∞). Then, the following statements are equivalent: ∑∞ i) i=1 xi = ∞. ii) For every sequence (yi )∞ i=1 ⊂ R with a limit in R ∪ {−∞, ∞}, it follows that ∑n xi yi = lim yi . lim ∑i=1 n n→∞ i→∞ i=1 xi In particular, if (yi )∞ i=1 ⊂ R has a limit in R ∪ {−∞, ∞}, then 1∑ yi = lim yi . n→∞ n i→∞ i=1 n

lim

Source: [2294, p. 89]. Remark: This is Cesaro’s lemma. ∑∞ Fact 12.18.15. Let (xi )∞ let (ai )∞ i=1 xi exists, i=1 ⊂ C, assume that i=1 ⊂ (0, ∞), and assume that 1 ∑n a x = 0. Source: [75]. Remark: (ai )∞ is nondecreasing and lim a = ∞. Then, lim i→∞ i n→∞ an i=1 i i i=1 This is Kronecker’s lemma. Related: Fact 10.11.76. ∑∞ xi 1 ∑n Fact 12.18.16. Let (xi )∞ i=1 i exists. Then, limn→∞ n i=1 xi = 0. i=1 ⊂ C, and assume that Source: Fact 12.18.15 and [2527, p. 177]. ∑j 1 ∑i ∞ Fact 12.18.17. Let (ai )∞ j=1 l=1 al )i=1 converges. Then, i=1 ⊂ R, and assume that ( i

lim x↑1

∞ ∑ i=1

1 ∑∑ al . ai x = lim i→∞ i j=1 l=1 i

i

j

960

CHAPTER 12

Source: [116, pp. 599, 600]. Remark: This result shows that Cesaro summability implies Abel summability. Example: For all i ≥ 1, let ai = (−1)i+1 . Then, both limits exist and equal 21 . For all

i ≥ 1, let ai = (−1)i+1 i. Then, the first limit equals 14 , but the second limit does not exist. ∑∞ ∑∞ ∞ Fact 12.18.18. Let (ai )∞ i=1 ai and i=1 bi converge, for i=1 ⊂ C and (bi )i=1 ⊂ C, assume that ∑ △ ∑ all i ≥ 1, define ci = ij=1 a j bi+1− j , and assume that ∞ c converges. Then, i=1 i ∞  ∞ ∞ ∑ ∑  ∑ ci =  ai  bi . i=1

i=1

i=1

Source: [116, p. 600]. ∞ Fact 12.18.19. Let (ai )∞ i=1 ⊂ R and (bi )i=1 ⊂ R. Then, the following statements hold:

i) Assume that one of the following statements holds: a) (bi )∞ i=1 ⊂ R is decreasing, and limi→∞ ai = limi→∞ bi = 0. b) (bi )∞ i=1 ⊂ R is increasing, and limi→∞ bi = ∞. If ai+1 − ai △ ∈ R ∪ {−∞, ∞}, α = lim i→∞ bi+1 − bi then lim

i→∞

ai = α. bi △

△

ii) Assume that (bi )∞ i=1 ⊂ R is increasing, limi→∞ bi = ∞, α = limi→∞ ai /bi exists, β = limi→∞ bi /bi+1 exists, and β , 1. Then, ai+1 − ai = α. lim i→∞ bi+1 − bi Source: [1103, pp. 263–266] and [2294, p. 7]. Remark: i) is the Stolz-Cesaro lemma, which is a discrete analogue of l’Hˆopital’s rule given by Fact 12.17.13. ii) is a converse. Remark: In ii), α

and β are real numbers. Fact 12.18.20.

lim inf cos i = −1, i→∞

lim sup cos i = 1. i→∞

Source: [968, p. 287]. Fact 12.18.21. Let ϕ denote Euler’s totient function, and let σ(n) denote the sum of the divisors

of n ≥ 1. If α ≥ 0, then n 6 1 ∑ α−1 lim α+1 i ϕ(i) = 2 , n→∞ n π (1 + α) i=1 In particular,

n 1 ∑ 3 ϕ(i) = 2 , n→∞ n2 π i=1

lim

Furthermore,

lim

n→∞

n ∑ i=1

σ(i) π2 = log(1 + α). n2 + αi2 12α

1 ∑ ϕ(i) 6 = 2. n→∞ n i π i=1 n

lim

n 1 ∑ π2 . σ(i) = n→∞ n2 12 i=1

lim

Source: [109, pp. 382, 383]. Related: Fact 1.20.3 and Fact 1.20.4. Fact 12.18.22. Let α and p be real numbers. Then, the following statements hold: p

i) If α > 0 and p > 0, then limn→∞ α1/n = 1. p ii) If p > 0, then limn→∞ n1/n = 1.

961

FUNCTIONS, LIMITS, SEQUENCES, SERIES, INFINITE PRODUCTS, AND DERIVATIVES p

iii) If p > 1, then limn→∞ (n!)1/n = 1. Fact 12.18.23. Let a, b, c, d, f, g ∈ R, and assume that ac > 0. Then,    0, f (|c| − |a|) > 0,    ( ) f n+g    f (|c| − |a|) < 0,  an + b ∞, ( )g lim = a  , f = 0,  n→∞ cn + d  c      f (b−d)/a e , a = c and f , 0. Fact 12.18.24. Let a be a real number. Then,

ean a2 /2 lim ( )n2 = e , n→∞ 1 + an

lim n→∞ (

ean 1+

a n

2

+

3

)n3

a2

= ea /6 , 3

lim n→∞ (

2n2

ean 1+

a n

+

a2 2n2

a /24 . )n4 = e 4

+

a3 6n3

Source: [1563]. Fact 12.18.25. Let f : (0, 1] 7→ (0, ∞), assume that f is differentiable, assume that f ′ (1) > 0,

and assume that f ′ / f is nonincreasing. Furthermore, let a ∈ (0, ∞) and b ∈ [0, ∞), and, for all n ≥ 1, define n ∑ △ xn = [ f ( ni )]ai+b . i=1

Then,

  0,    ′    ea f (1) lim xn =  ,   n→∞  ea f ′ (1) − 1   ∞,

f (1) < 1, f (1) = 1, f (1) > 1.

Now, let z be a complex number. Then, n ( n ( ∑ ∑ z + i )ai+b z + i )an+b ea(z+1) lim = lim = a . n→∞ n→∞ n n e −1 i=1 i=1 In particular, lim

n→∞

Furthermore,

n ( )ai ∑ i i=1

n

= lim

n→∞

n ( )an ∑ i i=1

n

=

ea . −1

ea

n 1 ∑ ni 1 = . n→∞ en i! 2 i=0

lim

Source: [1103, pp. 2, 3, 32], [1564, 1733], [2294, p. 56], and [2546]. Credit: The first result is due to O. Furdui. Related: Fact 12.18.26. Fact 12.18.26. Let a and b be real numbers such that 0 ≤ a < b. Then, )n n ( ∑ e1−a/b bi − a = . lim n→∞ bn e−1 i=1

In particular, lim

n→∞

Source: [1565]. Related: Fact 12.18.25.

n ( )n ∑ i i=1

n

=

e . e−1

962

CHAPTER 12

Fact 12.18.27. Let (αi )∞ i=1 ⊂ R. Then, the following statements are equivalent:

i) For all a, b ∈ [0, 1) such that a < b, limn→∞ 1n card({i ≤ n : αi − ⌊αi ⌋ ∈ [a, b]}) = b − a. ∑ ii) For all k ≥ 1, limn→∞ 1n ni=1 e2πkαi ȷ = 0. If (αi )∞ i=1 ⊂ [0, 1), then the following statement is equivalent to i) and ii): ∑ 1 iii) For all k ≥ 1, limi→∞ 1i ij=1 αkj = k+1 . Furthermore, the following sequences satisfy i) and ii): iv) (ie)∞ i=1 . p ∞ v) (i )i=1 , where p ∈ (0, 1). vi) (log p i)∞ i=1 , where p ∈ (1, ∞). p ∞ vii) (i log i)i=1 , where p ∈ (−∞, 0). Source: [968, pp. 269–275, 287]. Remark: The equivalence of i) and ii) is Weyl’s criterion. Fact 12.18.28. Let z ∈ C, and assume that |z| < 1. Then, 1 1∑ 2 1∑ 2 sin arg(z + e2π ȷk/n ) = lim cos arg(z + e2π ȷk/n ) = , n→∞ n n→∞ n 2 k=0 k=0 n−1

n−1

lim

1∑ [sin arg(z + e2π ȷk/n )] cos arg(z + e2π ȷk/n ) = 0. n→∞ n k=0 n−1

lim

Credit: N. Crasta. Fact 12.18.29. Let k ≥ 1. Then,

lim

n→∞

n 1 ∑

nk+1

ik =

i=1

1 . k+1

Source: [141] and Fact 1.12.1. Fact 12.18.30. Let x be a positive number. Then,

lim

n→∞

n ∑ 2 (xi/n − 1) =

1 2

log x.

i=1

Source: [1566, p. 61]. Fact 12.18.31. Let p be a nonzero real number, and let q be a positive number. Then,

 √ n  ∑  1 iq−1  p  . lim  1 + q − 1 = n→∞ n pq i=1

In particular,

√  n  ∑  1 i  lim  1 + 2 − 1 = , n→∞ 4 n i=1

Source: [1566, pp. 32, 61] and [2294, p. 15]. Fact 12.18.32. Let n ≥ 1. Then,

√  n  ∑  1 i2  3 lim  1 + 3 − 1 = . n→∞ 9 n i=1

n ∑ 1 1 1 i+1 ≤ (−1) − log 2 < , 2n + (2 log 2 − 1)/(1 − log 2) i=1 i 2n + 1 n ∑ 1 1 1 1 i+1 n+1 ≤ (−1) < − log 2 − (−1) . i 2n 4n2 + (6 − 8 log 2)/(2 log 2 − 1) 4n2 + 2 i=1

963

FUNCTIONS, LIMITS, SEQUENCES, SERIES, INFINITE PRODUCTS, AND DERIVATIVES

Source: [2463, 2690]. Fact 12.18.33. If n ≥ 1, then

1 ∑ B2i − . 2n i=1 2in2i ∞

Hn = γ + log n + For all k ≥ 1, as n → ∞,

Hn ∼ γ + log n − In particular, Hn ∼ γ + log n + Furthermore, lim Hn =

n→∞

∞ ∑ 1 i=1

i

= ∞,

1 , 2n

k ∑ Bi . ini i=1

Hn ∼ γ + log n +

  Hn Hn lim =  lim n→∞ γ + log n + n→∞ log n

1 1 . − 2n 12n2

  γ + log n +  lim 1  n→∞ log n 2n

1 2n

= 1,

[ ] △ γ = lim Hn − log(n + 1) = lim (Hn − log n) n→∞ n→∞ ( )] ( )] ∞ [ ∞ [ ∑ ∑ 1 1 1 1 = − log 1 + =1+ + log 1 − i i i i i=1 i=2 ⌊ ⌋ ∞ ∞ ∑ 1 log i ζ(i) ∑ = (−1)i = (−1)i i i log 2 i=1 i=2 ≈ 0.5772156649015328606065120900824024310421593359399235. Furthermore, let k ≥ 2. Then, γ= In particular,

1 lim (kHn − Hnk ). k − 1 n→∞

γ = lim (2Hn − Hn2 ) = n→∞

1 lim (3Hn 2 n→∞

− Hn3 ).

Source: The first equality is given in [721]. The third expression for γ is given in [1566, p.

69]. The fourth expression for γ is given in [516, p. 327]. The fifth expression for γ is given in [2294, p. 112]. The sixth expression for γ is given in [506, pp. 140, 141]. The second to last expression for γ is given in [1920]. Remark: γ is the Euler constant. See [880, 1350]. Remark: π ≈ 0.5778. Related: γ ≈ 323007/559595 ≈ 0.5772156649005, γ ≈ π/(2.002e) ≈ 0.577286, and 2e Fact 13.5.47. Fact 12.18.34. Let n ≥ 2. Then, Hn − log(n + 1) < γ < Hn − log n ≤ 1, where the (first, third) term is (an increasing, a decreasing) function of n. Furthermore, ( ) 1 1 1 1 − < H − log n + + , −γ 0, let π(x) denote the number of prime numbers that are either less than or equal to x. Then, the following statements hold: i) limi→∞ pi = ∞. 1 ∑n pn 1 ∑n n i=1 pi n ii) If n ≥ 10, then p2n − 9n < p < − . Thus, lim = 12 . n→∞ i=1 i 4 n 2 12 pn ∑ ∑ 1 2 iii) log log(n + 1) ≤ ni=1 p1i + log π6 , and thus ∞ i=1 pi = ∞. iv) If n ≥ 2, then there exists a prime between n and 2n. v) If n ≥ 25, then there exists a prime between n and 6n/5. vi) If n ≥ 26, then pn < (1.2)n . vii) If n ≥ 6, then pn < n log n + log log n. If n ≥ 7022, then pn ≤ n log n + n[log log n − 0.9385]. viii) For all m ≥ 1, there exists n ≥ 2 such that n/π(n) = m. ix) If n ≥ 2, then ( ) n 3 π(n) < 1+ . log n 2 log n

966

CHAPTER 12

x) If n ≥ 67, then n log n −

1 2

< π(n)
0, and define the sequences (ai )∞ i=1 and (bi )i=1 by a1 = a, b1 = b,

√ △ ∞ and, for all i ≥ 1, ai+1 = 21 (ai + bi ) and bi+1 = ai bi . Then, (ai )∞ i=1 is decreasing, (bi )i=1 is increasing, △ △ and thus α = limi→∞ ai and β = limi→∞ bi exist. Furthermore, α=β=

where △

∫

π/2

K(γ) =

√

0

1

α=β= △

∫

π/2

I(a, b) =

dx,

1 − γ2 sin2 x

Equivalently,

where

π(a + b) , 4K(γ)

√

△

γ=

a−b . a+b

π , 2I(a, b) 1

dx. a2 cos2 x + b2 sin2 x Source: [1566, p. 24]. Remark: Computational techniques based on iteration are discussed in [518, 519, 634] and [2013, pp. 68–70]. Remark: The limit is the arithmetic-geometric mean of a and b. K(γ) is a complete elliptic integral of the first kind. See [116, pp. 132–135] and [511, p. 88]. Remark: Note that √  ( ) 1  a2 − b2  2 a−b I(a, b) = K  K .  = a a a+b a+b 0

Related: Fact 14.3.14.

969

FUNCTIONS, LIMITS, SEQUENCES, SERIES, INFINITE PRODUCTS, AND DERIVATIVES △

1 ∞ Fact 12.18.56. Let a > 0 and b > 0, and define the sequences (ai )∞ i=1 and (bi )i=1 by a1 = 2 (a+b),

√ √ △ b1 = ab, and, for all i ≥ 1, ai+1 = 12 (ai + bi ) and bi+1 = ai+1 bi . Then,  a−b    , a , b,  log a − log b lim ai = lim bi =    i→∞ i→∞ a, a = b. △

Source: [635]. Related: Fact 2.2.63. Fact 12.18.57. Let a2 = 21 and b2 =

∞ and define the sequences (ai )∞ i=2 and (bi )i=2 by ai+1 = ai bi and bi+1 = 12 (ai+1 + bi ). Then, the following statements hold: i) For all i ≥ 3, 1/ai is the area of a regular 2i -sided polygon inscribed in a circle of radius 1. ii) For all i ≥ 3, 1/bi is the area of a regular 2i -sided polygon that circumscribes a circle of radius 1. iii) limi→∞ ai = limi→∞ bi = π. Remark: As discussed in [518], Archimedes calculated the areas of regular 96-sided inscribed and circumscribed polygons to obtain

√

1 4,

29376 25344 ≈ 3.1409 < π < ≈ 3.1428. 8069 9347 √ Fact 12.18.58. As n → ∞, n! ∼ 2nπ(n/e)n , ( )n+1/2 )n+1/2 ( n )n 1 ( √ √ n + 1/2 1 n! ∼ 2π = 2nπ √ 1+ e e 2n e ) ( ) ( n ( ) √ n n 1 1 5 1 1 1 + − + ··· . 1+ − = 2nπ √ 1+ e 2n 4n 32n2 128n3 2048n4 e )n ) ( n )n ( ( n )n ( √ √ 1 1 n−1 1 1 n! ∼ 2nπ 1+ 1 + + = + · · · + + , 2nπ e e 12n 288n3 12n2 12n−1 n2n−1 12n n2n ) ( n )n √ n ( n )n ( √ √ 1 1 5 35 = 2nπ 1+ + + + + ··· . n! ∼ 2nπ e n − 1/6 e 12n 96n2 3456n3 165888n4 For all k ≥ 1, as n → ∞,

( n )n √ n! ∼ 2nπ e

v t k

∞ ∑ Pi , ni i=0

△

where P0 = 1 and, for all i ≥ 1, △

Pi =

⌊(i+1)/2⌋ k ∑ B2 j Pi−2 j+1 . i j=1 2 j

Therefore, for all k ≥ 1, as n → ∞, ( 3 ) ]1/k ) ( 4 ( n )n [ √ k k2 1 k2 k 1 k k − + . n! ∼ 2nπ 1+ 2 + 5 2 2 + 7 4 − 3 2 3 11 5 5 3 e 2 3n 2 3 n 2 3 2 35 n 2 3 2 3 5 n4 In particular, as n → ∞,

) ( n )n ( √ 1 1 139 571 n! ∼ 2nπ 1+ + − − , e 12n 288n2 51840n3 2488320n4 ( n )n √ √ 1 1 31 139 n! ∼ 2nπ 1+ + − − , e 6n 72n2 6480n3 155520n4

970

CHAPTER 12

( n )n √ √ 11 59 1 1 3 − − , 1+ n! ∼ 2nπ + 2 3 e 4n 32n 1920n 30720n4 √ ( n )n √ 1 2 31 1 4 n! ∼ 2nπ 1+ − − , + e 3n 18n2 405n3 9720n4 ( n )n √ √ 1 1 11 1 6 + n! ∼ 2nπ + − , 1+ e 2n 8n2 240n3 1920n4 √ ( n )n √ 2 19 8 16 2 24 1+ + 2 + + + n! ∼ 2nπ . 3 4 e n n 15n 15n 105n5 As n → ∞, n! ∼

√

2nπ

( n )n e

v t 1/(12n)

e

n

∞ ∑ Q2i , n2i i=0

△

where Q0 = 1 and, for all i ≥ 1, i jB2 j+2 1 ∑ Q2i−2 j . Q2i = 2i j=1 ( j + 1)(2 j + 1) △

Hence,

( n )n √ e1/(12n) n! ∼ 2nπ e For all k ≥ 1, as n → ∞,

√ n

1−

1 1447 1170727 + · · ·. + − 360n2 1814400n4 1959552000n6

 √  n + n! ∼ 2π  e

1 1 n+ 2   2



v t∞ ∑ k

i=0

Ri , (n + 12 )i

△

where R0 = 1 and, for all i ≥ 1, △

Ri =

⌊(i+1)/2⌋ k ∑ (21−2 j − 1)B2 j Ri−2 j+1 . i j=1 2j

Hence, as n → ∞,

n √  √  n + 12  1 23 1  6 1 − + + + · · ·, n! ∼ 2π  e 4(n + 21 ) 32(n + 12 )2 1920(n + 12 )3  n √ √  n + 21  1 1 1   12 1 − n! ∼ 2π  + + + · · ·, e 2(n + 12 ) 8(n + 21 )2 120(n + 21 )3 n √  √  n + 12  1 13  24 1 − 1 +  n! ∼ 2π  − + · · ·. 1 1 2 e n + 2 2(n + 2 ) 120(n + 21 )3

For all k ≥ 1, as n → ∞,

Bi n! ( e )n ∑ ∼ log √ . n i(i − 1)ni−1 2nπ i=2

Hence, as n → ∞,  2k  ( n )n ∑  √ B i  exp  n! ∼ 2nπ i−1 e i(i − 1)n  i=2

2k

FUNCTIONS, LIMITS, SEQUENCES, SERIES, INFINITE PRODUCTS, AND DERIVATIVES

971

) B6 B2 B4 B2k + + + · · · + e 2n 12n3 30n5 2k(2k − 1)n2k−1 ( ) ) ( √ 1 1 B2k n n 1 = 2nπ exp − + · · · + + e 12n 360n3 1260n5 2k(2k − 1)n2k−1 [ ( ) √ 1 n n 1 1 B2k + = 2nπ 1+ − + ··· + e 12n 360n3 1260n5 2k(2k − 1)n2k−1 )2 ( 1 1 B2k 1 1 + + ··· + + − 2 12n 360n3 1260n5 2k(2k − 1)n2k−1 ( )3 1 B2k 1 1 1 − + · · · + + + 6 12n 360n3 1260n5 2k(2k − 1)n2k−1 ( )4 B2k 1 1 1 1 + ··· + − + + 24 12n 360n3 1260n5 2k(2k − 1)n2k−1  )5 (  1 B2k 1 1 1  + · · · + + + + · · · − 3 2k−1 120 12n 360n 2k(2k − 1)n 1260n5 ) ( n )n ( √ 139 571 163879 1 1 − − + = 2nπ 1+ + + ··· . e 12n 288n2 51840n3 2488320n4 209018880n5 =

Finally,

√

2nπ

( n )n

(

exp

1 ∑n e n 1 i=1 i = , n! = lim , lim √n = 0, lim √n = e, lim n √n n→∞ n→∞ n→∞ i→∞ (n + 1)i 2 n! n! n! n 2 √ √n √ log n! 4 (n!) 1 n+1 = 1, lim lim ( (n + 1)! − n!) = , lim √ = π. n n→∞ n→∞ n→∞ e log n (2n)! n

i!in

Source: [968, pp. 60–66], [985], [2502, Chapter 1], and Fact 1.13.14. The asymptotic approxi-

mation involving the Bernoulli numbers is given in [2013, pp. 125–127] and is based on the Euler summation formula given by Fact 13.1.6 with f (n) = log n!. Remark: The first asymptotic approximation for n! is Stirling’s formula. The second asymptotic approximation for n! is Burnside’s formula. See [2085]. The next three asymptotic approximations are given in [2086]. Remark: ∑ Bi For all n ≥ 1, the series ∞ i=2 i(i−1)ni−1 diverges. For all k ≥ 1, the last series (which depends on k) converges for all n ≥ 1 and provides an asymptotic approximation for n! as n → ∞. This series is an asymptotic expansion. See [116, p. 611]. Consequently, for all k ≥ 1, every truncation of this series provides an asymptotic approximation for n! as n → ∞. Stirling’s formula is the first term of this series. However, for all n ≥ 1, this series diverges as k → ∞. The coefficients of this series are discussed in [563], [771, p. 267], and [2117]. The approximation involving Pi is also an asymptotic expansion. See [985]. Remark: Refinements are discussed in [846]. Related: Fact 1.13.14. Fact 12.18.59.

lim

n→∞

(2n − 1)!!(2n + 1)!! 2 = ≈ 0.6367788676. π [(2n)!!]2

Source: [644, p. 106]. Related: Fact 1.13.15 and Fact 13.10.3. Fact 12.18.60. √ √

1 . e Source: [1158, pp. 106, 107] and [2294, p. 9]. Remark: A generalization is given in [1103, p. 13]. Credit: T. Lalescu. lim (

n→∞

n+1

(n + 1)! −

n

n!) =

972

CHAPTER 12

Fact 12.18.61. Let (ai )∞ i=1 ⊂ [0, ∞), and, for all i ≥ 1, define

√

△

√

a1 +

βi =

a2 +

√

a3 + · · · +

√ ai .

i

Then, limi→∞ βi exists if and only if (a1/2 )∞ i i=1 is bounded. Source: [1910]. Fact 12.18.62.

√

√

1+

√

√

1+

√

1+

√ 1 + · · · = 12 (1 + 5),

√

√

√ 1 + 3 1 + 4 1 + · · · = 3.

1+2

Source: The first equality is given in [1035, p. 24]. The second equality, which is due to S.

Ramanujan, is given in [977]. Fact 12.18.63. If a > 1, then √ √ a(a − 1) + Equivalently, if a > 0, then √

√ a+

In particular,

√

√

√

√

2+

2+

√

√

12 +

12 +

√

a(a − 1) +

√ a+

a+

a(a − 1) +

√ √ 1 a + · · · = (1 + 4a + 1). 2 √

√ 2 + 2 + · · · = 2,

12 +

√ a(a − 1) + · · · = a.

√

√

6+ 6+ √ √

√ 12 + · · · = 4,

20 +

6+

√

6 + · · · = 3,

√

20 +

20 +

√ 20 + · · · = 5.

Source: [3009]. Fact 12.18.64. Let n ≥ 2 and 1 ≤ m ≤ n − 1. Then,

√

√ n(n − m) + m

In particular, √

√

3+2 √

√

√

√

√ 3 + 2 3 + 2 3 + · · · = 3, √

√

√

n(n − m) + m n(n − m) +

√

n(n − m) + · · · = n.

√

5+4 √

√ √ 5 + 4 5 + 4 5 + · · · = 5,

√ √ 8 + 2 8 + 2 8 + 2 8 + · · · = 4, √ √ √ √ √ √ √ √ 15 + 2 15 + 2 15 + 2 15 + · · · = 10 + 3 10 + 3 10 + 3 10 + · · · = 5. 4+3

Source: [3009].

√ 4 + 3 4 + 3 4 + ··· =

FUNCTIONS, LIMITS, SEQUENCES, SERIES, INFINITE PRODUCTS, AND DERIVATIVES

973

Fact 12.18.65. Let a > 0. Then,

√

√ a−

√

a−

a−

√ 1 √ a − · · · = ( 4a + 1 − 1). 2

Source: [2219]. √ △ Fact 12.18.66. Let x be a positive number, let b ∈ (0, 21 ( 5 + 1)x), and define a = x2 + bx.

√

Then,

√

a−b In particular,

√

√

√ a − b a − b a − · · · = x.

√ √ √ √ √ 2 − 2 − 2 − 2 − · · · = 1, 6 − 6 − 6 − 6 − · · · = 2, √ √ √ √ √ √ √ √ 8 − 2 8 − 2 8 − 2 8 − · · · = 10 − 3 10 − 3 10 − 3 10 − · · · = 2. √

√

Source: [3009]. Fact 12.18.67. Define

2 △ α= √ ( ≈ 1.9903, √ √ √ √ ) 3 25 3 25 3 3 4 3 2+ 2 − 2 69 + 2 + 2 69 − 3 − 1 △

let x be a positive number, let b ∈ (0, αx), and define a = x2 + bx + b2 . Then, √ √ √ √ √ √ √ √ a − b a + b a − b a + · · · = x, a + b a − b a + b a − · · · = x + b. In particular,

√ √

3−

√ 3+

√ 7−

√ √

√

√ √ 3 − 3 + · · · = 1,

√ 7+

7−

√ 7 + · · · = 2,

√

√ √ 12 + 2 12 − 2 12 + · · · = 2,

12 − 2

√ 19 − 3 √ 13 −

Source: [3009].

√ √ 19 + 3 19 − 3 19 + · · · = 2, √

√ 13 +

13 −

√ 13 + · · · = 3,

√

√

√ 3 + 3 − 3 + 3 − · · · = 2, √ √ √ √ 7 + 7 − 7 + 7 − · · · = 3, √ √ √ √ 12 + 2 12 − 2 12 + 2 12 − · · · = 4, √ √ √ √ 19 + 3 19 − 3 19 + 3 19 − · · · = 5, √ √ √ √ 13 + 13 − 13 + 13 − · · · = 4.

974

CHAPTER 12

Fact 12.18.68.

v u u u t

6+

v u t

√

3

4

−7 −

√

3−

√

6+

Source: [1786]. Fact 12.18.69.

3

√4 −7 − 3 − · · · = · · ·

√

v u u u t

v u t

4

√

√

3

14 +

10 −

6−

4

√ 14 +

3

10 −

√ 6 = 2.

√

√ √ π 2 − 2 + 2 + 2 − · · · = 2 sin , 18   √ √   √ √ √ √   ∞ ∏  1 √ √  = 2 .  lim 2i 2 − 2 + · · · + 2 + 2 = π, 2 + 2 + · · · + 2 + 2   2 i→∞ | {z } {z } π i=1   | i square roots i square roots

Source: [977] and [2425]. Fact 12.18.70.

√

√ 1 + F2

√ 1 + F4

√ 1 + F6 1 + · · · =

√

√ F22

+

F42 +

√

F82 +

√

1 + · · · = 3.

Source: √ [2158, 2147]. Remark: Fn is the nth Fibonacci number. Remark: It is shown in [1676]

√

that

F1 +

F2 +

√

F3 +

√

F4 + · · · exists.

12.19 Notes In more standard terminology, f is holomorphic on D if f ′ (z) exists on D, and f is analytic on D if f has infinitely many derivatives on D and is equal to its power series expansion in a neighborhood of every point in D. Then, f is holomorphic on D if and only if D is analytic on D. The convergence of a power series on the boundary of its domain of convergence is discussed in [733, pp. 151, 152]. Differentials of functions of a complex matrix are studied in [1424]. Generating functions are often studied as formal power series [166, pp. 10–12], [133], where the variable is not viewed as a complex number and convergence is not considered. The relationship between power series and formal power series is discussed in [1054, pp. 223, 224].

Chapter Thirteen Infinite Series, Infinite Products, and Special Functions 13.1 Facts on Series for Subset, Eulerian, Partition, Bell, Ordered Bell, Bernoulli, Genocchi, Euler, and Up/Down Numbers Fact 13.1.1. Let n ≥ 1, let D ⊆ C, let f : D 7→ C, assume that there exists r > n such that △ f is analytic on D0 = {z ∈ C : |z| < r} ⊆ D, let (βi )∞ i=0 ⊂ C, and assume that, for all z ∈ D0 , ∑∞ i f (z) = i=1 βi z . Then, ( ) { } n ∞ ∑ ∑ i i n n (−1) f (i) = (−1) n! βi . n i i=n i=0 Source: [555]. Fact 13.1.2. Let n ≥ 1. Then, for all k ≥ 1 and z ∈ OIUD,

) ∞ ( ∑ i+n−k i zk = z. n (1 − z)n+1 i=k Now, define An ∈ F[s] by

⟩ n ⟨ ∑ n An (s) = si . i−1 △

i=1

Then, for all z ∈ OIUD,

∑ An (z) = in z i . (1 − z)n+1 i=1 ∞

⟨

⟩ ⟨ ⟩ n n Source: In [2225], is denoted by An,i . Remark: is an Eulerian number. See Fact i−1 i 1.19.5. Fact 13.1.3. For all n ≥ 1, let pn denote the nth partition number. Then,   √ √  sinh 6π x − 1  ∞ ∑ √ d   3i 24  1   , pn = √ αn,i i √    dx  1 2π i=1 x − 24 x=n

where, for all i ≥ 1, △

αn,i =

i ∑

truth(gcd {i, j} = 1)e(βi, j −2n j/i)π ȷ

j=1

and, for all i, j ≥ 1 such that j ≤ i, △

βi, j =

i−1 ( ∑ l l=1

i

−

⌊ ⌋ )( ⌊ ⌋ ) l 1 jl jl 1 − − . − i 2 i i 2

976

CHAPTER 13

Furthermore, as n → ∞,

√

3 √ 2n π e 3 , 12n √ Furthermore, limn→∞ n pn = 1. Source: [114, p. 70], [118, p. 63], and [747]. Remark: The infinite series for pn is Rademacher’s formula. Remark: The asymptotic expression corrects a misprint in [505, p. 67]. Related: Fact 1.20.1. Fact 13.1.4. For all n ≥ 0, let Bn denote the nth Bell number. Then, for all n ≥ 0, dn ez −1 Bn = n e . dz z=0 pn ∼

Therefore, for all z ∈ C, ee −1 = z

∞ ∑ Bi i=0

Consequently,

∞ ∑ Bi

ee−1 =

i=0

i!

,

e

∞ ∑ Bi , i i! 2 i=0

Bn =

1 ∑ in . e i=0 i!

i=1

i!

e

√ 3

e−1

=

∞ ∑ Bi . i i! 3 i=0

∞

∞ 2 ∑ i

∞ ∑ i = e, i! i=1 ∞ 5 ∑ i

zi .

=

√ e−1

Furthermore, for all n ≥ 1,

In particular,

i!

= 52e,

i=1 ∞ 6 ∑ i i=1

i!

i!

= 2e,

= 203e,

∞ 3 ∑ i

i!

∞ 4 ∑ i

= 5e,

i=1 ∞ 7 ∑ i=1

i=1

i = 877e, i!

i!

= 15e,

∞ 8 ∑ i i=1

i!

= 4140e.

Source: [34, pp. 159, 160] and [571, p. 623]. Remark: The series for Bn is Dobinski’s formula. Related: Fact 1.19.6. Fact 13.1.5. For all n ≥ 0, let On denote the nth ordered Bell number. Then, for all n ≥ 0, dn 1 . On = n dz 2 − ez z=0

Therefore, for all z ∈ C such that |z| < log 2, ∑ Oi 1 = zi . 2 − ez i! i=0 ∞

Consequently,

∑ Oi 1 , √ = 2 − e i=0 2i i! ∞

Finally, as n → ∞, On ∼

n! , 2(log 2)n+1

Source: [899, 2100]. Related: Fact 1.19.7.

∑ Oi 1 . √3 = 2 − e i=0 3i i! ∞

nOn−1 = log 2. n→∞ On lim

977

INFINITE SERIES, INFINITE PRODUCTS, AND SPECIAL FUNCTIONS

Fact 13.1.6. Define f : R 7→ R by

 x     ex − 1 , f (x) =    1,

x , 0,

△

x = 0,

△

(i) and define the sequence (Bi )∞ i=0 ⊂ R by Bi = f (0). Then, the following statements hold: i) For all z ∈ C such that |z| < 2π, ∞ ∑ Bi i z = z. z e − 1 i=0 i! 1 1 1 1 1 5 −691 7 −3617 43867 −174611 ii) (Bi )21 i=0 = (1, − 2 , 6 , 0, − 30 , 0, 42 , 0, − 30 , 0, 66 , 0, 2730 , 0, 6 , 0, 510 , 0, 798 , 0, 330 , 0). iii) For all n ≥ 1, B2n+1 = 0 and (−1)n+1 B2n > 0. iv) For all n ≥ 1, ∑ 1 B2n + p p−1|2n

is an integer, where the sum is taken over all primes p such that p − 1 divides 2n. v) For all z ∈ C such that |z| < 2π, ∞ z ∑ B2i 2i z = 1 − + z . ez − 1 2 i=1 (2i)! Therefore, 1 ∑ B2i 1 = + , e − 1 2 i=1 (2i)!

3 ∑ 1 B2i = + , √ e − 1 2 i=1 22i−1 (2i)!

∞

∞

5 ∑ B2i 1 = + . √3 e − 1 2 i=1 32i−1 (2i)! ∞

vi) For all z ∈ C such that |z| < π, z z ∑ |B2i | 2i cot = z . 2 2 i=1 (2i)! ∞

1− vii) For all n ≥ 1, ) n ( ∑ n+1 i=0

i

Bi =

n ( ) ∑ n i=1

i

Bn−i =

n ( ) ∑ n i=0 n ∑

i

(n + i)Bn−1+i = 0,

Bn = (−1)

n

n ( ) ∑ n i=0

i

Bi ,

) ( ) ) n ( n ( ∑ ∑ 2n + 1 2n + 1 2n + 1 (1 − 22i−1 )B2i = 0, 22i−1 B2i = n, B2i = n − 12 , 2i 2i 2i i=0 i=1 i=1 ) ) n ( ) n ( n ( ∑ ∑ n Bi n+1 Bi 1 ∑ n+2 Bn+1 = (n + 1) =− =− Bi , i n+2−i i n+2−i n + 2 i=0 i i=0 i=0 ) n−1 ( ∑ n+1 1 B2n = − (n + 1 + i)Bn+i , (n + 1)(2n + 1) i=0 i ) ) n−1 ( n−1 ( ∑ 1 1 ∑ 2n + 1 1 1 2n + 2 B2n = − B2i = − B2i . 2 2n + 1 i=0 2i 2n + 1 (n + 1)(2n + 1) i=0 2i viii) For all n ≥ 0,

( ) { } n i n ∑ ∑ 1 ∑ n j i n i i! Bn = (−1) j = (−1) . i i + 1 j i + 1 i=0 j=0 i=1

978

CHAPTER 13

For all n ≥ 1,

(n+1) ( ) i n+1 n i ∑ 1 n+1 ∑ n 1 ∑∑ i− j Bn = (−1) j jn (n) = − (−1)i j , n + 1 i=1 j=1 i i i=1 j=1 i

Bn = (−1)n

n 2n − 1

( ) i−1 1 ∑ j k−1 (−1) ( j + 1)n−1 , 2i j=0 j

n ∑ i=1

(

) i ( ) n+1 ∑ i n! (−1)i− j jn+i , Bn = (−1)i (n + i)! i + 1 j j=0 i=0 n ∑

ix) For all n, m ≥ 0, (−1)

m

m ( ) ∑ m i=0

x) For all n ≥ 1,

i

n

n ( ) ∑ n i=0

i

Bi = −

2n+1 ∑

(−1)i−1

i=2

Bn+i = (−1)

n [ ] ∑ n i=0

B2n =

i

( ) i−1 1 2n + 1 ∑ 2n j . i i j=1

Bm+i .

(n − 1)! . n+1

xi) For all n ≥ 2, B2n

( ) n−1 ( ) n−1 2i ∑ (2n)!ζ(2n) 1 ∑ 2n 2 − 1 2n B2i B2n−2i = (−1)n−1 2n−1 2n . B2i B2n−2i = − =− 2n − 1 2i 2n + 1 2i 2 2 π i=1 i=1

Therefore, as n → ∞, B2n ∼ (−1)n+1 xii) As n → ∞,

√ ( n )2n |Bn | ∼ 4 nπ , πe

2(2n)! . (2π)2n

B2n ∼

√ (−1)n+1 4n2n nπ . (πe)2n

xiii) For all n ≥ 0, n ∑ i=0

2n+1 ∑ 1 B4n−2i+2 B2i 4i B2i = , (−1)i = 0, (2i)!(2n + 1 − 2i)! (2n)! (2i)!(4n − 2i + 2)! i=0 [ ] n ∑ B4n−4i+2 B4i B4n−4i B4i+2 i (−4) +2 = 0. (4i)!(4n − 4i + 2)! (4n − 4i)!(4i + 2)! i=0

xiv) For all n ≥ 1, ζ(−n) = − Furthermore, ζ(0) = B1 = − 21 . xv) For all n ≥ 4, ( )] n−2 [ ∑ n Bi Bn−i 2Hn Bn 1− = , i i(n − i) n i=2

n−2 [ ∑ i=2

Bn+1 . n+1

( )] n+2 n+2−2 Bi Bn−i = n(n + 1)Bn . i

979

INFINITE SERIES, INFINITE PRODUCTS, AND SPECIAL FUNCTIONS

xvi) For all n ≥ 1,   1   1  2!   . Bn = n! det  ..   1  n!   1 (n+1)!

B2n

0

1

··· .. .

..

..

..

0

.

1 (n−1)! 1 n!

..

0

.

.

.

1

···

1 2!

  1  1  3!   1 (2n)! det  5!. =− n 4 −2  .  .  1  (2n−1)! 1 (2n+1)!

   1        0     ..   n .  = (−1) n! det     0       0 0 1

0 0

1 3!

1 .. .

..

 1  3!   1  5!  (2n)!  1 = (−1)n n det  7! 4 −2  ..  .  1  (2n−1)! 1 (2n+1)!

xvii) For all n ≥ 1,

∫ B2n = (−1)

n+1 2n+1

2

.

1 (2n−3)! 1 (2n−1)!

∞

1 3!

1

1 5!

1 3!

−∞

(

1 2!

.. .

..

1 n!

1 (n−1)!

1 (n+1)!

1 n!

0 0 0 .. . 1

. ··· ··· 0

.

1 3!

..

1

1 (2n−3)! 1 (2n−1)!

1

··· ··· .. .

1 (2n−5)! 1 (2n−3)!

..

1 2!

..

.

1 (2n−5)! 1 (2n−3)!

dn−1 sech2 x dxn−1

1 3!

··· .. . .. . .. . ··· ···

.

..

.

0

..

.

0

..

.

..

.

1 2!

···

1 3!

..

.

  0    0   ..  , .    1    1  2!

 1   0   0  ..  .   0   0 0 0 0 .. . 1 3! 1 5!

 0   0   0  . ..  .   1   1  3!

)2 dx.

xviii) For all n ≥ 1,

( ∑n )∏ ( ∑n )∏ )ki n+1 ( )ki−1 n ( ∑ ∑n ∑n 1 (2n)! ∑ 1 i=1 ki i=1 ki (−1) i=1 ki = n (−1) i=1 ki , k1 , . . . , kn i=2 i! 4 −2 k1 , . . . , kn i=1 (2i + 1)! ∑ where both sums are taken over all (k1 , . . . , kn ) such that ni=1 iki = n. xix) For all n ≥ 1, n ∑ 4i (4i − 1)dn−1,i−1 |B2i | = 2(2n − 1)!, i i=1

Bn = n!

∑ ∏n−i 2 △ △ where dn,n = 1 and, for all 0 ≤ i < n, dn,i = 4 j=1 ji , where the sum is taken over all subsets { j1 , . . . , jn−i } of {1, . . . , n}. xx) Let n, m ≥ 1, and assume that m is odd. Then, )( ) n+m ( ∑ n+m n+m+i Bn+i = 0. i m i=0

Source: [968, Chapter 11]. iv) is given in [155, p. 275]; vi) is given in [2504, pp. 142, 143]; vii) is

given in [155, pp. 265, 275], [348], [511, p. 100], [2468], and [2513, p. 129]; viii) is given in [155,

980

CHAPTER 13

p. 275], [555], [968, p. 308], and [1209]; ix) is given in [2267]; x) is given in [2880, pp. 137, 138]; xi) is given in [155, p. 267], [511, p. 131], and [2513, p. 403]; xii) is given in [155, p. 267] and [2504, pp. 142, 143]; xiii) is given in [155, p. 275] and [2792]; xiv) is given in [155, p. 266]; xv) is given in [2572]; xvi) is given in [702, 1930, 2016]; xvii) is given in [2068, p. 384]; xviii) is given in [2016]; xix) is given in [1149]; xx) is given in [348]. Remark: Bn is the nth Bernoulli number. See [1219, pp. 283–290, 367]. Related: Fact 1.12.1, Fact 3.16.29, and Fact 13.2.1. Fact 13.1.7. Define f : R 7→ R by 2x , +1

△

f (x) =

ex

△

(i) and define the sequence (Gi )∞ i=0 ⊂ R by G i = f (0). Then, the following statements hold: i) For all z ∈ C such that |z| < π, ∞ ∑ Gi i 2z = z. z e + 1 i=0 i!

ii) iii) iv) v)

(Gi )21 i=0 = (0, 1, −1, 0, 1, 0, −3, 0, 17, 0, −155, 0, 2073, 0, −38227, 0, 929569, 0, −28820619). For all n ≥ 0, Gn = 2(1 − 2n )Bn , where Bn is the nth Bernoulli number. If n ≥ 0, then G2n+1 = 0 and (−1)nG2n is an odd positive integer. If n ≥ 2, then ) n−1 ( ) n−1 ( ∑ 1 ∑ 2n 2n G2i G2n = −n − . G2i , G2n = −1 − 2 i=1 2i 2i − 1 2i i=1

Source: [2513, pp. 123, 124]. Related: Fact 14.6.45. Fact 13.1.8. Define f : R 7→ R by △

f (x) =

2e x 1 = sech x, = + 1 cosh x

e2x

△

(i) and define the sequence (Ei )∞ i=0 ⊂ R by E i = f (0). Then, the following statements hold: i) For all z ∈ C such that |z| < π, ∞ ∑ Ei i 2ez = z. 2z e + 1 i=0 i!

ii) (Ei )15 i=0 = (1, 0, −1, 0, 5, 0, −61, 0, 1385, 0, −50521, 0, 2702765, 0, −199360981, 0). 60

60

iii) For all n ≥ 0, E2n+1 = 0. For all n ≥ 1, E4n ≡ 5 and E4n+2 ≡ −1. iv) For all n ≥ 1, ( ) n n ( ) ∑ ∑ n 2n rem2 (n − i + 1) Ei = E2i = 0. i 2i i=0 i=0 v) For all n ≥ 1,

E2n

  1  1  2!   1 = (2n)! det  4!.  .  .  1  (2n−2)! 1 (2n)!

0 1

0 0

1 2!

1 .. .

..

.

1 (2n−4)! 1 (2n−2)!

1 (2n−5)! 1 (2n−4)!

··· ··· .. . ..

. ··· ···

0 0 0 .. . 1 1 2!

 1   0   0   . ...   0   0

981

INFINITE SERIES, INFINITE PRODUCTS, AND SPECIAL FUNCTIONS

( i ) △ vi) For all n ≥ 1, let An ∈ R2n×2n , where, for all i, j ∈ {1, . . . , 2n}, A(i, j) = j−1 cos 12 (i − j + 1)π. n Then, for all n ≥ 1, E2n = (−1) det An . vii) For all n ≥ 1, ( )∑ ( ) 2n i ( ) 2i 2n ∑ ∑ 2n i 1 ∑ 2i 1 (2 j − i)2n = (−1)i i (i − j)2n . (−1) j E2n = (2n + 1) (−1)i i 2 (i + 1) i j 2 j i=1 j=0 j=0 i=1 viii) For all n ≥ 1,

{ } i ( ) ( ){ } n n n−i j ∑ ∑ (i + 1)! n ∑ j+1 (−1)i ∑ (i + j + 1)! i + 1 n j 2 En = 1+ = 1+ . (−1) i i+ j 2i j+1 i− j i + 1 j=0 2j j i=1 i=1 j=1

ix) For all n ≥ 1,

n ∑

en,i |E2i | = (2n)!,

i=0

△ △ ∑ ∏n−i 2 where en,n = 1 and, for all i ∈ [0, n − 1], en,i = k=1 (2 jk − 1) , where the sum is taken over all subsets { j1 , . . . , jn−i } of {1, . . . , n}. Source: [771, pp. 48, 49], [1149], [1219, p. 559], [1412] [1524, p. 40], and [1930, 2849]. Remark: En is the nth Euler number. Related: Fact 13.2.3. Fact 13.1.9. Define f : {x ∈ R : |x| < π2 } 7→ R by △

f (x) = sec x + tan x, △

(i) and define the sequence (Ui )∞ i=0 ⊂ R by U i = f (0). Then, the following statements hold: π i) For all z ∈ C such that |z| < 2 ,

sec z + tan z =

∞ ∑ Ui i=0

i!

zi .

ii) = (1, 1, 1, 2, 5, 16, 61, 272, 1385, 7936, 50521, 353792, 2702765, 22368256). iii) For all i ≥ 1, Ui is the number of permutations (i1 , . . . , in ) of (1, . . . , n) such that i1 < i2 < i3 < i4 · · · . Source: [1368]. Remark: Un is the nth up/down number. Example: (1, 3, 2) and (2, 3, 1) are up/down permutations of (1, 2, 3). (Ui )13 i=0

13.2 Facts on Bernoulli, Euler, Chebyshev, Legendre, Laguerre, Hermite, Bell, Ordered Bell, Harmonic, Fibonacci, and Lucas Polynomials Fact 13.2.1. For all n ≥ 0, define Bp,n ∈ R[s] by

dn ze xz , Bp,n (x) = n z dz e − 1 z=0 △

xz △ where ezez −1 z=0 = 1. Then, the following statements hold: i) For all x, z ∈ C such that |z| < 2π, ∞ ∑ Bp,i (x) i ze xz = z. z e − 1 i=0 i!

982

CHAPTER 13

ii) For all n ≥ 0 and z ∈ C, △

Bp,n (z) =

n ∑ i=0

( ) ( ) ( ) i n n n i ∑ n n−i ∑ 1 ∑ j i (z + j)n . (−1) Bn−i z = Bi z = i + 1 j i i j=0 i=0 i=0

In particular, Bp,0 (z) = 1,

Bp,1 (z) = z − 21 ,

Bp,4 (z) = z4 − 2z3 + z2 −

1 30 ,

Bp,2 (z) = z2 − z + 61 ,

Bp,3 (z) = z3 − 23 z2 + 12 z,

Bp,5 (z) = z5 − 52 z4 + 53 z3 − 16 z,

Bp,6 (z) = z6 − 3z5 + 25 z4 − 12 z2 +

iii) For all n ≥ 1, Bp,n (0) = Bn , Bp,2n ( 12 ) = (21−2n − 1)Bn , and Bp,2n+1 ( 21 ) = 0. For all n ≥ 2, Bp,n (1) = Bp,n (0) = Bn . iv) For all n ≥ 1 and z ∈ C, ) n ( ) n ( ∑ n 1 1 ∑ n+1 n Bp,i (z) = Bp,i (z), z = i n−i+1 n + 1 i=0 i i=1 Bp,n (1 − z) = (−1)n Bp,n (z),

Bp,n (z + 1) = Bp,n (z) + nzn−1 ,

v) Let n ≥ 0 and a, b ∈ R. Then, ∫ b Bp,n+1 (b) − Bp,n+1 (a) , Bp,n (x) dx = n+1 a vi) For all n, m ≥ 1,

∫

1

∫

a+1

B′p,n (z) = nBp,n−1 (z).

Bp,n (x) dx = an .

a

Bn+m Bp,n (x)Bp,m (x) dx = (−1)n+m−1 (n+m) .

0

n

vii) Let n, m ≥ 1 and z ∈ C. Then, Bp,n (mz) = mn−1

m−1 ∑

Bp,n (z + mi ).

i=0

In particular,

Bp,n (2z) = 2n−1 [Bp,n (z) + Bp,n (z + 21 )].

viii) Let n ≥ 0 and x, y ∈ C. Then, Bp,n (x + y) =

n ( ) ∑ n

i

i=0

ix) For all n ≥ 0 and z ∈ C, Bp,n+1 (z) = Bn+1 +

{ } n ∑ n+1 n i=0

x) Let n, k ≥ 1. Then,

i+1 n ∑ i=1

ik =

i

i+1

z

,

Bp,i (x)yn−i .

n+1

z

=

[ ] n ∑ n+1 n i=0

i+1 i

[Bp,i+1 (z) − Bi+1 ].

Bp,k+1 (n + 1) − Bp,k+1 (0) . k+1

xi) Let k, l be integers, where k < l, let f : [k, l] → 7 R, let m be a positive integer, and assume that f is Cm . Then, ∫ l l−1 m ∑ ∑ Bi (i−1) f (i) = f (x) dx + [f (l) − f (i−1) (k)] + Rm , i! k i=1 i=k

1 42 .

983

INFINITE SERIES, INFINITE PRODUCTS, AND SPECIAL FUNCTIONS

∫

where △

Rm = (−1)

l

m+1 k

Bp,m (x − ⌊x⌋) (m) f (x) dx. m!

Source: ii) is given in [645, p. 100]; iii) is given in [155, pp. 265, 274], [771, p. 48], and [2068,

p. 361]; iv) is given in [155, p. 274] and [771, pp. 48, 49, 165]; vi) is given in [2068, p. 369]; vii) and viii) are given in [155, p. 275]; xi) is given in [1219, pp. 469, 470]. Remark: Bp,n is the nth Bernoulli polynomial. See [1219, pp. 283–290, 367] and [2264, pp. 112–117]. Bn denotes the nth Bernoulli number. x) is the Euler summation formula. See [645, pp. 246–248] and [1219, pp. 469, 470]. Related: Fact 1.12.1 and Fact 13.1.6. Fact 13.2.2. For all n ≥ 1, define Rn : (0, 1) 7→ R by n−1 1 △ d . Rn (x) = n−1 z dz e + 1 z=log(1/x−1) Then, for all n ≥ 1, Rn is a polynomial on [0, 1] with real coefficients. Hence, let Rn ∈ R[s] denote the extension of Rn to C. Then, the following statements hold: i) For all n ≥ 0 and z ∈ C, n+1 ∑ Rn+1 (z) = an,i zi , i=1

where an,i

( ) { } i−1 ∑ n+1 j i−1 n n+k−1 = (−1) (−1) ( j + 1) = (−1) (k − 1)! . k j j=0 △

n

ii) For all z ∈ C and n ≥ 1,

Rn+1 (z) = (z2 − z)R′n (z).

In particular, R1 (z) = z,

R2 (z) = z2 − z,

R3 (z) = 2z3 − 3z2 + z,

R5 (z) = 24z5 − 60z4 + 50z3 − 15z2 + z,

R4 (z) = 6z4 − 12z3 + 7z2 − z,

R6 (z) = 120z6 − 360z5 + 390z4 − 180z3 + 31z2 − z.

iii) If n ≥ 2 and z ∈ [0, 1], then Rn (z) = (−1)n Rn (1 − z). Hence, if n ≥ 3 is odd, then Rn ( 21 ) = 0. iv) If n ≥ 2, then Rn has n nonrepeated roots, all of which are contained in [0, 1]. √ △ x v) Let x ∈ (0, 1), and define α = π2 + log2 1−x . Then, for all z ∈ C such that |z| < α, ∑ Ri+1 (x) x = zi . z x + (1 − x)e i! i=0 ∞

vi) For all n ≥ 1,

∫

1

Rn (x) dx = −Bn .

0 △

vii) For all x ∈ R, define f (x) = 1/(e x + 1). Then, for all n ≥ k ≥ 1, ∫ ∞ ∫ 1 Rn−k+1 (x)Rk+1 (x) k dx = (−1) B , f (n−k) (x) f (k) (x) dx = (−1)k+1 Bn . n x2 − x −∞ 0 Source: [2354, 2355]. Fact 13.2.3. For all n ≥ 0, define Ep,n ∈ R[s] by

dn 2e xz . Ep,n (x) = n z dz e + 1 z=0 △

984

CHAPTER 13

Then, the following statements hold: i) For all x, z ∈ C such that |z| < π, ∞ ∑ Ep,i (x) i 2e xz = z. z e + 1 i=0 i!

ii) For all n ≥ 0 and z ∈ C, △

Ep,n (z) =

n ( ) ∑ n En

i 2i

i=0

(z − 21 )n−i =

( ) i n ∑ 1 ∑ j i (z + j)n . (−1) i 2 j j=0 i=0

In particular, Ep,0 (z) = 1,

Ep,1 (z) = z − 21 ,

Ep,4 (z) = z4 − 2z3 + z,

Ep,2 (z) = z2 − z,

Ep,5 (z) = z5 − 25 z4 + 52 z2 − 12 ,

Ep,3 (z) = z3 − 32 z2 + 41 , Ep,6 (z) = z6 − 3z5 + 5z3 − 3z.

iii) For all n ≥ 1, En = 2n Ep,n ( 21 ). iv) For all n ≥ 1 and z ∈ C,

n−1 ( ) 1∑ n z = Ep,n (z) + Ep,i (z), 2 i=0 i n

Ep,n (1 − z) = (−1)n Ep,n (z), ′ Ep,n (z) = nEp,n−1 (z).

Ep,n (z + 1) = Ep,n (z) + 2zn , v) Let n ≥ 0 and x, y ∈ C. Then, Ep,n (x + y) =

n ( ) ∑ n i=0

vi) Let n ≥ 0 and a, b ∈ R. Then, ∫ b

Ep,n (x) dx =

a

i

Ep,i (x)yn−i .

Ep,n+1 (b) − Ep,n+1 (a) . n+1

vii) Let n, m ≥ 1 and x ∈ C. If n is odd, then Ep,n (mx) = mn

m−1 ∑ (−1)i Ep,n (x + mi ). i=1

If n is even, then Ep,n (mx) =

m−1 −2mn ∑ (−1)i Bp,i+1 (x + mi ). n + 1 i=1

Remark: Ep,n is the nth Euler polynomial. See [771, pp. 48, 49], [1219, p. 559], and [1524, p. 40]. En denotes the nth Euler number. Related: Fact 13.1.8. Fact 13.2.4. For all n ≥ 0, define Bn ∈ R[s] by △

Bn (x) =

dn x(ez −1) . e dzn z=0

Then, the following statements hold: i) For all x, z ∈ C, e x(e −1) = z

∞ ∑ Bi (x) i=0

i!

zi .

985

INFINITE SERIES, INFINITE PRODUCTS, AND SPECIAL FUNCTIONS

ii) For all n ≥ 0 and z ∈ C, Bn (z) =

n { } ∑ n i=0

i

zi .

In particular, B0 (z) = 1,

B1 (z) = z,

B3 (z) = z3 + 3z2 + z,

B2 (z) = z2 + z,

B4 (z) = z4 + 6z3 + 7z2 + z,

B5 (z) = z5 + 10z4 + 25z3 + 15z2 + z,

B6 (z) = z6 + 15z5 + 65z4 + 90z3 + 31z2 + z.

iii) For all n ≥ 0, Bn = Bn (1). iv) For all n ≥ 0 and z ∈ C, n ( ) ∑ n Bi (z) = z[Bn (z) + B′n (z)], Bn+1 (z) = z i i=0

( ) ( ) n n−1 ∑ ∑ n n (−1)i−1 B′i (z). (−1)i Bi (z) = i i i=1 i=0

Remark: Bn is the nth Bell polynomial. See [898]. Fact 13.2.5. For all n ≥ 0, define On ∈ R[s] by

1 dn . On (x) = n dz 1 − x(ez − 1) z=0 △

Then, the following statements hold: i) For all x, z ∈ C such that |z| < log(1 + 1/|x|), ∑ Oi (x) 1 = zi . z 1 − x(e − 1) i=0 i! ∞

ii) For all n ≥ 1 and z ∈ C, On (z) =

{ } n ∑ n i i! z. i i=1

In particular, O0 (z) = 1,

O1 (z) = z,

O2 (z) = 2z2 + z,

O4 (z) = 24z4 + 36z3 + 14z2 + z,

O3 (z) = 6z3 + 6z2 + z,

O5 (z) = 120z5 + 240z4 + 150z3 + 30z2 + z,

O6 (z) = 720z6 + 1800z5 + 1560z4 + 540z3 + 62z2 + z. iii) For all n ≥ 0 and z ∈ C, On = On (1),

On+1 (z) = zOn (z) + z(z + 1)O′n (z) = z

) n ( ) n ( ∑ ∑ n ′ n zOi (z) = Oi (z), i i−1 i=0 i=1

∫

∞

On (z) =

) n ( ∑ n+1 Oi (z), i i=0 e−t Bn (zt) dt.

0

Source: [899, 2100]. Remark: On is the nth ordered Bell polynomial. Related: Fact 1.19.7 and

Fact 13.1.5. Fact 13.2.6. For all n ≥ 0, define T n ∈ R[s] by △

T n (x) = Then, the following statements hold:

1 dn 1 − xz . n! dzn z2 − 2xz + 1 z=0

986

CHAPTER 13 △

i) For all x, z ∈ C such that |z| < ρmax (p), where p(s) = s2 − 2xs + 1, ∑ 1 − xz = T i (x)zi . z2 − 2xz + 1 i=0 ∞

ii) For all n ≥ 0, T n+2 (z) = 2zT n+1 (z) − T n (z). Furthermore, T 0 (z) = 1,

T 1 (z) = z,

T 2 (z) = 2z2 − 1,

T 4 (z) = 8z4 − 8z2 + 1 T 5 (z) = 16z5 − 20z3 + 5z iii) For all n ≥ 0 and z ∈ C, T n (z) = 21 [(z −

√

T 3 (z) = 4z3 − 3z,

T 6 (z) = 32z6 − 48z4 − 18z2 − 1.

z2 − 1)n + (z +

√ z2 − 1)n ],

2 T n (cos z) = cos nz, T n+1 (z) = T n (z)T n+2 (z) + 1 − z2 , ⌊n/2⌋ ⌊n/2⌋ ∑ ∑ (n) (n − i − 1)! n−2i (z2 − 1)i zn−2i = 2n−1 n (−1)i i T n (z) = z , 2i 4 i!(n − 2i)! i=0 i=0

(1 − z2 )T n′′ (z) − zT n′ (z) + n2 T n (z) = 0,

′ (z) = (n + 1)T n (z). (n + 1)zT n+1 (z) + (1 − z2 )T n+1

iv) For all n ≥ 0, 2T n2 = 1 + T 2n ,

T n (1) = 1,

v) For all n, m ≥ 0,

T n (−1) = (−1)n ,

T 2n (0) = (−1)n ,

T 2n+1 (0) = 0.

T nm = T n ◦ T m = T m ◦ T n .

vi) For all n, m ≥ 0 and z ∈ C, (z2 − 1)[T n+m (z) − T n (z)T m (z)] = [T m+1 (z) − zT m (z)][T n+1 (z) − zT n (z)]. vii) For all n ≥ m ≥ 0, viii) For all n ≥ 0,

ix) For all n, m ≥ 1,

T n+m + T n−m = 2T m T n . 1 π 2 π

∫

1

−1

∫

1

−1

1 T 0 (x)T n (x) dx = δ0,n . √ 1 − x2 1 T n (x)T m (x) dx = δm,n . √ 1 − x2

x) For all n ≥ 1, √ F4n T 2n ( 5/2) = 12 (F2n−1 + F2n+1 ) = , 2F2n

√ T 2n+1 ( 5/2) =

√ 5 F2n+1 . 2

xi) let x ∈ C, let z ∈ C, and assume that |xz| is sufficiently small. Then, 2i+1 ∑ 2xz i T 2i+1 (x)z = 2 (−1) . 2i + 1 1 − z2 i=0 ∞

atan

Remark: T n is the nth Chebyshev polynomial of the first kind. See [352, pp. 187–196], [666],

[1158, p. 58], and [1976]. x) and xi) are given in [649]. Fact 13.2.7. For all n ≥ 0, define Un ∈ R[s] by △

Un (x) =

1 1 dn . n! dzn z2 − 2xz + 1 z=0

987

INFINITE SERIES, INFINITE PRODUCTS, AND SPECIAL FUNCTIONS

Then, the following statements hold: △ i) For all x, z ∈ C such that |z| < ρmax (p), where p(s) = s2 − 2xs + 1, ∑ 1 = Ui (x)zi . 2 z − 2xz + 1 i=0 ∞

ii) For all n ≥ 0 and z ∈ C, Un+2 (z) = 2zUn+1 (z) − Un (z). Furthermore, △

U0 (z) = 1,

U1 (z) = 2z,

U4 (z) = 16z4 − 12z2 + 1,

U2 (z) = 4z2 − 1,

U5 (z) = 32z5 − 32z3 + 6z,

U3 (z) = 8z3 − 4z, U6 (z) = 64z6 − 80z4 + 24z2 − 1.

iii) For all n ≥ 0 and z ∈ C,

Un (z) = det(2zIn + Nn + NnT ), √ √ (z + z2 − 1)n+1 − (z − z2 − 1)n+1 Un (z) = , √ 2 z2 − 1 (1 − z2 )Un′′ (z) − 3zUn′ (z) + n(n + 2)Un (z) = 0, ⌊n/2⌋ n ( ∑ (n + 1) ∏ iπ ) Un (z) = (z2 − 1)i zn−2i = 2n x − cos , 2i + 1 n+1 i=0 i=1

sin (n + 1)z , (1 − z2 )Un′ (z) = zUn (z) − (n + 1)T n+1 (z), sin z T n+1 (z) = Un+1 (z) − zUn (z), T n+2 (z) = zT n+1 (z) − (1 − z2 )Un (z).

Un (cos z) =

iv) For all n ≥ 0, 2 Un+1 = Un Un+2 + 1,

v) For all n, m ≥ 0,

vi) For all n, m ≥ 0,

′ T n+1 = (n + 1)Un ,

2 π

∫

1 −1

T n+2 = 12 (Un+2 − Un ),

Un (− ȷ/2) = (− ȷ)n Fn .

√ 1 − x2 Un (x)Um (x) dx = δm,n .

Um+n+2 = Um+1 Un+1 − Um Un .

vii) For all n ≥ 1, √ F4n+2 , U2n ( 5/2) = F2n + F2n+2 = 2F2n+1

√ √ U2n−1 ( 5/2) = 5F2n .

Source: vi) is given in [666]; vii) is given in [649]. Remark: Un is the nth Chebyshev polynomial of the second kind. Remark: Fn is the nth Fibonacci number. See Fact 1.17.1. Fact 13.2.8. For all n ≥ 0, define Pn ∈ R[s] by n 1 △ 1 d Pn (x) = √ . n! dzn z2 − 2xz + 1 z=0

Then, the following statements hold: △ i) For all x, z ∈ C such that |z| < ρmax (p), where p(s) = s2 − 2xs + 1, ∑ 1 = Pi (x)zi . √ z2 − 2xz + 1 i=0 ∞

ii)

√1 z2 +1

=

∑∞

i=0

Pi (0)zi .

988

CHAPTER 13

iii) For all n ≥ 0 and z ∈ C, Pn+2 (z) = P0 (z) = 1,

1 n+2 [(2n

P1 (z) = z,

+ 3)zPn+1 (z) − (n + 1)Pn (z)]. Furthermore,

P2 (z) = 12 (3z2 − 1),

P4 (z) = 18 (35z4 − 30z2 + 3), P6 (z) =

6 1 16 (231z

P3 (z) = 21 (5z3 − 3z),

P5 (z) = 18 (63z5 − 70z3 + 15z), − 315z4 + 105z2 − 5).

iv) For all n ≥ 0 and z ∈ C,

) n ( )( 1 ∑ n (n + i − 1)

( )( ) ⌊n/2⌋ 1 ∑ i n 2n − 2i n−2i z , (−1) i n 2n i=0 i n i=0 ∫ n ( )2 √ 1 ∑ n 1 dn 2 1 π n z2 − 1 cos θ)n dθ, Pn (z) = n (z − 1)i (z + 1)n−i = n (z − 1) = (z + 2 i=0 i 2 n! dzn π 0 Pn (z) = 2n

2

zi =

(1 − z2 )P′′n (z) − 2zP′n (z) + n(n + 1)Pn (z) = 0, 1 1 [zP′ (z) − P′n (z)] = [P′ (z) − zP′n+1 (z)], Pn+1 (z) = n + 1 n+1 n + 2 n+2 ⌊n/2⌋ n ∑ n! ∑ 2n − 4i + 1 (1 − z) (2i + 1)Pi (z) = (n + 1)[Pn (z) − Pn+1 (z)], zn = n Pn−2i (z). 2 i=0 i!( 3 )n−i i=0 2 v) For all n ≥ 0, Pn+1 = Pn (1) = 1,

Pn (−1) = (−1) , n

P′n (1)

P2n (0) = (−1)n vi) For all n ≥ 1,

∫

1 −1

vii) For all n ≥ 1 and k ≥ 0,

∫

1 −1

viii) For all n, m ≥ 0, 2n + 1 2

∫

1 −1

n ∑ (2i + 1)Pi = P′n+1 + P′n ,

1 [P′ − P′n ], 2n + 1 n+2 =

i=0 1 2 n(n

(2n)! , 4n (n!)2

+ 1),

P2n+1 (0) = 0.

xPn (x)Pn−1 (x) dx =

Pn (x)xn+2k dx =

Pn (x)Pm (x) dx =

2n + 1 2n(n + 1)

ix) For all n ≥ 1 and z ∈ C, 2 Pn (z) = √ πn!

∫

∞

P′n (−1) = (−1)n−1 12 n(n + 1),

2n . 4n2 − 1

(2k + 1)n 2n (k + 21 )n+1 ∫

1 −1

.

(1 − x)2 P′n (x)P′m (x) dx = δn,m .

xn e−x Hn (zx) dx. 2

0

x) Let n ≥ 1 and θ ∈ (0, π). Then, √ ∫ π √ ∫ θ cos (n + 12 )x sin (n + 21 )x 2 2 dx = dx. Pn (cos θ) = √ √ π 0 π θ cos x − cos θ cos θ − cos x xi) Let x ∈ (−1, 1) and n ≥ 1. Then, Pn−1 (x)Pn+1 (x) ≤ P2n (x).

989

INFINITE SERIES, INFINITE PRODUCTS, AND SPECIAL FUNCTIONS

xii) For all n ≥ 1 and z ∈ C,

( (1 − z) Pn n

) ∑ n ( )2 n i 1+z z. = i 1−z i=0

xiii) Let n ≥ 0 and z ∈ C. Then, Pn (z) = 1 + (z − 1)

n−1 ∑ (2i + 1)(Hn − Hi )Pi (z). i=0

xiv) Let n ≥ 0 and 0 ≤ m ≤ n − 1. Then, ∫ 1 1 − Pn (x) Pm (x) dx = 2(Hn − Hm ). 1−x −1 In particular,

∫

1

−1

xv) Let n ≥ 1. Then,

∫

1

1 − Pn (x) dx = 2Hn . 1−x

Pn (2x − 1) log x dx =

0

xvi) Let n ≥ 1 and x ∈ R. Then,

∫

1

Pn (t) dt =

x

(−1)n+1 . n(n + 1)

(1 − x2 )P′n (x) . n(n + 1)

xvii) Let n ≥ 1. Then,

( ) n ( )3 ∑ dn (1 − x2 )n 1 + x n Pn , = dxn n! 1 − x x=0 i=0 i

( ) n ( )4 ∑ dn (1 − x)2n 2 1 + x n Pn . = dxn n! 1 − x x=0 i=0 i

xviii) Let n ≥ 1 and x ∈ (−1, 1]. Then, n ∑ 3n−i (i + 12 )Pi (x) > 0. (n − i)! i=0

Remark: Pn is the nth Legendre polynomial. See [116, pp. 313, 342, 343], [178, p. 4], [352, pp. 42–55, 90, 91, 164], [771, p. 165], and [2068, pp. 169, 170, 388, 393, 396–400]. Remark: In ix), Hn is the nth Hermite polynomial. See Fact 13.2.7. In xiii) and xiv), Hn is the nth harmonic number. Remark: xi) is Turan’s inequality. See [116, p. 342]. Related: Fact 1.16.13. Fact 13.2.9. For all n ≥ 0, define Ln ∈ R[s] by n xz/(z−1) △ 1 d e . Ln (x) = n! dzn 1 − z z=0

Then, the following statements hold: i) For all x, z ∈ C such that |z| < 1, e xz/(z−1) ∑ = Li (x)zi . 1−z i=0 ∞

ii) For all n ≥ 0 and z ∈ C, Ln+2 (z) = L0 (z) = 1,

1 n+2 [(2n

+ 3 − z)Ln+1 (z) − (n + 1)Ln (z)]. Therefore,

L1 (z) = −z + 1,

L3 (z) = 16 (−z3 + 9z2 − 18z + 6),

L2 (z) = 21 (z2 − 4z + 2),

L4 (z) =

1 4 24 (z

− 16z3 + 72z2 − 96z + 24),

990

CHAPTER 13

L5 (z) = L6 (z) =

1 720 (z

iii) For all n ≥ 0 and z ∈ C,

5 1 120 (−z + 6 5

25z4 − 200z3 + 600z2 − 600z + 120),

− 36z + 450z4 − 2400z3 + 5400z2 − 4320z + 720).

( ) n ∑ ez dn −z n i1 n i Ln (z) = (−1) (e z ), z = i! i n! dzn i=0

zLn′′ (z) + (1 − z)Ln′ (z) + nLn (z) = 0,

′ zLn+1 (z) = nLn+1 (z) − nLn (z),

Ln(n) (z) = (−1)n .

iv) For all n, m ≥ 0 and z ∈ C, (m+2) (m+1) (m) zLn+m (z) + (m + 1 − z)Ln+m (z) + nLn+m (z) = 0.

v) For all n ≥ 0, n ∑

′ Li = −Ln+1 ,

i=0

Ln (0) = 1, ∫

∞

e−x Ln3 (x) dx

0

vi) For all n, m ≥ 0

Ln′ (0) = −n, = (−1)

n

n ( )3 ∑ n i=0

∫

∞

Ln′′ (0) = 12 n(n − 1),

i

.

e−x Ln (x)Lm (x) dx = δn,m .

0

vii) For all n ≥ m ≥ 0,

∫

∞ 0

   m < n, 0, x e Ln (x) dx =   n (−1) n!, m = n. m −x

Remark: Ln is the nth Laguerre polynomial. See [352, pp. 168–185]. Related: Fact 1.18.3. Fact 13.2.10. For all n ≥ 1, define Hn ∈ R[s] by

dn 2xz−z2 . e dzn z=0

△

Hn (x) = Then, the following statements hold: i) For all x, z ∈ C, 2

e2xz−z =

∞ ∑ Hi (x) i z. i! i=0

ii) For all n ≥ 0 and z ∈ C, Hn+2 (z) = 2zHn+1 (z) − 2(n + 1)Hn (z). Furthermore, H0 (z) = 1,

H1 (z) = 2z,

H2 (z) = 4z2 − 2,

H4 (z) = 16z4 − 48z2 + 12,

H3 (z) = 8z3 − 12z,

H5 (z) = 32z5 − 160z3 + 120z,

H6 (z) = 64z6 − 480z4 + 720z2 − 120. iii) For all n ≥ 1 and z ∈ C, Hn (z) = n!

⌊n/2⌋ ∑ i=0

2

(−1)i

n (−2 ȷ)n ez 1 2 d 2 (2z)n−2i = (−1)n ez n e−z = √ i!(n − 2i)! dz π

Hn′′ (z) − 2zHn′ (z) + 2nHn (z) = 0,

Hn+1 (z) = 2zHn (z) − 2nHn−1 (z),

∫

∞

−∞

e−x xn e2zx ȷ dx, 2

Hn′ (z) = 2nHn−1 (z),

991

INFINITE SERIES, INFINITE PRODUCTS, AND SPECIAL FUNCTIONS

1 (2n)! ∑ H2i (z), 4n i=0 (2i)!(n − i)!

(2n + 1)! ∑ 1 H2i+1 (z). 22n+1 i=0 (2i + 1)!(n − i)!

n

z2n =

iv) For all n ≥ 0,

n

z2n+1 =

   n odd, 0, Hn (0) =   (−1)n/2 2n/2 (n − 1)!!, n even.

v) For all n ≥ 0, ∫ ∞ √ 2 e−2x Hn2 (x) dx = 2n−1 2Γ(n + 12 ), −∞

vi) For all n, m ≥ 0, ∫

∞ −∞

1 √ 2n n! π

∫

∞ −∞

∫

∞

−∞

√ 2 x2 e−x Hn2 (x) dx = 2n n!(n + 21 ) π.

e−x Hn (x)Hm (x) dx = δn,m , 2

√ √ 2 xe−x Hn (x)Hm (x) dx = 2n−1 n! πδn−1,m + 2n (n + 1)! πδn+1,m .

vii) Let 0 ≤ l ≤ m ≤ n, and assume that l + m + n is even and n ≤ l + m. Then, √ ∫ ∞ 2(l+m+n)/2 l!m!n! π 2 e−x Hl (x)Hm (x)Hn (x) dx = 1 . [ 2 (l + m − n)]![ 21 (m + n − l)]![ 12 (n + l − m)]! −∞ viii) For all n, m ≥ 0 and z ∈ C, Hm (z)Hn (z) =

min {m,n} ∑ i=0

ix) For all n ≥ m ≥ 0, Hn(m) =

2i i!

( )( ) m n Hm+n−2i (z). i i

2m n! Hn−m , (n − m)!

Hn(n) = 2n n!.

x) Let x ∈ R, and assume that x , 0. Then, ∞ ∑ √ 1 H2i+1 (x) = sign(x) π. (−1)i i 4 (2i + 1)i! i=0 xi) Let x, y ∈ C and n ≥ 0. Then, n ( ) ∑ n Hn (x + y) = Hi (x)(xy)n−i , i i=0

Ln (x2 + y2 ) =

n (−1)n ∑ H2i (x)H2n−2i (y) . 4n i=0 i!(n − i)! △

xii) Let n ≥ 1. Then, the n roots r1 , . . . , rn of Hn are real. Now, define r = [r1 · · · rn ]T and f : Rn 7→ R by   n  1 ∑  ∏ 2   f (x) = exp − x(i) |x(i) − x( j) |, 2 i=1 where the product is taken over all i, j ∈ {1, . . . , n} such that i < j. Then, for all x ∈ Rn such that x , r, f (x) < f (r). xiii) Let m ≥ 1 and z1 , . . . , zm ∈ C. Then,   m m   1 ∑ n! ∑ ∏ Hi j (z j )  zi  = n/2 , Hn  √ i j! m m i=1 j=1

992

CHAPTER 13

where the sum is taken over all m-tuples (i1 , . . . , im ) of nonnegative integers such that ∑m j=1 i j = n. Remark: Hn is the nth Hermite polynomial. See [116, pp. 278, 279, 318, 328, 339, 340–342, 418, 419], [352, pp. 156–167], and [771, p. 165]. xiii) is given in [2799]. Remark: Ln is the nth Laguerre polynomial. See Fact 13.2.9. Fact 13.2.11. For all n ≥ 0, define Hp,n ∈ R[s] by △

Hp,n (x) = Then, for all n ≥ 1, Hn = Hp,n (1) =

i=1

∑n

1 i=1 i .

n ∑

n ∑ 1

i

xi .

If n ≥ 1 and x ∈ C, where x , 1, then

Hi xi =

i=1

Hn xn+1 − Hp,n (x) . x−1

If n ≥ 1 and x ∈ C, where x < {0, 1}, then n ∑ Hp,n (1/x)xn+1 − Hn Hi xn−i = . x−1 i=1 Furthermore, if n ≥ 1, then n ∑

(−1)n−i Hi =

i=1

1 [Hn + (−1)n Hp,n (−1)]. 2

Source: [899]. Remark: Hp,n is the nth harmonic polynomial, and Hn is the nth harmonic number. Fact 13.2.12. For all n ≥ 0, define Fp,n ∈ R[s] by

−z 1 dn . Fp,n (x) = n! dzn z2 + xz − 1 z=0 △

Then, the following statements hold: △ i) For all x, z ∈ C such that |z| < ρmax (p), where p(s) = s2 + xs − 1, ∑ −z = Fp,i (x)zi . 2 z + xz − 1 i=0 ∞

ii) For all n ≥ 2, Fp,n (z) = zFp,n−1 (z) + Fp,n−2 (z). In particular, Fp,0 (z) = 0,

Fp,1 (z) = 1,

Fp,4 (z) = z3 + 2z,

Fp,2 (z) = z,

Fp,5 (z) = z4 + 3z2 + 1,

Fp,3 (z) = z2 + 1, Fp,6 (z) = z5 + 4z3 + 3z.

iii) For all n ≥ 1, Fn = Fp,n (1).

√ √ △ △ iv) For all x ∈ R, define α(x) = 21 (x + x2 + 4) and β(x) = 12 (x − x2 + 4). Then, for all n ≥ 0 and x ∈ R, αn (x) − βn (x) Fp,n (x) = . α(x) − β(x) v) For all n ≥ 1 and z ∈ C, Fp,n (z) =

⌊(n−1)/2⌋ ∑ ( i=0

vi) For all n ≥ 2, roots(Fp,n ) = {2(cos

iπ n ) ȷ:

) n − i − 1 n−2i−1 z . i

1 ≤ i ≤ n − 1}.

993

INFINITE SERIES, INFINITE PRODUCTS, AND SPECIAL FUNCTIONS

vii) For all n ≥ 0, xn = viii) If n ≥ 4 is even, then

∫

∞ −∞

ix) If n ≥ 3 is odd, then

∫

∞ −∞

( ) n ∑ n (−1)i Fp,n+1−2i (x). i i=0

π( π) 1 dx = 1 + sec . Fp,n (x) n n

( ) π π 3π x dx = tan + tan . Fp,n (x) n 2n 2n

Source: [595], [2068, pp. 124, 125], and [2683]. Remark: Fp,n is the nth Fibonacci polynomial. Fact 13.2.13. For all n ≥ 0, define Lp,n ∈ R[s] by △

Lp,n (x) =

1 dn xz − 2 . n! dzn z2 + xz − 1 z=0

Then, the following statements hold: △ i) For all x, z ∈ C such that |z| < ρmax (p), where p(s) = s2 + xs − 1, ∑ xz − 2 = Lp,i (x)zi . z2 + xz − 1 i=0 ∞

ii) For all n ≥ 2, Lp,n (z) = zLp,n−1 (z) + Fp,n−2 (z). In particular, Lp,0 (z) = 2,

Lp,1 (z) = z,

Lp,2 (z) = z2 + 2,

Lp,3 (z) = z3 + 3z,

Lp,4 (z) = z4 + 4z2 + 2,

iii) iv) v) vi)

Lp,5 (z) = z5 + 5z3 + 5z, Lp,6 (z) = z6 + 6z4 + 9z2 + 2. √ √ For all n ≥ 1, Lp,n (z) = 21n [(z − z2 + 4)n + (z + z2 + 4)n ]. For all n ≥ 1, Ln = Lp,n (1) and Lp,n (0) = 1 + (−1)n . √ √ △ △ For all x ∈ R, define α(x) = 12 (x + x2 + 4) and β(x) = 12 (x − x2 + 4). Then, for all n ≥ 0 and x ∈ R, Lp,n (x) = αn (x) + βn (x). For all n ≥ 1 and z ∈ C, ⌊n/2⌋ ∑ n ( n − i) Fp,n (z) = zn−2i . n − i i i=0

′ vii) For all n ≥ 1, Lp,n (z) =

n [zLp,n (z) + 2Lp,n−1 (z)]. z2 +4 roots(Lp,n ) = {2(sin iπn ) ȷ: 1 ≤ i ≤ n −

viii) For all n ≥ 1, 1}. ix) Lp,m divides Lp,n if and only if n/m is an odd integer. x) If n is prime, then roots(Lp,n ) ⊂ IA . Source: [2683]. Remark: Lp,n is the nth Lucas polynomial. Fact 13.2.14. Let n ≥ 0. Then, there exists Qn ∈ R[s] such that n ∑ dn sec x = (sec x)Q (tan x) = (sec x) En,i tani x. n dxn i=0 Furthermore, En,0 En,n = n!,

   dn (−1)n/2 En , n even, = Qn (0) = sec x = 0, x=0   dxn n odd. En+1,0 = En,1 ,

En+1,n = nEn,n−1 ,

En+1,n+1 = (n + 1)En,n .

994

CHAPTER 13

If, in addition, 1 ≤ m ≤ n − 1, then En+1,m = mEn,m−1 + (m + 1)En,m+1 . For all x ∈ R,

Qn+1 (x) = xQn (x) + (x2 + 1)Q′n (x).

In particular, Q0 (x) = 1, Q1 (x) = x, Q2 (x) = 2x2 + 1, and Q3 (x) = 6x3 + 5x. For all t, x ∈ R such that cos t − x sin t , 0, ∞ ∑ Qi (x) i 1 = t. cos t − x sin t i=0 n! Furthermore, there exists Pn ∈ R[s] such that ∑ dn tan x = P (tan x) = Dn,i tani x. n dxn i=0 n

For all t, x ∈ R such that cos t − x sin t , 0, sin t + x cos t ∑ Pi (x) i = t. cos t − x sin t i=0 n! ∞

Source: [630, 1430]. Remark: Qn and Pn are derivative polynomials. En is the nth Euler number. See Fact 13.1.8. Related: Fact 14.8.19. Fact 13.2.15. Let An ∈ C[s](n+1)×(n+1) be the Hankel matrix whose (i, j) entry is pi+ j−2 (x), △ △ where, for all k ∈ {0, . . . , 2n}, pk (x) = C0 xk + · · · + Ck−1 x + Ck , and define Hn (x) = det An (x). Then, for all x ∈ C, ( ) n ∑ n+i i Hn (x) = (−1)i x. n−i i=0

In particular, H(0) = 1. Alternatively, for all x ∈ C and k ∈ {0, . . . , 2n}, define ) k ( ∑ 2k − 2i i △ pk (x) = x. k−i i=1 Then, Hn (x) = 2n

( ) n ∑ n+i i (−1)i x. n−i i=0

In both cases, for all x ∈ C, Hn (x) satisfies x(x − 4)Hn′′ (x) + 2(x − 1)Hn′ (x) − n(n + 1)Hn (x) = 0. ∞ Source: [979]. Remark: (Hi (x))∞ i=0 is the Hankel transform of (pi (x))i=0 .

13.3 Facts on the Zeta, Gamma, Digamma, Generalized Harmonic, Dilogarithm, and Dirichlet L Functions Fact 13.3.1. For all z ∈ C such that Re z > 1, define △

ζ(z) = Then, the following statements hold:

∞ ∑ 1 . iz i=1

995

INFINITE SERIES, INFINITE PRODUCTS, AND SPECIAL FUNCTIONS

i) The series is given by

∑∞

1 i=1 iz

△

converges for all z ∈ D0 = {z ∈ C : Re z > 1}. Furthermore, ζ : D0 7→ C

∫ ∞ z−1 ∫ 1 ∞ 1 1 x logz−1 1/x 1 ∑ = dx = dx. ζ(z) = 1 − 2−z i=1 (2i − 1)z Γ(z) 0 e x − 1 1−x 0 △

ii) ζ has an analytic continuation to D1 = ORHP \{1}. Furthermore, ζ : D1 7→ C is given by ∫ ∞ z−1 ∫ ∞ ∞ ∑ x 1 1 z ⌊x⌋ − x i−1 1 (−1) dx. ζ(z) = = dx = + z z x 1−z 1−z i z−1 1−2 (1 − 2 )Γ(z) 0 e + 1 xz+1 1 i=1 △

iii) ζ has an analytic continuation to D = C\{1}. Furthermore, ζ : D 7→ C is given by () ∑i ∫ ∞ j i −z ∞ ∑ 1 2z−1 sin(z atan x) j=0 (−1) j ( j + 1) z dx = ζ(z) = . −2 2 z/2 πx z−1 1 − 21−z i=0 2i+1 0 (1 + x ) (e + 1) iv) For all z ∈ D,

( ) ∞ i ∞ ∑ 1 γi 1 ∑ 1 ∑ 1 j i + (−1) (−1)i (z − 1)i , ζ(z) = = z − 1 i=0 i + 1 j=0 i! j ( j + 1)z−1 z − 1 i=0 △

where γ0 = γ and, for all i ≥ 0,

 j  ∑ logi k logi+1 j    . γi = lim  −  j→∞ k i + 1 k=1 ( ) 1 Hence, limz→1 (z − 1)ζ(z) = 1 and limz→1 ζ(z) − z−1 = γ. v) For all z ∈ D, ( ) ∫ ∞ z z 1 πz/2 −1 − −1 1 2 + ζ(z) = z + x 2 )ψ(x) dx , 2 (x Γ( 2 ) z(z − 1) 1 △

where, for all x > 0, △

ψ(x) =

∞ ∑

e−i

2

πx

.

i=1

vi) Define f : C 7→ C and g : C 7→ C by    z , 1, △ (z − 1)ζ(z), f (z) =   1, z = 1,

   ζ(z) − g(z) =   γ, △

1 z−1 ,

Then, f and g are analytic. 1 vii) Let n ≥ 1. Then, ζ(1 − 2n) = − 2n B2n , ζ(−2n) = 0, and ζ ′ (−2n) = (−1)n

(2n)! ζ(2n + 1). 2(2π)2n

π viii) ζ(0) = − 12 , ζ ′ (0) = − 12 log 2π, and ζ ′′ (0) = γ1 + γ2 − 24 − 21 log2 2π. ix) For all z ∈ C\N, ( πz ) ζ(z) = 2(2π)z−1 sin Γ(1 − z)ζ(1 − z). 2 2

2

z , 1, z = 1.

996

CHAPTER 13

x) For all z ∈ C\(−P ∪ {0, 1}), ζ(1 − z) =

2 ( πz ) cos Γ(z)ζ(z). (2π)z 2

xi) For all n ≥ 1, 22n−1 π2n |B2n |, (2n)!

2 lim Γ(z)ζ(2z) = (−1)n ζ ′ (−2n), n! √ 2π2n+1/2 ′ π(2n − 1)!! lim Γ( 12 − z)ζ(1 − 2z) = (−1)n ζ (−2n) = ζ(2n + 1). z→−n n! 2n ζ(2n) =

z→−n

△

xii) The function ξ(z) = 21 z(z − 1)π−z/2 Γ( 2z )ζ(z) is analytic on C and, for all z ∈ C, satisfies ξ(z) = ξ(1 − z). xiii) If z ∈ C and Re z = 12 , then π−z/2 Γ( 2z )ζ(z) is real. xiv) For all α > 1, limn→∞ Hn,α = ζ(α). ∑ xv) For all n ≥ 2, (n + 12 )ζ(2n) = n−1 i=1 ζ(2i)ζ(2n − 2i). ∑∞ d(i) 2 xvi) For all x > 1, ζ (x) = i=1 nx , where d(i) is the number of positive divisors of i. (Example: d(12) = 6.) ∑ σ(i) xvii) For all x > 2, ζ(x)ζ(1 − x) = ∞ i=1 n x , where σ(i) is the sum of the positive divisors of i. (Example: σ(12) = 28.) ∑∞ ϕ(i) xviii) For all x > 2, ζ(x−1) i=1 n x , where ϕ(i) is the number of positive integers j ≤ i such that ζ(x) = {i, j} is coprime. (Example: ϕ(12) = 4.) xix) If z ∈ −2P, then ζ(z) = 0. xx) If z ∈ D, z < −2P, and ζ(z) = 0, then 0 ≤ Re z < 1, ζ(z) = 0, and ζ(1 − z) = 0. xxi) card{z ∈ C : Re z = 21 and ζ(z) = 0} = ∞. xxii) For all i ≥ 1, let pi denote the ith prime number, where p1 = 2. Then, for all z ∈ C such that Re z > 1, ∞ ∏ 1 ζ(z) = . 1 − p−z i i=1 In particular,

∞ ∏ i=1

1 π2 . = 6 1 − p−2 i

xxiii) The probability that k random positive integers are coprime is 1/ζ(k). In addition, the following statements are equivalent: xxiv) If z < −2P and ζ(z) = 0, then Re z = 12 . ∫x xxv) For all ε > 0 there exists c > 0 such that, for all x ≥ 2, |π(x) − 2 log1 t dt| < cx1/2+ε . ∑ ∑∞ i1 i1 xxvi) If a, b ∈ R and ∞ i=1 (−1) ia sin(b log i) = i=1 (−1) ia cos(b log i) = 0, then either a = 1/2 or a = 1. xxvii) For all n ≥ 1, let ω(n) denote the number of prime divisors of n. (Example: ω(1) = 0 and 1 ∑n ω(i) = 0. ω(24) = 4.) Then, for all ε > 0, limn→∞ n1/2+ε i=1 (−1) Source: [116, p. 16], [525, pp. 6, 13, 14], [723], [968, pp. 276–288], [1350, pp. 68, 207], [2106, pp. 352–360], [2486], [2513, pp. 164, 167, 226], and [2528, Chapter 7]. Remark: ζ is the zeta function. Remark: Analytic continuation of ζ is discussed in [1004, 2491]. Remark: xxi) is Hardy’s theorem. See [525, pp. 24–47]. Remark: xxii) is Euler’s product formula. Remark: xxiv)– xxvii) are equivalent statements of the Riemann hypothesis. See [2493]. An additional equivalent

INFINITE SERIES, INFINITE PRODUCTS, AND SPECIAL FUNCTIONS

997

statement is given by Fact 1.11.47. Further additional statements are given in [525, Chapter 5]. Remark: iv) is the Laurent series of ζ at z = 1. Remark: xxv) is a strengthened version of the prime number theorem given by Fact 12.18.41, where π(x) denotes the number of primes that are either less than or equal to x. Remark: ϕ is the Euler totient function. See Fact 1.20.4. √ Remark: i is counted The definitions of d(i) and σ(i) are different from the definitions in Fact 1.20.3, where √ twice in the case where i is an integer. Remark: γn is the nth Stieltjes constant. Remark: ψ(x) = △ ∑∞ 1 −i2 πx is the Jacobi theta function. For all x > 0, x1/2 θ(x) = θ( 1x ). i=−∞ e 2 (θ(x) − 1), where θ(x) = See [525, p. 13]. Remark: On the line {z ∈ C : Re z = 12 }, ξ(z) is real, and this can be used to compute the zeros of ζ whose real part is 12 . Note that ζ(z) is not necessarily real for all z such that Re z = 21 . See [975, pp. 119–135]. Remark: The zeros of ζ are related to the eigenvalues of random matrices. See [525, p. 40]. Remark: Historical and background material on Riemann’s contribution to the zeta function is given in [525, 975]. Fact 13.3.2. For all z ∈ ORHP, define )z−1 ∫ ∞ ∫ 1( 1 △ z−1 −x dx. Γ(z) = x e dx = log x 0 0 Then, the following statements hold: i) Both integrals exist for all z ∈ ORHP. ii) If z ∈ C and Re z > 1, then ∫ ∞ z−1 ∫ ∞ z−1 x 1 1 x dx = dx. Γ(z) = ζ(z) 0 e x − 1 (1 − 21−z )ζ(z) 0 e x + 1 If x ∈ R\{1, 0, −1, −2, . . .}, then Γ(x) =

1 ζ(x)

∫ 0

∞

t x−1 dt. et − 1

△

iii) Γ has an analytic continuation to D = C\(−N). In particular, for all z ∈ D and n ≥ 1, Γ(z) =

Γ(z + n) . zn

iv) If z ∈ D, then z ∈ D and Γ(z) = Γ(z). v) If n ≥ 0, then √ √ (2n)! √ π 1 n 1 π = ( 2 ) π = n (2n − 1)!!. Γ(n + 2 ) = n 4 n! 2 √ √ √ In particular, Γ( 21 ) = π, Γ( 32 ) = 2π , and Γ( 25 ) = 3 4 π . vi) If n ≥ 0, then √ n (−4)n n! √ (−1)n √ π n 2 1 Γ( 2 − n) = π= 1 π = (−1) . n (2n)! (2n − 1)!! (2) √ √ √ In particular, Γ(− 21 ) = −2 π, Γ(− 32 ) = 4 3 π , and Γ(− 52 ) = − 815π . vii) If n ≥ 1, then Γ(n) = (n − 1)!. In particular, Γ(1) = Γ(2) = 1. viii) If n ≥ 0, then lim x↓−n Γ(x) = (−1)n ∞ and lim x↑−n Γ(x) = (−1)n+1 ∞. ix) If z ∈ D, then ∞ 1 i!iz i!iz−1 1 ∏ (1 + i )z Γ(z) = lim = lim = . i→∞ zi+1 i→∞ zi z i=1 1 + zi

998

CHAPTER 13

x) If z ∈ D, then Γ(z) = xi) limz→0 zΓ(z) = 1. xii) If z ∈ D and 1 − z ∈ D, then

∞ 1 ∏ ez/i . zeγz i=1 1 + zi

Γ(z)Γ(1 − z) =

xiii) If z ∈ D and z + 1 ∈ D, then

π . sin πz

Γ(z + 1) = zΓ(z).

In particular, Γ(− 13 ) = −3Γ( 23 ),

Γ(− 14 ) = −4Γ( 43 ),

Γ( 14 ) = 4Γ( 54 ), √ Γ( 12 ) = 2Γ( 23 ) = − 12 Γ(− 12 ) = π.

xiv) If z ∈ D and −z ∈ D, then

xv) If

1 2

+ z ∈ D and

1 2

Γ(z)Γ(−z) =

−π . z sin πz

− z ∈ D, then Γ( 12 + z)Γ( 12 − z) =

π . cos πz

In particular, Γ( 16 )Γ( 56 ) = 2π, xvi) If x ≥ 0, then

Γ( 13 ) = 3Γ( 34 ),

Γ( 41 )Γ( 34 ) =

√ 2π,

Γ( 13 )Γ( 32 ) =

√ 2 3π . 3

⌊x⌋!x x−⌊x⌋ ≤ Γ(x + 1) ≤ ⌊x⌋!(⌊x⌋ + 1) x−⌊x⌋ .

xvii)

√ Γ(x + 1)e x = 2π. x+1/2 x→∞ x lim

xviii) If a ∈ (0, 1), then Γ(a)Γ(1 − a) = xix) Let n ≥ 1. Then,

n−1 ( ) ∏ Γ ni = i=1

xx) If z ∈ D, z +

1 2

xxi) If n ≥ 1, then

π = sin πa √

∫

∞ 0

xa−1 dx. x+1

(2π)n−1 . n

∈ D, and 2z ∈ D, then

√ Γ(z)Γ(z + 21 ) = 21−2z πΓ(2z).

( ) 4n Γ(n + 12 ) 2n Γ(2n + 1) = 2 , = √ n Γ (n + 1) πΓ(n + 1) ( ) 4n Γ(n + 21 ) 1 2n Γ(2n + 1) Cn = = = √ . n+1 n Γ(n + 1)Γ(n + 2) πΓ(n + 2)

999

INFINITE SERIES, INFINITE PRODUCTS, AND SPECIAL FUNCTIONS

xxii)

∞ ∑ Γ(i + 12 ) √ = π log 4. i2 Γ(i) i=1

xxiii) If a > 0, then

∫

∞

xa−1 e−x log x dx = Γ′ (a).

0

In particular, Γ′ (1) = −γ. xxiv) If z ∈ D, then

∑ 1 d2 log Γ(z) = . 2 dz (z + i)2 i=0 ∞

xxv) Let a, b ∈ C, and assume that Re a > 0 and Re b > 0. Then, ∫ 1 Γ(a)Γ(b) xa−1 (1 − x)b−1 dx = . Γ(a + b) 0 xxvi) If z ∈ ORHP, then ) ) ∫ ∞ ( ∫ ∞ ( 1 z − 1 e−x − e−zx 1 z − 1 (x + 1)−1 − (x + 1)−z − dx = − dx log Γ(z) = x ex log(x + 1) x ex 1 − e−x 0 0 ) ∫ ∞ −zx ( e 1 1 1 = (z − 12 ) log z − z + 21 log 2π + − + x dx x 2 x e −1 0 ∫ ∞ atan xz = (z − 12 ) log z − z + 21 log 2π + 2 dx. 2πx − 1 0 e xxvii) If x ∈ (0, 1), then 1 ∑ log i sin 2πix. π i=1 i ∞

log Γ(x) =

1 2

log 2π − 12 log(2 sin πx) + 12 (γ + log 2π)(1 − 2x) +

xxviii) For all i ∈ {1, . . . , n}, let 0 < αi ≤ βi ≤ 1. Then, ∑ ∑ Γ(1 + ni=1 βi ) Γ(1 + ni=1 αi ) ≤ ≤ n!. 1 ≤ ∏n ∏ n 1 1 i=1 Γ(1 + αi ) i=1 Γ(1 + βi ) xxix) Let x > 0. Then,

Γ(x + 1) 2x + 1 < √ , Γ(x + 12 ) 4x + 3

xxx) Let n ≥ 2. Then,

( Γ

In particular,

[Γ′ (x)]2 < Γ(x)Γ′′ (x).

)∏ ) n−1 ( i 2 + 2n − 1 Γ = 2n−1 πn/2 . 2n+1 − 2 i=1 2n+1 − 2 1

Γ( 61 )Γ( 56 ) = 2π,

2 1 19 23 Γ( 30 )Γ( 17 30 )Γ( 30 )Γ( 30 ) = 8π ,

3/2 1 9 Γ( 14 )Γ( 14 )Γ( 11 , 14 ) = 4π 5/2 1 33 39 47 Γ( 62 )Γ( 62 )Γ( 35 . 62 )Γ( 62 )Γ( 62 ) = 16π

△

△

xxxi) Let n ≥ 2, define Φ(n) = {i ∈ {1, . . . , n} : gcd {n, i} = 1}, and define ϕ(n) = card Φ(n). Then, ∏ Γ( ni ) = (2π)ϕ(n)/2 . i∈Φ(n)

In particular,

3 1 3 5 9 13 Γ( 14 )Γ( 14 )Γ( 14 )Γ( 14 )Γ( 11 14 )Γ( 14 ) = (2π) .

1000

CHAPTER 13

xxxii) If z ∈ C/(−P), then

n!(n + 1)z . n→∞ zn

Γ(z + 1) = lim xxxiii) (

)1/(n log n) )1/(n2 log n) ( πn/2 1 πn/2 = 0, lim = √ , lim = 1. n→∞ Γ(n/2 + 1) e n→∞ Γ(n/2 + 1) √∏ xxxiv) limn→∞ [n − Γ( 1n )] = γ and limn→∞ 1n n ni=1 Γ( 1i ) = 1e . πn/2 lim n→∞ Γ(n/2 + 1)

)1/ log n

(

xxxv) Let x be a nonzero real number. Then, π |Γ( ȷx)|2 = , x sinh πx xxxvi) Let x be a real number. Then,

|Γ(1 + ȷx)|2 =

πx . sinh πx

π . cosh πx xxxvii) Let x and y be real numbers such that 1 + x + ȷy, 1 + x − ȷy, 1 − x + ȷy, 1 − x − ȷy ∈ D. Then, |Γ( 12 + ȷx)|2 =

Γ(1 + x + ȷy)Γ(1 + x − ȷy)Γ(1 − x + ȷy)Γ(1 − x − ȷy) =

2π2 (x2 + y2 ) . cosh 2πx − cos 2πx

xxxviii) Let x be a positive number, and let n ≥ 1. Then, n−1 nnx−1/2 ∏ Γ(x + ni ). (2π)(n−1)/2 i=0

Γ(nx) = In particular,

22x−1 Γ(2x) = √ Γ(x)Γ(x + 12 ), π xxxix)

8 ∏ i=1

Γ(3x) =

Γ( 3i )

33x−1/2 Γ(x)Γ(x + 13 )Γ(x + 32 ). 2π

√ 640 3π3 . = 6561

Source: [116, Chapter 1], [134], [208], [516, pp. 202, 206], [352, p. 41], [969], [1217, pp. 892–

898], [1350, p. 60], [1568, pp. 35, 41, 42], [1890], and [2500, Chapter 43]. xiv) is given in [1145], see Fact 5.5.14; xxx) is given in [2135]; xxxi) is given in [667]; xxxiii) is given in [78]; xxxiv) is given in [1103, p. 33]; xxxvii) is given in [1217, p. 896]. Remark: xxvii) is Kummer’s expansion. See 3 5 3/2 [116, pp. 29–32]. Remark: Combining xxx) and xxxi) yields Γ( 14 )Γ( 14 )Γ( 13 . Remark: Γ 14 ) = 2π is the Gamma function. Related: For xxix), see Fact 1.13.15. △ Fact 13.3.3. Let D = C\−N, and, for all z ∈ D, define ψ(z) =

d Γ′ (z) log Γ(z) = . dz Γ(z)

Then, the following statements hold: i) If z ∈ D\Z, then ψ(1 − z) = ψ(z) + π cot πz. ∑ 1 1 ii) Let z ∈ D and n ≥ 1. Then, ψ(z + n) = ψ(z) + n−1 i=0 z+i . In particular, ψ(z + 1) = ψ(z) + z . ( ) ∑ i iii) Let z ∈ D and n ≥ 2. Then, ψ(nz) = n1 n−1 i=0 ψ z + n + log n. iv) If z ∈ C, then limn→∞ [ψ(z + n) − log n] = 0.

1001

INFINITE SERIES, INFINITE PRODUCTS, AND SPECIAL FUNCTIONS

v) If z ∈ D, then

) ∞ ( ∑ z−1 1 1 = −γ + − . (i + 1)(i + z) i+1 i+z i=0 i=0 ∑ i i vi) If z ∈ OIUD, then ψ(z + 1) = −γ − ∞ i=1 (−1) ζ(i + 1)z . vii) If z ∈ ORHP, then ) ∫ 1 ∫ ∞( e(1−z)x 1 − xz−1 1 + dx = dx − γ = Hz−1 − γ. ψ(z) = xe x 1 − e x 1−x 0 0 ψ(z) = −γ +

∞ ∑

viii) If n ≥ 1, then ψ(n) = Hn−1 − γ. In particular, ψ(1) = −γ, ψ(2) = 1 − γ, and ψ(3) = 32 − γ. ix) If k ≥ 0 and 1 ≤ m < n, then ) ⌊(n−1)/2⌋ k−1 ∑ ∑ ( n 2miπ iπ π mπ m ψ( n − k) = +2 cos log sin − cot − log 2n − γ. n(i + 1) − m n n 2 n i=0 i=1 x) If k ≥ 0 and 1 ≤ m < n, then ψ( mn

+ k) =

k−1 ∑ i=0

) ⌊(n−1)/2⌋ ∑ ( iπ π mπ 2miπ n +2 log sin − cot − log 2n − γ. cos ni + m n n 2 n i=1

∑ 2 1 xi) If n ∈ Z, then ψ(n + 12 ) = −2 log 2 + |n| i=1 2i−1 − γ. In particular, ψ( 2 ) = −2 log 2 − γ, ψ( 23 ) = ψ(− 12 ) = 2 − 2 log 2 − γ, and ψ( 25 ) = ψ(− 32 ) = 83 − 2 log 2 − γ. √ ∑ ∑n−1 3 3 log 3 √3π 3 log 3 3π 3 1 xii) If n ≥ 0, then ψ(n+ 31 ) = n−1 − − −γ and ψ( −n) = i=0 3i+1 i=0 3i+2 − 2 − 6 −γ 2 6 3 √ √ ∑n−1 3 ∑n−1 3 3 3π 3 log 3 2 2 xiii) If n ≥ 0, then ψ(n+ 3 ) = i=0 3i+2 + 6 − 2 −γ and ψ( 3 −n) = i=0 3i+1 + 63π − 3 log 2 −γ. ∑ ∑n−1 4 4 π 1 π xiv) If n ≥ 0, then ψ(n+ 14 ) = n−1 i=0 4i+1 − 2 −3 log 2−γ and ψ( 4 −n) = i=0 4i+3 − 2 −3 log 2−γ. ∑ ∑n−1 4 4 π 3 π xv) If n ≥ 0, then ψ(n+ 43 ) = n−1 i=0 4i+3 + 2 −3 log 2−γ and ψ( 4 −n) = i=0 4i+1 + 2 −3 log 2−γ. π2 π2 π2 π2 5 ′ ′ ′ 6 −Hn−1,2 . In particular, ψ (1) = 6 , ψ (2) = 6 −1, and ψ (3) = 6 − 4 . ∑ 2 4 π2 π2 ′ 1 ′ 3 If n ≥ 0, then ψ′ (n + 21 ) = π2 − ni=1 (2i−1) 2 . In particular, ψ ( 2 ) = 2 , and ψ ( 2 ) = 2 − 4. ∑ 2 4 π2 3 1 ′ ′ If n ≥ 0, then ψ′ ( 12 − n) = π2 + ni=1 (2i−1) 2 . In particular, ψ (− 2 ) = 2 + 4, and ψ (− 2 ) = π2 40 2 + 9 . ∑ ∑n 16 16 ′ 1 2 If n ≥ 0, then ψ′ (n − 14 ) = π2 − 8G + 16 − n−1 i=0 (4i−1)2 and ψ ( 4 − n) = π + 8G + i=1 (4i−1)2 . ∑ ∑n−1 16 16 2 ′ 3 If n ≥ 0, then ψ′ (n + 14 ) = π2 + 8G − 16 − n−1 i=1 (4i+1)2 and ψ ( 4 − n) = π − 8G + i=0 (4i+1)2 .

xvi) If n ≥ 1, then ψ′ (n) = xvii) xviii) xix) xx)

Source: [1217, pp. 902–905], [138], and [2513, pp. 24–37]. Remark: ψ is the digamma function. Remark: Hz,p is the generalized harmonic function. See Fact 13.3.4. Related: Fact 13.5.71. Fact 13.3.4. For all z ∈ C such that Re z > −1, define △

∫

1

Hz = 0

1 − xz dx. 1−x

Then, the following statements hold: ∑ i) If n ≥ 1, then Hn = ni=1 1i . ii) If z ∈ C and Re z > −1, then Hz = ψ(z + 1) + γ. iii) If p > 0, then H p = H p−1 + 1p . iv) If p > −1 and p is not an integer, then H p − H1−p =

1 p

−

1 1−p

− π cot pπ.

1002

CHAPTER 13

v) If p > − 21 , then H2p = 21 (H p + H p− 21 ) + log 2. vi) If p > − 23 , then H3p = 31 (H p + H p− 31 + H p− 23 ) + log 3. vii) If p > 0, then ∞ ∑ 1 Hp = p . i(p + i) i=1 viii) If p > 0 and n ≥ 1, then

∫

n

H x dx = nγ + log n!.

0 △

Next, define H p,1 = H p , and, for all p > −1 and n ≥ 1, define △

H p,n+1 = ζ(n + 1) −

1 d H p,n . n dp

Then, the following statements hold: ∑ ix) If n ≥ 1 and k ≥ 1, then Hn,k = ni=1 i1k . x) For all n ≥ 1, H p,n+1 = ζ(n + 1) + (−1)n+1 In particular, π2 + 6

H p,2 = H p,4 =

π 1 + 90 6 4

∫

1 0

∫

1 0

x p log x dx, 1−x p

3

x log x dx, 1−x

1 n!

∫

1 0

x p logn x dx. 1−x

H p,3 = ζ(3) −

1 2

H p,5 = ζ(5) −

∫

1 0

1 24

∫

x p log2 x dx, 1−x 1 0

x p log4 x dx. 1−x

xi) Let n ≥ 1 and z ∈ C, and assume that Re z > −1. Then, Hz,n+1 = ζ(n + 1) + (−1)n

1 (n) ψ (z + 1). n!

π2 5π2 ′ 1 6 − ψ (z + 1). In particular, H 4 ,2 = 16 − 8G − 6 , 2 5π2 H 21 ,2 = 4 − π3 , and H 34 ,2 = 8G + 16 9 − 6 . If z ∈ C and Re z > −1, then Hz,3 = ζ(3) + 21 ψ′′ (z + 1). In particular, H 14 ,3 = 64 − π2 − 27ζ(3), 3 H 12 ,3 = 8 − 6ζ(3), and H 43 ,3 = 64 27 + π − 27ζ(3).

xii) If z ∈ C and Re z > −1, then Hz,2 = xiii)

xiv) If z ∈ C and Re z > −1, then Hz,4 = ζ(4) + 12 ψ′′′ (z + 1). In particular, H 21 ,4 = 16 − 7π 45 . Remark: Hz,p is the generalized harmonic function. Related: Fact 13.3.3 and Fact 14.2.7. Fact 13.3.5. For all z ∈ CIUD, define ∫ 1 z log x △ dx. Li2 (z) = 0 zx − 1 4

Then, the following statements hold: i) The integral exists for all z ∈ CIUD . ii) Let z ∈ CIUD . Then, Li2 (z) = and the series converges absolutely.

∞ ∑ zi , i2 i=1

1003

INFINITE SERIES, INFINITE PRODUCTS, AND SPECIAL FUNCTIONS △

iii) Li2 has an analytic continuation on D = C\[1, ∞). In particular, for all z ∈ C\[1, ∞), ∫ z log(1 − t) Li2 (z) = − dt. t 0 where the integral is along an arbitrary path from 0 to z and contained in C\[1, ∞). If the path is the line segment connecting 0 to z, then ∫ 1 log(1 − tz) dt. Li2 (z) = − t 0 iv) Let x ∈ [0, 1]. Then,

∫

x

Li2 (x) = − 0

log(1 − t) dt = −(log x) log(1 − x) + t

∫

x 0

log t dt. t−1

v) Let x ∈ (0, 1). Then, π2 − (log x) log(1 − x), Li2 (x) + Li2 (1 − x) = 6 ( x ) = − 21 log2 (1 − x). Li2 (x) + Li2 x−1

Li2 (x) + Li2 (−x) = 21 Li2 (x2 ),

π π vi) Li2 (−1) = − 12 , Li2 (0) = 0, Li2 ( 21 ) = 12 − 12 log2 2, √ √ △ △ △ vii) Let α = 21 ( 5 + 1) and β = 1/α = 12 ( 5 − 1). Then, 2

2

Li2 (−α) = −

Li2 (1) =

π2 6.

π2 − log2 β, 10 π2 π2 1 Li2 (1 − β) = − log2 β = − log2 (1 − β). 15 15 4

π2 − log2 α, 10

Li2 (β) =

π2 1 + log2 β, 15 2 viii) Let x ∈ [−1, 1]. Then, ∫ 1 π2 Li2 (t) Li2 (x) log2 (1 − x) dt = − − . 2 6(1 − x) x 2x 0 (1 − xt) Li2 (−β) = −

In particular,

∫

1 0

∫

1 Li2 (t) dt = log2 2, 2 2 (1 + t)

0

1

π2 Li2 (t) dt = . 24 (2 − t)2

Source: [116, pp. 102–106], [324], [1158, p. 179], and [2513, pp. 175–179]. Remark: Li2 is the dilogarithm function. Related: Fact 13.5.35. Fact 13.3.6. For all x ∈ (0, 1), the series △

L(x) =

∞ ∑ (−1)i i=0

1 (2i + 1) x

converges. In addition, L can be extended to C by analytic continuation. Furthermore, for all z ∈ C, ( )z ( 2 πz ) L(1 − z) = sin Γ(z)L(z). π 2 Finally, 1 L(0) = , 2

π L(1) = , 4

′

L (0) =

log Γ( 41 )

−

log Γ( 34 )

− log 2,

πγ π L (1) = + log 4 2 ′

Source: [2783]. Remark: L is the Dirichlet L function. Related: Fact 14.6.23.

√

2πΓ( 43 ) Γ( 14 )

.

1004

CHAPTER 13

13.4 Facts on Power Series, Laurent Series, and Partial Fraction Expansions Fact 13.4.1. Consider the power series △

f (z) =

∞ ∑ i=1

zi ,

△

g(z) =

∞ i ∑ z i=1

i

,

△

h(z) =

∞ ∑ zi . i2 i=1

Then, the following statements hold: i) The radius of convergence of f, g, and h is 1. ii) The domain of convergence of f is {z ∈ C : |z| < 1}. iii) The domain of convergence of g is {z ∈ C : |z| ≤ 1}\{1}. iv) The domain of convergence of h is {z ∈ C : |z| ≤ 1}. Source: [2113, p. 113]. Fact 13.4.2. Let z ∈ C. Then, the following statements hold: i) Let α ∈ C, and assume that one of the following conditions is satisfied: a) |z| < 1. b) z = −1 and α ∈ ORHP ∪{0}. c) z ∈ UC \{−1} and Re α > −1. d) |z| > 1 and α ∈ N. Then, ∞ ∑ αi i (1 + z)α = z. i! i=0 Furthermore, as z → 0, ( ) ( ) ( ) ( ) ( ) α α α 2 α 3 α 4 (1 + z)α = + z+ z + z + z + O(z5 ) 0 1 2 3 4 α(α − 1) 2 α(α − 1)(α − 2) 3 α(α − 1)(α − 2)(α − 3) 4 = 1 + αz + z + z + z + O(z5 ). 2! 3! 4! ii) Let α ∈ C. For all |z| < 1 and, as z → 0, ( ) ( ) ( ) ( ) ( ) ∞ ∑ 1 αi i α−1 α α+1 2 α+2 3 α+3 4 = z = + z+ z + z + z + O(z5 ) (1 − z)α i! 0 1 2 3 4 i=0 (α + 1)α 2 (α + 2)(α + 1)α 3 (α + 3)(α + 2)(α + 1)α 4 z + z + z + O(z5 ). 2! 3! 4! iii) For all |z| < 1 and, as z → 0, ∞ ∞ ∑ ∑ 1 1 = zi = 1+z+z2 +z3 +z4 +O(z5 ), = (−1)i zi = 1−z+z2 −z3 +z4 −O(z5 ). 1 − z i=0 1 + z i=0 = 1 + αz +

iv) For all |z| < 1 and, as z → 0, ∞ ∑ 1 = (i + 1)zi = 1 + 2z + 3z2 + 4z3 + 5z4 + O(z5 ), (1 − z)2 i=0 ∑ 1 = (−1)i (i + 1)zi = 1 − 2z + 3z2 − 4z3 + 5z4 − O(z5 ). 2 (1 + z) i=0 ∞

INFINITE SERIES, INFINITE PRODUCTS, AND SPECIAL FUNCTIONS

v) Let k ≥ 1. Then, for all |z| < 1, ) ( ) ∞ ( ∞ ∑ ∑ i+k−1 i 1 1 i i+k−1 i z , z, = = (−1) i i (1 − z)k (1 + z)k i=0 i=0 ) ) ∞ ( ∞ ( ) ∞ ( ∑ ∑ ∑ 1 k+i i zk i i (k + 1)z k+i i = z, = z, = i z. i k i (1 − z)k+1 (1 − z)k+1 (1 − z)k+2 i=0 i=1 i=k vi) For all |z| < 1 and, as z → 0, ∑ z = zi = z + z2 + z3 + z4 + z5 + O(z6 ), 1 − z i=1 ∞

∑ z = (−1)i+1 zi = z − z2 + z3 − z4 + z5 − O(z6 ), 1 + z i=1 ∞

∑ z = izi = z + 2z2 + 3z3 + 4z4 + 5z5 + O(z6 ), (1 − z)2 i=1 ∞

∑ z = (−1)i+1 izi = z − 2z2 + 3z3 − 4z4 + 5z5 − O(z6 ), (1 + z)2 i=1 ∞

∑ z i 2 3 4 5 6 1 = 2 i(i + 1)z = z + 3z + 6z + 10z + 15z + O(z ), 3 (1 − z) i=1 ∞

∑ z = (−1)i+1 12 i(i + 1)zi = z − 3z2 + 6z3 − 10z4 + 15z5 − O(z6 ), (1 + z)3 i=1 ∞

z(z + 1) ∑ 2 i = i z = z + 4z2 + 9z3 + 16z4 + O(z5 ). (1 − z)2 i=i ∞

vii) For all |z| > 1 and, as z → ∞, ∑1 1 1 1 1 z = = 1 + + 2 + 3 + 4 + O(z−5 ), i z − 1 i=0 z z z z z ∞

∑ z 1 1 1 1 1 = (−1)i i = 1 − + 2 − 3 + 4 − O(z−5 ). z + 1 i=0 z z z z z ∞

viii) For all |z| > 1 and, as z → ∞, ∞ ∑ i+1 4 5 2 3 z2 = = 1 + + 2 + 3 + 4 + O(z−5 ), i z z (z − 1)2 z z z i=0 ∑ z2 2 3 i+1 4 5 = (−1)i i = 1 − + 2 − 3 + 4 − O(z−5 ). 2 z z z (z + 1) z z i=0 ∞

ix) For all |z| > 1 and, as z → ∞, ∑ i 3 4 5 1 2 z = = + 2 + 3 + 4 + 5 + O(z−6 ), i z z (z − 1)2 z z z z i=1 ∞

∑ z i 1 2 3 4 5 = (−1)i+1 i = − 2 + 3 − 4 + 5 − O(z−6 ). 2 z z z (z + 1) z z z i=1 ∞

1005

1006

CHAPTER 13

x) Let a, b ∈ C. Then, for all |z| < min {1/|a|, 1/|b|} and, as z → 0, ∞ ∑ 1 + bz =1+ (−1)i ai−1 (a − b)zi = 1 + (b − a)z + a(a − b)z2 + a2 (b − a)z3 + O(z4 ). 1 + az i=1 xi) Let n ≥ 1. Then, as z → 0, )n ( 1 + bz = 1 + n(b − a)z + 21 n(a − b)[(n + 1)a − (n − 1)b]z2 1 + az + [na2 (b − a) + 16 n(n − 1)(n − 2)(b − a)3 − n(n − 1)a(a − b)2 ]z3 + O(z4 ). In particular, as z → 0, )2 ( 1 + bz = 1 + 2(b − a)z + (a − b)(3a − b)z2 + 2a(b − a)(2a − b)z3 + O(z4 ). 1 + az xii) For all |z| < 1 and, as z → 0, ( )3 ∞ ∑ 1+z =1+ (4i2 + 2)zi = 1 + 6z + 18z2 + 38z3 + 66z4 + 102z5 + O(z6 ). 1−z i=1 xiii) For all |z| < 1 and, as z → 0,

⌋ ∞ ⌊ ∑ z3 z3 + z4 i−1 i = = z = z3 + z4 + 2z5 + 2z6 + O(z7 ). 2 (1 − z)2 (1 + z) (1 − z2 )2 i=3

xiv) For all |z| < 1, ∞ ∑ √ z+1=1+ (−1)i+1 i=1

1 = √ 1−z

∞ ( 1) ∑ − 2

i=0

i

(2i)! zi , 4i (i!)2 (2i − 1)

(−z)i =

∞ ∑ (2i − 1)!! i=0

(2i)!!

xv) For all |z| < 41 , √ xvi) For all |z|
1 and, as z → 0, ∞ ∑ 11 11 11 11 11 1 z = = z + 2 + 3 + 4 + 5 + 6 + O(z−7 ). log i z − 1 i=1 iz 2z 3z 4z 5z 6z v) Let a, b ∈ C. For all z ∈ C such that |z| ≤ min {1/|a|, 1/|b|} and z < {−1/a, −1/b} and, as z → 0, ∞ 1 + bz ∑ 1 1 ai − bi i log = z = (b − a)z + (a2 − b2 )z2 + (b3 − a3 )z3 + O(z4 ). (−1)i 1 + az i=1 i 2 3 In particular, if z ∈ CIUD\{−1, 1} and, as z → 0, 1 + z ∑ 2 2i+1 2 2 2 2 = z = 2z + z3 + z5 + z7 + z9 + O(z11 ). 1 − z i=0 2i + 1 3 5 7 9 ∞

log

vi) For all z ∈ C such that Re z > 0, log z =

∞ ∑ i=0

( )2i+1 2 z−1 . 2i + 1 z + 1

vii) For all z ∈ C such that |z| < 1 and, as z → 0, ∞ ∑ 1 1 1 1 1 1 i 1 = z = z + z2 + z3 + z4 + z5 + z6 + O(z7 ). log 1 − z i=1 i 2 3 4 5 6 viii) For all z ∈ C such that |z| < 1 and, as z → 0, ∞ log(1 − z) ∑ 3 11 25 137 5 49 6 = Hi zi = z + z2 + z3 + z4 + z + z + O(z7 ). z−1 2 6 12 60 20 i=1 ix) For all z ∈ C such that |z| < 1 and, as z → 0, ∞ ∑ Hi i+1 11 5 136 6 7 7 log2 (1 − z) = 2 z = z2 + z3 + z4 + z5 + z + z + O(z8 ). i + 1 12 6 180 10 i=1 x) For all z ∈ C such that |z| < 1 and, as z → 0,  i  ∞ ∑ 3 7 15 29 1 ∑ H j  i+2 3   z = −z3 − z4 − z5 − z6 − z7 − O(z8 ). log (1 − z) = −6 i + 2 j + 1 2 4 8 15 j=1 i=1 xi) For all z ∈ C such that |z| < 41 and, as z → 0, √ ( ) ( ) ( ) ∞ 1 − 1 − 4z 2 1 ∑ 1 2i i log = log = z √ 2z 2 i=1 i i 1 + 1 − 4z

1019

INFINITE SERIES, INFINITE PRODUCTS, AND SPECIAL FUNCTIONS

3 10 35 126 5 1716 7 = z + z2 + z3 + z4 + z + 77z6 + z + O(z8 ). 2 3 4 5 7 xii) For all z ∈ C such that |z| < 1 and, as z → 0, [ ] ∞ i ∑ 1∑ 1 1 1 19 4 3 5 z i i i− j 1 = z = 1 + z − z2 + z3 − z + z − O(z6 ). (−1) j log(1 + z) i=0 i! j=0 j+1 2 12 24 720 160 xiii) For all z ∈ C such that |z| < 1 and, as z → 0, ( ) ∞ ∑ 1 2i 5 47 6 319 8 1 Hi − H2i − z = −z2 − z4 − z − z − O(z10 ). [log(1 + z)] log(1 − z) = i 2i 12 180 1680 i=1 Source: [1524, pp. 11, 12]. For x ∈ R such that |x| < 1, it follows that

d −1 log(1 − x) = = −[1 + x + x2 + x3 + x4 + O(x5 )]. dx 1−x (

) 1 2 1 3 4 log(1 − x) = − x + x + x + O(x ) , 2 3

Integrating yields

which yields ii). xii) is given in [2276]. Remark: iii) is Mercator’s series, while iv) and v) are equivalent forms of Gregory’s series. See [1391, p. 273]. Remark: viii) is the generating function for the harmonic numbers. Related: Setting a = β and b = α in v) yields a series involving the Fibonacci numbers. See Fact 13.9.3. Fact 13.4.15. The following statements hold: i) For all A ∈ Fn×n, sin A = A − 3!1 A3 + 5!1 A5 − 7!1 A7 + · · · . ii) For all A ∈ Fn×n,

cos A = I −

1 2 2! A

+

1 4 4! A

−

1 6 6! A

+ ··· .

iii) For all A ∈ Fn×n such that ρmax (A) < π2 , tan A = A + 31 A3 +

2 5 15 A

+

17 7 315 A

+

9 62 2835 A

+ ··· .

iv) For all A ∈ Fn×n , then eA = I + A +

1 2 2! A

+

1 3 3! A

+

1 4 4! A

+ ··· .

v) For all A ∈ Fn×n such that ρmax (A − I) < 1, log A = −[I − A + 12 (I − A)2 + 31 (I − A)3 + 41 (I − A)4 + · · · ]. vi) For all A ∈ Fn×n such that ρmax (A) < 1, log(I − A) = −(A + 21 A2 + 13 A3 + 41 A4 + · · · ). vii) For all A ∈ Fn×n such that ρmax (A) < 1, log(I + A) = A − 12 A2 + 31 A3 − 14 A4 + · · · . viii) For all A ∈ Fn×n such that spec(A) ⊂ ORHP, log A =

∞ ∑ i=0

2 [(A − I)(A + I)−1 ]2i+1 . 2i + 1

1020

CHAPTER 13

ix) For all A ∈ Fn×n, x) For all A ∈ F xi) For all A ∈ F

sinh A = sin ȷA = A +

1 3 3! A

+

1 5 5! A

+

1 7 7! A

+ ··· .

cosh A = cos ȷA = I +

1 2 2! A

+

1 4 4! A

+

1 6 6! A

+ ··· .

,

n×n

n×n

such that ρmax (A)
1, then

∑ z2 z . = 1 − z i=0 1 − z2i+1 ∞

i

i

∑ z2 1 = . 1 − z i=0 1 − z2i+1 ∞

i

Source: [1647, p. 267]. Fact 13.5.2. Let n ≥ 1, let z ∈ C, and assume that |z| < 1. Then, ∞ ∑ i=1

i

4i zn2 4z2n = . (1 + zn2i )2 (1 − z2n )2

Source: [2150]. Related: Fact 2.1.14. Fact 13.5.3. Let n ≥ 1 and z ∈ OIUD . Then,

∑ zn zi = α , n,i (1 − z)n 1 − zi i=n ∞

where △

αn,m = card {(k1 , . . . , km ) ∈ Pn : k1 ≤ · · · ≤ km , gcd {k1 , . . . , km } = 1, and Source: [771, p. 123]. Remark: α3,2 = 1, α5,3 = 2, α7,3 = 3, and α10,2 = 2. Fact 13.5.4. Let z ∈ OIUD . Then, ∞ ∑ i=1

∑ zi z2i+1 = (−1)i . 2i 1+z 1 − z2i+1 i=0 ∞

Source: [2013, p. 130]. Fact 13.5.5. Let z ∈ OIUD . Then,

 ∞ 4 ∞ ∑  ∑  i i2   (−1) z  = 1 + 8 i=−∞

i=1

zi . [zi + (−1)i ]2

Source: [117, 2970]. Related: Fact 1.11.25 and Fact 2.1.17.

∑m

i=1 ki

= n}.

1022

CHAPTER 13 △

Fact 13.5.6. For all n ≥ 1 and z ∈ C, define gn (z) = 2 ∞ ∞ ∏ ∑ 1 zi = , 5i−4 )(1 − z5i−1 ) g (z) (1 − z i=1 i=0 i

∏n

i=1 (1

− zi ). Then, for all z ∈ OIUD,

2 ∞ ∞ ∑ 1 zi +i ∏ = . 5i−3 )(1 − z5i−2 ) g (z) (1 − z i=1 i=0 i

Source: [2970]. Remark: These are the Rogers-Ramanujan identities. Related: Fact 2.1.19 and

Fact 13.10.27. Fact 13.5.7. Let z ∈ C, and assume that z < {0, −1, −1/2, −1/3, . . .}. Then, ∞ ∑

1 i = . z j=1 ( jz + 1)

∏i

i=1

In particular,

∞ ∑ i=1

i = 1. (i + 1)!

Source: [1647, p. 267]. ∑∞ ∑∞ ∞ Fact 13.5.8. Let (xi )∞ i=1 yi are convergent, i=1 xi and i=1 ⊂ R and (yi )i=1 ⊂ R, assume that

assume that at least one of these series is absolutely convergent, and define (zi )∞ i=1 , where, for all ∑ △ ∑ z converges. Furthermore, i ≥ 1, zi = ij=1 x j yi− j+1 . Then, ∞ i=1 i ∞  ∞ ∞ ∑ ∑  ∑ yi . xi  zi =  i=1

i=1

i=1

Source: [1566, p. 102]. Remark: This is Mertens’s theorem. The series

∑∞

∑∞

i=1 xi and i=1 yi . Fact 13.5.9. Let (xi )∞ i=1 ⊂ (0, ∞). Then,

product of

∞ ∑

i

∑i

xj

j=1

i=1

≤4

∞ ∑ 1 . x i=1 i

Source: [506, pp. 177, 178]. Related: Fact 2.11.45. Fact 13.5.10. Let (xi )∞ i=1 ⊂ (0, ∞). If, for all i ∈ P, xi+1 < xi , then ∞ ∑ 1 (xi − xi+1 ) = ∞. x i=1 i

Alternatively, if, for all i ∈ P, xi < xi+1 , then ∞ ∑ 1 (xi+1 − xi ) = ∞. x i=1 i+1

Source: [2801]. Fact 13.5.11. Let (xi )∞ i=1 ⊂ R. Then, the following statements hold:

i) Assume that

∏∞

i=1 (1

+ xi ) converges to α ∈ (0, ∞). Then, ∞ ∑

ii) Assume that limn→∞

∏n

i=1 (1

xi 1 =1− . α j=1 (1 + x j )

∏i

i=1

+ xi ) = ±∞. Then, ∞ ∑ i=1

xi = 1. (1 + x j) j=1

∏i

∑∞

i=1 zi

is the Cauchy

1023

INFINITE SERIES, INFINITE PRODUCTS, AND SPECIAL FUNCTIONS

Source: [1566, p. 65]. ∑∞ Fact 13.5.12. Let (xi )∞ i=1 i=1 ⊂ (0, ∞), and assume that ∞ ∑

∑i

1

j=1

i=1

xj

≤2

1 xi

converges. Then,

∞ ∑ 1 . x i=1 i

Source: [2294, p. 93]. Fact 13.5.13. Let x > 2. Then, ∞ ∑

1 i! = . x − 2 j=0 (x + j)

∏n−i

i=1

Source: [1566, p. 65]. Fact 13.5.14. ∞ ∑ 1 = e, i! i=0

∞ ∑ 2i i2 i=1

i!

∞ ∑

∞ ∑ 1 4i (−1)i = 4 , i! e i=0 ∞ ∑ i=0

∞ ∑ 1 1 (−1)i = , i! e i=0

= 6e2 ,

i=1

2

∞ ∑

√ 4i + 3 = 2 e, i 4 (2i + 1)! i=0 ( ) ∞ ∑ 1 1 1 = e− , (2i + 1)! 2 e i=0

(3i)!

i=0

∞ ∑

= e,

∞ ∑ i=1

i = 1, (i + 1)!

∞ ∑

1 1 1 i = , (−1)i i = √4 , (2i + 1)! 2e i!4 e i=1 i=0 ( ) ( ) ∞ ∞ ∑ 1 ∑ i 1 1 1 1 = e+ , = e− , (2i)! 2 e (2i)! 4 e i=0 i=1

i = 5e, (i − 1)!

2 = e, (i4 + i2 + 1)i!

∞ ∑ 9i2 + 1

∞ ∞ ∑ ∑ (i + 1)3 1 9i2 − 6i + 1 1 (−1)i+1 = , (−1)i = , i! e (3i)! e i=0 i=0 ( ) ∞ ∞ ∑ ∑ 1 3 i2 1 = e− , = 3 − e. (2i + 1)! 8 e i(i + 1)(i + 1)! i=1 i=1

Source: [506, pp. 112, 129, 130, 140], [579], [1217, p. 13], and [1566, pp. 49, 64]. Fact 13.5.15. Let 0 ≤ m < n. Then, ∞ ∑ i=1

n 1 ∑ 1 1 = , (i + m)(i + n) n − m i=m+1 i

∞ ∑ i=1

Hn 1 = . i(i + n) n

For example, ∞ ∑ i=1 ∞ ∑ i=1

∞ ∑

1 = 1, i(i + 1)

i=1 ∞ ∑

25 1 = , i(i + 4) 48

i=1

1 3 = , i(i + 2) 4

1 137 = , i(i + 5) 300

∞ ∑

1 11 = , i(i + 3) 18

i=1 ∞ ∑ i=1

1 49 = . i(i + 6) 120

Source: [1217, p. 12]. Remark: The second equality follows from the penultimate equality in ∑ ∑∞ 1 1 n−1 1 Fact 13.7.8 with n = 1. Remark: n−1 i=n i(i+1) = n . i=1 i(i+1) = n implies Fact 13.5.16. ∞ ∑ Hi+1 = 2, i(i + 1) i=1 ∞ ∑ i=1

∞ ∑ i=1

Hi π2 = , i(i + 1) 6

Hi π 7 = + , i(i + 3) 18 12 2

∞ ∑ i=1

∞ ∑ i=1

Hi = 1, (i + 1)(i + 2)

Hi π 85 = + , i(i + 4) 24 144 2

∞ ∑ i=1

∞ ∑ i=1

Hi π2 1 = + , i(i + 2) 12 2

Hi π2 83 = + , i(i + 5) 30 144

1024

CHAPTER 13

∑ (2i + 1)Hi 1 ∑ Hi ∑ Hi = = = ζ(3), 2 i=1 i2 (i + 1)2 i2 (i + 1)2 i=1 i=1 ∞

∞

∞ ∑ i=1

∞ ( ∑ Hi )2 i=1 ∞ ∑

i

17π , 360

∞ ∑

i=1

i=1

π Hi Hi+1 = 2ζ(3) + , i(i + 1) 6

i=1 ∞ ∑ i=1 ∞ ∑ i=1

∞ ∑ i=1

Hi,2 59 5π π = − + , 12 120 (i + 3)2 16 ∞ ∑

Hi,2 = ζ(3), i(i + 1) ( 21 )i+1 Hi

(i + 1)(i + 1)!

=

i=1

4

i=1

∞ ∑ i=1

∞ ∑ i=1

Hi,2 π4 π2 + , =3− 2 3 120 (i + 2)

Hi,2 1717 49π2 π4 = − + , 432 108 120 (i + 4)2

Hi,2 1 π2 1 = ζ(3) + − , i(i + 2) 2 12 2

π + 2 log2 2, 6

∞ ∑

∞ ∑ Hi π6 1 = − ζ 2 (3), 5 540 2 i i=1

∞ ∑ Hi2 97π6 − 2ζ 2 (3), = 4 22680 i i=1

Hi,2 π4 , = 2 120 (i + 1) 2

2

( 12 )i+1 Hi,2 (i + 1)(i + 1)!

∞ ∑ i=1

=

i=1

Hi2 π2 = + 1, (i + 1)(i + 2) 6

∞ ∑ H2i ζ(2i) π2 = , i(2i + 1) 6 i=1

∞ ∑ π2 Hi = 3ζ(5) − ζ(3), 4 6 i i=1

∞ ∑ Hi,2 7π4 , = 2 360 i i=1

∞ ∑

∞ ∑ π2 5 Hi Hi+2 = ζ(3) + + , i(i + 2) 8 4 i=1

2

∞ ∑ Hi2 7 π2 ζ(5) − ζ(3), = 2 6 i3 i=1

∞ ∑

∞ ( ∑ Hi )2 11π4 = 2ζ(3) − 3 + , i+2 360 i=1

π2 H2i = − log2 2 + 2 log 2, i(2i − 1) 6

∞ ∑ Hi π4 = , 3 72 i i=1

i=1

Hi2 3 π2 1 = ζ(3) + + , i(i + 2) 2 12 2

i=1 ∞ ∑

∞ ∑ π4 Hi Hi+1 = , 2 30 (i + 1) i=1 ∞ ∑

i=1

π2 Hi = ζ(3) + − 2, 6 (i + 2)2

11π2 395 Hi = ζ(3) + − , 2 36 108 (i + 4)

∞ ( ∑ Hi )2 11π4 = , i+1 360 i=1

4

Hi2 = 3ζ(3), i(i + 1)

i=1

∞ ∑

π2 Hi = ζ(3) + − 3, 2 4 (i + 3) =

∞ ∑

∞

( 12 )i+1 Hi2 (i + 1)(i + 1)!

∞ ∑ i=1

Hi,2 π2 = − 1, (i + 1)(i + 2) 6

7 4 = π2 log 2 + ζ(3) + log3 2, 2 3

π2 1 4 log 2 − ζ(3) − log3 2. 3 2 3

Let n ≥ 2. Then,

  n−2 ∞ ∑ ∑  Hi 1  = (n + 2)ζ(n + 1) + ζ(n − i)ζ(i + 1) , n i 2 i=1 i=1 ∞ ∑ i=1

Let p > 1. Then,

∞ ∑ Hi,n 1 2 = 2 [ζ (n) + ζ(2n)], in i=1

Hi 1 = . (i + 1)n (n − 1)!(n − 1)2 ∞ ( ∑ Hi ) p i=1

i

≤

p ζ(p). p−1

Source: [506, pp. 126, 134, 135], [511, p. 78], [515, 650, 1101, 2220, 2483], [2513, p. 354], and [2859]. Related: Fact 13.5.41.

1025

INFINITE SERIES, INFINITE PRODUCTS, AND SPECIAL FUNCTIONS

Fact 13.5.17. Let n ≥ 2. Then, ∞ ∑ n n Hi = log , i n n − 1 n − 1 i=1

Furthermore, ∞ ∑ Hi π2 = , 2i i 12 i=1 ∞ ∑ i=1

∞ ∑ Hi = 2 log 2, 2i i=1

∞ ∑ Hi+1 π2 = + 1 − log 2, 2i i 12 i=1

Hi = log2 2, 2i (i + 1)

∞ ∑ i=1

∞ ∑ 3 Hi 3 = log . i 3 2 2 i=1

∞ ∑ Hi+2 π2 27 5 = + − log 2, 2i i 12 12 2 i=1 ∞ ∑ Hi2 π2 = log2 2 + , i 2 6 i=1

Hi = 2 log2 2 + 2 log 2 − 2, 2i (i + 2)

∞ ∑ Hi π2 log 2 , = ζ(3) − 12 2i i2 i=1

∞ ∑ Hi,2 π6 2 , = ζ (3) − 2835 i4 i=1

∞ ∑ Hi,2 5 = ζ(3). 2i i 8 i=1

Source: [2483, 2859]. Fact 13.5.18.

√ √ ( ) ( ) ∞ ∞ ∑ ∑ √ √ 1+ 2 2 2 Hi 2i i+1 Hi 2i = 2 2 log , (−1) = 2 log √ , 8i i 2 4i i 1+ 2 i=0 i=0 2 ( ) ( ) ∞  ∞ ∞ ∑ ∑ ∑ H2i − 12 Hi 2i  H2i − 21 Hi  π4 π2 Hi+1 2i   = = 3, , = ,   4i (i + 1) i i 32 4i i i 4 i=1 i=1 i=1 ( ) ( ) ∞ ∞ ∑ ∑ i(H2i − 21 Hi ) 2i 2i Hi π log 2 = 2, , = i (2i − 1) i i (2i − 1)2 4 2 i 4 i=1 i=1  i  i (  i ( ) ∑ ) ∞ ∑  i Hi ∑ 1  ∏ 1 1  ∏ 3  3 1− 1− −      = π , i 4 j − 1 4 j 4 j − 3 4 j i=1 j=1 j=1 j=1 j=1    2 ( ) i ∞ i  π4 ∑ Hi 2i ∑ 1  ∑ 1  = ,   −      i i i  2  4 2 j − 1 4 (2 j − 1) j=1 i=1 j=1 √   ( ) ∞ ∑ √   2 2 Hi+1 2i  , − 1 = 5 + 4 2 log (−1)i+1 i √ 4 (i + 1) i 1+ 2 i=1 ∞ ∞ ∑ ∑ √ √ √ Ci Hi Ci Hi = 4 log 2, (−1)i+1 i+1 = (1 + 3 2/2) log 2 − (1 + 2) log(1 + 2). i 4 4 i=0 i=0

( ) ∞ ∑ Hi 2i π2 = , 4i i i 3 i=1

Source: [62]. Fact 13.5.19. Let n ≥ 1. Then,

lim (Hni − Hi ) = log n.

i→∞

In particular,

log 2 = lim (H2i − Hi ) =

1 1

−

1 2

+

1 3

−

1 4

+

1 5

−

1 6

+ ··· ,

log 3 = lim (H3i − Hi ) =

1 1

+

1 2

−

2 3

+

1 4

+

1 5

−

2 6

+ ··· .

i→∞

i→∞

Source: [506, p. 125], [1158, pp. 153, 154], and [1566, p. 48].

1026

CHAPTER 13

Fact 13.5.20. Let n ≥ 2. Then, ∞ ( ∑ i=1

) n−1 n−1 1 π ∑ n−1 iπ = − log n − 2 log n + Hi − Hni − i cot . 2in 2n 2 n 2n i=1

In particular, ) ∞ ( ∑ 1 1 1 log 2 + Hi − H2i − = − log 2, 4i 4 2 i=1

√ ) ∞ ( ∑ 3π 1 1 1 log 3 + Hi − H3i − = − log 3 + . 3i 3 2 54 i=1

Source: [1103, pp. 150, 214–216]. Fact 13.5.21. Let n ≥ 2. Then, ∞ ∑ 1 i=1

i

(n − 1)(n + 2) 2 1 1 ∑ 2( iπ ) π − log2 n − . log 2 sin 24n 2 2 i=1 n n−1

(log n + Hi − Hni ) =

In particular, ∞ ∑ 1 i=1

i

(log 2 + Hi − H2i ) =

∞ ∑ 1

π2 − log2 2, 12

i

5π2 3 − log2 3, 36 4

√ 5 7π2 5 1 2 2 1+ (log 5+Hi −H5i ) = − log 5− log . i 30 8 2 2

∞ ∑ 1

∞ ∑ 1

i=1

i=1

3π2 11 (log 4+Hi −H4i ) = − log2 2, i 16 4

i=1

(log 3 + Hi − H3i ) =

Source: [1682]. Fact 13.5.22. Let n ≥ 2. Then, n ∞ ∑ 1 π ∑ iπ n−1 log 2 + log n − cot (−1)i+1 (log n + Hi − Hni ) = 2n 2 4n 2n i=1 i=1

=

⌊n/2⌋ 1 π ∑ n−1 (2i − 1)π log 2 + log n − 2 . (n + 1 − 2i) cot 2n 2 2n 2n i=1

In particular, ∞ ∑ 3 π (−1)i−1 (log 2 + Hi − H2i ) = log 2 − , 4 8 i=1

√ ∞ ∑ 3π 1 1 (−1)i−1 (log 3 + Hi − H3i ) = log 2 + log 3 − , 3 2 9 i=1

∞ ∑ √ π 11 log 2 − (1 + 2 2) . (−1)i−1 (log 4 + Hi − H4i ) = 8 16 i=1

Furthermore,

∞ ∑ π2 π 7 G (−1)i−1 (log 2 + Hi − H2i )2 = − log 2 − log2 2 + . 48 8 8 2 i=1

Source: [1682, 1683]. The last equality is given in [1103, pp. 146, 196–199]. Fact 13.5.23. ∞

∑ π2 1 (log 2 + Hi − H2i )2 = + log 2 − log2 2. 48 2 i=1

Source: [1103, pp. 143, 184–186].

1027

INFINITE SERIES, INFINITE PRODUCTS, AND SPECIAL FUNCTIONS

Fact 13.5.24.

∞ ∞ ∑ ∑  1  (−1) j+1 i+ j=1

i=0

∞ 2 ∞ ∑ ∑ 1  π2 j+1 i   = (−1)  (−1) . i+ j 24 j=1 i=0

2   = log 2, j

Source: [1103, pp. 143, 146, 186, 199, 200]. Fact 13.5.25. Let n ≥ 2. Then,

∫ ∞ ∑ i+1 (−1) (Hni − Hn(i−1) ) =

1 0

i=1

In particular,

⌊n/2⌋ xn − 1 log 2 π ∑ (2i − 1)π dx = − . (n + 1 − 2i) cot n 2 (x + 1)(x − 1) n 2n n i=1

∫ ∞ ∑ i+1 (−1) (Hi − Hi−1 ) =

1 0

i=1

∫

∞ ∑

1

1 dx = log 2, x+1

x2 − 1 dx = + 1)(x − 1)

1 log 2 + π, 4 0 i=1 √ ∫ ∞ 1 ∑ 2 3 1 x3 − 1 i+1 π, dx = log 2 + (−1) (H3i − H3(i−1) ) = 3 3 9 0 (x + 1)(x − 1) i=1 √ ∫ 1 ∞ ∑ 1 x4 − 1 1+2 2 i+1 dx = log 2 + (−1) (H4i − H4(i−1) ) = π. 4 4 8 0 (x + 1)(x − 1) i=1 (−1)i+1 (H2i − H2(i−1) ) =

Finally,

(x2

1 2

( ) ∞ ∑ 1 2n 1 π2 i+1 +O 4 . (−1) (Hni − Hn(i−1) ) = log + γ + log 2 − 2 π n 72n n i=1

Source: [1202, 1683]. Fact 13.5.26. Let n, m ≥ 1. Then, ∞ ∑

1 1 = ∏n−1 . n j=0 (m + j) j=0 (i + j)

∏n

i=m

In particular,

∞ ∑ i=1

1 1 = . nn! j=0 (i + j)

∏n

Hence, ∞ ∑ i=1

1 = 1, i(i + 1)

∞ ∑ i=1

1 1 = , i(i + 1)(i + 2) 4

∞ ∑ i=1

1 1 = . i(i + 1)(i + 2)(i + 3) 18

Source: [113], [506, pp. 123, 124], and [2577]. Related: Fact 13.5.27 and Fact 14.2.11. Fact 13.5.27. Let m, n ≥ 1 and l ≥ 0. Then, ∞ ∑ i=1

In particular,

∏l+1

1

j=0 [n

+ m(i + j − 1)]

∞ ∑ i=1

=

(l + 1)m

1 ∏l

j=0 (n

1 1 = , [n + m(i − 1)](n + mi) mn

+ m j)

.

1028

CHAPTER 13 ∞ ∑ i=1

1 1 = , [n + m(i − 1)](n + mi)[n + m(i + 1)] 2mn(n + m) ∞ ∑

∏l+1

1

j=0 (i

i=1

+ j)

=

(l + 1)

1 ∏l

j=0 (1

+ j)

.

Furthermore, ∞ ∑ i=1 ∞ ∑ i=1

∞ ∑

1 = 1, i(i + 1)

1 1 = , (i + 2)(i + 3) 3

i=1 n ∑ i=1

∞ ∑

1 1 = , 2 4i − 1 2

i=1

1 1 = , (i + 1)(i + 2) 2 ∞ ∑

1 1 = , (3i + 1)(3i − 2) 3

i=1

1 1 = . i(i + 1)(i + 2) 4

Source: [1217, p. 12]. Related: Fact 1.12.25, Fact 1.12.27, Fact 13.5.26, and Fact 13.5.67. Fact 13.5.28. ∞ ( ∑ 1 i=1

) i+1 − log = γ, i i

( ) ∞ ∑ i+1 4 i+1 1 (−1) − log = log , i i π i=1 ) ( ∞ ∑ 1 = 12 (γ + 1 − log 2π), Hi − log i − γ − 2i i=1

∞ ∑ log i log2 2 (−1)i+1 = − γ log 2, i 2 i=1 ) ∞ ( ∑ 1 1 lim − = γ, x↓1 i x xi i=1

lim [x − Γ( 1x )] = γ.

x→∞

Source: [1103, pp. 144, 192–195]. The last two equalities are given in [1350, p. 109]. Fact 13.5.29. Let n be a positive integer. Then, ∞ ∑

∑ 1 1 3 = + . i2 − n2 4n2 i=1 n2 − i2 i=n+1 n−1

In particular, ∞ ∑ i=2

3 1 = , i2 − 1 4

∞ ∑ i=3

∞ ∑

25 1 = , i2 − 4 48

i=4

49 1 = . i2 − 9 120

Source: [1217, p. 10]. Related: Fact 1.12.27 and Fact 13.5.59. Fact 13.5.30. Let n be an even positive integer. Then, ∞ ∑ i=n+1

In particular,

∞ ∑ (−1)i i=3

∑ 1 3 1 = 2+ (−1)i 2 2 . 2 2 i −n 4n n −i i=1 n−1

(−1)i

1 7 =− , 48 i2 − 4

∞ ∑ (−1)i i=5

1 533 =− . 6720 i2 − 16

Source: [1217, p. 10]. Related: Fact 13.5.61. Fact 13.5.31. Let n be an odd positive integer. Then, ∞ ∑ i=n+1

∑ 1 1 1 = + (−1)i 2 2 . 2 2 2 i −n 4n n −i i=1 n−1

(−1)i

1029

INFINITE SERIES, INFINITE PRODUCTS, AND SPECIAL FUNCTIONS

In particular,

∞ ∑ (−1)i i=2

1 1 = , i2 − 1 4

∞ ∑ (−1)i i=4

37 1 = . i2 − 9 360

Fact 13.5.32. Let n be an even positive integer. Then, ∞ ∑ i=0

1 = 0. 4i2 + 4i + 1 − n2

Source: Fact 13.5.29 and Fact 13.5.30. Fact 13.5.33. Let n be an odd positive integer. Then, (n−3)/2 ∑ 1 1 1 = + . 2 2 2 2 − (2i + 1)2 (2i + 1) − n 4n n i=0 i=(n+1)/2 ∞ ∑

In particular, ∞ ∑ k=1

∞ ∑

1 1 = , (2i + 1)2 − 1 4

k=2

∞ ∑

1 11 , = (2i + 1)2 − 9 72

k=3

1 137 . = (2i + 1)2 − 25 1200

Source: Fact 13.5.29 and Fact 13.5.31. Fact 13.5.34. Let z be a complex number, and assume that |z| < 1. Then, ∞ ∑

zi =

i=0

1 , 1−z

∞ ∑

izi =

i=1

z , (1 − z)2

∞ ∑

i2 z i =

i=1

z2 + z , (1 − z)3

∞ ∑

i3 zi =

i=1

z3 + 4z2 + z . (1 − z)4

In particular, ∞ ∑ 1 = 1, 2i i=1 ∞ ∑ 1 1 = , i 3 2 i=1

∞ ∑ i = 2, 2i i=1

∞ 2 ∑ i = 6, 2i i=1

∞ ∑ 3 i = , i 3 4 i=1

∞ 2 ∑ 3 i = , i 3 2 i=1

∞ 3 ∑ i = 26, 2i i=1 ∞ 3 ∑ 33 i = . i 3 8 i=1

Furthermore, ∞ 4 ∑ i = 150, i 2 i=1

∞ 5 ∑ i = 1082, i 2 i=1

∞ 6 ∑ i = 9366, i 2 i=1

∞ 4 ∞ 5 ∞ 6 ∑ ∑ ∑ i i 273 i 1491 = 15, = , = . i i i 3 3 4 3 4 i=1 i=1 i=1 ∑ 2 i 2 3 Source: Fact 1.19.3. Remark: ∞ i=1 i z = 1/(1 − z) − 3/(1 − z) + 2/(1 − z) .

Fact 13.5.35. ∞ ∑ i=1 ∞ ∑ i=1

1 = 1, 2i

∞ ∑ 1 = log 2, 2i i i=1

1 = 2 log 2 − 1, 2i (i + 1)

∞ ∑ 1 π2 1 = − log2 2, i 2 12 2 2 i i=1

∞ ∑

∞ ∑ 1 1 = , i 3 2 i=1

i=1

1 = 1 − log 2, 2i i(i + 1)

∞ ∑ 1 3 = log , i 3i 2 i=1

∞ ∑ i=1

∞ ∑ 1 7ζ(3) log3 2 π2 log 2 = + − , 8 6 12 2i i3 i=1

π2 1 1 = + log 2 − log2 2 − 1, 2 2i i2 (i + 1) 12

∞ ∑ 1 = Li2 ( 13 ) ≈ 0.366213, i i2 3 i=1

1030 ∞ ∑ i=1

CHAPTER 13

1 27 = log − 1, 3i (i + 1) 8

∞ ∑ i=1

9 1 = 1 − log , 3i i(i + 1) 4

∞ ∑ i=1

9 1 = Li2 ( 31 ) + log − 1. 4 3i i2 (i + 1)

Remark: Li2 is the dilogarithm function defined by Fact 13.3.5. Fact 13.5.36. Let n ≥ 1. Then, the following statements hold:

i) ζ(2n) = ii) iii)

∞ ∑ 22n−1 π2n 1 = |B2n |. 2n (2n)! i i=1

∞ ∑ 1 (22n−1 − 1)π2n |B2n |. (−1)i+1 2n = (2n)! i i=1 ∞ ∑ i=1

1 (4n − 1)π2n |B2n |. = 2n 2(2n)! (2i − 1)

In particular, ∞ ∑ 1 π2 = , 2 6 i i=1 ∞ ∑ π2 1 (−1)i+1 2 = , 12 i i=1 ∞ ∑ i=1

∞ ∑ 1 π4 , = 4 90 i i=1

∞ ∑ 1 π6 , = 6 945 i i=1

∞ ∑ 7π4 1 , (−1)i+1 4 = 720 i i=1 ∞ ∑

π2 1 = , 2 8 (2i − 1)

i=1

1 π4 , = 4 96 (2i − 1)

Finally, if n ≥ 2, then

∞ ∑ 1 π8 , = 8 9450 i i=1

∞ ∑ 1 31π6 , (−1)i+1 6 = 30240 i i=1 ∞ ∑ i=1

1 π6 = , 6 960 (2i − 1)

∞ ∑ 1 π10 , = 10 93555 i i=1 ∞ ∑ 127π8 1 , (−1)i+1 8 = 1209600 i i=1 ∞ ∑ i=1

1 17π8 . = 8 161280 (2i − 1)

2 ∑ ζ(2i)ζ(2n − 2i). 2n + 1 i=1 n−1

ζ(2n) =

Source: [968, p. 297] and [2513, p. 167]. Remark:

∑ 1∑ 1 ∑ π2 π2 π2 1 1 π2 ∑ 1 ∑ 1 = + = + = = + = . 2 2 2 2 2 6 4 i=1 i 24 8 6 i (2i) (2i − 1) (2i − 1) i=1 i=1 i=1 i=1 ∞

∞

∞

Fact 13.5.37. Let p > 1. Then,

∞

∞

∑ 1 ζ(p) = , i p ζ(2p)

where the sum is taken over all positive integers i that are not divisible by a square that is greater than 1. In particular, with the same summation index, ∑1 ∑1 ∑1 ∑1 15 105 675675 34459425 = , = , . = , = 2 2 4 4 6 6 6 i π i π 3617π8 i 691π i Furthermore, with the same summation index, ∑ 1 9 (−1)i+1 2 = 2 . i π Source: [1406, 2414]. Fact 13.5.38. Let n ≥ 2. Then, ∑1 1 ζ(2n) = ζ(n) − , in 2 2ζ(n)

1031

INFINITE SERIES, INFINITE PRODUCTS, AND SPECIAL FUNCTIONS

where the sum is taken over all positive integers that have an odd number of prime divisors. Furthermore, ∑1 ζ(n) = − 1, in ζ(2n) where the sum is taken over all positive integers that have distinct prime divisors. In addition, ∑1 ζ(n) 1 = − , n i 2ζ(2n) 2ζ(2n) where the sum is taken over all positive integers that have an odd number of distinct prime divisors. Finally, ∑1 ζ(n) = ζ(n) − , in ζ(2n) where the sum is taken over all positive integers that have at least two equal prime divisors. Source: [1317, pp. 20, 21]. Related: Fact 13.10.24. △ ∑ x Fact 13.5.39. For x ∈ (1, ∞), define ζ(x) = ∞ i=1 1/i . Then, 1 x ≤ ζ(x) ≤ . x−1 x−1 Furthermore,

lim ζ(x) = 1,

x→∞

Source: [2294, p. 67]. Fact 13.5.40. Let n ≥ 1. Then,

lim ζ(x) = ∞, x↓1

lim (x − 1)ζ(x) = 1. x↓1

[ ] 1 i ζ(n) = . i(i!) n − 1 i=n−1 ∞ ∑

In particular,

[ ] ∑ [ ] ∑ ∞ ∞ ∞ ∞ ∑ 1 i π2 ∑ 1 i 1 Hi−1 , ζ(3) = , = = = 2 1 2 6 i(i!) i(i!) i i2 i=2 i=1 i=1 i=2 [ ] ∞ ∞ 2 − Hi−1,2 π4 ∑ 1 i 1 ∑ Hi−1 = = , 90 i=3 i(i!) 3 2 i=3 i2 [ ] ∞ ∞ 3 ∑ − 3Hi−1 Hi−1,2 + 2Hi−1,3 1 ∑ Hi−1 1 i = ζ(5) = . 4 i(i!) 6 i2 i=4 i=4

Source: [610]. Related: Fact 1.19.1 and Fact 13.5.16. Fact 13.5.41.

[ ] ∞ ∞ ∞ ∞ ∞ ∞ 1 ∑ ∑ 1 8∑ 1 1 ∑ Hi ∑ Hi 1 i 4 ∑ H2i − 2 Hi ζ(3) = = = = = = i(i!) 2 7 i=0 (2i + 1)3 7 i=1 i3 2 i=1 i2 (i + 1)2 i2 i=1 i=1 i=2 ∞ ∞ ∞ 2 ∑ 4∑ 1 5∑ 7π3 1 i+1 (i!) (−1)i = (−1) = − 2 = −4π2 ζ ′ (−2) 3 3 3 2iπ 3 i=0 2 i=1 (i + 1) i (2i)! 180 i (e − 1) i=1   ( ) ∞ ∞ 1 ∑   H H − 1 ∑ 1 2i 4  2 i i 2i 1  = (H2i − 2 Hi ) = πG + (−1) i 2 7 i=1 4 i i 7 i i=1

=

ζ(2i) 8π2 ∑ ζ(2i) 2π2 ∑ ζ(2i) 8π2 ∑ =− =− i i 9 i=0 (2i + 1)(2i + 3)4 5 i=0 (2i + 1)(2i + 2)(2i + 3)4 7 i=0 (i + 1)(2i + 1)4i ∞

=−

∞

∞

1032

CHAPTER 13

6π2 ∑ (98i + 121)ζ(2i) 23 i=0 (2i + 1)(2i + 2)(2i + 3)(2i + 4)(2i + 5)4i   ∞ ∞ ∑  4 π ∑ ζ(2i) 16i (16i3 + 12i2 + 6i + 1)(i!)4 π2  3 1  = + 1 =  − log + 6 4 2 3 i=1 i(2i + 1)(2i + 2)36i 7 14 i=1 i3 (2i + 1)3 [(2i)!]2   ∞ ∑  π2  11 1 ζ(2i) =  − log π +  i 2 18 3 i(i + 1)(2i + 1)(2i + 3)4  i=1 ∞

=−

=−

∞ (8576i2 + 24286i + 17283)ζ(2i) 120π2 ∑ 1573 i=0 (2i + 1)(2i + 2)(2i + 3)(2i + 4)(2i + 5)(2i + 6)(2i + 7)4i

∞ ∞ 2 4 3 2 1 ∑ 1∑ i+1 56i − 32i + 5 i 5265i + 13878i + 13761i + 6120i + 1040 ( )( ) ( )( ) = (−1) (−1) = 4 i=1 72 i=0 (2i − 1)2 i3 3ii 2ii (4i + 1)(4i + 3)(i + 1)(3i + 1)2 (3i + 2)2 3ii 4ii ( ) ∫ ∞ π ∞ 2 4 1 37π3 2 ∑ 1 4 sech πx − x tanh πx = + dx = − 900 5 i=1 i3 eiπ − 1 e4iπ − 1 (x2 + 14 )2 0  2  ( )  i i ∞ ∞  2π2  ∑  2 ∑ 1 2i ∑ 1  ∑ 1 ζ(2i)   log 2 + 2   =  − =  i i 2  7 i=1 4 i i 2j − 1 9 (2i + 3)4  (2 j − 1) j=1 j=1 i=0   i   i 2 i ∞ 2  9 ∑ 1 ∑ ∏ 3 j − k  ∑ 1  ∑ 1      − = 52 i 3j  3j − k (3 j − k)2  i=1

k=1

j=1

j=1

j=1

≈ 1.202056903159594285399738161511449990764986292340498881792271555341838. Furthermore, ζ(3) is an irrational number. Source: [62, 66, 323], [511, pp. 78, 236], [645, p. 99], [1352, Chapter 5], [2513, pp. 246, 405, 431, 432, 443], and [2792]. Remark: ζ(3) is Apery’s constant. See [2389]. Related: Fact 13.5.16. Fact 13.5.42.

ζ(5) =

∞ ∞ π2 8π4 ∑ 3π2 4π4 ∑ ζ(2i) ζ(2i) ζ(3) − = ζ(3) + i 13 13 i=0 (2i + 3)(2i + 4)(2i + 5)4 31 93 i=0 (2i + 3)(2i + 4)4i

=

π4 ∑ 7π2 8π4 ∑ ζ(2i) ζ(2i) π2 ζ(3) + = ζ(3) + i 11 33 i=0 (2i + 4)(2i + 5)4 75 255 i=0 (2i + 3)(2i + 5)4i

=

∞ 4π2 8π4 ∑ ζ(2i) ζ(3) + 31 13 i=0 (2i + 1)(2i + 2)(2i + 3)(2i + 4)4i

∞

∞

∞ 2π2 4π4 ∑ ζ(2i) ζ(3) − 27 9 i=0 (2i + 2)(2i + 3)(2i + 4)(2i + 5)4i   ∞ ∑  2π4  ζ(2i) 47  . = − 30 log π − i 45 60 i(i + 2)(2i + 3)(2i + 5)4 i=1

=

Source: [2513, pp. 429–431]. Remark: At least one of the numbers ζ(5), ζ(7), ζ(9), ζ(11) is

irrational. See [1350, p. 42]. Fact 13.5.43. Let n ≥ 2. Then, n−2 ∞ ∑ 1∑ Hi n + 2 = ζ(n + 1) − ζ(i + 1)ζ(n − i), in 2 2 i=1 i=1

∞ ∑ i=1

Hi n 1∑ = ζ(n + 1) − ζ(i + 1)ζ(n − i). n (i + 1) 2 2 i=1 n−2

1033

INFINITE SERIES, INFINITE PRODUCTS, AND SPECIAL FUNCTIONS

In particular, ∞ ∑ Hi = 2ζ(3), i2 i=1 ∞ ∑ i=1

∞ ∑ 1 2 π4 Hi 5 ζ(4) − ζ (2) = , = 2 2 72 i3 i=1 ∞ ∑

Hi = ζ(3), (i + 1)2

i=1

∞ ∑

Hi π4 , = (i + 1)3 360

Now, let n ≥ 1. Then, ∞ ∑ Hi 1 2 = [2ζ(2) + Hn−1 + Hn−1,2 ], i(i + n) 2n i=1

∞ ∑ i=1

i=1

∞ ∑ Hi π2 ζ(3), = 3ζ(5) − 6 i4 i=1

Hi π2 ζ(3). = 2ζ(5) − 6 (i + 1)4

Hi = ζ(3) + ζ(2)Hn−1 − Hn−1 Hn−1,2 − Hn−1,3 , (i + n)2

2n ∞ ∑ Hi 1∑ (−1)i ζ(i)ζ(2n − i + 2), = 2n+1 2 i i=2 i=1 ∞ n+1 ∑ ∑ Hi,2 1 (n + 2)(2n + 1)ζ(2n + 3) + 2 (i − 1)ζ(2i − 1)ζ(2n + 4 − 2i). = ζ(2)ζ(2n + 1) − 2 i2n+1 i=1 i=2

In particular, ∞ ∑ i=1 ∞ ∑

∞ ∑

π2 Hi = , i(i + 1) 6

i=1

Hi 1 π2 = + , i(i + 2) 2 12

Hi π = ζ(3) + − 2, 6 (i + 2)2

∞ ∑ i=1

Hi 7 π2 = + , i(i + 3) 12 18

∞ ∑

∞ ∑ i=1

Hi 85 π2 = + , i(i + 4) 144 24

∞ ∑

π 11π2 395 Hi Hi = ζ(3) + = ζ(3) + − 3, − . 4 36 108 (i + 3)2 (i + 4)2 i=1 i=1 i=1 ∑ Hi Source: [2485] and [2513, pp. 228, 229]. Remark: ∞ i=1 i = ∞. Credit: L. Euler. Fact 13.5.44. If n ≥ 1, then ( ) ∞ ∑ 1 Hi+n 1 = + Hn . n+1 n(n!) n i=1 i 2

2

In particular, ∞ ∑ Hi+1 = 2, i(i + 1) i=1

If n ≥ 2, then

∞ ∑ i=1

Hi+2 1 = , i(i + 1)(i + 2) 2 ∞ ∑ Hi i=1

in+1

=

∞ ∑ i=1

Hi+3 13 = . i(i + 1)(i + 2)(i + 3) 108

( ) 1 π2 − Hn−1,2 . n! 6

In particular, ∞ ∑ i=1

π2 1 Hi = − , i(i + 1)(i + 2) 12 2

Furthermore, ∞ ∑ Hi+2 i=1

i(i + 1)

=

9 , 4

∞ ∑ Hi+3 22 = , i(i + 1) 9 i=1

∞ ∑ i=1

Hi π2 5 = − . i(i + 1)(i + 2)(i + 3) 36 24

∞ ∑ Hi+4 125 = , i(i + 1) 48 i=1

∞ ∑ Hi+5 137 = . i(i + 1) 50 i=1

1034

CHAPTER 13

Finally,

∞ ∞ ∑ Hi ∑ 1 7 = ζ(4), 2 i 4 j i=1 j=i+1

∞ ∑ 1 i=1

i

(log 3 + Hi − H3i ) =

5π2 3 log2 3 − . 36 4

Source: [1099] and [1103, pp. 149, 208–214]. Fact 13.5.45. Let n ≥ 1. Then,

( ) n−1 ∞ ∑ (2i + 1)π Hni (n2 + 1)π2 1 ∑ 2 = − . log 2 sin (−1)i−1 i 24n 2 i=0 2n i=1

In particular, ∞ ∑ Hi π2 1 (−1)i−1 = − log2 2, i 12 2 i=1 ∞ ∑ H3i 5π2 1 (−1)i−1 = − log2 2, i 36 2 i=1

∞ ∑ H2i 5π2 1 (−1)i−1 = − log2 2, i 48 4 i=1

∞ ∑ √ H4i 17π2 1 1 (−1)i−1 = − log2 2 − log2 (1 + 2). i 96 8 2 i=1

Furthermore, ∞ ∞ ∞ ∑ ∑ ∑ 5 5 π2 Hi π2 i+1 Hi+1 i 2 i+1 Hi = = ζ(3), (−1) ζ(3) + − 2 log 2 + 1, (−1) = log 2 − , (−1) 8 8 12 i(i + 1) 12 i2 i2 i=1 i=1 i=1 ∞ ∑ (−1)i i=1 ∞ ∑

∞ ∑ (−1)i+1

1 π2 Hi = − −log 2+log2 2, i(i + 1)(i + 2) 2 24

i=1

1 π2 Hi 3 3 ζ(3)+ log 2− log 2, = 3 12 i2 (i + 1)2 4

∞ ∑ Hi2 π2 2 = log 2 − log3 2 − ζ(3). (−1)i+1 i(i + 1) 6 3 i=1

H2 3 1 π2 (−1)i+1 i = ζ(3) + log3 2 − log 2, i 4 3 12 i=1

Source: [1682] and [2513, pp. 357, 358]. Remark: The last expression corrects (41) given in

[2513, p. 358]. Fact 13.5.46. Let x be a real number. If |x| > 1, then ∞ ∑ ζ(2i) − 1 i=1

x2i

=

1 1 π π + − cot . 2 1 − x2 2x x

In particular, ∞ ∑ ζ(2i) − 1

4i

i=1

1 = , 6

∞ ∑ ζ(2i) − 1 i=1

If |x| < 1, then ∞ ∑ ζ(i)xi−1 = −ψ(1 − x) − γ, i=2 ∞ ∑

16i ∞ ∑

i=1 ∞ ∑ ζ(i)

i

xi = log Γ(1 − x) − γx,

∞ ∑ ζ(2i + 1) i=1

∞ ∑ ζ(2i) − 1 i=1

64i

√ 61 (1 + 2)π = − . 126 16

ζ(2i)x2i−1 = 12 [ψ(1 + x) − ψ(1 − x)] =

i=1

ζ(2i)x2i = ψ(1 + x) + ψ(1 − x) = log

i=2

13 π = − , 30 8

2i + 1

πx , sin πx

∞ ∑ ζ(2i) i=1

i

∞ ∑

1 π − cot πx, 2x 2

ζ(2i + 1)x2i = − 12 [ψ(1 + x) + ψ(1 − x)] − γ,

i=1

x2i = log Γ(1 + x) + log Γ(1 − x) = log

x2i+1 = 12 [log Γ(1 − x) − log Γ(1 − x)] − γx,

∞ ∑ ζ(2i) i=1

i2

πx , sin πx

x2i = log(πx csc πx),

1035

INFINITE SERIES, INFINITE PRODUCTS, AND SPECIAL FUNCTIONS ∞ ∑ π 1 (−1)i+1 ζ(2i)x2i−1 = coth πx − . 2 2x i=1

If 0 < |x| ≤ 1, then ∞ ∑ 3x2 + 1 π (−1)i+1 [ζ(2i) − 1]x2i−1 = coth πx − , 2 2x(x2 + 1) i=1

∞ ∑ sinh πx ζ(2i) 2i x = log . (−1)i+1 i πx i=1

In particular, ∞ ∑ ζ(2i) π 1 (−1)i+1 i = coth π − , 4 4 2 i=1

∞ ∑ π (−1)i+1 [ζ(2i) − 1] = coth π − 1, 2 i=1 ∞ ∑ 2 sinh π2 ζ(2i) (−1)i+1 i = log . 4i π i=1

∞ ∑ sinh π ζ(2i) = log , (−1)i+1 i π i=1

If z ∈ C and |z| < 2, then

In particular,

∞ ∑ ζ(2i) πz πz (−1)i+1 i z2i = + . πz − 1) 4 4 2(e i=0

∞ ∑ π πeπz π ζ(2i) π = − . (−1)i+1 i = + π − 1) πz − 1) 4 4 2(e 2(e 4 i=0

Source: [2513, pp. 268, 270, 271, 366, 423]. Remark: ψ is the digamma function. Fact 13.5.47.

√ ∞ ∑ 3π 1 π ζ(i) 3 log 3 − , = log 2 − , i i i 2 3 2 18 4 4 8 i=2 i=2 i=2 √ ∞ ∞ ∞ ∑ ∑ ∑ ζ(i) 1 ζ(i) π 3 3π ζ(i) (−1)i i = 1 − log 3 − , (−1)i i = 1 − − log 2, (−1)i i = 1 − log 2, 2 3 2 18 4 8 4 i=2 i=2 i=2 √ ∞ ∞ ∞ ∞ ∑ ζ(2i) 1 ∑ ζ(2i + 1) ∑ ζ(2i) 1 ∑ ζ(2i + 1) 3 3π = , = 2 log 2−1, = − , = (log 3−1), i i i 4 2 4 9 2 18 9i 2 i=1 i=1 i=1 i=1 ∞ ∑ ζ(i)

∞ ∑ 3 [ζ(2i) − 1] = , 4 i=1

∞ ∑ ζ(i)

= log 2,

∞ ∑ 1 [ζ(2i + 1) − 1] = , 4 i=1 ∞ ∑ iζ(2i + 1)

4i

i=1 ∞ ∑ ζ(i) − 1 i=2

i

= 1 − γ,

i=2

2i

i=1

i

1 , 2 i

i=1

1 = log 2 − , 2

∞ ∑ ζ(4i) − 1

=

∞ ∑ [ζ(i) − 1] = 1, i=2

∞ ∑ (2i + 1)ζ(2i + 1)

4i

i=1

∞ ∑ ζ(2i) − 1

= log 2,

= 2 log 2,

∞ ∑ ζ(3i) − 1 i=1

∞ ∑ ζ(i) − 1

∞ ∑ 7 π [ζ(4i) − 1] = − coth π, 8 4 i=1

i

(

) 3π = log 3π sech , 2

∞ ∑ 3 γ 1 ζ(2i) − 1 3 − − log 2π, = − 2 log π, i+1 2 2 2 i+1 2 i=2 i=2 √ ∞ ∞ ∑ ζ(i) − 1 1 ∑ ζ(i) − 1 3 1 3π 1 π = log 3 − − , = log 2 − − , i i 3 2 6 18 4 4 12 8 i=2 i=2

∞ ∑ ζ(i) − 1 = γ + log 2 − 1, (−1)i i i=2 ∞ ∑ ζ(i) − 1

=

= log(4π csch π),

∞ ∑ i=2

=

(−1)i

ζ(i) = γ, i

∞ ∑ ζ(2i) (−1)i = log(π csch π), i i=1

1036

CHAPTER 13

√   ∞ ∑  3π  i ζ(3i)  , (−1) = log π sech i 2 i=1 ∞ ∑ ζ(2i)

4i i

i=1

π = log , 2

∞ ∑ [2( 3 )i − 3][ζ(i) − 1] 2

i

i=2 ∞ ∑

1 ζ(2i) = , (i + 1)(2i + 1) 2

i=1 ∞ ∑ i=1

ζ(2i) = log π − 1, 4i i(2i + 1) ∞ ∑ i=1

4i (i ∞ ∑ i=1

i=2

∞ ∑ (Im[(1 + ȷ)n − 1 − ȷn ])[ζ(i) − 1] π = , i 4 i=2

= log π,

∞ ∑ 1 ζ(2i) = log 2π − , i(i + 1) 2 i=1

∞ ∑

i+1

i=1

∞ ∑ ζ(2i) = log 2π − 1, i(2i + 1) i=1 ∞ ∑

ζ(2i) 9ζ(3) 1 1 − log 2 + , = 2 + 3) 2 6 4π

∞ ∑ 4 ζ(i + 1) = log − γ, (−1)i i 2 (i + 1) π i=1

∞ ∑ ζ(2i + 1) − 1

= log 2 − γ,

i=1 ∞ ∑

ζ(2i + 1) = log 2 − γ, (2i + 1)4i

i=1

ζ(2i) 1 = (1 − log 2), 4i (2i + 1) 2

4i (2i

i=1

ζ(3) ζ(2i) =1− 2, 4i (i + 1)(2i + 3) 2π

∞ ∑

i=1

2i i √ ∞ ∞ ∑ ∑ ζ(2i) 2 3π ζ(2i) 2π = log , = log , i i 9i 9 16 i 4 i=1 i=1

ζ(2i) 1 7ζ(3) , = − + 1)(2i + 1) 2 2π2

i=1

√ π 1 γ = − + log , 2 2 2

∞ ∑ ζ(2i) 1 7ζ(3) , = − log 2 + i (i + 1) 4 2 2π2 i=1

∞ ∑ ζ(2i + 1) − 1

∞ ∑

∞ ∑ ζ(i) − 1

i=1

2i + 1

=1−

1 log 2 − γ, 2

3 ζ(2i + 1) − 1 = 1 − log − γ, (2i + 1)4i 2

∞ ∑ ζ(2i + 1) − 1 = 2 − 2 log 2 − γ, (i + 1)(2i + 1) i=1

ζ(2i + 1) = 1 − γ, (i + 1)(2i + 1)

∞ ( )2i ∑ 3 ζ(2i + 1) − 1 1 8 = 1 + log − γ, 2 (2i + 1) 3 15 i=1

∞ ∑ 2i [ζ(i) − 1] (−1)i = 2γ + log 6 − 2. i i=2

Now, let a ∈ R, and assume that |a| > 1. Then, ∞ ∑ 1 π ζ(2i) − 1 1 π cot . = + − 2i 2 2 1−a 2a a a i=1 In particular, ∞ ∑ ζ(2i) − 1 i=1

4i

=

1 , 6

∞ ∑ ζ(2i) − 1 i=1

16i

=

13 π − , 30 8

∞ ∑ ζ(2i) − 1 i=1

64i

=

√ 61 π − (1 + 2). 126 16

Source: [511, pp. 131, 248], [559, 650, 700], and [2513, pp. 272–314, 365, 556, 557]. Related:

Fact 12.18.33. Fact 13.5.48. Let n ≥ 1, and define △

Sn =

∑

1 ∏n

2 j=1 i j

,

1037

INFINITE SERIES, INFINITE PRODUCTS, AND SPECIAL FUNCTIONS

where the sum is taken over all n-tuples (i1 , . . . , in ) such that i1 ≤ · · · ≤ in . Then, S n = (2−41−n )ζ(2n). Furthermore, n n ∑ ∑ 2 − 41−i 1 Si = ζ(2i) = 0. (−1)i 2i (−1)i 2i π (2n − 2i + 1)! π (2n − 2i + 1)! i=0 i=0 Source: [2468]. Fact 13.5.49. Let n ≥ 1. Then, ∞ ∑ i=1

( ) 2n (2π)4n−1 ∑ 2 i 4n = −ζ(4n − 1) + B2i B4n−2i . (−1) (4n)! i=0 2i i4n−1 (e2iπ − 1)

Furthermore,

( ) ∞ ∞ ∞ 4 π5 72 ∑ 37π3 2 ∑ 1 1 1 1 2 ∑ , ζ(5) = ζ(3) = − + − − , 900 5 i=1 i3 eiπ − 1 e4iπ − 1 294 35 i=1 i5 (e2iπ − 1) 35 i=1 i5 (e2iπ + 1) ( ) ( ) ∞ ∞ ∑ 1 4 1 π3 4 5 1 409π7 2 ∑ 1 + − + , = − . ζ(7) = 94500 5 i=1 i7 eiπ − 1 e4iπ − 1 180 i=1 i3 eiπ − 1 e2iπ − 1 e4iπ − 1 Source: [2792]. Fact 13.5.50. Let n ≥ 1. Then, ∞ ∑ B4n+2 i4n+1 = , 2iπ 8n +4 e − 1 i=1

∞ ∑ (2i − 1)4n+1 i=1

e(2i−1)π

+1

=

(24n+1 − 1)B4n+2 . 8n + 4

In particular, ∞ ∑ i=1

i5 1 , = 2iπ e − 1 504

∞ ∑ i=1

i13 1 , = 2iπ e − 1 24

∞ ∑ (2i − 1)5 31 , = (2i−1)π e + 1 504 i=1

∞ ∑ (2i − 1)9 511 . = (2i−1)π e + 1 264 i=1

Source: [1317, p. xxvi] and [2792]. Fact 13.5.51. Let n ≥ 1. Then, ∞ ∑ i=1

⌊n/2⌋ ∑ 4i−n (4i − 1)|B2i |π2i (2n − 2i − 1) 1 1 n = (−1) + (−1)n+1 . 2 n (2i)! n−1 2 (4i − 1) i=1

In particular,

∞ ∑ i=1 ∞ ∑ i=1

1 1 = , 4i2 − 1 2

1 1 3π2 , = − 2 3 2 64 (4i − 1)

∞ ∑ i=1 ∞ ∑ i=1

π2 1 1 = − , (4i2 − 1)2 16 2 (4i2

1 5π2 1 π4 + − . = 4 768 128 2 − 1)

Source: [1888]. Fact 13.5.52. Let n ≥ 1. Then, ∞ ∑ i=1

( ) ⌊n/2⌋ ∑ 4i |B2i |π2i (2n − 2i − 1) 1 n n+1 2n − 1 = (−1) + (−1) . in (i + 1)n (2i)! n−1 n i=1

In particular, ∞ ∑ i=1

1 = 1, i(i + 1)

∞ ∑ i=1

1 π2 = − 3, 3 i2 (i + 1)2

∞ ∑ i=1

1 = 10 − π2 , i3 (i + 1)3

1038

CHAPTER 13 ∞ ∑ i=1

∞ ∑ i=1

∞ ∑

π4 10π2 1 = + − 35, 3 i4 (i + 1)4 45

2π6 7π4 1 + + 42π2 − 462, = 6 6 945 15 i (i + 1)

i=1

π4 35π2 1 = 126 − − , 9 3 i5 (i + 1)5

∞ ∑ i=1

i7 (i

2π6 28π4 1 = 1716 − − − 154π2 . 7 135 15 + 1)

Source: [1888]. Fact 13.5.53. Let x be a nonzero real number. Then, ∞ ∑ i=1

In particular,

1 π coth πx 1 = − 2. 2x i2 + x 2 2x

∞ ∑

1 π coth π 1 1 + π + e2π (π − 1) − = = . 2 2 +1 2(e2π − 1) i=1 ( ) 2 πx − 2x12 = π6 . Source: Fact 13.4.12. Remark: lim x→0 π coth 2x Fact 13.5.54. Let x be a nonzero real number. Then, ∞ ∑ 1 π2 x2 csch2 πx + πx coth πx − 2 = . (i2 + x2 )2 4x4 i=1 i2

π coth πx−2 = 90 . Remark: lim x→0 π x csch πx+πx 4x4 Fact 13.5.55. Let x be a nonzero real number. Then, 2 2

2

4

∞ ∑ (−1)i π csch πx 1 = − 2. 2 2 2x i +x 2x i=1

In particular,

(

)

∞ ∑ π csch π 1 (−1)i = − . 2 2 2 i +1 i=1

πx π = 12 . Remark: lim x→0 2x12 − π csch 2x Fact 13.5.56. Let x be a nonzero real number. Then, 2

∞ ∑ 1 π(πx coth πx + 1) csch πx (−1)i+1 = 4− . 2 2 2 (i + x ) 2x 4x3 i=1 ( ) csch πx 7π4 Remark: lim x→0 2x14 − π(πx coth πx+1) = 720 . 4x3 Fact 13.5.57. Let x be a nonzero real number. Then, ∞ ∑ π tanh πx 1 2 = . 2 + x2 4x (2i − 1) i=1 π tanh

πx

Source: [1167, p. 301]. Remark: lim x→0 4x 2 = π8 . Fact 13.5.58. Let x be a nonzero real number. Then, 2

) π(sinh πx − πx) 1 π ( πx πx 2 πx − sech = 3 = tanh . 2 2 2 3 2 2 2 [(2i − 1) + x ] 8x 8x (cosh πx + 1) i=1 ) ( 2 πx πx π4 Source: [1167, p. 301]. Remark: lim x→0 8xπ3 tanh πx 2 − 2 sech 2 = 96 . ∞ ∑

1039

INFINITE SERIES, INFINITE PRODUCTS, AND SPECIAL FUNCTIONS

Fact 13.5.59. Let x be a complex number, and assume that x is not an integer. Then, ∞ ∑ i=1

∞ ∑

1 1 π cot πx = − 2x i2 − x2 2x2

i=−∞

1 = π cot πx. x−i

In particular, ∞ ∑ i=1

1 1 = , 4i2 − 1 2

∞ ∑

1

i=1

16i2 − 1

Now, let n ≥ 2, and assume that

=

1 π − , 2 8

∞ ∑

1

i=1

64i2 − 1

=

√ 1 (1 + 2)π − . 2 16

√ n is not an integer. Then, √ √ ∞ ∑ 1 − nπ cot nπ 1 = . 2n i2 − n i=1

In particular,

√

√ √ √ ∞ ∑ 1 − 3π cot 3π 1 2π cot 2π = , . 4 6 i2 − 3 i=1 i=1 ) ( 2 πx = π6 . Related: Fact 13.5.29. Source: [116, p. 11]. Remark: lim x→0 2x12 − π cot 2x Fact 13.5.60. Let x be a real number, and assume that x is not an integer. Then, ∞ ∑

1− 1 = i2 − 2

∞ ∑ i=1

In particular,

∞ ∑ i=1

π2 x2 csc2 πx + πx cot πx − 2 1 = . (i2 − x2 )2 4x4 ∞ ∑

1 π2 1 = − , (4i2 − 1)2 16 2

2 2 2 cot πx−2 lim x→0 π x csc πx+πx 4x4

i=1

1 π2 + 2π 1 = − . 32 2 (16i2 − 1)2

π4 90 .

Remark: = Fact 13.5.61. Let x ∈ C, and assume that x is not an integer. Then, ∞ ∑ (−1)i i=1 ∞ ∑

1 1 π − 2, = 2 2 2x sin πx 2x i −x

(−1)i

i=−∞

In particular,

∞ ∑ (−1)i i=1

1 i+x+

1 2

=

π , cos πx

1 π 1 = − , 2 4i − 1 2 4

∞ ∑

i=−∞

1 = πsin πx, x−i

∞ ∑

1 i − x + i=−∞

∞ ∑ (−1)i i=1

(−1)i

1 2

= πtan πx.

1 = − 2 16i − 1 2 1

√ 2π . 8

Source: [116, p. 11] and [968, p. 236]. Related: Fact 13.5.30. Fact 13.5.62. Let x be a real number, and assume that x is not an integer. Then, ∞ ∑ 1 − cos 2πx − πx sin πx − π2 x2 cos πx (−1)i+1 = . (i2 − x2 )2 4(sin2 πx)x4 i=1

In particular,

∞ ∑ (−1)i+1 1 π = − , 2 − 1)2 2 8 (4i i=1

∞ ∑ i=1

1 (−1)i+1 = − (16i2 − 1)2 2

√ 2 2(π + 4π) . 64

1040

CHAPTER 13

Fact 13.5.63. Let x be a nonzero real number. Then, ∞ ∑ i=1

√ √ √ 1 1 2π sinh 2πx + sin 2πx − = . √ √ i4 + x4 4x3 cosh 2πx − cos 2πx 2x4

In particular,

∞ ∑ i=1

Source: [645, p.

(√

lim x→0

π coth π − 1 1 = . 8 i4 + 4

Remark: As y → 0,

97].

√ √ 2π sinh √2πx+sin √2πx 4x3 cosh 2πx−cos 2πx

−

1 2x4

)

y4 sinh y+sin y y cosh y−cos y −2

= 90 +

3 4 14 y

+ O(y6 ). Hence,

π4 90 .

=

Fact 13.5.64. Let x be a real number, and assume that x is not an integer. Then, ∞ ∑ i=1

In particular,

1 π cot πx + π coth πx 1 = 4− . 4 −x 2x 4x3

i4

∞ ∑

1

i=1

16i4 − 1

)

(

π 1 π coth 2 − . 2 8

=

coth πx π . = 90 Remark: lim x→0 2x14 − π cot πx+π 4x3 Fact 13.5.65. Let x be a positive number. Then, ∞ ∑ (−1)i+1 i=1

4

( ) i 1 1 = 2(x + 2 ) log 1 + . (i + 1)(i + 2)xi x

In particular, ∞ ∑ (−1)i+1 i=1

i = 2 log 2 − 2, (i + 1)(i + 2)

∞ ∑ (−1)i+1 i=1

5 i = 4 log − 2. (i + 1)(i + 2)(3/2)i 3

Source: [2085]. Fact 13.5.66. Let x and y be complex numbers such that division by zero does not occur in the

expressions below. Then, ∞ ∑ i=0

1 1 = , (xi + y)(xi + x + y) xy ∞ ∑ i=0 ∞ ∑ i=0

∞ ∑ i=0

i=0 ∞ ∑ i=0

i=0

x + 2y 1 = , (xi + y)(xi + 2x + y) 2xy(x + y)

1 2x2 + 6xy + 3y2 = , (xi + y)(xi + 3x + y) 3xy(x + y)(2x + y)

3x3 + 11x2 y + 9xy2 + 2y3 1 = , (xi + y)(xi + 4x + y) 3xy(x + y)(2x + y)(3x + y)

1 1 = , (xi + y)(xi − x + y) xy − x2 ∞ ∑

∞ ∑

∞ ∑ i=0

1 3x − 2y = , (xi + y)(xi − 2x + y) 2x(2x − y)(y − x)

1 11x2 − 12xy + 3y2 = , (xi + y)(xi − 3x + y) 3x(3x − y)(2x − y)(y − x)

1 25x3 − 35x2 y + 15xy2 − 2y3 = . (xi + y)(xi − 4x + y) 2x(4x − y)(3x − y)(2x − y)(y − x)

1041

INFINITE SERIES, INFINITE PRODUCTS, AND SPECIAL FUNCTIONS

Fact 13.5.67. Let n, k ≥ 1. Then, ∞ ∑ i=0

1 1 = . (ki + n)(ki + n + k) kn

In particular, ∞ ∑ i=0 ∞ ∑

∞ ∑

1 1 = , (ki + 1)(ki + 1 + k) k

i=0

i=0 ∞ ∑

1 1 = , (i + n)(i + n + 1) n

i=0

∞ ∑

1 1 = , (ki + 2)(ki + 2 + k) 2k

1 1 = , (2i + n)(2i + n + 2) 2n

i=0 ∞ ∑ i=0

1 1 = , (ki + 3)(ki + 3 + k) 3k

1 1 = . (3i + n)(3i + n + 3) 3n

Source: Fact 1.12.25, Fact 13.5.27, and Fact 13.5.66. Fact 13.5.68. Let n, k ≥ 1. Then, ∞ ∑ i=0

Hn+k−1 − Hn−1 1 = . (i + n)(i + n + k) k

In particular, ∞ ∑ i=0 ∞ ∑

1 2Hk+2 − 3 = , (i + 3)(i + 3 + k) 2k

i=0

Hence,

∞ ∑ i=0 ∞ ∑ i=0 ∞ ∑ i=0

1 Hk = , (i + 1)(i + 1 + k) k

1 = 1, (i + 1)(i + 2)

1 1 = , (i + 2)(i + 3) 2 1 1 = , (i + 3)(i + 4) 3

Furthermore,

∞ ∑

i=0 ∞ ∑ i=0

∞ ∑ i=0 ∞ ∑ i=0

Hence,

∞ ∑ i=0 ∞ ∑ i=0

∞ ∑ i=0

1 6Hk+3 − 11 = . (i + 4)(i + 4 + k) 6k

5 1 = , (i + 2)(i + 4) 12

i=0 ∞ ∑

i=0

Hk+1 − 1 1 = , (i + 2)(i + 2 + k) k

1 3 = , (i + 1)(i + 3) 4

i=0 ∞ ∑

∞ ∑

1 7 = , (i + 3)(i + 5) 24

∞ ∑

1 11 = , (i + 1)(i + 4) 18

i=0 ∞ ∑

13 1 = , (i + 2)(i + 5) 36

i=0 ∞ ∑ i=0

1 47 = . (i + 3)(i + 6) 180

1 1 = , (i + n)(i + n + 1) n

2n + 1 1 = , (i + n)(i + n + 2) 2n(n + 1)

1 3n2 + 6n + 2 = . (i + n)(i + n + 3) 3n(n + 1)(n + 2)

1 1 = , (i + 2)(i + 3) 2

1 5 = , (i + 2)(i + 4) 12

∞ ∑ i=0 ∞ ∑ i=0

1 1 = , (i + 3)(i + 4) 3 1 7 = , (i + 3)(i + 5) 24

∞ ∑ i=0 ∞ ∑ i=0

1 1 = , (i + 4)(i + 5) 4 1 9 = , (i + 4)(i + 6) 40

1042

CHAPTER 13 ∞ ∑ i=0

∞ ∑

1 13 = , (i + 2)(i + 5) 36

i=0

47 1 = , (i + 3)(i + 6) 180

∞ ∑ i=0

37 1 = . (i + 4)(i + 7) 180

Related: Fact 1.12.26. Fact 13.5.69. Let n ≥ 0. Then, ∞ ∑ i=0

In particular,

∞ ∑ i=0 ∞ ∑ i=0

∑ 2 2n − 1 = − log 4. (i + 1)(2i + 2n + 1) i=0 2i + 1 n−1

1 = log 4, (i + 1)(2i + 1)

1 8 log 4 = − , (i + 1)(2i + 5) 9 3

∞ ∑

1 = 2 − log 4, (i + 1)(2i + 3)

i=0 ∞ ∑ i=0

46 log 4 1 = − . (i + 1)(2i + 7) 75 5

Fact 13.5.70. Let x and y be complex numbers such that division by zero does not occur in the △

expressions below, and define α = cot πy/x. Then, ∞ ∑ i=0

απ 1 =− , (xi − y)(xi + x + y) x(x + 2y) ∞ ∑ i=0

∞ ∑ i=0

11x3 + 12x2 y + 3xy2 + y2 − απ(6x3 + 11x2 y + 6xy2 + y3 ) 1 = , (xi − y)(xi + 4x + y) x(x + y)(2x + y)(3x + 2y)

i=0 ∞ ∑ i=0

i=0

i=0

1 x2 − 2xy + απ(xy − y2 ) = , (xi − y)(xi − x + y) xy(x − 2y)(x − y)

2x3 − 6x2 y + 3xy2 + απ(2x2 y − 3xy2 + y3 ) 1 = , (xi − y)(xi − 2x + y) 2xy(2x − y)(x − y)2

6x4 − 22x3 y + 18x2 y2 − 4xy3 + απ(6x3 y − 11x2 y2 + 6xy3 − y4 ) 1 = . (xi − y)(xi − 3x + y) xy(3x − 2y)(x − y)(2x − y)(3x − y)

In particular, ∞ ∑

i=0

x − απ(x + y) 1 = , (xi − y)(xi + 2x + y) 2x(x + y)2

3x2 + 2xy − απ(2x2 + 3xy + y2 ) 1 = , (xi − y)(xi + 3x + y) x(x + y)(2x + y)(3x + 2y)

∞ ∑

∞ ∑

∞ ∑

√ ∞ ∑ 3π 1 1 1 π = + , = , (3i + 1)(3i + 5) 8 36 (4i + 1)(4i + 3) 8 i=0 i=0 √ √ √ √ ∞ ∞ ∑ ∑ 1 π 2 5 1 π 2 5 = 1+ , = 1− , (5i + 1)(5i + 4) 15 5 (5i + 2)(5i + 3) 5 5 i=0 i=0 √ √ ∞ ∞ ∑ ∑ 1 3π 1 1 3π = , = − , (6i + 1)(6i + 5) 24 (6i + 5)(6i + 7) 2 12 i=0 i=0 √ √ ∞ ∞ ∑ ∑ 1 (1 + 2)π 1 ( 2 − 1)π = , = , (8i + 1)(8i + 7) 48 (8i + 3)(8i + 5) 16 i=0 i=0 √ √ ∞ ∞ ∑ ∑ 1 1 (1 + 2)π 1 ( 2 − 1)π 1 = − , = − . (8i + 7)(8i + 9) 2 16 (8i + 11)(8i + 13) 16 15 i=0 i=0

1 = (3i + 1)(3i + 2)

√ 3π , 9

∞ ∑

1043

INFINITE SERIES, INFINITE PRODUCTS, AND SPECIAL FUNCTIONS

Fact 13.5.71. Let a, b ∈ (0, ∞), and assume that a , b. Then, ∞ ∑ i=1

In particular,

1 ψ(a + 1) − ψ(b + 1) = . (i + a)(i + b) a−b

√ 3π 3 log 3 1 = + , (i + 1)(3i + 1) 12 4

∞ ∑ i=0

∞ ∑ i=0

π 1 = + log 2, (i + 1)(4i + 1) 6

∞ ∑

∞ ∑ i=0 ∞ ∑

3 log 3 1 = − (i + 1)(3i + 2) 2

√ 3π , 6

1 π = 3 log 2 − , (i + 1)(4i + 3) 2

i=0 ∞ ∑

1 π 4 π 1 = 4 − 3 log 2 − , = + − log 2, (i + 1)(4i + 5) 2 (i + 1)(4i + 7) 9 6 i=0 i=0 √ √ ∞ ∞ ∑ ∑ 1 1 3π 3 log 3 3π 3 log 3 = + − 2 log 2, = − + 12 log 2, (2i + 1)(3i + 1) 6 2 (2i + 1)(3i + 2) 6 2 i=0 i=0 ∞ ∑

∞ ∑ π log 2 π log 2 1 1 = + , = − , (2i + 1)(4i + 1) 4 2 (2i + 1)(4i + 3) 4 2 i=0 i=0 √ √ ∞ ∞ ∑ ∑ 3π + 3 log 3 3π − 3 log 3 1 1 = , = , (2i + 1)(6i + 1) 8 (2i + 1)(6i + 5) 8 i=0 i=0 √ √ √ ∞ ∑ 2 log( 2 + 1) log 2 ( 2 + 1)π 1 = + + , (2i + 1)(8i + 1) 12 6 3 i=0 √ √ ∞ ∑ √ ( 2 − 1)π 1 2 = − − log( 2 + 1) + log 2, (2i + 1)(8i + 3) 4 2 i=0 √ √ √ ∞ ∑ ( 2 − 1)π 1 2 log( 2 + 1) = + − log 2, (2i + 1)(8i + 5) 4 2 i=0 ∞ ∑ i=0 ∞ ∑

1 1 √ = ( 3π − 3 log 3), (3i + 1)(3i + 2)(3i + 3) 12

∞ ∑ 1 1 π log 2 1 4 π log 2 = + + , = + + , (4i + 1)(2i + 3) 5 20 10 (4i + 1)(2i + 5) 27 36 18 i=0 i=0 √ √ ∞ ∞ ∑ ∑ 3π log 2 1 3π log 2 1 = + , = − , (6i + 1)(6i + 4) 27 9 (6i + 2)(6i + 5) 27 9 i=0 i=0 √ √ ∞ ∞ ∑ ∑ 1 3π log 3 3π log 2 1 = − , = − , (6i + 3)(6i + 5) 24 8 (6i + 4)(6i + 5) 18 3 i=0 i=0 √ √ √ √ √ ∞ ∞ ∑ ∑ 1 π + 2 log(3 + 2 2) 1 2π + 2 log(3 + 2 2) = , = , (8i + 1)(8i + 3) 16 (8i + 1)(8i + 5) 32 i=0 i=0 √ √ √ √ √ ∞ ∞ ∑ ∑ π − 2 2 log(1 + 2) 1 1 2π − 2 2 log(1 + 2) = , = , (8i + 3)(8i + 7) 32 (8i + 5)(8i + 7) 16 i=0 i=0

1044

CHAPTER 13 ∞ ∑ i=0

1 = (5i + 2)(5i + 4)

√

5[log(5 −

√ √ 5) − log(5 + 5)] . 20

Source: [116, pp. 58, 59], [593], and [1311, p. 17]. Remark: ψ is the digamma function. See Fact

13.3.3. Fact 13.5.72. ∞ ∑ i=0

π2 1 , = 6 (i + 1)2 ∞ ∑ i=0

∞ ∑

∞ ∑ i=0

1 π2 , = 8 (2i + 1)2

π G 1 = + , 2 16 2 (4i + 1) 2

∞ ∑ i=0

∞ ∑ i=0

1 1 d2 , = Γ(z) z=1/3 (3i + 1)2 9 dz2

π2 G 1 = − , (4i + 3)2 16 2

∞ ∑ 31ζ(5) π4 1 1 = = , , 4 5 96 32 (2i + 1) (2i + 1) i=0 i=0 i=0 √ √ ∞ ∞ ∑ ∑ 13ζ(3) 2 3π3 1 13ζ(3) 2 3π3 1 = + , = − , 27 243 27 243 (3i + 1)3 (3i + 2)3 i=0 i=0 √ √ ∞ ∞ ∑ ∑ 1 121ζ(5) 2 3π5 121ζ(5) 2 3π5 1 = = + , − , 243 2187 243 2187 (3i + 1)5 (3i + 2)5 i=0 i=0

1 7ζ(3) = , 8 (2i + 1)3

∞ ∑ i=0 ∞ ∑

∞ ∑

7ζ(3) π3 1 = + , 16 64 (4i + 1)3 5

∞ ∑ i=0

7ζ(3) π3 1 = − , 16 64 (4i + 3)3

∞ ∑

5π 5π5 1 31ζ(5) 31ζ(5) 1 + , − , = = 5 5 64 3072 64 3072 (4i + 1) (4i + 3) i=0 i=0 √ √ ∞ ∞ ∑ ∑ 1 1 91ζ(3) + 2 3π3 11(1023ζ(5) + 2 3π5 ) , . = = 216 23328 (6i + 1)3 (6i + 1)5 i=0 i=0 Source: [2545]. Related: Fact 13.3.2, Fact 13.5.86, and Fact 13.5.95. Fact 13.5.73. Let a, b, c be real numbers. Then, ∞ ( ∑ i=0

) b c a+b+c log 2 π a + + − = (3a + 2b + 3c) + (a − c) . 4i + 1 4i + 2 4i + 3 4i + 4 4 8

In particular,

) ∞ ( ∑ 1 1 2 3 8i + 5 = + − = log 2, (4i + 1)(4i + 3)(2i + 2) i=0 4i + 1 4i + 3 4i + 4 2 i=0 ( ) ∞ ∞ ∑ ∑ 1/6 1 1/2 1/2 1/6 log 2 π = − + − = − . (4i + 1)(4i + 2)(4i + 3)(4i + 4) 4i + 1 4i + 2 4i + 3 4i + 4 4 24 i=0 i=0 ∞ ∑

Source: [506, pp. 128, 129], [1647, p. 268], and [1686]. Fact 13.5.74. ∞ ∑ i=0 ∞ ∑ i=0

1 1 = , (2i + 1)(2i + 3)(2i + 5) 12

11 1 = , (2i + 1)(2i + 3)(2i + 7) 180

∞ ∑ i=1 ∞ ∑ i=1

i 1 = , (2i + 1)(2i + 3)(2i + 5) 24 i 13 = , (2i + 1)(2i + 3)(2i + 5) 360

INFINITE SERIES, INFINITE PRODUCTS, AND SPECIAL FUNCTIONS ∞ ∑

1 7 = , (2i + 1)(2i + 5)(2i + 7) 180

i=0 ∞ ∑ i=0

1 1 = , (2i + 3)(2i + 5)(2i + 7) 60

∞ ∑ i=1 ∞ ∑ i=1

1045

11 i = , (2i + 1)(2i + 5)(2i + 7) 360 i 1 = . (2i + 3)(2i + 5)(2i + 7) 40

Fact 13.5.75. ∞ ∑ i=0 ∞ ∑ i=0 ∞ ∑

1 π 1 = − , (4i + 1)(4i + 3)(4i + 5) 16 8

∞ ∑ i=1

1 π 1 = − , (4i + 1)(4i + 3)(4i + 7) 48 72

∞ ∑

7 π 1 = − , (4i + 1)(4i + 5)(4i + 7) 72 48

∞ ∑

i=0 ∞ ∑

5 3π i = − , (4i + 1)(4i + 3)(4i + 5) 32 64

i=1

i 7 3π = − , (4i + 1)(4i + 3)(4i + 5) 288 192 7π 31 i = − , (4i + 1)(4i + 5)(4i + 7) 192 288

i=1 ∞ ∑

1 5 π 5π 23 i = − , = − . (4i + 3)(4i + 5)(4i + 7) 24 16 (4i + 3)(4i + 5)(4i + 7) 64 96 i=0 i=1 √ △ △ Fact 13.5.76. Define α = 12 (1 + 5) and β = −1/α. Then, √ ) ∑ ∞ ∞ ( ∑ 1 1 1 2 5 10(2i + 1) 1 − − + = = log α, 5i + 1 5i + 2 5i + 3 5i + 4 (5i + 1)(5i + 2)(5i + 3)(5i + 4) 5 i=0 i=0 √ ) ( ∞ ∑ 6 4 α β2 β4 β5 1 i 5i − − − =π , (−1) β 5i + 1 5i + 3 5i + 4 5i + 5 55 i=0 √ ( ) ∞ ∑ 10 1 1 β2 β4 4 α i 5i (−1) β + − − =π , 5i + 1 5i + 2 5i + 4 5i + 5 55 i=0 √ ) ( 3 ∞ ∑ 14 α 1 1 α 4 α i 5i + − − =π (−1) β , 5i + 2 5i + 3 5i + 4 5i + 5 55 i=0 √ ) ( ∞ ∑ 2 β2 β3 β3 1 4 α i 5i , + + + = 2π (−1) β 5i + 1 5i + 2 5i + 3 5i + 4 55 i=0 √ ( ) ∞ ∑ 6 4 α 1 2 β2 β β2 i 5i , (−1) β + + + − =π 3 5i + 1 5i + 2 5i + 3 5i + 4 5i + 5 5 i=0 ( ) ∞ ∑ 1 4 2 1 1 − − − = π, 16i 8i + 1 8i + 4 8i + 5 8i + 6 i=0 ] [ ∞ ∑ α α2 α5 2α5 π2 1 α2 − − + + = , 2 2 2 2 2 5i 6 (5i + 2) (5i + 3) (5i + 4) (5i + 5) α (5i + 1) i=0 [ ] ∞ ∑ 1 16 16 8 16 4 4 2 − − − − − + = π2 , 16i (8i + 1)2 (8i + 2)2 (8i + 3)2 (8i + 4)2 (8i + 5)2 (8i + 6)2 (8i + 7)2 i=0 ) ∞ ( ∑ 1 1 1 1 1 1 + − + − − 7i + 1 7i + 2 7i + 3 7i + 4 7i + 5 7i + 6 i=0

1046

CHAPTER 13

√ ∞ ∑ 7π 7(2401i4 + 4802i3 + 3437i2 + 1036i + 108) = = . (7i + 1)(7i + 2)(7i + 3)(7i + 4)(7i + 5)(7i + 6) 7 i=0 Source: [666, 1229, 1230, 1682]. Fact 13.5.77. ∞ ∑ i=1 ∞ ∑ i=1

1 = 2 log 2 − 1, i(4i2 − 1)

∞ ∑ i=1

∞ ∑

3 1 = log 3 + 2 log 2 − 3, 2 i(36i − 1) 2

∞ ∑ i=1

π 1 1 = − , (4i2 − 1)2 16 2 2

∞ ∑ i=1 ∞ ∑ i=1

i(4i2

∞ ∑ i=1

i=1

i=1

π i = , 2 3 256 (4i − 1) 2

∞ ∑ i=1

π2 i2 = , (4i2 − 1)2 64

∞ ∑ 12i2 − 1 = 2 log 2, i(4i2 − 1)2 i=1

1 3 = − 2 log 2, 2 2 − 1)

2

π2 1 = 4 − − 2 log 2, 4 i(2i + 1)2

1 i = , (4i2 − 1)2 8

1 3π2 1 , = − 2 3 2 64 (4i − 1)

∞ ∑

1 3 = [log 3 − 1], i(9i2 − 1) 2

∞ ∑ i=1 ∞ ∑

i 7ζ(3) − 6 , = 64 (4i2 − 1)3

7ζ(3) + 2 i3 = , 2 3 256 (4i − 1)

i=1 ∞ ∑

∞ ∑ 2i + 1 3i2 + 3i + 1 1 = 3 − 4 log 2, = = 1, 2 2 i(i + 1)(2i + 1) i (i + 1) i3 (i + 1)3 i=1 i=1 i=1 ] ∞ [ ∑ 1 1 1 log 2 − = − , (4i + 1)(4i + 2)(4i + 3) (4i + 3)(4i + 4)(4i + 5) 2 2 i=0 ] ∞ [ ∑ 1 1 π 3 − = − . (4i + 2)(4i + 3)(4i + 4) (4i + 4)(4i + 5)(4i + 6) 4 4 i=0

∞ ∑

Source: Fact 1.12.27, [1217, pp. 9–12], [1524, p. 75], and [1787]. The first equality is given in

[506, p. 119]. The third equality is given in [1757, pp. 235, 236]. The first expression in the third to last equality is given in [1566, p. 63]. The last two equalities are given in [1647, p. 269]. Related: Fact 1.12.26 and Fact 1.12.27. Fact 13.5.78. ∞ ∑ 3i2 − 1 3 = , 3 2 8 (i − i) i=2 ∞ ∑ i=0

∞ ∑ i=0

1 π(π − 2) = , 2 2 32 [4(2i + 1) − 1]

6i + 3 3 = , 4i4 + 8i3 + 8i2 + 4i + 3 2

∞ ∑

∞ ∑ i=1

∞ ∑ i=1

i4

i 1 = , 2 +i +1 2

i + 3i + 3 1 =− , 2 i4 + 2i3 − 3i2 − 4i + 2 2

∞ ∑

1 3 1 1 = − log 2, = log 2 − , 2i(2i + 1)(2i + 2) 4 (2i + 1)(2i + 2)(2i + 3) 2 i=1 i=0 √ ∞ ∞ ∑ ∑ 1 3π log 3 9i + 5 = − , = log 3, (3i + 1)(3i + 2)(3i + 3) 12 4 (3i + 1)(3i + 2)(3i + 3) i=0 i=0 ∞ ∑ i=0

1 1 = , (i + 1)(i + 2)(i + 3)(i + 4) 18

∞ ∑ i=0

1 13 = , (i + 1)(i + 2)(i + 3)(i + 5) 288

INFINITE SERIES, INFINITE PRODUCTS, AND SPECIAL FUNCTIONS ∞ ∑ i=0 ∞ ∑ i=1

∞ ∑

1 5 = , (i + 1)(i + 2)(i + 4)(i + 5) 144

i=0

1047

211 1 = , (i + 1)(i + 2)(i + 4)(i + 6) 7200 ∞ ∑

i2 5 = , (i + 1)(i + 2)(i + 3)(i + 4) 36 i=1 √ ∞ ∑ 1 3π log 3 1 = + − , (3i + 1)(3i + 2)(3i + 3)(3i + 4) 6 36 4 i=0

1 1 = , (2i + 1)(2i + 2)(2i + 4)(2i + 5) 360

∞ ∑ i=0

∞ ∑

1 = 10 − π2 , 3 (i + 1) (i + 2)3

i=0

(i +

1)3 (i

1 29 3ζ(3) − . = 3 3 32 4 + 2) (i + 3)

Source: The first equality is given in [109, pp. 272, 273]. The second equality is given in [645,

p. 99]. The third equality is given in [2294, p. 70]. The sixth and seventh equalities are given in [1568, p. 199]. The eighth equality is given in [1647, p. 268]. The ninth equality is given in [506, p. 125], and is given by Fact 13.5.80. The 14th equality is given in [1647, p. 268]. The 15th equality is given in [1566, p. 64]. The 16th equality is given in [1217, p. 10]. The 17th equality is given in [1647, p. 272]. Fact 13.5.79. Let x be a real number. Then, ∞ ∑ 3(x2 + 2) i(i4 + 4 − x2 ) = . 8 2 4 2 2 2 2 2 i + 2(x + 4)i + 16x i + (x + 4) 4(x + 1)(x2 + 4) i=1 In particular,

∞ ∑ i=1 ∞ ∑ i=1

i 3 = , 4 i +4 8

∞ ∑ i=1

5

i 9 = , i8 + 16i4 + 64i2 + 64 80

i8

i(i4 + 3) 9 , = 4 2 + 10i + 16i + 25 40 ∞ ∑ i=1

33 i(i4 − 5) = . i8 + 26i4 + 144i2 + 169 520

Source: [512]. Fact 13.5.80. Let k ≥ 2. Then,

  k−1 ∞ ∑ ∑  k − 1  1  = log k.  − ki + j ki + k  i=0 j=1

In particular,

∞ ( ∑ i=0

) 1 1 − = log 2, 2i + 1 2i + 2

∞ ( ∑ i=0

) 1 1 2 + − = log 3. 3i + 1 3i + 2 3i + 3

Source: [1787]. Fact 13.5.81. Let m and n be positive integers. Then, ∞ ( ∑ i=1

In particular,

∞ ( ∑ i=1

Source: [1787].

) 1 π mπ 1 − = cot . (n + m)i − n (n + m)i − m n+m n+m

) 1 1 π 2π − = cot , 7i − 5 7i − 2 7 7

∞ ( ∑ i=1

) 1 1 π 3π − = cot . 11i − 8 11i − 3 11 11

1048

CHAPTER 13

Fact 13.5.82.

∞ ∑ i=0

∞ ∑ i=0

(2i)! = log 2, (2i + 2)!

2 log 2 5 (2i)! = − , (2i + 4)! 3 12 ∞ ∑ i=0

2

∞ ∑

√

i=0

(i!) 2 3π = , (2i + 1)! 9

∞ ∑ i=0

(2i)! 1 = log 2 − , (2i + 3)! 2

(2i)! log 2 2 = − , (2i + 5)! 3 9 ∞ ∑ π 2i (i!)2 = , (2i + 1)! 2 i=0

∞ ∑

(2i)! 2 log 2 131 = − , (2i + 6)! 15 1440 i=0 √ ∞ ∑ 4 3π 3i (i!)2 = , (2i + 1)! 9 i=0

∞ ∑

∞ ∞ ∞ ∑ ∑ ∑ (i!)2 π2 2i (i!)2 π2 3i (i!)2 2π2 4i (i!)2 π2 = , = , = , = , (2i + 2)! 18 (2i + 2)! 16 (2i + 2)! 27 (2i + 2)! 8 i=0 i=0 i=0 i=0 √ √ ∞ ∞ ∞ ∑ ∑ ∑ π2 2 3π π2 π 2π2 4 3π 4 2i (i!)2 3i (i!)2 (i!)2 = + − 4, = + − 2, = + − , (2i + 3)! 18 3 (2i + 3)! 16 2 (2i + 3)! 27 27 3 i=0 i=0 i=0 √ ∞ ∞ ∞ ∑ (i!)2 ∑ 2i (i!)2 ∑ 4i (i!)2 π2 π2 π2 3π 7 3π 42 = − 1, = + − , = + − , (2i + 3)! 8 (2i + 4)! 12 2 12 (2i + 4)! 16 8 4 i=0 i=0 i=0 √ [ ] ∞ ∞ ∞ 2 ∑ ∑ ∑ 3i (i!)2 5π2 4i (i!)2 3π2 7 i! π2 3π 7 = + − , = − , = , (2i + 4)! 81 9 6 (2i + 4)! 32 8 (i + 1)! 6 i=0 i=0 i=0 ]2 ]2 ]2 ∞ [ ∞ [ ∞ [ ∑ ∑ ∑ π2 π2 39 5π2 197 i! i! i! = − 3, = − , = − , (i + 2)! 3 (i + 3)! 4 16 (i + 4)! 54 216 i=0 i=0 i=0 √ ∞ ∞ ∞ ∑ ∑ ∑ √ (2i)! π 2 (2i)! i (2i)! = = = 1 − 2, − 1, (−1) − 1, (−1)i i 2 i (i!)2 (2i + 1) i (i!)2 2 2 4 4 4 (i!) (2i − 1) i=1 i=1 i=1 ∞ ∑ 16i (i!)4 = 8πG − 14ζ(3), 3 i [(2i)!]2 i=1

∞ ∑ i=1

7 16i (i!)4 = ζ(3) − πG. 3 2 2 (2i + 1) [(2i)!]

Source: [323], [968, p. 103], and [1647, p. 269]. Fact 13.5.83. ∞ ∑ π (i + 1)! = , (2i + 1)!! 2 i=1 ∞ ∑ i=0

∞ ∑ i=0

π (2i − 1)!! = , (2i)!!(2i + 1) 2

∞ ∑ i=0

(2i − 1)!! = (2i)!!(3i + 1)

√ πΓ( 43 ) Γ( 56 )

,

∞ ∞ ∑ [(2i + 1)!!]2 8G ∑ [(2i + 1)!!]2 4 (2i + 1)!! = log 4, = 4 log 2 − , = , 2 (i + 1) 2 (i + 2) (2i + 2)!!(i + 1) π π [(2i + 2)!!] [(2i + 2)!!] i=0 i=0 ∞ ∑ i=0

(2i − 1)!! π = log 2, 2 2 (2i)!!(2i + 1) ∞ ∑ i=0

∞ ∑ i=0

i=0

i=0

π (2i − 1)!! π3 2 = log 2 + , 48 (2i)!!(2i + 1)3 4

π (2i − 1)!! π3 πζ(3) = log3 2 + log 2 + , 4 12 48 8 (2i)!!(2i + 1)

π2 (2i + 1)!! = − 2 log2 2, 6 (2i + 2)!!(i + 1)2 ∞ ∑

∞ ∑

∞ ∑ i=0

Γ2 ( 1 ) (2i − 1)!! = √4 , (2i)!!(4i + 1) 4 2π

(2i + 1)!! 4 log3 2 π2 log 2 = − + 2ζ(3), 3 3 (2i + 2)!!(i + 1)3 ∞ ∑ i=0

1 (2i − 1)!! π Γ2 ( 4 ) = √ , (2i)!!(4i + 1)2 4 4 2π

1049

INFINITE SERIES, INFINITE PRODUCTS, AND SPECIAL FUNCTIONS

( 2 ( 3 ) ) ∞ 1 1 ∑ π 5π (2i − 1)!! (2i − 1)!! G Γ2 ( 4 ) πG Γ2 ( 4 ) = = , , + + √ √ 32 2 4 2π 384 8 4 2π (2i)!!(4i + 1)3 (2i)!!(4i + 1)4 i=0 i=0 ]3 [ ]5 [ ∞ ∞ ∑ ∑ 2 2 (2i − 1)!! 2 (2i − 1)!! i = , (−1) (4i + 1) = = , (−1)i (4i + 1) 2( 1 ) 4( 3 ) (2i)!! π (2i)!! Γ Γ i=0 i=0 2 4 ]3 [ ]2 [ ∞ ∑ Γ(9/8) i (2i − 1)!! = . (−1) (2i)!! Γ(5/4)Γ(7/8) i=0 ∞ ∑

Source: [116, p. 182], [650, 1206], [1317, pp. 16, 17], [1647, pp. 268, 269], and [1675, pp. 51,

58, 59]. Fact 13.5.84. Let x ∈ (1, ∞). Then, ∞ ∑ 1 x = log . i xi x−1 i=1

In particular,

∞ ∑ 1 = log 3, 3 i i=1 ( 2 ) i

∞ ∑ 1 = log 2, 2i i i=1

∞ ∑ 1 10 = log . i 10 i 9 i=1

Source: [516, p. 129]. Fact 13.5.85. ∞ ∑ 2i − 1 i=0

2i i! ∞ ∑ i=0

= 0,

3 1 = log 2, 9i (2i + 1) 2 ∞ ∑ i=0

∞ ∑ i=1

∞ ∑ 2i (i!)2 π = , (2i + 1)! 2 i=0 ∞ ∑ i=0

∞ ∑ 1 π2 1 − log2 2, = i i2 12 2 2 i=1

256 1 = 8 log , 16i (i + 1) 225

1 1 1 = log 3 + atan , 16i (4i + 1) 2 2

i = 1, (i + 1)!

∞ 2 ∑ i +i−1 i=0

(i + 2)!

∞ 3 ∑ i + 6i2 + 11i + 5 i=0

(i + 3)!

=

5 , 2

= 0,

∞ ∑ i=0

i=0

5 1 = 2 log , 16i (2i + 1) 3

(i + 2)! (2i + 4)!

∞ ∑ (4i + 1)i!

= 2,

∞ ∑ (i + 1)(i + 2)(i + 3) i=1

i=0

1 = 8 log 3, + 1)

4i (2i

1 1 = 2 log 3 − 4 atan , 16i (4i + 3) 2

∞ 2 ∑ i + 3i + 1 i=0

∞ ∑

∞ ∑

i=0

=

(2i + 1)!

= 2,

e2 − 4e + 5 , 16e

∞ ∞ ∑ ∑ i 1 e2 − 1 1 1 ( )= , , = , ∏ ∏ i 2 i 1 2 (2i + 4)! 2 2 8e j=1 (2 j + 1) i=1 i=0 i=2 i j=2 1 − j2 √ ∞ ∞ ∑ ∑ 1 (4i)!(1103 + 26390i) 9801 2 (6i)!(13591409 + 545140134i) = , (−1)i = , 4 3964i 3 6403203i+3/2 4π 12π (i!) (3i!)(i!) i=0 i=0  ∏i−1 3  ∏i−1 4 √ ∞ ∞ ∑ ∑  j=0 (2 j + 1)   j=0 (4 j + 1)  2 2 2 k     1+ (−1) (4i + 1)  (8i + 1)   = , 1 +  = √ 2 3 . 2i i! π 4i i! πΓ ( 4 ) i=1 i=1 ∞ ∑ (i + 1)(i + 2)(2i + 3)

=

Source: [213], [516, p. 108], [1566, pp. 28, 64, 65], and [1581]. The last two equalities are given

in [1317, pp. xxv, xxvi].

1050

CHAPTER 13

Fact 13.5.86. Define △

G=

∞ ∑ (−1)i i=0

1 ≈ 0.9159655941772190150546035149323841107741493742816721. (2i + 1)2

Then,

∞ ∑ i=0 ∞ ∑ i=0

π2 G 1 = + , (4i + 1)2 16 2

2k + 1 G = , 8 (4i + 1)2 (4k + 3)2

∞ ∑ i=0

π2 G 1 = − , (4i + 3)2 16 2

∞ ∑ 16i (12i3 + 12i2 + 6i + 1)(i!)4

i3 (2i + 1)3 [(2i)!]2

i=1

= 4πG − 4.

Remark: G is Catalan’s constant. See Fact 14.9.1. Source: [323]. Related: Fact 13.5.72. Fact 13.5.87.

) ) ∞ i ( i ( ∞ ∑ ∑ 1 π 1∏ 3 π 1∏ 1− = 3 log 2 + , 1− = 3 log 2 − , i 4 j 2 i 4 j 2 i=1 j=1 j=1 i=1     i ∞ i ∞ i i ∑ ∑ 1 ∏ 4 j − 1 ∏ 4 j − 3  1 ∏ 4 j − 1 ∏ 4 j − 3    = 6 log 2,   = π, + − i 4j 4j  i 4j 4j  i=1

j=1

i=1

j=1

j=1

j=1

   i  i ∞ ∞ i i ∑ ∑ ∏ ∏ 1 ∏ 4 j 1 ∏ 4 j 4 j  4 j  2  = 2π ,  = 16G,   + − i2  j=1 4 j − 1 j=1 4 j − 3  i2  j=1 4 j − 1 j=1 4 j − 3  i=1 i=1  i  i (  i ( ) ∑ ) ∞ ∑  i 1 1  ∏ 3  π2 1 ∑ 1  ∏     = , 1− 1− +  i  j=1 4 j − 1  j=1 4j 4 j − 3  j=1 4j  2 j=1 i=1       ( ( ) ) i i i i ∞ ∑ ∑ 1 1  ∏ 3  1 ∑ 1  ∏     = 4G, 1− 1− −  i j=1 4 j − 1 j=1 4j 4 j − 3 j=1 4j  j=1 i=1  i (  i ( ) ∏ ) ) ∏ ) √ ∞ ∞ i ( i ( ∑ ∑ 1 ∏ 3π 1 1 ∏ 1 2  2    = 3 log 3,   = 1− 1− + − , 1− 1− i j=1 3j 3j i j=1 3j 3j 3 i=1 i=1 j=1 j=1  i  i  i   i ∞ ∑  4π2 ∏ 3 j − 2  ∑ 1 1 ∏ 3 j − 1  ∑ 1     = +  , i j=1 3 j 3j − 1 3j 3j − 2 9 j=1 j=1 j=1 i=1  i  i  i 2  √ ∞ 2 ∑ ∑ 1   8 3π3 ∏ 3 j − k  ∑ 1 1∑   k          = −  (−1)  , i k=1 3 j   j=1 (3 j − k)2  j=1 3 j − k   81 j=1 i=1   i   i 2 i ∞  ∑ 1 Hi ∑ ∏ 4 j − k  ∑ 1  ∑  = π4 .     − i 4j  4j − k (4 j − k)2  i=2

k=1,3

If, in addition, x ∈ (0, 1), then

j=1

j=1

j=1

∞ ∑ (1 − x)i − xi i=1

i(i!)

= π cot πx.

Source: [62]. Fact 13.5.88. ∞ ∑ i=1

2i 1 ∏ 2j − 1 √ = 2 − 1, 2i + 1 j=1 2 j

∞ ∑ i=1

√3 2i 1 ∏ 3j − 2 3 4 = − 1. 2i + 1 j=1 3 j 4

1051

INFINITE SERIES, INFINITE PRODUCTS, AND SPECIAL FUNCTIONS

Source: [2461]. Fact 13.5.89. Let n ≥ 1. Then, ∞ ∑ 1 (−1)i ∏i+n j=i

i=1

2n log 2 1 ∑ 2n−i + . n! n! i=1 i n

j

=−

In particular, ∞ ∑ (−1)i i=1 ∞ ∑ (−1)i i=1

∞ ∑ (−1)i

1 = 1 − 2 log 2, i(i + 1)

i=1 ∞ ∑ (−1)i

8 4 log 2 1 = − , i(i + 1)(i + 2)(i + 3) 9 3

i=1

1 5 = − 2 log 2, i(i + 1)(i + 2) 4

1 131 2 log 2 = − . i(i + 1)(i + 2)(i + 3)(i + 4) 288 3

Source: [107, pp. 37, 210, 211]. Fact 13.5.90. Let n ≥ 1, let m ∈ Z, and assume that n + m ≥ 1. Then, ∞ ∑ (−1)i+1 i=1

Therefore, for all n ≥ 1,

1 = in + m

1 1∑ (−1)i+1 = n i=1 i ∞

∫ 0

1

∫

0

1

xn+m−1 dx. xn + 1

xn−1 log 2 dx = . n x +1 n

In particular, ∞ ∑ 1 (−1)i+1 = 1 − log 2, i+1 i=1 ∞ ∑ 7 1 = log 2 − , (−1)i+1 i + 4 12 i=1 ∞ ∑ (−1)i+1 i=1 ∞ ∑ (−1)i+1 i=1

∞ ∑ 1 1 (−1)i+1 = log 2 − , i+2 2 i=1 ∞ ∑ 1 47 (−1)i+1 = − log 2, i + 5 60 i=1 ∞ ∑ (−1)i+1

1 1 = log 2, 2i − 1 2

1 13 π = − , 2i + 5 15 4

i=1 ∞ ∑ (−1)i+1 i=1

π 76 1 = − , 2i + 7 4 105

∞ ∑

(−1)i+1

i=1 ∞ ∑ (−1)i+1 i=1 ∞ ∑

π 2 1 = − , 2i + 3 4 3

(−1)i+1

i=1

1 37 = log 2 − , i+6 60

263 π 1 = − , 2i + 9 315 4

√ 1 3π log 2 3π log 2 + , = − , 9 3 3i − 1 9 3 i=1 i=1 √ √ ∞ ∞ ∑ ∑ 1 1 1 3π log 2 3π log 2 i+1 i+1 =1− − , (−1) = − + , (−1) 3i + 1 9 3 3i + 2 2 9 3 i=1 i=1 √ √ ∞ ∞ ∑ ∑ 1 1 3 3π log 2 1 3π log 2 i+1 i+1 (−1) = + − , (−1) = − − , 3i + 4 9 3 4 3i + 5 9 3 10 i=1 i=1 √ √ √ √ √ √ ∞ ∞ ∑ ∑ 2π 2 log(3 − 2 2) 2π 2 log(1 + 2) 1 1 i+1 i+1 (−1) = + , (−1) = + , 4i − 1 8 8 4i − 3 8 4 i=1 i=1 √ √ √ √ √ √ ∞ ∞ ∑ 2 log(3 − 2 2) − 2π ∑ 2 log(3 − 2 2) − 2π 1 1 1 (−1)i+1 = 1+ , (−1)i+1 = − , 4i + 1 8 4i + 3 3 8 i=1 i=1 ∞ ∑ (−1)i+1

1 = 3i − 2

√

π 1 =1− , 2i + 1 4

∞ ∑ 1 5 (−1)i+1 = − log 2, i+3 6 i=1

∞ ∑ (−1)i+1

1052

CHAPTER 13

√ √ √ 2π − 2 log(3 − 2 2) 4 − , 8 5 i=1 √ √ √ ∞ ∑ 2 log(3 − 2 2) + 2π 4 1 = − . (−1)i+1 4i + 7 8 21 i=1 ∞ ∑ (−1)i+1

1 = 4i + 5

Source: [712]. Fact 13.5.91. Let n ≥ 2 and 1 ≤ m ≤ n − 1. Then, ∞ ∑ (−1)i+1 i=1

1 = in + m

In particular,

∫

1 0

1 π mπ 2 xn+m−1 dx = − csc + n x +1 m 2n n n

⌊(n−2)/2⌋ ∑ ( i=0

) (2i + 1)mπ (2i + 1)π cos log sin . n 2

∫ 1 π x2 1 dx = 1 − , = 2 2i + 1 4 0 x +1 i=1 √ ∫ ∞ 1 ∑ x3 3π log 2 1 = − , dx = 1 − (−1)i+1 3 3i + 1 9 3 x + 1 0 i=1 √ ∫ 1 ∞ ∑ 1 1 3π log 2 x4 i+1 (−1) dx = − = + . 3+1 3i + 2 2 9 3 x 0 i=1 ∞ ∑ (−1)i+1

Source: [628]. Fact 13.5.92. ∞ ∞ ∑ ∑ 1 i 1 (−1) = = log 2, i + 1 (2i + 1)(2i + 2) i=0 i=0

∞ ∑ (−1)i

∑ 1 2 π = = , 2i + 1 (4i + 1)(4i + 3) 4 i=0 i=0 √ ∞ ∞ ∑ ∑ 3π log 2 1 3 (−1)i = = + , 3i + 1 i=0 (6i + 1)(6i + 4) 9 3 i=0 √ √ √ √ √ ∞ ∞ ∑ ∑ 2π 2 log(1 + 2) 2π 2 asinh 1 4 1 = = + = + . (−1)i 4i + 1 (8i + 1)(8i + 5) 8 4 8 4 i=0 i=0 ∞

Fact 13.5.93. If n ≥ 2, then ∞ ∞ ∑ 1 2n − 2 ∑ 1 (−1)i+1 n = . i 2n i=1 in i=1

If n ≥ 1, then

∞ ∑ (22n−1 − 1)π2n 1 (−1)i+1 2n = |B2n |. (2n)! i i=1

If n ≥ 0, then ∞ ∑ (−1)i+1 i=1

π2n+1 4n π2n+1 1 = |E | = |B2n+1 (1/4)|. 2n (2n + 1)! (2i − 1)2n+1 22n+2 (2n)!

In particular, ∞ ∑ 1 π2 (−1)i+1 2 = , 12 i i=1

∞ ∑ 1 7π4 (−1)i+1 4 = , 720 i i=1

∞ ∑ 1 31π6 (−1)i+1 6 = , 30240 i i=1

1053

INFINITE SERIES, INFINITE PRODUCTS, AND SPECIAL FUNCTIONS ∞ ∑ (−1)i+1 i=1 ∞ ∑ (−1)i+1

1 π = , 2i − 1 4

5π5 1 , = 5 1536 (2i − 1)

i=1

∞ ∑ (−1)i+1 i=1 ∞ ∑

π3 1 = , (2i − 1)3 32

(−1)i+1

i=1

61π7 1 = . 7 184320 (2i − 1)

Source: The expression involving the Bernoulli polynomial is given in [116, p. 56] and [705]. Remark: Each alternating series can be rewritten as a nonalternating series. For example, ∞ ∞ ∑ 1 ∑ π2 4i + 3 . (−1)i+1 2 = = 2 2 12 i (2i + 1) (2i + 2) i=1 i=0

Related: Fact 14.6.15. Fact 13.5.94. ∞ ∑ (−1)i+1 i=1

1 = 2 log 2 − 1, i(i + 1) ∞ ∑ (−1)i+1 i=1

∞ ∑ (−1)i+1 i=1

i3 (i

i=1

i=1 ∞ ∑ (−1)i+1 i=1

i=1

i=1

i=1

∞ ∑ (−1)i+1 i=1

i3 (i

i3 (i

1 = 3 − 4 log 2, + 1)2

∞ ∑ (−1)i+1 i=1

5 1 π2 − , = 2 2 24 16 i (i + 2)

∞ ∑ (−1)i+1 i=1

i2 (i

17 1 4 = − log 2, 2 108 27 + 3)

2 1 61 π2 1 = log 2 + ζ(3) − − , 2 27 12 972 324 + 3)

1 π2 3 = + 2 log 2 + ζ(3) − 3, 4 i(i + 1)3 12 ∞ ∑ (−1)i+1 i=1

i=1

i2 (i

1 3 π2 3 = − + ζ(3), 2 16 32 16 + 2)

1 π2 2 41 = + log 2 − , 2 36 9 108 i(i + 3)

i=1

π2 2 5 1 = − log 2 + , 54 + 3) 36 9

π2 1 3 ζ(3) − − 4, = 6 log 2 + 4 12 i3 (i + 1)2

1 1 π2 − , = i(i + 2)2 2 24

∞ ∑ (−1)i+1

i2 (i

∞ ∑ (−1)i+1

1 π2 = + 2 log 2 − 2, 2 12 i(i + 1)

∞ ∑ (−1)i+1

∞ ∑ (−1)i+1

∞ ∑ (−1)i+1

1 π2 5 2 1 log 2 + ζ(3) − − , = 4 108 162 i3 (i + 3) 27

∞ ∑ (−1)i+1

i=1

1 π2 3 = 2 log 2 − + ζ(3) − 1, 12 4 + 1)

1 2 5 = log 2 − , i(i + 3) 3 18 ∞ ∑ (−1)i+1

∞ ∑ (−1)i+1

i=1

π2 1 = − 2 log 2 + 1, i2 (i + 1) 12

∞ ∞ 1 ∑ 1 1 π2 3 π2 1 ∑ 1 1 = , (−1)i+1 2 − , (−1)i+1 3 − + ζ(3), = = i(i + 2) 4 i=1 i (i + 2) 24 8 i=1 i (i + 2) 16 48 8

∞ ∑ (−1)i+1

i=1

∞ ∑ (−1)i+1

∞ ∑ i=1

(−1)i+1

1 π2 3 = 6 − − 6 log 2 − ζ(3), 12 4 i2 (i + 1)3

1 3 = 12 log 2 + ζ(3) − 10, 3 3 2 i (i + 1)

1054

CHAPTER 13 ∞ ∑ (−1)i+1 i=1 ∞ ∑ (−1)i+1 i=1

∞ ∑ (−1)i+1 i=1

11 π2 3 1 = − − ζ(3), i(i + 2)3 16 48 8

∞ ∑ (−1)i+1 i=1

∞ ∑ (−1)i+1

11 π2 1 = − , 3 3 32 32 i (i + 2)

i=1

π2 3 1 1 = + ζ(3) − , 2 i2 (i + 2)3 32 16

π2 1 2 1 31 = + log 2 + ζ(3) − , 3 108 27 4 72 i(i + 3)

∞ 2 ζ(3) ∑ 1 ζ(3) 503 1 127 π 4 − − log 2 − , (−1)i+1 3 log 2 + − . = = 12 i=1 18 5832 i2 (i + 3)3 648 324 27 i (i + 3)3 81 2

Fact 13.5.95. Let x be a real number, and assume that x is not an integer. Then, ∞ ∑

1 π2 = , (i + x)2 sin2 πx i=−∞

∞ ∑ i=−∞

(

( πx )2 )2 = sin πx . +1 1

i x

In particular, ∞ ∑

π2 1 = , 2 4 (2i + 1) i=−∞

∞ ∑

∞ ∑

π2 1 = . 2 8 (4i + 1) i=−∞

4π2 1 = , 2 27 (3i + 1) i=−∞

Source: Fact 13.4.10. Related: Fact 13.5.72. Fact 13.5.96. Let n ≥ 1, and define Qn ∈ R[s] and Pn ∈ R[s] as in Fact 13.2.14. If n is odd,

then

∞ ∑ πn+1 Pn (0) 1 = , in+1 2(2n+1 − 1)n! i=1

If n is even, then ∞ ∑ (−1)i i=0

∞ ∑ (−1)⌊(i+1)/2⌋ i=0

1 πn+1 Qn (0) = , (2i + 1)n+1 2n+2 n!

∞ ∑

(−1)

1 = (2i + 1)n+1

⌊i/2⌋

i=0

√ n+1 2π Qn (1) . 4n+1 n!

1 = (2i + 1)n+1

√ n+1 2π Qn (1) . 4n+1 n!

In particular, ∞ ∑ (−1)i

1 π = , 2i + 1 4

∞ ∑ (−1)i

π3 1 = , (2i + 1)3 32

∞ ∑ (−1)i

5π5 1 = , (2i + 1)5 1536 i=0 i=0 i=0 √ 2 √ ∞ ∞ ∑ ∑ 2π 1 1 11 2π4 ⌊(i+1)/2⌋ ⌊(i+1)/2⌋ (−1) = , (−1) = , 16 6144 (2i + 1)2 (2i + 1)4 i=0 i=0 √ √ ∞ ∞ ∑ ∑ 1 2π 1 361 2π6 ⌊i/2⌋ ⌊(i+1)/2⌋ , = , (−1) (−1) = 6 491520 2i + 1 4 (2i + 1) i=0 i=0 √ √ ∞ ∞ ∑ ∑ 1 3 2π3 19 2π5 1 ⌊i/2⌋ ⌊i/2⌋ , (−1) . = = (−1) 128 8192 (2i + 1)3 (2i + 1)5 i=0 i=0

Source: [1430]. Fact 13.5.97. ∞ ∑ 1 1 (−1)i = , i! e i=0

∞ ∑ 1 (−1)i+1 = log 2, i i=1

∞ ∑ Hi log2 2 π2 = − , (−1)i i 2 12 i=1

∞ ∑ 1 (−1)i = cos 1, (2i)! i=0

∞ ∑ Hi log2 2 (−1)i+1 = , i+1 2 i=1

∞ ∑ 2i + 1 (−1)i+1 = 1, i(i + 1) i=1

∞ ∑ (−1)i+1 i=0

1 = sin 1, (2i − 1)!

1055

INFINITE SERIES, INFINITE PRODUCTS, AND SPECIAL FUNCTIONS ∞ ∑ (−1)⌊(i+5)/3⌋ i=1

1 5π = , 2i − 1 12

∞ ∑ (−1)i+1 i=1

∞ ∑ (−1)⌊(i+3)/2⌋ i=1

1 5 π = − , (2i + 1)(2i + 3) 6 4

√ 2π , 4

1 = 2i − 1

∞ ∑ (−1)i+1 i=1

∞ ∑

(−1)i+1

i=1

1 = 1 − log 2, i(4i2 − 1)

1 2 π = − , (2i + 1)(2i + 3)(2i + 5) 5 8

∞ ∑ 83 π 1 1 2 log 2 1 = − , (−1)i+1 = − , (2i + 1)(2i + 3)(2i + 5)(2i + 7) 630 24 (3i + 1)(3i + 2) 2 3 i=1 i=1 √ √ ∞ ∞ ∑ ∑ 1 3π 2 log 2 2π 1 2i + 1 i+1 i (−1) = + − , = , (−1) (3i + 1)(3i + 2)(3i + 3) 6 18 3 (4i + 1)(4i + 3) 16 i=1 i=0 √ ∞ ∞ ∑ ∑ 3π log 2 1 π2 i i+1 − log 2, (−1) = − , = (−1)i+1 2 12 3i − 1 9 3 (i + 1) i=1 i=1 √ ∞ ∞ ∞ ∑ ∑ ∑ 1 1 3 3π log 2 i i+1 (−1) (−1) = = = + , 3i + 1 i=1 3i − 2 i=0 (6i + 1)(6i + 4) 9 3 i=0

∞ ∑ (−1)i+1

∞ ∑ (−1)i+1 i=1

1 = log 4 − 1, i(i + 1) ∞ ∑ (−1)i+1 i=1

∞ ∑ (−1)i+1 i=1 ∞ ∑ (−1)i+1 i=1 ∞ ∑ (−1)i+1 i=1

i=1

i=1 ∞ ∑ (−1)i i=1 ∞ ∑ (−1)i i=0

2 131 1 = log 2 − , i(i + 1)(i + 2)(i + 3)(i + 4) 3 288

1 1 = , i(i + 2) 4

1 2 log 2 5 = − , i(i + 3) 3 18 ∞ ∑

(−1)i+1

i=1

1 5 = log 4 − , i(i + 1)(i + 2) 4

1 4 8 = log 2 − , i(i + 1)(i + 2)(i + 3) 3 9

11 1 = , i(i + 2)(i + 4)(i + 6) 1440

∞ ∑ (−1)i+1

∞ ∑ (−1)i+1

∞ ∑ (−1)i+1 i=1 ∞ ∑ (−1)i+1 i=1 ∞ ∑ (−1)i+1 i=1

1 5 = , i(i + 2)(i + 4) 96 31 1 = , i(i + 2)(i + 4)(i + 6)(i + 8) 35840

2 log 2 137 1 = − , i(i + 3)(i + 6) 9 1080

√ 1 1 = [4 − 8 log 2 + ( 3 − 1)π], i(2i + 1)(3i − 1) 5

1 = 3 − π, i(i + 1)(2i + 1)

π log 2 1 = − , (2i + 1)(2i + 2) 4 2

∞ ∑ (−1)i i=2

∞ ∑ (−1)i i=0

∞ ∑ 1 5 − π log 2 (−1)i ∏4 = − , 12 6 (2i + j) j=1 i=0 ∞ ∑ 23 1 3π log 2 = (−1)i ∏6 − + , 140 380 60 j=1 (2i + j) i=0

1 = π − 3, i(i − 1)(2i − 1)

1 1 log 2 = − , (2i + 1)(2i + 2)(2i + 3) 2 2

∞ ∑ i=0

(−1)i ∏5

1

j=1 (2i + j)

=

π 5 − , 36 24

∞ ∑ 1 log 2 79 (−1)i ∏7 = − , 180 21600 j=1 (2i + j) i=0

1056

CHAPTER 13

( ) ∞ ∑ 1 π 109 1 i = − , (−1) ∏9 2520 4 140 j=1 (2i + j) i=0

∞ ∑ π 1 log 2 67 = (−1)i ∏8 − − , 2520 1260 37800 j=1 (2i + j) i=0 ∞ ∑ (−1)i i=0

1 5 log 2 = − , (2i + 1)(2i + 2)(2i + 4)(2i + 5) 36 6

∞ ∑ (−1)i

π 149 1 = − , (2i + 1)(2i + 2)(2i + 6)(2i + 7) 60 3600 i=0 √   ∞ 2 ∑ 1  3π  i+1 i  , (−1) 3 = 1 − log 2 + π sech 2 i +1 3 i=1

∞ ∞ ∑ ∑ π log 2 64i2 + 80i + 22 ⌊(i+3)/2⌋ 1 = = + , (−1) i (4i + 1)(4i + 2)(4i + 3)(4i + 4) 4 2 i=0 i=1 ∞ ∑ (2i + 1)3 = 0, (−1)i (2i + 1)4 + 4 i=0 ∞ ∑ (−1)i i=1

1 1 = − log 2, i(4i4 + 1) 2

∞ ∑ (−1)i

1 π , = 4 (2i + 1)[(2i + 1) + 4] 16 i=0 [ ] ∞ ∑ 1 1 log 7 (−1)i . + =1− 3i+2 3i+3 2 (6i + 5)3 (6i + 7)3 i=0

Source: [116, p. 60], [506, p. 25], [712], [1217, pp. 10–12], [1566, pp. 69, 104], [1568, p. 199],

[1647, pp. 268, 269], [1757, p. 181], [2481], and [2577]. Fact 13.5.98. Let a ≥ 1. Then, ∞ ∞ ∑ ∑ 1 a+1 a+1 1 (−1)i+1 i = log , = a log . (−1)i i a i a a (i + 1) a i=1 i=0 Fact 13.5.99. Let a ≥ 1. Then ∞ ∑ (−1)i i=0

√ √ 1 = a acot a. + 1)

ai (2i

In particular, ∞ ∑ (−1)i i=0

π 1 = , 2i + 1 4

Fact 13.5.100.

∞ ∑ (−1)i i=0

√ √ 1 = 2 acot 2, i 2 (2i + 1)

√ ∞ ∑ i − i2 − 1 = 1, √ i(i + 1) i=1

∞ ∑ i=1

∞ ∑ (−1)i i=0

1 = i 3 (2i + 1)

1 = 1. √ √ √ ( i + i + 1) i(i + 1)

Source: [1566, p. 63]. Fact 13.5.101. ∞ ∑

i2 + 2i + 1 = log 2, i2 + 2i

∞ ∑

i(2i + 1) = log 2, (i + 1)(2i − 1) i=1 i=1 ) ∞ ∞ ( ∑ ∑ (i + 1)(3i + 1) 4 2i + 1 1 log = log , i log − 1 = (1 − log 2), i(3i + 4) 3 2i − 1 2 i=1 i=1 ( ) ∞ ∞ ∑ ∑ log i 1 1 π (−1)i = γ log 2 − log2 2, (−1)i+1 log 1 + = log , i 2 i 2 i=1 i=1 log

log

√ 3π . 6

1057

INFINITE SERIES, INFINITE PRODUCTS, AND SPECIAL FUNCTIONS

) π2 1 = log , 8 (i + 1)2

( ∞ ∑ i (−1) log 1 − i=1

n ( ∑

lim

n→∞

i=0

) 1 1 γ − log n = + log 2. 2i + 1 2 2

Furthermore, for all n ≥ 1,

n−1 ( ) ∞ n ∑ logn+1 2 ∑ n i−1 log i γi logn−i 2. = − (−1) i i n + 1 i=0 i=1

Source: [968, p. 68] and [1566, pp. 64, 69, 70]. Remark: γi is defined in Fact 13.3.1. Fact 13.5.102. Let x be a real number such that x > 1. Then, ∞ ∑

xi = 1. j j=1 (1 + x )

∏i

i=1

Source: Fact 13.10.1 and [1566, p. 117]. Fact 13.5.103. Let n ≥ 1. Then, ∞ ∑

1

∑i

j3

j=1

i=1

4π2 − 12, 3

=

∞ ∑ 1 (−1)i+1 ∑i j=1

i=1

j3

= 12 − 16 log 2.

Source: [108, pp. 34, 203–205]. Related: Fact 1.12.28.

13.6 Facts on Series of Trigonometric and Hyperbolic Functions Fact 13.6.1. Let x ∈ C and n ≥ 0. Then, ∞ ∑ n sin nx = n sin x + (−1)i

∏i

j=1 [n

2

− (2 j − 1)2 ]

(2i + 1)!

i=1

sin2i+1 x.

Source: [1745, 1756]. Remark: If n is odd, then the series is finite. Fact 13.6.2. ∫ ∞ iπ ) (

∑ sin

i2

i=1

If x ∈ (0, 2π), then

∞ ∑ sin ix

i ∫

i=1 ∞ ∑ sin ix i=1 ∞ ∑ sin ix i=1

i3

i2 =

x

=− 0

=

2

π−x , 2

∞ ∑ cos ix i=1

i

( t) dt, log 2 sin 2

∞ ∑ sin ix

i5

log 2 sin 0

1 3 π 2 π2 x − x + x, 12 4 6

i=1

π/2

=−

=−

( x) = − log 2 sin , 2

∞ ∑ cos ix i=1

∞ ∑ cos ix i=1

x dx = G. 2

i4

i2 =−

=

π2 1 2 π x − x+ , 4 2 6

1 4 π 3 π2 2 π4 x + x − x + , 48 12 12 90

1 5 π 4 π2 3 π4 x + x − x + x. 240 48 36 90

If x ∈ (0, 2π) and k ≥ 1, then ∞ ∑ sin ix i=1

i2k+1

∞ ∑ cos ix i=1

i2k

∑ 4k π2k+1 π ζ(2k − 2i) 2i+1 B2k+1 [x/(2π)] = (−1)k x2k + (−1)i x , (2k + 1)! 2(2k)! (2i + 1)! i=0 k

= (−1)k+1

∑ 4k π2k π ζ(2k − 2i) 2i B2k [x/(2π)] = (−1)k x2k−1 + (−1)i x . 2(2k)! 2(2k − 1)! (2i)! i=0 k

= (−1)k+1

1058

CHAPTER 13

If x ∈ (−π, π), then ∞ ∞ ( ∑ ∑ sin ix x cos ix x) (−1)i =− , (−1)i = − log 2 cos , i 2 i 2 i=1 i=1 ∫ ∞ ∞ ( π−x ∑ ∑ t) cos ix 1 π2 sin ix log 2 sin dt, (−1)i 2 = x2 − , (−1)i 2 = 2 4 12 i i 0 i=1 i=1 ∞ ∑ 1 3 π2 sin ix x − x. (−1)i 3 = 12 12 i i=1

If x ∈ (0, π), then ∞ ∑ sin (2i − 1)x i=1

π , 4

=

2i − 1

∞ ∑ sin (2i − 1)x

(2i − 1)2

i=1

∞ ∑ i=1 ∞ ∑ cos (2i − 1)x

=

2i − 1

i=1

=−

1 2

∫

x

0

∫ ( ( 1 π−x t) t) dt − dt, log 2 sin log 2 sin 2 2 0 2

π 2 π2 sin (2i − 1)x x + x, = − 8 8 (2i − 1)3 ∞ ∑ cos (2i − 1)x

1 x log cot , 2 2

(2i − 1)2

i=1

π π2 =− x+ . 4 8

If n ≥ 0 and x ∈ (nπ, (n + 1)π), then ∞ ∑ sin (2i − 1)x

2i − 1

i=1

If x ∈

[− π2 , π2 ],

π = (−1)n . 4

then ∞ ∑ cos (2i − 1)x π (−1)i+1 = , 2i − 1 4 i=1

∞ ∑ sin (2i − 1)x π = x, (−1)i+1 4 (2i − 1)2 i=1

∞ ∑ cos (2i − 1)x π 2 π3 x + . (−1)i+1 = − 8 32 (2i − 1)3 i=1

If x ∈ [−π, π] and α ∈ R\Z, then ∞ ∑

(−1)i

i=1

cos ix 1 π cos αx − = . α2 − i2 2α sin απ 2α2

Source: [116, p. 371], [154, p. 338], [672], [968, p. 210], and [2163, pp. 324–326]. Remark:

The indefinite integral is Clausen’s integral. Expressions for Catalan’s constant G are given in Fact 14.9.1. Fact 13.6.3. ∞ ∑ sin i

i

i=1 ∞ ∑ sin2 i i=1 ∞ ∑ i=1

i4

=

=

∞ ∑ sin2 i i=1

(π − 1)2 , 6

i2

=

∞ ∑ sin2 2i i=1

sin3 i ∑ sin4 i π = = , i 4 i2 i=1 ∞

∞ ∑ sin 2i

π−1 , 2 i2

i

=

π−2 , 2

∞ ∑ sin 3i i=1

i

=

π−3 , 2

∞ ∞ ∑ sin2 i 1 sin i ∑ = (−1)i+1 2 = , (−1)i+1 i 2 i i=1 i=1

= π − 2,

∞ ∑ sin5 i i=1

i

i=1

=

∞ ∑ sin6 i i=1

i2

=

3π , 16

∞ ∑ sin2 2i i=1

16i4

=

(π − 2)2 , 24

1059

INFINITE SERIES, INFINITE PRODUCTS, AND SPECIAL FUNCTIONS ∞ ∑ sin(2i − 1)

2i − 1

i=1

=

∞ ∑ sin2 (2i − 1)

(2i − 1)2

i=1 ∞ ∑ i=1

∞ ∞ ∑ sin(2i − 1) ∑ 1 π = (−1)i−1 (−1)i−1 = , 2 2i − 1 4 (2i − 1) i=1 i=1

=

∞ ∑ 4 1 π tan i+2 = . i 2 π 2 i=0

sin2 (2i − 1) 3π2 − 4π = , 24 (2i − 1)4

If a, b are real numbers such that b < a < (2π − b)/3, then ∞ ∞ ∑ ∑ (sin3 ai) cos bi 3a2 π − b2 π − 4a3 (sin3 ai) cos bi π = = , . i 4 8 i3 i=1 i=1 If n ∈ {0, 1, 2, 3}, then

∞ ∑ (sinn i) sin 3i i=1

If n ∈ {1, 2, 3}, then

∞ ∑ (sinn i) cos i

in

i=1

If x ∈ [0, 1], then

∞ ∑ (sin i) sin ix

∞ ∑ (sin i) sin ix

∞ ∑ sin 2ix i=1 ∞ ∑ i=1

2i

=

π−2 . 4

=

π−x . 2

∞ ∑ (sin 2i) sin 2ix

4i2

i=1

sin (2i − 1)x = 2i − 1

(π − 1)x . 2

=

i2

i=1

If x ∈ (1, π − 1), then

π−3 . 2

=

i2

i=1

If x ∈ [1, 2π − 1), then

=

in+1

=

π − 2x , 4

∞ ∑ [sin(2i − 1)] sin (2i − 1)x

(2i − 1)2

i=1

=

∞ ∞ ∑ sin ix ∑ (sin i) sin ix x (−1)i+1 = . = (−1)i+1 i 2 i2 i=1 i=1

If x ∈ (0, 2π 3 ), then

∞ ∑ sin3 ix

i

i=1

If x ∈ [0,

π 2 ],

then

∞ ∑ sin4 ix i=1

If x ∈ (0,

2π 5 ),

then

∞ ∑ sin5 ix i=1

If x ∈ [0,

π 3 ],

then

i2

i

∞ ∑ sin6 ix i=1

i2

=

π . 4

=

πx . 4

=

3π . 16

=

3πx . 16

π , 4

1060

CHAPTER 13

If x ∈ (−π, π), then ∞ ∑ sin ix x (−1)i+1 = , i 2 i=1

∞ ∑ cos ix π2 − 3x2 (−1)i+1 2 = . 12 i i=1

Source: [216], [516, p. 316], [968, p. 240], and [1647, p. 268]. Remark: These equalities are

Fourier series. Fact 13.6.4.

∞ ∑ (−1)⌊i/2⌋ i=0

1 = 2i + 1

√

∞ ∑

2π , 4

i=0

1 π2 = . 2 8 (2i + 1)

Source: [216]. Fact 13.6.5. Let x ∈ R. Then, ∞ ∑

3i−1 sin3

i=1 ∞ ∑

1 x = (x − sin x), 3i 4 4i sin4

i=1

∞ ∑ x x 1 3 sin3 1−i = sin , i 3 4 3 3 i=0

x = x2 − sin2 x, 2i

∞ ∑ cos3 3i x 3 = cos x, (−1)i 3i 4 i=0

2 4 ∑ cos 2ix 8 ∑ sin2 ix = . − π π i=1 4i2 − 1 π i=1 4i2 − 1 ∞

| sin x| =

∞

If 2x/π is not an integer, then ∞ ∑ x 1 1 1 tan i = − 2 cot 2x = + tan x − cot x, i 2 2 x x i=0

If |x| < π, then ∞ ( ∑ x )2 x x x 2i 1 − cos i csc i−1 = tan − , 2 2 2 2 i=1

∞ ( ∑

∞ ∑ x 1 1 tan i = − cot x. i 2 2 x i=1

1 − cos

i=1

x x x) 1 csc i−1 = tan . 2i 2 2 2

Source: [311], [516, p. 213], [968, p. 236], [1566, p. 66], and [1757, p. 173]. Fact 13.6.6. Let x ∈ (0, 1) and θ ∈ (0, 2π). Then, ∞ ∑ xi sin iθ x sin θ = atan , i 1 − x cos θ i=1

∞ ∑ √ xi cos iθ = − log x2 − 2(cos θ)x + 1. i i=1

Source: [1757, p. 189]. Fact 13.6.7. Let a ∈ (−1, 1) and x ∈ R. Then, ∞ ∑

a|i| eix ȷ = 1 + 2

i=−∞

∞ ∑

ai cos ix =

1 − a2 1 + ae x ȷ = Re ≥ 0, 1 − ae x ȷ 1 + − 2a cos x

ai cos ix =

a cos x − a2 . 1 + a2 − 2a cos x

i=1 ∞ ∑ i=1

a2

Source: [116, pp. 243, 601]. Related: Fact 14.4.54. Fact 13.6.8. If x be a real number. Then, ∞ ∑ i=1

In particular,

atan

x = atan x, 2 i + i + x2 ∞ ∑ i=1

∞ ∑

atan

i=1

1 π atan 2 = , i +i+1 4

∞ ∑ i=1

i4

4ix x = atan x + atan . 2 2 +4+x √

atan

3 π = , i2 + i + 3 3

1061

INFINITE SERIES, INFINITE PRODUCTS, AND SPECIAL FUNCTIONS ∞ ∑

π 4i 1 = + atan , atan 2 2 i +5 4

i=1

√ 4 2i π = . atan 2 i +6 2

∞ ∑ i=1

Source: [512, 2180]. Related: Fact 2.16.9. Fact 13.6.9. Let a, b be real numbers, and assume that a ≥ 0 and a + b ≥ 0. Then, ∞ ∑

atan

i=1

a2 i2

π a = − atan(a + b). 2 2 + a(a + 2b)i + 1 + ab + b

In particular, ∞ ∑

atan

i=1

∞ ∑

1 π = , 2 i +i+1 4

atan

i=1

π 2 = , 2 2 (2i + 1)

∞ ∑

atan

i=1

i2

π 1 = . −i+1 2

Source: [512], [1566, p. 68], and [2180]. Fact 13.6.10. Let a, b, c be real numbers, and assume that b2 ≤ 4ac. Then, ∞ ∑

atan

+ 2a(a + + + 3ab + π = − atan(a + b + c). 2 In particular, i=1

a2 i4

b)i3

∞ ∑

atan

i=1 ∞ ∑

b2

2ai + a + b + 2ac)i2 + (ab + b2 + 2ac + 2bc)i + 1 + ac + bc + c2

π a 8ai = − atan , 2 4a2 i4 + a2 + 4 2

atan

i=1

(a2

8i π 1 = − atan , 2 +5 2

4i4

∞ ∑

∞ ∑

atan

i=1

atan

i=1

i4

π 2ai = , a2 i4 − a2 i2 + 1 2

π 8i = + atan 2. 2 − 2i + 5 2

Source: [512], [1566, p. 68], and [2068, p. 349]. Fact 13.6.11. Let x ∈ R. Then, ∞ ∑ (−1)i atan i=0

If n ≥ 1, then

1 π πx 1 x = − atan e− 2 πx = atan sinh . 2i + 1 4 2 2

∞ ∑ (−1)i (2i + 1) atan i=0

Furthermore,

8n2 = (−1)n−1 4n atan e−πn . (2i + 1)2

∞ ∑ π 2 (−1)i atan 2 = − . 4 i i=0

Source: [1311, pp. 276, 277] and [1317, p. 42]. Fact 13.6.12. Let x, y ∈ R, and assume that x , 0 and, for all n ≥ 1, n2 + y2 , x2 . Then, ∞ ∑ i=1

atan

i2

2xy y tanh πy = atan − atan . 2 2 x tan πx +y −x

In particular, ∞ ∑ i=1

atan

2x2 π tanh πx = − atan , 2 4 tan πx i

∞ ∑ i=1

atan

1 π = , 2 4 2i

∞ ∑ i=1

atan

2 3π = , 2 4 i

1062 ∞ ∑ i=1

CHAPTER 13 ∞ ∑

1 2 π atan = + atan , 2 (i + 1)2 4

i=1

√

tanh 22π tan 1 π atan 2 = − atan = atan √ 4 i tan 2π tan 2

Furthermore, ∞ ∑ 1 sinh 2i x tanh x atan = atan , i ix 2 sin 2 tan x i=1 ∞ ∑ i=1

√ 2π 2 √ 2π 2

− tanh + tanh

√

2π 2 √ 2π 2

.

∞ ∑ π 2x2 sinh πx (−1)i+1 atan 2 = − + atan , 4 sin πx i i=1

∑ i2 π coth π π(sin 2πx − sinh 2πx) i2 = = , , 4 4 4 4x(cos 2πx − cosh 2πx) 4 i + 4x i +4 i=1 √ √ ∞ ∑ π(sinh 3πx − 3 sin πx) i2 = √ . √ i4 + i2 x2 + x4 2 3x(cosh 3πx − cos πx) i=1 ∞

Source: [506, pp. 105–107], [512], [1566, p. 68], and [2068, p. 349]. Remark: Note that πx π lim x↑1 atan tanh tan πx = − 2 .

Fact 13.6.13. ∞ ∑ 2(i2 − i + 1) = π, csch 2i π = coth π − 1, 2 + 1)(i2 − 2i + 2) (i i=1 i=1 √ √ ∞ ∞ ∑ ∑ 2 1 + (i2 + 2i)(i2 + 4i + 3) π 1 + 4i − 1 π = , acos acos = . √ 4 (i + 1)(i + 2) 6 2 i2 + i i=1 i=1 ∞ ∑

asin

Source: [512, 2145, 2180, 2573]. Fact 13.6.14. If x ∈ (−1, 1), ∞ ∑

atanh

i=1

i2

x = atanh x. + i − x2

Source: [2180]. Related: Fact 2.16.9. Fact 13.6.15. Let n ≥ 1. Then, ∞ ∑ coth iπ

i2n+1

i=1

= 4n π2n+1

n+1 ∑ (−1)i+1 i=0

B2i B2n−2i+2 . (2i)!(2n − 2i + 2)!

Consequently, ∞ ∑ coth iπ i=1

In particular,

i4n−1

∑ B2i B4n−2i 1 (2π)4n−1 (−1)i+1 , 2 (2i)!(4n − 2i)! i=0

∞ ∑ coth iπ i=1

∞ ∑ coth iπ

2n

=

i3

=

7π3 , 180

∞ ∑ coth iπ i=1

i9

∞ ∑ coth iπ i=1

= 0,

i5

i=1

∞ ∑ coth iπ

= 0,

i=1

∞ ∑ coth iπ

i11

i=1

=

i7

=

i4n+1

= 0.

19π7 , 56700

1453π11 . 425675250

Source: [645, p. 99], [1311, p. 285], and [1317, p. xxvi]. Related: Fact 13.1.6. Fact 13.6.16. If x , 0, then ∞

∑ i=1

csch 2i x =

2 . e2x − 1

1063

INFINITE SERIES, INFINITE PRODUCTS, AND SPECIAL FUNCTIONS

In particular,

∞ ∑

1 = coth 1 − 1. sinh 2i

i=0

Source: [2159]. Fact 13.6.17.

∞ ∑ (−1)i i=0

1 (2i + 1)5 cosh

π5 . 768

=

(2i+1)π 2

Now, let n be an odd integer. Then, ∞ ∑

(−1)i

i=0

1 (2i + 1)[cos

+ cosh

(2i+1)π 2n

(2i+1)π 2n ]

π . 8

=

Source: [1317, pp. xxvi, xxviii]. Fact 13.6.18. Let z ∈ C, and assume that z , 0. Then,

) z z i . −2 =1− sinh 2zi tanh z

∞ ( ∑ i=0

Furthermore, ∞ ∑ 2i − coth i=0

2i sinh

1 2i

1 2i

=

∞ ∑ 2 − 4i + csch2

1 + 4e2 − e4 , 1 − 2e2 + e4

4i

i=0

Source: [512]. Fact 13.6.19. Let z ∈ C, and assume that

z 2π

1 2i

− sech2

1 2i

sinh 21−i

=

e12 − 17e8 − 17e4 + 1 . e12 − 3e8 + 3e4 − 1

is not an integer. Then,

1 1 ∑∑ cos iz = − . lim n→∞ n 2 j=1 i=1 j

n

Source: Use Fact 2.16.9.

13.7 Facts on Series of Binomial Coefficients Fact 13.7.1. Let p be a real number, and assume that |p| > 4. Then, ∞ ∑

(2i) pi

i=0

In particular,

∞ ∑ i=0

i

(2i) i

8i

√ =

√ = 2,

p , p−4

∞ ∑

2i p2i

i=0

∞ ∑ (−1)i i=0

(4i)

(2i) i

8i

1 = 2 √

=

√

(√

2 , 3

p + p−4

∞ ∑ i=0

(4i) 2i 82i

√

) p . p+4

√ √ 3 2+ 6 = . 6

Source: Use the power series for 1/ 1 − 4z given by Fact 13.4.2. See [1781]. Fact 13.7.2. Let p be a real number, and assume that |p| ≥ 4. Then, ∞ ∑ i=1

(2i)

√     p  p − 4     , = 2 log  1 − ipi 2 p  i

In particular, ∞ ∑ i=1

(2i) i

i4i

= log 4,

∞ ∑ (−1)i i=1

(2i) i

i4i

∞ ∑ i=1

(4i)

2i ip2i

  p2 = 2 log  4

√ = 2 log 2( 2 − 1),

√  √     1 − p − 4   p + 4 − 1 .     p p

∞ ∑ i=1

(4i) 2i

i42i

√ = 2 log 4( 2 − 1).

1064

CHAPTER 13

( Source: Use the power series for 2 log

) √ 1− 1−4z 2z

given by Fact 13.4.2. See [1781].

Fact 13.7.3. Let p be a real number, and assume that |p| ≥ 4. Then, ∞ ∑ i=0

(2i)

√   p − 4  p   , = 1 − (i + 1)pi 2 p 

∞ ∑

i

In particular, (2i) ∞ ∑ i

i=0

(4i)

√ √  p  p + 4 p − 4   . =  − 4 p p  (2i + 1)p2i 2i

(4i) √ √ √ ∞ ∑ 77 − 7 77 − 21 2i = , = , . = (i + 1)7i 2 (i + 1)7i 2 4 (2i + 1)72i i=0 i=0 i=0 √ Source: Use the power series for (1 − 1 − 4z)/(2z) given by Fact 13.4.2. See [1781]. Fact 13.7.4. Let p be a real number, and assume that |p| ≥ 1. Then, (2i) ∞ ∑ 1 i = p asin . i 2i p 4 (2i + 1)p i=0 7−

In particular, ∞ ∑ i=0

√

21

(2i)

∞ ∑ (−1)i

(2i)

∞ ∑

π = , 4i (2i + 1) 2 i

i=0

i

(2i)

√ 2 3π = , 16i (2i + 1) 9 3i

i

(2i)

∞ ∑ i=0

i

16i (2i + 1)

=

π . 3

Source: Use the power series for asin z given by Fact 13.4.9. Fact 13.7.5. Let p be a real number, and assume that |p| > 1. Then, ∞ ∑ i=1

1 2 asin |p| 4i (2i) . = √ i i p2i p2 − 1

In particular,

√ ∞ ∑ 1 3π (2i) = , 9 i=1 i i

√ √ ∞ ∞ ∞ ∑ ∑ ∑ 3i 2i 1 2 3π π 3π (2i) = (2i) = , (2i) = , . 3 2 9 i=1 i i i=1 i i i=1 i i √ Source: Use the power series for (z asin z)/ 1 − z2 given by Fact 13.4.9. Fact 13.7.6. Let p be a real number, and assume that |p| > 21 . Then, √ 1 ∞ ∑ 4p2 − 1 + 4p2 asin 2|p| 1 (2i) = . (4p2 − 1)3/2 p2i i=1 i

In particular,

Source: Use the power series for

√ ∞ ∑ 1 1 2 3π (2i) = + . 3 27 i=1

i √ z2 4−z2 +4z asin(z/2) √ (4−z2 ) 4−z2

given by Fact 13.4.9.

Fact 13.7.7. Let k ≥ 0, and define

△

fk (z) =

∞ ∑ ik (2i) zi . i=1

i

1065

INFINITE SERIES, INFINITE PRODUCTS, AND SPECIAL FUNCTIONS

Then, the radius of convergence of f is 4. Furthermore, √ √ √ ∞ ∞ ∞ ∑ ∑ ∑ i i2 1 2 3π 2 2 3π 4 10 3π 1 (2i) = + (2i) = + (2i) = + , , , 3 27 3 27 3 81 i=1 i=1 i=1 i

i

∞ ∑ 2i π (2i) = 1 + , 2 i=1 i

i

∞ ∑ 2i i (2i) = 3 + π, i=1

i

√ 4 3π 3i (2i) = 3 + , 3 i=1 i √ ∞ ∑ 3 i i2 172 3π (2i) = 156 + , 3 i=1

√ ∞ ∑ i3 10 74 3π (2i) = + , 3 243 i=1 i

∞ ∑ 2i i2 7π (2i) = 11 + , 2 i=1

∞ ∑ 35π 2i i3 (2i) = 55 + , 2 i=1 i i √ ∞ ∑ 3i i 20 3π (2i) = 18 + , 3 i=1 i √ ∞ ∑ 3i i3 2084 3π (2i) = 1890 + . 3 i=1

∞ ∑

i

i

Source: [971, 1781], Remark: A closed-form expression for fk (z) is given in [971]. Fact 13.7.8.

√ ∞ ∑ 1 3π (2i) = , 9 i=1 i i

∞ ∑ i=1

∞ ∑

1 π2 (2i) = , 18 i2

i=1

i

1 17π4 (2i) = , 3240 i4 i

∞ ∑ 2i π (2i) = + 2, 2 i=0 i

∞ ∞ ∞ ∑ ∑ ∑ π π2 35ζ(3) π2 log 2 2i 2i 2i (2i) = , (2i) = , (2i) = πG − + , 2 3 2 8 16 8 i=1 i i i=1 i i i=1 i i √ √ √ ∞ ∞ ∞ ∑ ∑ ∑ 1 3π 3π π2 4 3π π2 1 1 1 (2i) = − ( )= (2i) = , + − 1, − , 2 2i 2 18 6 36 9 9 i=2 (i − 1) i i=2 (i − 1) i i=0 (i + 1) i √ ∞ ∞ ∑ ∑ √ 2 3π 8 1 1 π (2i) = (2i) = G − log(2 + 3), , 2 9 3 3 i=0 (2i + 1) i=0 (2i + 1) i

√ ∞ ∑ 3i 4 3π (2i) = 3 + , 3 i=1

i

√ ∞ ∑ 20 3π 3i i (2i) = 18 + , 3 i=1

√ ∞ ∑ 172 3π 3i i2 (2i) = 156 + , 3 i=1 i i i √ ( ) √ √ √ i √ ∞ ∞ ∞ ∞ 3i 2 i i ∑ ∑ ∑ ∑ 2π 5−2 5 2 3π 2π 3 3 5 3 (5 − 5) i (2i) = (2i) = (2i) = , , , = , 2 3 9 5 8i 5 2i i i i=1 i i i=1 i i i=1 i=1 ( ) ( ) √ ∞ 3i ∞ ∞ 3i π ∑ ∑ ∑ 2 3 cos 18 π 1 6 log 2 11π 2 i i (3i) = = 2 cos − 1, = − 1, − + , i i i 9 9 27 3 25 125 250 i=1 i=1 2 i=1 i

∞ ∑ i=0 ∞ ∑ i=1

∞ ∑ 50i − 6 ( ) = π + 6, i 3i i=0 2 i

i 81 18 log 2 79π (3i) = − + , i 625 3125 3125 2 i

1 π2 log2 2 (3i) = − , 24 2 2i i2 i

i

i=0

1 (2i + 1)2

(2i) = i

1 π log 2 ( )= − , 10 5

i 3i i=1 2 i i

√ 8G π − log(2 + 3), 3 3

∞ ∑ i−2 ∑ 1 3 (i) = , 2 i=4 j=2 j

√ √ √  36 23τ 18 3τ(τ2 + 1)  3  atan + 2τ − 1 529(τ2 − τ + 1) 23(τ2 − τ + 1)2 √    9τ(t4 − 2τ3 − 2τ + 1) 6 69τ(τ − 1)  τ3 + 1 108τ3  log +  + + , 3 2 3 3 23(τ + 1) 529(τ + 1) (τ + 1) 23(τ3 + 1)

 ∞ ∑  1 (3i) =  i=1

∞ ∑

∞ ∑

1066

CHAPTER 13 △

where τ =

(

25 2

+

3 2

√ )1/3 69 ,

√ 2  1 12(9 + 69) 2 , √ √  − log 2 2 1 − (100 + 12 69)1/3 [2 + (100 + 12 69)1/3 ]3 i=1 i i √ ∞ ∑ 3 6i 1/3 2/3 1/2 (3i) = 2[240 + 96(2 ) + 75(2 )] atan 4/3 + 21/3 [4(21/3 ) − 5] log(21/3 − 1) + 8, 2 − 1 i=1 i √ ∞ ∑ √ 3 6i 4/3 1/3 (3i) = 2 (1 + 2 ) 3 atan 4/3 − 21/3 (1 − 21/3 ) log(21/3 − 1), 2 − 1 i i=1 i √ ∞ ∞ i ∑ ∑ 3 2π2 27i 6 1 (3i) = 6 atan2 4/3 (3i) = − 2 log2 2, − log2 (21/3 − 1), 2 i 2 2 3 2 − 1 i=1 4 i i i=1 i i ) ) ( ( ( ) ∞ ∞ ∞ ∑ ∑ ∑ 1 π log 2 7π3 2i 2i 2i 1 1 = = = 1, , , i (i + 1) i i (2i + 1)2 i i (2i + 1)3 i 2 216 4 4 16 i=0 i=0 i=1 ( ) ( ) ∞ ∞ ∞ ∞ ∑ ∑ ∑ ∑ π2 7ζ(3) 1 2i 1 2i π 4i 4i ( ) (2i) = π2 log 2 − = = log 4, = − 1, , , ii i i (2i + 1) i 2i 2 3 4 4 2 2 2 i=1 i=1 i=1 i i i=1 i i √ ∞ ∞ ∞ ∑ ∑ ∑ 3π 1 11 3π2 4i 4i (2i) = π2 − 4, (2i) = (2i) = 4 − − , , 2 2 2 2 2 8 4 8 i=2 i (i − 1) i i=2 (i − 1) i i=2 i (i − 1) i √ √ ( ) ∞ ∞ ∑ ∑ √ H2i+2 − 12 Hi+1 2i 16 3π 2 5 1 1 ( ) = = π log 2 − 1, + − log (1 + 5), 2 i 4i 4 (2i + 1) i 15 27 25 i=0 2i i=1 √√ √ √ √ √ √ √ ∞ ∑ 17 + 1 − 2 2 34( 17 + 2) 16 4 34( 17 − 2) 2 i 1 + atan √ + log √ , (−1) (4i) = √√ √√ √ √ √ 17 i=0 2i 289 17 − 1 17 − 1 289 17 + 1 17 + 1 + 2 ( ) ( ) ∞ ∞ ∞ i ∑ ∑ √ √ √ ∑ 1 2i π2 i+1 1 2i i 4 i 1 ( ) = (−1) = log , 2), 2 log( 2 − 1), (1 + (−1) (−1) = 2 4i i i 10 8i (2i + 1)2 i i 2ii i=1 i=1 i=0 √ ∞ ∞ ∑ ∑ √ √ √ 2 4i 4i (2i) = − (−1)i log( 2 − 1), (−1)i (2i) = 2 log2 ( 2 − 1), 2 i2 (2i + 1) i=0 i=1 √ 3

  1 (3i) = 6 atan

∞ ∑

i

∞ ∑ (−1)i i=0

i

√ √ √ 4 (2i) = − 2 log( 2−1)−log2 ( 2−1), (i + 1) i i

∞ ∑ (−1)i

i−1

4

(2i) =

√

∞ ∑ (−1)i i=2

√ √ (2i) = −3 2 log( 2−1)−2, (i − 1) i 4i

√ √ 2 log( 2 − 1) + log2 ( 2 − 1) + 1,

(i − 1)2 i √ √ √ √ ∞ ∞ ∑ ∑ 4 5 log 21 (1 + 5) 1 4 5 log 21 (1 + 5) 6 i+1 1 i+1 i (−1) (2i) = + , (−1) (2i) = + , 25 5 125 25 i=1 i=1 i i √ √ √ √ ∞ ∞ ∑ ∑ 56 5 log 12 (1 + 5) 4 5 log 21 (1 + 5) 4 i2 i3 2 (−1)i+1 (2i) = − , (−1)i+1 (2i) = + , 25 125 625 125 i=1 i=1 i i √ √ ∞ ∞ ∑ ∑ √ 2 5 log 12 (1 + 5) 1 i+1 1 , (−1)i+1 (2i) = 2 log2 21 (1 + 5) = 2 acsch2 2, (−1) (2i) = 5 i i2 i=1 i=1 i=2

i

i

1067

INFINITE SERIES, INFINITE PRODUCTS, AND SPECIAL FUNCTIONS

√ ∞ ∞ ∑ ∑ √ √ 2ζ(3) 8 5 1 i+1 1 i (2i) = (2i) = (−1) , log 12 (1 + 5) − 4 log2 12 (1 + 5), (−1) 3 5 5 i i (i + 1) i i=1 i=0 √ √ ∞ ∞ ∑ ∑ √ √ 4 5 3 5 1 1 1 i i 1 (2i) = (2i) = log 2 (1 + 5), log 12 (1 + 5) − , (−1) (−1) 5 5 2 (2i + 1) i (i − 1) i i=0 i=2 ∞ ∑ (−1)i i=2 ∞ ∑

(−1)

i=2

1 (i − 1)2 1

i

i2 (i2 − 1)

∞ ∑ (−1)i

(2i) = 1 −

√ √ √ 5 log 12 (1 + 5) + log2 21 (1 + 5),

i

(2i) = i

4 log2 21 (1

√ + 5) −

√ √ 3 5 log 12 (1 + 5) − , 2 8

√ √ √ 5 (2i) = log2 ( 2 − 1) − 2 log( 2 − 1) − 3, 2 − 1) i i=2 ) ( ) ( ) ( ∞ ∞ ∑ ∑ √ π2 log2 2 1 1 2i π2 2i i i (−1) i , (−1) i − , = = 2 10 12 4 16 (2i + 1)2 i 32 (2i + 1)2 i i=0 i=0 √ √ √ √ ∞ ∞ i i ∑ ∑ 3 + 3 log(2 + 3) 3 log(2 + 3) i+1 2 i+1 2 , , (−1) (2i) = (−1) (2i) = 9 3 i i i=1 i=1 i √ √ √ √ ∞ ∞ i 2 i 3 ∑ ∑ 3 − 3 log(2 + 3) 15 + 3 log(2 + 3) i+1 2 i i+1 2 i (−1) (2i) = , , (−1) (2i) = 27 81 i=1 i=1 i

4

i2 (i2

i

i

∞ ∞ ∞ ∑ 2 1 ∑ Hi 2 ∑ Hi 1 (−1)i+1 (2i) = ζ(3) = = , 5 5 i=1 i2 5 i=1 (i + 1)2 i3 i i=1 ∞ ∞ 2 ∑ ∑ 1 i+1 205i − 160i + 32 =2 (−1) , (2i)5 3 i i5 i i=1 i=1 i √ ∞ ∑ √ √ 1 3 39 7 i 1 acot(2 3 + 7) − − log 2, (−1) (3i) = i 784 28 32 4 i=1

∞ ∑ 1 2i i (−1)i+1 (2i) = , 3 i=1

i

∞ ∑ (−1)i

√ √ 1 1 (3i) = 6[acot(2 3 + 7)]2 − log2 2, i 2 2 4i i i=1 (i+3) ( ) √ √ i+3 ∞ ∞ ∑ ∑ 100 3π 10 10 40 log 2 160 3π i i i (3i+5) = − , (−1) (3i+5) = + − , 243 9 9 27 729 i=0 i=0 3i 3i (i+ 7 ) (i+ 7 ) √ ∞ ∞ 2 2 ∑ ∑ √ 9π 456 2559 2 552 i i i (2i+9) = − , (−1) (2i+9) = 9 log(1 + 2) + − , 2 35 35 5 i=0 i=0 2i 2i ( )3 ( )2 ( ) ( )3 ∞ ∞ ∞ ∑ ∑ ∑ 2 (42i + 5) 2i 16 4i 8 i (20i + 3) 2i i (4i + 1) 2i = , (−1) = , = , (−1) 10i 12i 6i i π i 2i π i π 2 2 2 i=0 i=0 i=0 (6i)(3i) (6i)(3i) √ √ ∞ ∞ ∑ ∑ 3 3 3 3 3i i 3i i (2i) (2i) = , = , 8 8 108i 108i i=0 2(2i + 1) i=0 2(2i + 1)(2i + 3) i

i

1068

CHAPTER 13 n n ∑ 1 n + 1 ∑ 2i (n) = lim lim = 2, n→∞ n→∞ 2n i+1 i=0 i=0

n ∑ lim (−1)i

n→∞

i=0

i

2

∞ ∑

5 1 (−1)i+1 (2i) − 5 2 i i i=1

∞ ∑

(−1)i+1

i=2

∞ ∑

∞

∞ ∑ 1 (i+n) = i=0

i

∞

i

(i+m−1) i

(i+n) =

i=0

In particular,

Hi−1,4 5∑ 25 ∑ 1 (−1)i+1 (2i) + (−1)i+1 (2i) = ζ(7). 7 2 i=1 2 i i3 i=2

Hi−1,2 ( ) = ζ(5), i3 2ii

Let n > m ≥ 1. Then,

√( ) √( ) n ∑ n i 3 n = lim = 0, (−1) n→∞ i i i=0

i

n . n−m

∞ ∑

n , n−1

i

(i+n−2) i

(i+n) = n.

i=0

i

Let n ≥ 2. Then, ∞ ∑ 1 (n+i) = i=1

i

∞ ∑ 1 (i) =

1 , n−1

i=n

n

Let n ≥ 1. Then, ∞ ∑ i=1

1

∞ ∑

i

i=1

1 (n+i) = , n i

Furthermore, ∞ ∑ π2 1 1 (i+1) = − , 3 2 i=1 i 2

∞ ∑ i=1

∞ ∑ i=1

i=1

∞ ∑ i=1

∞ ∑ i=1

i4

i=1

(2n − 1)!!π , (2i − 2)!!4

∞ ∑ i=1

3

∞ ∑

i

i=1

i=1

∞ ∑ i=1

n+1

1 5π2 1025 (i+4) = − , 6 144 i 5

449 11π2 1 (i+3) = 4ζ(3) + − , 54 9 i2 4

7π π4 9 87 1 (i+2) = + − ζ(3) − , 3 8 30 2 16 i 3

i

4

∞ ∑

∞ ∑ 1 n+1 ( i+n ) = . n i=1

2

13 π2 1 (2+i) = ζ(3) + − , 8 4 i3

π π 1 (1+i) = + − 1 − ζ(3), 6 90 i4 2

i

4

39 3π 1 (i+2) = 3ζ(3) + − , 2 8 4 i

∞ ∑

n . (n − 1)2

1 2π2 49 (i+3) = − , 3 9 i 2

5

i

n+i+1

i=1

21035 125π 1 (i+4) = 5ζ(3) + − , 2 1728 72 i

π 1 (1+i) = ζ(3) + 1 − , 3 6 i

i=1

i=1

n+i+1

2i (2n+2i+1) =

∞ ∑

2

2

∞ ∑

∞ ∑

i=1

( 2n+i )

i=0

3

2

i=1

∞ ∑

1 π2 15 (i+2) = − , 2 4 i

π 1 (i+1) = 2ζ(3) + 2 − , 2 3 i

∞ ∑

i=1

i

2

∞ ∑

∞ ∑

π2 1 (n+i) = − Hn,2 , 6 i2

∞ ∑ Hi (n+i) =

n , n−1

∞ ∑ i=1

449 11π2 1 (3+i) = ζ(3) + − , 216 36 i3 i

1 7π π 29 3ζ(3) (2+i) = + − − , 24 90 16 2 i4 2

4

i

1 3 (4i+2)3 = ζ(4) + 864ζ(3) + 32π + 840ζ(2) + 1024(π + G) + 5216 log 2 − 11184, 2

1 π2 π4 (1+i) = ζ(3) + ζ(5) + 1 − − , 6 90 i5 i

∞ ∑ i=1

1 7 61 5π2 π4 (2+i) = ζ(3) + ζ(5) + − − , 4 32 16 60 i5 i

Hi 680 1312 2560 2480 20480 55 (4i+3) = 3ζ(5)−ζ(3)ζ(2)− ζ(4)+ ζ(3)+ ζ(2)− π log 2− log2 2+ G, 6 9 9 9 3 27 i4 3

INFINITE SERIES, INFINITE PRODUCTS, AND SPECIAL FUNCTIONS ∞ ∑ i=1

1069

π4 3602 Hi 52 3 8 2 64 2 (4i+3)2 = 72 − 3 ζ(3)− 3 π − 3 π +(130π −32π+312 log 2) log 2+ 3 (7π+4+39 log 2)G, i3 3

∞ ∑ i=1 ∞ ∑ i=1 ∞ ∑ i=1

137 12019 874853 131891 Hi (i+5) = 3ζ(5) − ζ(3)ζ(2) − ζ(4) + ζ(3) − ζ(2) + , 48 1800 216000 172800 i4 5

311383 2296919 642641 Hi 137 (i+5)2 = 3ζ(5) − ζ(3)ζ(2) − 24 ζ(4) + 3600 ζ(3) + 108000 ζ(2) − 4800 , i4 5

269701 11312047 41259977 Hi 3209 (i+5)3 = 3ζ(5) − ζ(3)ζ(2) − 120 ζ(4) + 900 ζ(3) + 216000 ζ(2) − 34560 , i4 5

∞ ∑ 16π2 Hi (i+2)3 = 120 − 56ζ(3) − 3 , i=1

∞ ∑ 28π2 π4 iHi (i+2)3 = 88ζ(3) + 3 + 45 − 200, i=1

2

2

∞ ∑ π4 i2 Hi 2 (i+2)3 = 328 − 136ζ(3) − 16π − 15 , i=1 2

∞ ∑ 80π2 7π4 i3 H i (i+2)3 = 208ζ(3) − 528 + 3 + 45 . i=1 2

If n ∈ {0, 1, 2}, then ∞ ∑ in H i 2 2 n 1 (i+2)3 = (−2) [ 4 n(n − 5)ζ(4) − (n − 13n + 56)ζ(3) + (4n − 32)ζ(2) + (n − 21n + 120)]. i=1

2

Let n ≥ 1. Then,

( ) ∞ n−1 ∑ ∑ 1 n − 1 2n−1−i . (−1)i (n+i) = n2n−1 (log 2 − Hn−1 ) − n (−1)i i i i=0 i=1 i

In particular,

∞ ∑ 1 (−1)i = log 2, i + 1 i=0

Let n ≥ 1. Then,

∞ ∑ i=0

In particular,

∞ ∑ i=0

Let n ≥ 1. Then,

In particular,

∞ ∑ (−1)i i=0

1 = 2 log 2 − 1. (i + 1)(i + 2)

( )  n−1 1 + (−2)i n−1  ∑  1 i   (n+i) = 2n(−1)n−1 log 2 +  . i 2i i=1 i

1 = 2 log 2, 2i (i + 1)

∞ ∑ i=0

1 = 4 − 4 log 2. 2i−1 (i + 1)(i + 2)

( ) n ∞ i ∑ 1 2n log 2 2n ∑ i2 − 1 n i + (−1) . (−1) ∏n+1 = n! n! i=1 2i i i j=1 (i + j) i=0 ∞ ∑ (−1)i i=0

1 = 2 log 2 − 1. (i + 1)(i + 2)

1070

CHAPTER 13 △

Let n ≥ 2, and define ω = e(2π/n) ȷ . Then, ∞ n−1 ∑ ∑ 1 (ni) = −ωi (1 − ωi )n−1 log(1 − ω−i ). i=0

Let n ≥ 1. Then,

i

i=1

( ) ∞ ∑ 1 i n =n+ , i+2 i 2 2 i=n

( ) ∞ 2 ∑ i +i n i=n

2i+3 i

= (n + 1)2 .

Furthermore,

(2n−i) n ∑ 1 i n lim (−4) (2n) = , n→∞ 3 i=1 n

( ) i ( )2 ( ) ∞ ∑ 1054i + 233 2i ∑ 2j 520 j 2 j−i i = . (−1) 8 i 480 i j=0 j i π i=0

Source: [42, 69], [217, pp. 163, 164, 167], [324, 408], [506, pp. 127, 136, 137], [511, p. 56],

[516, p. 239], [700], [517, pp. 20, 25, 26, 126, 136–139], [520, 521], [570, pp. 166, 167, 175], [1207, 1563], [1566, p. 56], [1675, pp. 72, 84, 86, 87], [1781], [2013, p. 178], [2325, 2481, 2484, ( )3 2485, 2486, 2509, 2571, 2577, 3021]. Credit: The equality involving (4i + 1) 2ii and the next two equalities are due to S. A. Ramanujan. Fact 13.7.9. Let n ≥ 1, and let αn be the positive root of xn − xn−1 − · · · − x − 1 = 0. Then, ( ) ( ) ∞ ∞ ∑ 1 1∑ 1 i(n + 1) − 2 1 i(n + 1) 1 = + , , αn = 2 − 2 αn 2 2 i=1 2i(n+1) i i − 1 i−1 2i(n+1) i i=1 ( ) ∞ i(n + 1) 1 n 1∑ 1 . = 2n − − 2 − αn 2 2 i=1 2i(n+1) i i − 1 √ In particular, α2 = 12 (1 + 5), √ √ ( ) √ √ 3 3 1 1 + 19 + 3 33 + 19 − 3 33 ≈ 1.83929, α3 = 3 √   1  26  2 α4 = 1 + a + 11 − a +  ≈ 1.92756, 4 a √ √ √ √ √ √ 3 3 3 a= 11 + 2 12 1689 − 260 − 2 12 1689 + 260. 3

where

△

Therefore, ( ) ( ) ( ) ∞ ∞ ∞ ∑ ∑ ∑ √ √ √ 1 3i 1 3i 1 3i − 2 1 = (3 − 5), = 5 − 2, = 3 − 5, i i i 8i i−1 4 8i i−1 8i i+1 i=1 i=1 i=1 √ √ ) ( ) ( ∞ ∑ 1 4i − 2 √ √ 3 3 1 19 + 3 33 − 19 − 3 33 . = 5 − 16i i i − 1 6 i=1 Source: [1320, 1854]. Fact 13.7.10. Let z, w ∈ C, and assume that z < −N and Re(z − w) > 1. Then, ∞ ∑ i=1

(w+i) i

i ∑ w+1 1 (z+i) = , z + j (z − w − 1)2 j=1

∞ ∑

i

i=1

Source: [2516].

(w+i) i

(z+i) i

∑

1 w+1 = . 2 + ( j + l)z + jl z (z − w − 1)3 1≤ j≤l≤i

1071

INFINITE SERIES, INFINITE PRODUCTS, AND SPECIAL FUNCTIONS

Fact 13.7.11. Let x be a real number, assume that 0 < |x| < 27/4, and let y be the real root of (y − 1)2 y = 1/x; that is, √ √  1/3  −1/3 1  27 − 2x + 3 81 − 12x  2 1  27 − 2x + 3 81 − 12x   + . y =   +  3 2x 3 2x 3

Then, ∞ ∑ i=0

   y(y − 1)  3 y y 3y − 2  2 − y  + √ ( )= + atan √ atan √  .  log 3y − 1  2 y − 1 (3i + 1) 3ii 3y2 − 4y 3y2 − 4y 3y2 − 4y xi

Source: [324, 326].

13.8 Facts on Double-Summation Series Fact 13.8.1.

π 1 1 ∑∑ (−1) j log j = log . n→∞ n 2 2 i=2 j=2 n

i

lim

Source: [1675, p. 55]. Fact 13.8.2.

∞ ∑ i ∑ i=1 j=1

1 = e − 1. j( j + 1)(i − j + 1)!

Source: [1566, p. 103]. Fact 13.8.3. ∞ 2i−1 ∑ ∑

(−1)n+i

i=1 j=1

1

27 ζ(3), 16

= πG −

i2 j

∞ 2i−1 ∑ ∑

(−1)i−1

i=1 j=1

1 29 = πG − ζ(3). 16 i2 j

Source: [2513, p. 235]. Fact 13.8.4. Let z be a complex number, and assume that |z| < 1. Then, ∞ ∑

(−1)i+ j

i, j=1

Source: [1757, p. 182]. Fact 13.8.5.

∞ ∑

(−1)i+ j

i, j=−∞

zi+ j 1 = log(z + 1) + − 1. i+ j z+1

i2

√ 2π 1 = log(2 3 − 2). 2 9 + (3 j + 1)

Source: [3022]. Fact 13.8.6. ∞ ∑

(−1)i+ j

i, j=1

1 1 = ζ(3), i j(i + j) 4

∞ ∑ i, j=1

(−1)i

1 5 = ζ(3). i j(i + j) 8

Source: [2513, p. 228]. Fact 13.8.7. ∞ ∑

1 13π8 = , i6 (i2 + j2 ) 28350 i, j=0

∞ ∑

1 = 2 (i2 − i j + j2 ) i i, j=0

√ 4 3π , 30

where i = j = 0 is excluded from both summations. Source: [2513, p. 229].

1072

CHAPTER 13

Fact 13.8.8. Let n be a positive integer. Then,

∑ (

1 ∑n = n!, j=1 i j )(1 + j=1 i j )

∏n

∑ (

1 ∑n = n!ζ(n + 1), j=1 i j ) j=1 i j

∏n

where both sums are taken over all n-tuples (i1 , . . . , in ) of positive integers. In particular, ∞ ∑ i=1

∞ ∑

1 = 1, i(i + 1)

1 = 2, i j(i + j + 1) i, j=1 ∞ ∑

∞ ∑ 1 π2 = , 2 6 i i=1

1 = 2ζ(3), i j(i + j) i, j=1

Furthermore,

∑

∞ ∑

1 = 6, i jk(i + j + k + 1) i, j,k=1 ∞ ∑

1 π4 = . i jk(i + j + k) 15 i, j,k=1

1 ∏ ∑ = n!, ( nj=1 i j ) nj=1 i j

where the sum is taken over all n-tuples (i1 , . . . , in ) of positive integers such that gcd {i1 , . . . , in } = 1. Source: [1103, pp. 158, 239] and [2124]. Fact 13.8.9. Let n ≥ 1 and k ≥ 1. Then, ( ) k−1 ∑ ∑ 1 k−1 1 i ∏ ∑ = n! (−1) , ( nj=1 i j )(k + nj=1 i j ) (i + 1)n+1 i i=0 where the sum is taken over all n-tuples (i1 , . . . , in ) of positive integers. In particular, ∞ ∑

∞ ∑

∞ ∑

415 1 = , i j(i + j + 4) 288 i, j=1

∞ ∑

7 1 = , i j(i + j + 2) 4 i, j=1

1 = 2, i j(i + j + 1) i, j=1

85 1 = , i j(i + j + 3) 54 i, j=1

∞ ∑

∞ ∑

12019 1 = , i j(i + j + 5) 9000 i, j=1

13489 1 = . i j(i + j + 6) 10800 i, j=1

Source: [1103, pp. 158, 239] and [1757, p. 172]. Fact 13.8.10. ∞ ∑

1 = log 2, (2 j)i i=2, j=1 ∞ ∑

log 2 1 = , 2j 4 (4i − 1) i, j=1 ∞ ∑

i! j! = 1, (i + j + 1)! i, j=1 ∞ ∑

√ 1 log 3 3π = − , i (3 j) 2 18 i=2, j=1 √ ∞ ∑ log 3 1 3π = − , 2j 8 72 (6i − 1) i, j=1 ∞ ∑

1 3 log 2 π = − , i (4 j) 4 8 i=2, j=1

∞ ∑

∞ ∑

i2 5 = e, (i + j)! 6 i, j=1

∞ ∑

5 ij = e, (i + j + k)! 24 i, j,k=1 ∞ ∑

∞ ∑

∞ ∑

3 log 2 π 1 = − , 2j 16 32 (8i − 1) i, j=1

1 1 = (e2 − 1), i! j!(i + j + 1) 2 i, j=0

1 = 1, (i + j)! i, j=1

If a > −1, then

∞ ∑

∞ ∑

i! j! π2 = , (i + j + 2)! 6 i, j=0

∞ ∑

ij 2 = e, (i + j)! 3 i, j=1

i jk 31 = e. (i + j + k)! 120 i, j,k=1

1 1 = . j (a + i) a+1 i, j=2

INFINITE SERIES, INFINITE PRODUCTS, AND SPECIAL FUNCTIONS

1073

Source: [107, pp. 47, 259], [1103, pp. 158, 159, 163, 241, 242], and [1566, pp. 110, 111]. Fact 13.8.11. Let a > 1. Then, ∞ ∑

a2 i2 j = . ai ( jai + ia j 2(a − 1)4 i, j=1 Source: [387]. Fact 13.8.12. Let x be a real number. Then, ∞ ∑

1 xi+ j = (x − 1)e x + 1, (i + j)! i, j=1

( ) x 1 ij xi+ j = x2 e x + . (i + j)! 6 2 i, j=1 ∞ ∑

Now, let y be a real number such that y , x. Then, ∞ ∑

1 x(ey − e x ) xi y j = + 1 − ex . (i + j)! y−x i, j=1 Source: [1103, pp. 158, 241]. Fact 13.8.13. ∞ ∑ Hi log2 2 π2 = − , (−1)i i 2 12 i=1

∞ ∑ ∞ ∑ Hi+ j π2 log 2 log2 2 = − − , (−1)i+ j i + j 12 2 2 i=1 j=1

∞ ∑ ∞ ∑ ∞ ∑ Hi+ j+k π2 7 log 2 log2 2 1 (−1)i+ j+k =− + + − , i+ j+k 12 8 2 8 i=1 j=1 k=1 ∞ ∑ ∞ ∑ Hi+ j 1 2 2 1 (−1)i+ j = π log 2 − log3 2 − ζ(3), i j 6 3 4 i=1 j=1 ∞ ∑ ∞ ∑ Hi H j 1 1 1 = ζ(3) + log2 2 − log3 2 + log 2 − 1, (−1)i+ j i+ j 4 2 3 i=1 j=1 ∞ ∑ ∞ ∑ Hi H j 1 = 1 − log 2 − log2 2. (−1)i+ j i+ j+1 2 i=1 j=1

Source: [1100, 1102] and [1103, pp. 156, 231, 232]. Fact 13.8.14. ∞ ∑ ∞ ∑ Hi (H j+1 − 1) = 2ζ(5) + 4ζ(2)ζ(3) − 2ζ(3) − 4ζ(2). i j(i + j)( j + 1) i=1 j=1

Source: [2215]. Fact 13.8.15. Let n ≥ 2. Then, ∞ ∑ n! i=1

in+1

(Hi+n − Hn ) =

j ∞ ∑ ∑ (−1)i j=0 i=0

( ) 1 j 1 = 2. 2 (i + n + 1) i n

Source: [739]. Fact 13.8.16. ∞ ∑ ∞ ∑ log(i + j) 1 π (−1)i+ j = log , i + j 2 2 i=0 j=1

∞ ∑ ∞ ∑ log(i + j) 1 π 1 (−1)i+ j = log + log2 2 − γ log 2. i + j 2 2 2 i=1 j=1

1074

CHAPTER 13

Source: [976] and [1103, pp. 156, 231]. Remark: A generalization is given in [1100]. Fact 13.8.17. Let m ≥ 1. Then,

lim

n→∞

In particular,

n i 1 ∑∑

nm+2

jm =

i=1 j=1

n i 1 ∑∑ 1 j= , n→∞ n3 6 i=1 j=1

lim

1 . (m + 1)(m + 2)

n i 1 1 ∑∑ 2 j = . n→∞ n4 12 i=1 j=1

lim

Source: [1566, p. 28]. Fact 13.8.18. ∞ ∑ ∞ ∑ i=1

1 π4 , = (i j)2 120 j=i+1

∞ ∑ ∞ ∑ 1 = ζ(3), i j2 i=1 j=i+1

∞ ∑ ∞ ∑ 1 π4 , = i j3 360 i=1 j=i+1

∞ ∑ ∞ ∑

∞ ∑ ∞ ∑

∞ ∑ ∞ ∑

i=1

i=1

i=1

ζ 2 (3) 1 π6 = − , 2 1890 (i j)3 j=i+1

Source: [2174]. Fact 13.8.19.

π8 1 = , (i j)4 113400 j=i+1

∞ ∑ ∞ ∑ 1 = 2ζ(3), i2 j i=1 j=i

π10 1 ζ 2 (5) − . = 2 187100 (i j)5 j=i+1

∞ ∑ ∞ ∑ 5 1 = ζ(4). 3 i j 4 i=1 j=i

Source: [2068, p. 268]. Fact 13.8.20. Let n be a positive integer. Then, ∞ ∑ 1 ∑1 = 2, j i2 i=1

∞ ∑ 1 ∑1 5 = , j 4 i3 i=1

where the second summation in both equalities is taken over all positive integers j < i such that gcd {i, j} = 1. Source: [506, pp. 127, 128] and [2068, pp. 267, 268]. △ Fact 13.8.21. Define S = {nm : n, m ≥ 2}. Then, ∑ 1 = 1. i−1 i∈S Source: [723], [1566, p. 112], and [1647, p. 273]. Remark: The summation is over distinct

integers. Note that 1=

∞ ∑ i=2

∞ ∞ ∑ 1 1 ∑ 1 = < ≈ 1.128. i(i − 1) i, j=2 i j i, j=2 i j − 1

Credit: C. Goldbach.

13.9 Facts on Miscellaneous Series Fact 13.9.1. For all n ≥ 0, let Cn denote the nth Catalan number. Then, the following statements

hold:

√ √ ∞ ∞ ∑ ∑ 1 4 3π i 16 3π =2+ , =2+ , Ci 27 Ci 81 i=0 i=1 √ √ ∞ ∞ ∑ ∑ √ i(i + 1) 8 56 3π 14 24 5 i 1 = + , (−1) = − log 12 (1 + 5), Ci+1 3 243 Ci 25 125 i=1 i=0 Cn+1 = 4, n→∞ C n lim

INFINITE SERIES, INFINITE PRODUCTS, AND SPECIAL FUNCTIONS

√ ∞ i ∑ √ 3 1 i2 (−1) = − log(2 + 3), C 3 9 i i=0

1075

√ ∞ i ∑ √ 10 12 21 i3 (−1) = − log 12 (5 + 21). C 49 343 i i=0

As n → ∞, Cn ∼

4n √ . n3/2 π

Now, let x be a real number such that |x| < π2 . Then, ∞ ∑ 1 C sin2i x = 21 (1 − cos x), i i−1 4 i=1

∞ ∑ Ci−1 sin2i+1 x = sin x − 21 sin 2x, 2n−1 2 i=1

∞ ∑ 1 (8Ci−1 − Ci ) sin2i+3 x = i 4 i=1

1 2

sin 4x − 2 sin x + 5 sin3 x.

In particular,

√  √    )i ∞ ∞ ( ∞ ∑ ∑ ∑ 2  3  1 1 3 1 1  1   ,  , Ci−1 = , Ci−1 = 1 − Ci−1 = 1 − i i 16 4 8 2 2 16 2 2 i=1 i=1 i=1 )i ∞ ∞ ( ∞ ∞ ∑ ∑ ∑ √ 8Ci−1 − Ci 5 ∑ 8Ci−1 − Ci 3 18Ci−1 − Ci = 3, (8C − C ) = , = 1, = 2 3 − 3. i−1 i i i i 4 16 3 i=1 8 16 i=1 i=1 i=1

∞ ∑ 1 1 Ci−1 = , i 4 2 i=1

Source: [1678, 1744, 1746, 1748, 1756]. Related: Fact 1.18.4. The generating functions for ∞ (Ci )∞ i=1 and (1/C i )i=1 are given by Fact 13.4.2. Fact 13.9.2. For all respec[ n] ≥ 0, let Fn and Ln denote the nth Fibonacci and Lucas number, √ △ 1 1 △ 1 2 tively, and define A = 1 0 . Then, χA (s) = s − s − 1 and spec(A) = {α, β}, where α = 2 (1 + 5) ≈ √ △ 1.61803 and β = 21 (1 − 5) ≈ −0.61803. Furthermore, √ α2 = α + 1, α = 1 + 1/α, 2α2 = α3 + 1, 5α = α + 2 = α2 + 1, √ √ α3 = 2α + 1, α6 = 4α3 + 1, 5α = 12 (α3 + 1), 5 = 2α − 1, α = e(π/5) ȷ + e−(π/5) ȷ ,

β2 = β + 1, β − 1 = 1/β, αβ = −1, 1/α = −β, 1/β = −α, √ √ √ α + β = 1, α − β = 5, α2 + β2 = 3, α2 − β2 = 5, α3 + β3 = 4, α3 − β3 = 2 5, √ √ √ √ √ 1 4 4 3−β= 10 + 2 5, (β + 2)2 = 5β2 , α + β = 7, α 3 − α = α + 2, 2 { } β 1 π atan 2 3π 1 = , α = 2 cos = cot , β = 2 cos , α = max min x, , x∈[1,∞) β+2 β−α 5 2 5 x−1 1 1 1 1 atan α = atan 1 + atan , atan α3 = 2 atan 1 − atan . 2 2 2 2 For all k ≥ 0, atan α4k−1 = 3 atan 1 −

For all k ≥ 1,

1 1 F2k−1 1 1 F2k−2 atan − atan , atan α4k−3 = atan 1 + atan + atan , 2 2 F2k 2 2 F2k−1 1 2 atan 2k = atan √ . α 5F2k k ∑ i=1

1 1 1 atan √ = atan 2 − atan 2k+2 . α α 5F2k+1

1076

CHAPTER 13

[ ] For all k ≥ 3, αk−2 < Fk . Furthermore, α1 is an eigenvector of A associated with α. Now, for all k ≥ 0, consider the difference equation xk+1 = Axk . Then, for all k ≥ 0, xk+2(1) = xk+1(1) + xk(1) , where, for all k ≥ 2,

[

] Fk = Fk A + Fk−1 In . Fk−1

F A = k+1 Fk k

If x0 >> 0, then lim

△

In particular, if x0 =

[ ] 1 1

xk = Ak x0 ,

k→∞

xk(1) = α. xk(2)

, then, for all k ≥ 0,

[

] Fk+2 xk = , Fk+1

where F1 = F2 = 1 and, for all k ≥ 1, Fk satisfies √ ( √ ) αk − βk 2k 5 5 k k k π k 3π (α − β ) = = cos − cos , Fk+2 = Fk+1 + Fk , Fk = 5 α−β 5 5 5 k ( ) ∑ √ k 3i−2k k k k+1 k k+1 k+1 α − β = Fk 5, α Fk = α Fk+1 + (−1) , α = αFk+1 + Fk , α = 2k , i i=0 ( ) ⌊n/2⌋ ∑ √ n 2i Fk + Fk+2 = 5Fk+1 + 2βk+1 , (1 + α)k + (1 − α)k = 2 α = α(F2k − Fk ) + F2k−1 + Fk+1 . 2i i=0 For all k ≥ 2,

√k

√k αFk + Fk−1 + (−1)k+1 Fk+1 − αFk = 1.

For all k ∈ Z, αk+2 = αk+1 + αk ,

(α3 + 1)k = 2k α2k = 2k (αF2k + F2k−1 ),

αk+1 = αFk+1 + Fk ,

α−k = (−1)k (Fk+1 − αFk ),

(1 + α3 )k + (1 − α3 )k = 2k [αk + (−1)k αk ], 1 1 1 , = k + k+1 Fk Fk+1 α Fk α Fk+1 For all k, l ∈ Z,

√ ( 5α)k = 5k/2 (αFk + Fk−1 ),

(1 + α2 )k + (1 − α2 )k = 5k/2 αk + (−α)k ,

(1 + α3 )k − (1 − α3 )k = 2k [α2k − (−1)k αk ], 1 1 1 . = k + (−1)k+1 2k F2k α Fk α F2k

αl (α3 + 1)k = 2k α2k+l .

For all k > l ≥ 1, △

Alternatively, if x0 =

[ ] 3 1

αFk−l+1 + Fk−l = αk−l+1 . , then, for all k ≥ 0, xk =

[

] Lk+2 , Lk+1

where L1 = 1, L2 = 3 and, for all k ≥ 1, Lk satisfies √ k+1 Lk+2 = Lk+1 + Lk , 5α = αLk+1 + Lk , Consequently, lim

k→∞

Fk+1 Lk+1 = lim = α. k→∞ Lk Fk

Lk = α k + β k .

1077

INFINITE SERIES, INFINITE PRODUCTS, AND SPECIAL FUNCTIONS

Source: [649, 1240, 1239], [1241, pp. 59, 74], and [2156, 2480]. The limit follows from Fact 6.11.5. Remark: α is the golden ratio. The expressions for Fk and Lk involving powers of α and β are Binet’s formulas. Remark: α = eacsch 2 . Related: Fact 1.17.1. Fact 13.9.3. For all n ≥ 0, let Fn , Ln , and √ Pn denote the nth Fibonacci, Lucas number, and Pell △

number, respectively, and define α = 12 (1 +

5). Then,

∞ ∑ (−1)i = β. Fi Fi+1 i=1

√i Fk+1 = α, lim Fi = lim i→∞ k→∞ F k If |z| < 1/α, then

∑ −z = F i zi , z2 + z − 1 i=1 ∞

∑ −z = Pi z i , z2 + 2z − 1 i=0

∑ z−2 = Li zi , z2 + z − 1 i=0 ∞

∞

Consequently, if k ≥ 2, then

If |z| < 21 (3 −

√

∑ Li 1 = zi . i 1 − z − z2 i=1 ∞

log

∞ ∑ k Fi . = 2 i k k − k−1 i=1

5), then

∑ z = F2i zi , 2 1 − 3z + z i=1

∑ 1−z = F2i+1 zi , 2 1 − 3z + z i=0

∞

∑ 2 − 3z = L2i zi , 2 1 − 3z + z i=0

∞

∞

∑ z(1 − z) = F 2 zi . (1 + z)(1 − 3z + z2 ) i=1 i ∞

If |z|
b, then, π 8A − (16 − 5π)G − (3π − 8)H ≤ L ≤ [21A − 2G − 3H]. 8 Finally, for a − b → 0,   ( )2   a−b 3 a+b √    . L ∼ π[3(a + b) − (3a + b)(a + 3b)], L ∼ π(a + b) 1 + √ ( )   a−b 2  10 + 4 − 3 (

a+b

Source: [41, 130, 670, 2828] and [1568, p. 56]. Related: Fact 5.5.8 for the area of an ellipse.

13.10 Facts on Infinite Products Fact 13.10.1. Let (xi )∞ i=1 ⊂ R. Then, the following statements hold:

∑ ∑∞ 2 ∏∞ ∏∞ i) If two of the quantities ∞ i=1 xi , i=1 xi , i=1 (1 + xi ), and i=1 (1 − xi ) converge, then all four quantities converge. ∏ ii) If ∞ (1 + xi ) converges, then limi→∞ xi = 0. ∏ ∑ ∑∞i=1 2 xi converges, then ∞ iii) If i=1 xi = ∞ and ∞ i=1 (1 + xi ) = 0. ∑∞ ∏i=1 ∑∞ 2 (1 + x ) converges, then iv) If i=1 xi = ∞ and ∞ i i=1 xi = ∞. i=1 ∏∞ ∑ ∞ v) Assume that (xi )i=1 ⊂ (−1, ∞), and assume that ∞ i=1 (1 + xi ) i=1 xi converges. Then, ∏ ∑∞ 2 converges. Furthermore, i=1 xi = ∞ if and only if ∞ i=1 (1 + xi ) = 0. ∏∞ ∑∞ vi) Assume that (xi )∞ i=1 (1 + xi ) converges if and only if i=1 xi converges. i=1 ⊂ [0, ∞). Then, ∑∞ ∏ ∞ vii) Assume that (xi )∞ i=1 xi i=1 (1 − xi ) converges if and only if i=1 ⊂ [0, 1) ∪ (1, ∞). Then, converges. ∏ i viii) Assume that limi→∞ xi = 0 and, for all i ≥ 1, xi ≥ xi+1 ≥ 0. Then, ∞ i=1 [1 + (−1) xi ] ∑∞ 2 converges if and only if i=1 xi converges. ∏∞ ∏ ix) If there exist distinct nonzero real numbers α1 and α2 such that ∞ i=1 (1+ i=1 (1+α1 xi ) and ∏∞ α2 xi ) converge, then i=1 (1 + αxi ) converges for all α ∈ R. ∑∞ ∑ 2 x) If ∞ i=1 xi does not converge, then i=1 xi converges and ∏n (1 + xi ) lim i=1∑n x n→∞ e i=1 i exists. ∑∞ xi) Assume that (xi )∞ i=1 xi converges. Then, i=1 ⊂ (0, ∞), and assume that   ∞ ∞  ∏ xi  ∑  xi . x1 1 + ∑i−1  = j=1 x j i=1 i=2 ∏∞ xii) Assume that (xi )∞ i=1 (1 + xi ) converges. Then, i=1 ⊂ (−1, ∞), and assume that ∞ ∑ i=1

xi 1 = 1 − ∏∞ . i=1 (1 + xi ) j=1 (1 + x j )

∏i

1081

INFINITE SERIES, INFINITE PRODUCTS, AND SPECIAL FUNCTIONS

xiii) Assume that (xi )∞ i=1 ⊂ (0, ∞), and assume that ∞ ∑

i=1 (1

+ xi ) does not converge. Then,

xi = 1. j=1 (1 + x j )

∏i

i=1

∏n i=1 (1 + xi ) converges if and only if there exists k ≥ 1 such that limn→∞ i=k (1 + xi ) exists and is nonzero. If these conditions hold, then card({i ≥ 1 : xi = −1}) is finite. ∏∞ ∑∞ xv) Assume that (xi )∞ i=1 (1 − xi ) converges if and only if i=1 xi converges. i=1 ⊂ (0, 1). Then, ∑∞ ∏∞ ∞ x i xvi) Assume that (xi )i=1 ⊂ [0, ∞). Then, i=1 (1 + xi ) ≤ e i=1 . Source: [217, pp. 241–249], [1136, p. 356], [1566, pp. 113–117], and [2294, pp. 63–65, 86]. ∏∞ Fact 13.10.2. Let (zi )∞ i=1 |zi | exists. Then, i=1 ⊂ OIUD \{0}, and assume that ∏ ( ) ∞ ∞ ∞ ∞ 2 ∑ ∑ ∑ 1 − |zi |2 ∑ 1 1− ∞ i=1 |zi | ∏ . ≤ − |zi | = −2 sinh log |zi | ≤ −2 sinh log |zi | = ∞ |zi | |zi | i=1 |zi | i=1 i=1 i=1 i=1 xiv)

∏∞

∏∞

Source: [2089]. Fact 13.10.3. The following statements hold:

i) If n ≥ 1, then

( ) i2 2n = . 2 − n2 n i i=n+1 ∞ ∏

In particular, ∞ ∏ i=2

i2 = 2, 2 i −1

∞ ∏ i=3

ii) If n ≥ 2, then

i2 = 6, 2 i −4 ∞ ∏ i=1,i,n

iii) Let n ≥ 1. Then,

i2

∞ ∏ i=4

i2 = 20, 2 i −9

∞ ∏ i=5

i2

i2 = 70. − 16

i2 = (−1)n+1 2. − n2

( ) ∞ ∏ i2 − 1 2n = . n−1 i2 − n2 i=n+1

In particular, ∞ 2 ∏ i −1 i=3

i2 − 4

iv) Let n ≥ 2. Then,

= 4,

∞ 2 ∏ i −1 i=4

i2 − 9

∞ ∏ i=1

In particular,

= 15,

∞ ∏ i2 − 1 = 56, i2 − 16 i=5

∞ ∏ i2 − 1 = 210. i2 − 25 i=6

π n2 i2 = . 2 2 n i − 1 n sin π/n

√ √ ∞ ∞ ∏ ∏ 4i2 9i2 16i2 2π π 2 3π , , = , = = 2 2 2 9 4 4i − 1 2 9i − 1 16i − 1 i=1 i=1 i=1 √ √ √ ∞ ∞ ∞ ∏ ∏ ∏ √ π π π 25i2 5− 5 36i2 64i2 = = = , , 2 − 2. 2 25i2 − 1 10 36i2 − 1 3 64i2 − 1 16 i=1 i=1 i=1 ∞ ∏

1082

CHAPTER 13

v) Let n ≥ 1. Then,

∞ ( ∏ i=1

1+

) sinh2 nπ 4n4 = . 4 i 2n2 π2

vi) Let z be a complex number, and assume that |z| < 1. Then, ∞ ∏

i

(1 + z2 ) =

i=0

∞ ∑ i=0

zi =

1 , 1−z

∞ ∏

∑ 1 ∑ i 1 = z2i . z = 2 1 + z i=0 1−z i=0 ∞

i

(1 + z2 ) =

i=1

∞

Consequently, if p is a real number and |p| > 1, then ) ∞ ( ∏ 1 p 1 + 2i = . p − 1 p i=0 In particular,

) ) ∞ ( ∞ ( ∏ ∏ 1 1 3 1 + 2i = . 1 + 2i = 2, 2 2 3 i=0 i=0 ∏∞ vii) Let α be a real number. Then, i=1 (1 + 1/iα ) converges if and only if α > 1. In particular, √ ) ) ) ∞ ( ∞ ( ∞ ( ∏ ∏ ∏ cosh 23π 1 1 1 sinh π 1+ = ∞, , , 1+ 2 = 1+ 3 = i π π i i i=1 i=1 i=1 √ √ √  √  ) ∞ ( ∏ 1 1  2π 2π  cosh 2π − cos 2π  = 1 + 4 = 2 sin2 , + sinh2 2 2 i π 2π2 i=1 √ ) ∞ ( ∏ 1 (sinh π)(cosh π − cos 3π) 1+ 6 = . 2π3 i i=1 viii) Let pi denote the ith prime. Then, ) ∞ ( ∞ ∏ i ∏ ∑ pj − 1 1 = 0, = 1, 1− pi p j+1 i=1 i=1 j=1 ) ∏ ) ∞ ( ∞ ( ∏ 2 (−1)(pi −1)/2 sin pi π/2 = = , 1+ 1+ pi pi π i=2 i=1 ( ) ( )−1 −1 1 6eγ 1 1 ∏ 1 ∏ 1+ = 2 , lim 1− = eγ . lim i→∞ log i i→∞ p log i p π j j p ≤i p ≤i j

j

Source: i) and ii) are given in [968, p. 141]. iii) for n = 2 is given in [1136, p. 356]. To prove iv) for n = 2, Stirling’s formula given by Fact 12.18.58 implies that

√

en n! 22n (n!)2−n (2n)!! 22n (n!)2 √ = lim √ √ = lim n→∞ nn 2n n→∞ (2n)!!(2n − 1)!! n n→∞ (2n)! n √ (2n)!! (2n)!! (2n)!! = lim = 2 lim √ √ √ = lim n→∞ (2n − 1)!! n n→∞ n→∞ (2n − 1)!! 2n + 1 (2n − 1)!! 2n+1

π = lim

=

√ 2

√

2

v t∞ ( ∏ 2i ) ( 2i ) √ 2 2 4 4 6 6 · · · · · ··· = 2 . 1 3 3 5 5 7 2i − 1 2i + 1 i=1

INFINITE SERIES, INFINITE PRODUCTS, AND SPECIAL FUNCTIONS

1083

See [1693]. iv) for n = 3 is given in [1566, p. 113]; v) is given in [2013, p. 102]; vi) follows from [1757, p. 169]; vii) is given in [1136, p. 354]; the cases α = 2 and α = 4 are given in [2013, p. 102]; viii) is given in [1350, p. 109] and [2067, 2861]. Remark: iv) for n = 2 is due to J. Wallis. This can be written as π [(2i)!!]2 [(2i)!!]2 . = lim = lim 2 i→∞ (2i − 1)!!(2i + 1)!! i→∞ [(2i − 1)!!]2 (2i + 1) See [1206], Fact 1.13.15, and Fact 12.18.59. Remark: It is noted in [69] that n ∏ 4i2 24n 24n π ( (2n)(2n+1) = . )( ) = , lim 2−1 2n 2n+1 n→∞ 2 4i (n + 1) n n (n + 1) n n i=1 Credit: iv) for n = 4 is due to E. Catalan. Fact 13.10.4. Let z be a complex number. Then,

) ) ∞ ( ∞ ( ∏ ∏ z2 4z2 1 − 2 2 , cos z = 1− , i π (2i − 1)2 π2 i=1 i=1 ) ) ∞ ( ∞ ( ∏ ∏ z2 4z2 sinh z = z 1 + 2 2 , cosh z = , 1+ i π (2i − 1)2 π2 i=1 i=1 ) ) ∞ ( ∞ ( ∏ ∏ 16z4 4z2 1+ , cosh 2z − cos 2z = 4z2 1 + 4 4 , cosh 2z + cos 2z = 2 i π (2i − 1)4 π4 i=1 i=1 ) ∞ ( ∏ z2 z z/2 e − 1 = ze 1+ 2 2 . 4π i i=1 sin z = z

If |z| < 1, then

∏ 1 i = (1 + z2 ). 1 − z i=0 ∞

If z , 0, then log z = (z − 1)

∞ ∏ i=1

2 . 1 + z1/2i

Source: [217, p. 255], [635], [1136, pp. 356, 357], [1167, pp. 291–293], and [2173]. Fact 13.10.5. Let z ∈ C\N and n ≥ 2. Then, ∞ [ n ( z )n ] ∏ 1 ∏ 1 . 1− =− n (2iπ/n) ȷ ) i z Γ(−ze i=1 i=1

Now, let x be a real number. Then, ) ∞ ( ∞ ∞ ) ∏ ∏ x2 x x ∏( x sin x = x 1− 2 2 = x cos i = 4 cos2 i − 1 , 2 3 i=1 3 iπ i=1 i=1 ) ) ( ( ∞ ∞ ∏ 4x2 sin πx ∏ x2 1− , = 1 − 2 , cos x = πx i (2i − 1)2 π2 i=1 i=1 ) ) ∞ ( ∞ ∞ ( ∏ ∏ x2 x sinh πx ∏ x2 sinh x = x 1+ 2 2 = x cosh i , = 1+ 2 , 2 πx iπ i i=1 i=1 i=1 ( ) ( ) ∞ ∞ ∏ 4x2 (sin πx) sinh πx ∏ x4 cosh x = 1+ , = 1 − , (2i − 1)2 π2 π2 x2 i4 i=1 i=1

1084

CHAPTER 13

( ) ) ∏ ∞ ( 1 x4 2 πx 2 πx sin √ + sinh √ = 1+ 4 . π2 x2 i 2 2 i=1 If |x| < 1, then

∞ ∏

(1 + x ) = i

i=1

∞ ∏ i=1

If x > −1, then

1 . 1 − x2i−1

)3 n ( ∏ i2 x 1+ 3 . n→∞ n i=1

e x = lim If x ≥ 0, then ex =

∞ ∏

v t n ∏

n=1

i=1

n

i+1 n (ix + 1)(−1) ( i ) .

If y is a real number, then sin2 πx − sin2 πy = (sin2 πx)

∞ [ ∏ 1− i=−∞

] y2 . (i + x)2

Source: [1266], [1566, pp. 56, 113, 117], and [2013, pp. 99–104]. The first equality is given in

[1217, p. 896]. Fact 13.10.6. Let n > k ≥ 1. Then, ∞ ∏ i=0

n2 (i + 1)2 kπ kπ = csc . (ni + n − k)(ni + n + k) n n

In particular,

√ √ ∞ ∏ 2 3π 4 3π 9(i + 1)2 9(i + 1)2 = , = , (3i + 2)(3i + 4) 9 (3i + 1)(3i + 5) 9 i=0 i=0 i=0 √ √ ∞ ∞ ∏ ∏ 16(i + 1)2 16(i + 1)2 3 2π 2π = , = , (4i + 3)(4i + 5) 4 (4i + 1)(4i + 7) 4 i=0 i=0 √ √ √ √ ∞ ∞ ∏ ∏ √ √ 2 10 10 25(i + 1)2 25(i + 1)2 = = 5 + 5π, 5 − 5π, (5i + 4)(5i + 6) 25 (5i + 3)(5i + 7) 25 i=0 i=0 √ √ √ √ ∞ ∞ ∏ 25(i + 1)2 ∏ 25(i + 1)2 √ √ 3 10 4 10 = 5 − 5π, = 5 + 5π, (5i + 2)(5i + 8) 25 (5i + 1)(5i + 9) 25 i=0 i=0

∞ ∏

π 4(i + 1)2 = , (2i + 1)(2i + 3) 2

∞ ∏ i=0

∞ ∏

36(i + 1)2 π = , (6i + 5)(6i + 7) 3

∞ ∏ i=0

36(i + 1)2 5π = . (6i + 1)(6i + 11) 3

Source: [357]. Remark: The second equality is Wallis’s equality. Related equalities are given in [2495]. Related: Fact 13.10.4 and Fact 13.10.5. Fact 13.10.7. Let n > k ≥ 1. Then, ∞ ∏ i=0

n2 (2i + 1)2 kπ = sec . (2ni + n − k)(2ni + n + k) 2n

1085

INFINITE SERIES, INFINITE PRODUCTS, AND SPECIAL FUNCTIONS

In particular, ∞ ∏ i=0

∞ ∏

√ 4(2i + 1)2 = 2, (4i + 1)(4i + 3) ∞ ∏ i=0

i=0

16(2i + 1) = (8i + 3)(8i + 5) 2

√

√ 9(2i + 1)2 2 3 = , 4(3i + 1)(3i + 2) 3

√ 4 − 2 2,

∞ ∏

9(2i + 1)2 = 2, (6i + 1)(6i + 5) i=0 √ ∞ ∏ √ 16(2i + 1)2 = 4 + 2 2. (8i + 1)(8i + 7) i=0

Source: [357]. Related: Fact 13.10.4 and Fact 13.10.5. Fact 13.10.8. Let n ≥ 1. Then, ∞ ∏ n ∏ 4n2 (2i − 1)2 − 4( j − 1)2 i=1 j=1

4n2 (2i

−

1)2

− (2 j −

1)2

∞ ∏ n ∏ (2n + 1)2 (2i − 1)2 − (2 j − 1)2

(2n +

i=1 j=1

Source: [2137]. Fact 13.10.9. Let n ≥ 1. Then,

1)2 (2i

−

1)2

−

4 j2

= =

√ 2n, √ 2n + 1.

( ) ∞ ∏ (i + n)2 2n = . i(i + 2n) n i=1

In particular, ∞ ∏ (i + 1)2 i=1

i(i + 2)

Furthermore,

= 2,

∞ ∏ (i + 2)2 i=1

i(i + 4)

∞ ∏ (i + 3)2

= 6,

i=1

i(i + 6)

= 20,

∞ ∏ (i + 4)2 i=1

i(i + 8)

= 70.

∞ ∏ (i + n + 12 )2 Γ(2n + 2) . = i(i + 2n + 1) Γ2 (n + 32 ) i=1

In particular, ∞ ∏ (i + 32 )2 32 = , i(i + 3) 3π i=1

Finally,

∞ ∏ (i + 25 )2 512 = , i(i + 5) 15π i=1 ∞ ∏ i=1

∞ ∏ (i + 27 )2 4096 = , i(i + 7) 35π i=1

∞ ∏ (i + 92 )2 131072 = . i(i + 9) 315π i=1

(i + n + 1)2 n+2 = . (i + n)(i + n + 2) n + 1

Fact 13.10.10.

√ √ √ ∞ ∞ ∏ ∏ √ 1+ 3 (4i + 3)(12i + 5) (24i + 5)(24i + 7) = 6 + 3 3, = 1/4 3/8 , (24i + 1)(24i + 11) 8(2i + 1)(3i + 2) 2 3 i=0 i=0 √ √ √ √ ( )3 ∞ ∞ 15 + 5+2 5 ∏ ∏ (10i + 3)(30i + 19) (6i + 5)2 3i + 1 , = 1, = 20(3i + 1)(5i + 3) 3(2i + 1)(6i + 1) 3i + 2 219/20 31/20 51/3 i=0 i=0 ∞ ∏ (14i + 1)(14i + 9)(14i + 11) i=0

243(2i + 1)3

=

1 , 4

∞ ∏ i=1

i(i + 3)(i + 5)(i + 6) 7 = , (i + 1)(i + 2)(i + 4)(i + 7) 15

1086

CHAPTER 13 ∞ ∏ 3 ∏ (18i + 9 − 2k)(18i + 9 + 2k)

81(2i +

i=0 k=1

1)2

=

1 . 8

Furthermore, let k ≥ 1. Then,

)(−1)i ∞ ∏ k ( ∏ (2k + 1)i + 2 j − 1 i=1 j=1

Source: [357, 667]. Fact 13.10.11.

(2k + 1)i + 2 j

) ∞ ( ∏ 1 1− = 0, i i=2

√ 3π 2

∞ ( ∏ i=2

4k ( ). = √ 2k + 1 2kk

) 1 1 1− 2 = , 2 i

√ ) ∞ ( ∏ cosh 1 1 1 + cosh 3π 1 sinh π 1− 4 = 1− 6 = , , 1− 3 = , 3π 4π i i 12π2 i i=2 i=2 i=2 √ ) √ ) ∞ ( ∞ 2 ∞ ( ∏ ∏ ∏ 16 i −1 3 sinh 3π sinh 2π 8 , 1− 4 = , = 4, 1− 3 = 126π 120π i i i3 − 4 i=3 i=3 i=3 )i2 ( ∞ 2 ∞ ∞ 3 ∞ ∏ ∏ ∏ π π i − 1 1 ∏ i2 1 i −1 2 = = = , ≈ 0.2720, e 1 − 2 = 3/2 , 2 2 3 2 sinh π 3 i +1 i +1 i +1 i e i=2 i=2 i=2 i=2 √ √ ∞ ∞ 4 ∏ ∏ 3 2 cosh2 23π (i2 + i + 1)2 i −1 π sinh π ≈ 0.8480, , = = √ √ √ i4 + 1 cosh 2π − cos 2π i4 + 2i3 + 3i2 π sinh 2π i=1 i=2 √ ] ] ] ∞ [ ∞ [ ∞ [ ∏ ∏ ∏ 1 5π 2 1 1 1 1+ = 2, 1− = , 1− = − cos , i(i + 2) i(i + 1) 3 i(i − 1) π 2 i=1 i=2 i=2 √ √ ] ) √ ∞ [ ∞ ( ∏ ∏ π 1 1 2π 2π 3π i+1 i+1 = = , + cosh , 1 + (−1) 1 + (−1) 3 2i + 1 4 12 12 4 (2i + 1) i=1 i=1 √ ( )i+1/2 ∞ ∞ ∏ ∏ 1 e (2i − 1)2 1 2 1+ = √ , = , e i (4i − 1)(4i − 3) 4 2π i=1 i=1 √ ∞ ∞ ∏ ∏ √ 6π 100i(i + 1) + 25 100i(i + 1) + 49 = 1 + 5, = cosh , 100i(i + 1) + 9 100i(i + 1) + 25 5 i=0 i=0  √ √i ∞ ∞ ∞  ∏ ∏ ∏  e 5+1 1 1  i γ , = tan 1, =e , 1 + (−1) 2  = 1 2 −i 2 1 − tan 2 Fi i=1 i=1 1 + i i=1 )2i+2 ( )2i+1 )(−1)i+1 i ∞ ( 2n ( ∏ ∏ i+1 2i + 3 2 π = lim 1+ = ≈ 0.577863674895, n→∞ i+2 2i + 1 i 2e i=0 i=1 )2i ( )2i+1 )(−1)i i ∞ ( 2n+1 ∏ ∏( i+1 2i + 1 2 6 = lim 1+ = ≈ 0.70259797829182, n→∞ i 2i + 3 i πe i=1 i=2 ( )4i−2 ( )4i ) ∞ ( ∞ ∏ ∏ 1 1 4i − 1 4i − 1 1 eπ/2 + e−π/2 1 + = e4G/π−1 , + = , 2 1/i 4i − 3 4i + 1 i 2i πeγ e i=1 i=1 √ √4 ) ∞ ∞ ( ∏ ∏ √ √ 1 e 3π/18 2 −i 3π 1 + (2i+1)π = π/24 , (1 + 2e cosh i 3π/3) = √4 , e e 3 i=0 i=1 ∞ ( ∏

)

∞ ( ∏

)

1087

INFINITE SERIES, INFINITE PRODUCTS, AND SPECIAL FUNCTIONS

lim

n ∏ cosh(i2 + i + 21 ) + [sinh(i + 12 )] ȷ

n→∞

cosh(i2 + i + 21 ) − [sinh(i + 12 )] ȷ

i=1

∞ ∏ (4i − 1)2 [(4i + 1)2 − 1] i=1

=

(4i + 1)2 [(4i − 1)2 − 1]

=

e2 − 1 + 2e ȷ , e2 + 1

1 4 1 Γ ( 4 ). 16π2

Source: [109, pp. 288, 289], [112, pp. 39], [517, pp. 4–6], [522, 667], [1158, pp. 124, 125], [1217,

p. 897], [1410, 1433, 1434, 1563], [1566, pp. 112, 113], [2012], [2013, pp. 99–104], [2079], [2249, p. 421], and [2517, 2531, 2861]. Related: Fact 1.13.14 and Fact 13.5.86. Fact 13.10.12. 2n−1 ∏

lim

n→∞

i=0

2n + 1 + 3i √3 = 4, 2n + 3i

Source: [2461]. Fact 13.10.13. Let x > 1. Then, ∞ ( ∏

1 + (−1)

i+1

i=1

In particular, ) ∞ ( ∏ i+1 1 = 1, 1 + (−1) i i=1 Fact 13.10.14. ∞ ( ∏

1+

i=1

If n ≥ 2, then

1 i2

)(−1)i−1 =

∞ ( ∏ i=1

∞ ( ∏

1 + (−1)

i=1

lim

n−1 ∏ n + 1 + 7i

n→∞

i=1

n + 7i

=

√7 8.

) √ π 1 = . 1 ix Γ(1 − 2x )Γ( x+1 2x )

i+1

) √ 1 π = , 2 2i Γ ( 43 )

)(−1)i−1 ∞ ( 2 ∏ i +1

) √ ∞ ( ∏ π i+1 1 = 2 5 . 1 + (−1) 3i Γ( 3 )Γ( 6 ) i=1

)(−1)i−1 ∞ ( ∏ π2 2 π 1 = , tanh , 1 − 2 2 π 2 8 i −1 i i=1 i=2 √ √4 (−1)i−1 ∞ √ ∏  π (2i − 1)!! 2i + 1  8   = √ .  i i! 2 2 π i=1

π π tanh , 2 2

1 1 + √n ⌊ i⌋

)(−1)i−1 =

∞ ( ∏ i=2n

1 1 − √n ⌊ i⌋

=

)(−1)i−1 =

∞ ( ∏ i=1

1 1+ i

)(−1)i−1 =

If x is a real number such that x < πZ, then  )(−1)i−1 i ( ∞  ∏  x ∏ x2  2   1− = x. 2  sin x x sin ( jπ) 2 j=1 i=1 If z ∈ ORHP, then

∞ ( ∏

1+

i=1

In particular, if n ≥ 1, then ∞ ( ∏ i=1

2n 1+ i

)(−1)i−1

Source: [1103, pp. 146, 147, 201–204].

z )(−1) i

i−1

=

2Γ2 (z/2)2z−2 . Γ(z)

( ) ∞ ( ∏ i ) 2n + 2i − 1 4n = = (2n) . n+i 2i − 1 i=1 n

π . 2

1088

CHAPTER 13

Fact 13.10.15. If x ∈ [−1, 1), then

√

1− = acos x x2

v u u u t

∞ ∏ i=1

|

v u t 1 1 + 2 2

√ 1 1 + 2 2

1 1 + ··· + 2 2 {z i

If x > 1, then √

−1 = √ log(x + x2 − 1) x2

∞ ∏ i=1

v u u u t |

1 x + . 2 2 }

roots

v u t 1 1 + 2 2

√

√ 1 1 + 2 2

1 1 + ··· + 2 2 {z i

√

1 x + . 2 2 }

roots

In particular, v u v √ u u u √ t √ t √ √ ∞ ∞ ∞ ∏1 ∏ 1 1 1 1 1 √ 1 1 ∏ π 2 2 + 2 + 2 + ··· + 2 = + + + ··· + = cos i = , 2| 2 2 i=2 2 π {z } i=1 | 2 2 2 2{z 2 i=1 } i

∞ ∏ i=1

v u t |

roots

√ 1 1 + 2 2

1 1 + ··· + 2 2 {z i

√

√

3 3 3 = , 4 2π }

∞ ∏ i=1

roots

i

v u u u t |

roots

1 1 + 2 2

v u t

√ 1 1 + ··· + 2 2 {z i

√ 2+ 3 3 = . 4 π }

roots

Source: [1410] and [2107, p. 64]. Remark: The fifth equality is due to F. Vi`ete. The equalities of Wallis (see Fact 13.10.6) and Vi`ete are unified and generalized in [2172]. Fact 13.10.16. If x ∈ [−1, 1), then   v v u u   u u √ t t √   ∞  ∏   1 1 1 1 1 2x + 1 1 1 x   = + + + ··· + + −1 . 2 3 2 2 2 2 2 2 2 2    i=1  {z }   | i

If x > 2, then

roots

 √ √   √ ∞  ∏   √  x+1  2 + 2 + 2 + · · · + 2 + x −1 . = | 3 {z }  i=1  i

roots

Source: [1410]. Fact 13.10.17. Let n, m ≥ 1. Then, n ∏

[(2i−1)2 −4m2 ] = (−1)m

i=1

m 1 1 ∏ 4n 2n + 2i − 1 Γ(n − m + 2 )Γ(n + m + 2 ) . Γ(n−m+ 21 )Γ(n+m+ 21 ), = π 2n − 2i + 1 Γ2 (n + 12 ) i=1

Source: [1745]. Fact 13.10.18. Let a, b ∈ R. Then,

 ( )3  ∞  ∏ 1 + a + b  =  i+a  i=1

Γ3 (1 + a) √ √ . Γ(2a + b + 1)Γ[1 + 12 (a − b + (a + b) 3 ȷ]Γ[1 + 12 (a − b − (a + b) 3 ȷ]

1089

INFINITE SERIES, INFINITE PRODUCTS, AND SPECIAL FUNCTIONS

In particular,

∞ [ ∏ i=1

√ ( a )3 ] a2 Γ3 (a) sinh a 3π Γ3 (1 + a) = , = 1+8 √ √ √ i+a Γ(3a + 1)Γ(1 + a 3 ȷ)Γ(1 − a 3 ȷ) Γ(3a + 1) 3π √ √ ] ] ∞ [ ∞ [ ∏ ∏ 8 sinh 3π 64 sinh 2 3π 1+ = 1+ = , . √ √ (i + 1)3 (i + 2)3 6 3π 180 3π i=1 i=1

Source: [667]. ∑ ∑ Fact 13.10.19. Let n ≥ 1, let a1 , . . . , an , b1 , . . . , bn ∈ C\(−N), and assume that ni=1 ai = ni=1 bi .

Then,

n ∞ ∏ n ∏ j + ai ∏ Γ(bi ) = . j + bi Γ(ai ) i=1 j=0 i=1

In particular, ∞ ∏ (i + 9)(i + 10)(i + 12)(i + 15) i=0

=

(i + 8)(i + 11)(i + 13)(i + 14)

15 , 14

∞ ∏ (i + 3/14)(i + 5/14)(i + 13/14) i=0

(i + 1/14)(i + 9/14)(i + 11/14)

∞ ∏ (i + 15/62)(i + 23/62)(i + 27/62)(i + 29/62)(i + 61/62)

(i + 1/62)(i + 33/62)(i + 35/62)(i + 39/62)(i + 47/62)

i=0

= 8.

Source: [667]. Related: Fact 13.3.2. Fact 13.10.20. Let a ∈ R and p ∈ (1, ∞). Then,

lim

n ∏ n p + (a − 1)i p−1

n→∞

n p − i p−1

i=1

= ea/p .

In particular, lim

n→∞

n ∏ n2 + i i=1

n2 − i

= e,

n n ( ∏ ∏ n2 i ) √ = lim 1 + = e. n→∞ n2 − i n→∞ i=1 n2 i=1

lim

Source: [1103, pp. 3, 35], [1566, pp. 48, 61], and [2294, p. 230]. Fact 13.10.21. n ( ∏ i )1/i 2 1+ = eπ /12 , n→∞ n i=1

lim

n n3n/2 ∏ i sin 3/2 = e−1/18 , n→∞ n! n i=1

lim

Let x > 0. Then,

n ( ∏

lim

n→∞

In particular,

i=1

3 n ( ∏ i )n/i 2 1+ = eπ /6 , n→∞ n i=1

lim

2n 1 ∏ √n 2 2 n + i = 25e2 atan 2−4 . n→∞ n4 i=1

lim

) i2 x 1 + 3 = e x/3 . n

) n ( ∏ √ i2 1 + 3 = 3 e. n→∞ n i=1 lim

Let x ∈ (0, 1). Then, lim

n→∞

n ( ∏ i=1

1−

) i2 x = e−x/3 . n3

= 2,

1090

CHAPTER 13

In particular, lim

n ( ∏

n→∞

i=1

) i2 1 − 3 = e−1/6 . 2n

Source: [108, pp. 33, 199, 200], [1566, pp. 48, 56, 61, 231], [1757, pp. 230, 231], and [2294, p.

55]. Fact 13.10.22. Let n ≥ 1. Then,

∫

n + 2 ∏ [2(ni + 1) + n]i 1 dx = . √ n i=1 (2i + 1)(ni + 1) 1 − xn ∞

1

0

In particular,

∫

1

0

∏ 4(i + 1)i π 1 dx = 2 = . √ (2i + 1)2 2 1 − x2 i=1 ∞

Source: [1492]. Related: Fact 14.3.10. Fact 13.10.23. Let n ≥ 1. Then,

( )n ∞ 1+ 1 ∏ i i=1

Source: [2086]. Credit: L. Euler. Fact 13.10.24. Let n ≥ 2. Then,

1+

n i

= n!.

) ∏( 1 ζ(n) 1+ n = , i ζ(2n)

where the product is taken over all primes. In particular, ∏1 ∏1 15 105 = , = 4 . 2 2 4 i π i π Source: [1317, pp. 20, 21]. Related: Fact 13.5.38. △ Fact 13.10.25. For all n ≥ 0, let pn denote the nth partition number, where p0 = 1, and let z ∈ C, where |z| < 1. Then, ∞ ∞ ∏ ∑ 1 = pi zi . i 1 − z i=1 i=0 Furthermore,

∞ ∏ i=1

Remark: See [621, p. 210].

∏∞

∑ 1 zi = 1 + . ∏ i j 2 1 − zi j=1 (1 − z ) i=1 ∞

2

1 i=1 1−zi

is the generating function for the partition numbers. The last equality is the Durfee square identity. See [771, p. 119]. Related: Fact 1.20.1 and Fact 13.1.3. Fact 13.10.26. Let a and z be complex numbers, and assume that a , 0 and |z| < 1. Then, ∞ ∏

(1 − z2i )(1 + az2i−1 )(1 − a1 z2i−1 ) =

2

ai zi .

i=−∞

i=1

Equivalently,

∞ ∑

∞ ∞ ∑ ∏ ai zi(i+1)/2 . (1 − zi )(1 + azi )(1 + 1a zi−1 ) = i=1

i=−∞

1091

INFINITE SERIES, INFINITE PRODUCTS, AND SPECIAL FUNCTIONS

Source: [118, pp. 80, 81], [155, pp. 319, 320], and [621, pp. 216, 217]. Remark: This is Jacobi’s

triple product identity. Fact 13.10.27. Let z ∈ OIUD, and define (ei )∞ i=1 as in Fact 1.20.1. Then, ∞ ∏

(1 − zi ) =

∞ ∑

(−1)i zi(3i−1)/2 = 1 +

i=−∞

i=1

i=1

∞ ∏ (1 − z2i )(1 + zi )2

(1 + z2i )2

i=1

∞ ∏

∞ ∞ ∑ ∑ (−1)i [zi(3i−1)/2 + zi(3i+1)/2 ] = ei zi ,

∞ ∏ [1 − (−z)i ]2

=

1 − z2i

i=1

(1 − z2i )(1 + zi−1 ) =

=

i=0 ∞ ∑

∞ ∏ 1 − zi

2

zi ,

i=−∞ ∞ ∑

i=1

zi(i+1)/2 = 2

i=−∞

i=1

∞ ∑

1 + zi

=

∞ ∑

2

(−1)i zi ,

i=−∞

zi(i+1)/2 ,

i=0

  i ∞ ∞ ∏ ∞ ∏ ∑ ∏  (1 − z2i )2 1 − z2 j  2i+1 1 − z2i i(i+1)/2  z ,  = =1+z+ z = 1 − zi 1 − z2i−1 1 − z2 j+1  i=1 i=1 j=1 i=0 i=1   i ∞ ∞ ∞ ∞ ∏ ∑ ∑ ∑  1 − z2 j  2i+1 ∏ 2i 3 i(i+1)/2   z , (1 − z ) = (−1)i (2i − 1)zi(i+1) , z =  2 j+1  1 − z i=−∞ i=1 i=2 i=1 j=1 ∞ ∑

∞ ∏

∞ ∑

(1 − zi )3 =

(−1)i izi(i+1)/2 = 1 +

i=−∞

i=1

∞ ∑ (−1)i (2i + 1)zi(i+1)/2 . i=1

Source: These equalities are special cases of Fact 13.10.26. See [118, p. 81], [155, p. 321], [621, p. 215], and [2374, 2885]. Remark: The first equality, which is due to L. Euler, is the pentagonal

number theorem. This infinite product has the form ∞ ∏

(1 − zi ) = 1 − z − z2 + z5 + z7 − z12 − z15 + z22 + z26 − z35 − z40 + · · · ,

i=1

where the exponents (g0 , g1 , g2 , . . .) are the generalized pentagonal numbers defined by Fact 1.12.5. The last equality is the Gauss identity. See [1083, p. 54]. Related: Fact 13.4.7 and Fact 13.5.6. Fact 13.10.28. Let z ∈ OIUD . Then, ∞ ∞ ∞ ∏ ∏ ∏ (1 + z2i−1 )8 = (1 − z2i−1 )8 + 16z (1 + z2i )8 , i=1 ∞ ∏

(1−z2i )6 (1−z6i )6 =

i=1

∞ ∏

i=1

(1−zi )4 (1−z3i )4 (1−z4i )2 (1−z12i )2 +4z

i=1 ∞ ∏

(1 − z2i )4 (1 − z3i )9 =

i=1

∞ ∏

(1−zi )2 (1−z3i )2 (1−z4i )4 (1−z12i )4 ,

i=1 ∞ ∏

(1 − zi )8 (1 − z3i )(1 − z6i )4 + 8z

i=1 ∞ ∏

i=1 ∞ ∏

(1 − zi )3 (1 − z2i )(1 − z6i )9 ,

i=1

(1 − z2i )9 (1 − z3i )(1 − z12i )2 + 2

i=1

∞ ∏

(1 − zi )3 (1 − z4i )6 (1 − z6i )3

i=1

=3

∞ ∏

(1 − zi )2 (1 − z2i )2 (1 − z3i )3 (1 − z4i )3 (1 − z6i )(1 − z12i ).

i=1

Now, let n ≥ 1. Then, ∞ n n ( ) ∞ ∑ ∏  n j j∏ i 2n+ j 3i 2n−3 j 4i 3n− j 12i 3 j 2i 7 6i 2z (1 − z ) (1 − z ) (1 − z ) (1 − z ) =  (1 − z ) (1 + z ) . j j=0 i=1 i=1

1092

CHAPTER 13

Source: [27]. Remark: The first equality is Jacobi’s abstruse identity. Remark: Additional equalities are given in [28]. Fact 13.10.29. Let a ∈ (0, 1), let z ∈ C, and assume that a < |z| < 1/a. Then, ∞ ∞ ∏ (1 + a2i+2 )(1 + a2i+1 z)(z + a2i+1 ) ∑ 1 zi . = 2i+2 2i+1 2i+1 cosh i log a (1 − a )(1 − a z)(z − a ) i=−∞ i=0 Source: [2399]. Remark: The infinite product is analytic on C\({0} ∪ {a2i+1 : i ∈ Z}).

13.11 Notes An introductory treatment of limits and continuity is given in [2112]. An essential closure point is traditionally called either a limit point or an accumulation point. i) =⇒ iii) in Proposition 12.2.8 is given in [1647, p. 70]. A power series is also called a Taylor series. Generating functions are presented within the context of hypergeometric functions in [2514]. The derivative and the directional differential are typically called the Fr´echet derivative and the Gˆateaux differential, respectively [1060]. Differentiation of matrix functions is considered in [1343, 1924, 1981, 2247, 2324, 2416]. An extensive treatment of matrix functions is given in Chapter 6 of [1450]; see also [1456]. The identity theorem is discussed in [1498]. A chain rule for matrix functions is considered in [1924, 1989]. Differentiation with respect to complex matrices is discussed in [1554]. Extensive tables of derivatives of matrix functions are given in [835, pp. 586–593].

Chapter Fourteen Integrals 14.1 Facts on Indefinite Integrals Fact 14.1.1. Let x > 0 and a ∈ R. Then, the following statements hold:

∫

x

ta dt exists if and only if a > −1. ∫0∞ ii) x ta dt exists if and only if a < −1. Fact 14.1.2. Let x ∈ R and a > −1. Then ∫ x xa+1 . ta dt = a+1 0 i)

In particular,

∫

0 −1

∫

0

−1

1 √ dt = −2 ȷ, t √ 2ȷ t dt = , 3

∫

1 0

∫

1 √ dt = 2, t √

1

2 , 3

t dt =

0

∫

0

−(3/2)2/3

∫

(3/2)2/3

√ t dt = ȷ, √ t dt = 1.

0

Fact 14.1.3. Let x > 0. If a < −1, then

∫

∞

ta dt = −

x

If a > 1, then

∫

∞ x

x1−a 1 dt = , a t a−1

xa+1 . a+1

∫ 1

∞

1 1 dt = . a t a−1

Fact 14.1.4. Let x ∈ R and a > 0. Then,

∫

x 0

at dt =

ax − 1 . log a

Fact 14.1.5. Let a, b, c, x be real numbers. Then,

( )   2ax + b b 2    atan √ − atan √ , √     4ac − b2 4ac − b2 4ac − b2   ∫ x    1  4ax dt = ,   2  2abx + b2  0 at + bt + c   ( )    −2 2ax + b b    atanh √ − atanh √ ,  √ 2 b − 4ac b2 − 4ac b2 − 4ac In particular, √ √   ∫ x 1 3(2x + 1) π  2 3    , atan − dt =  2 3 3 6 0 t +t+1 √  √ √  ∫ 1 √  2 5 acoth 1 2 5  3 5  dt = − atanh 5 = atanh 2 5 5 5 0 t + 3t + 1

b2 < 4ac, b2 = 4ac , 0, 4ac < b2 .

√

5

.

1094

CHAPTER 14

Fact 14.1.6. Let x be a real number. Then,

∫

x 0

1 1 (x + 1)2 1 dt = + atan x. log 4 2 t3 + t2 + t + 1 x2 + 1

Related: Fact 14.2.31. Fact 14.1.7. Let a > 0 and x > −a. Then,

∫

x 0

√ 1 (x + a)2 1 3 dt = log + t3 + a3 6a2 x2 − ax + a2 3a2

√    3(2x − a) π  atan +  . 3a 6

In particular, if x > −1, then √  √  ∫ x 1 1 3  3(2x − 1) π  (x + 1)2  dt = log 2 + +  . atan 3 6 3 3 6 x −x+1 0 t +1 ∫x 1 Remark: A similar formula can be obtained for 0 t5 +a dt. Fact 14.1.8. Let x ∈ R. Then, ∫ x 4 2 4 t (t − 1)4 1 dt = x7 − x6 + x5 − x3 + 4x − 4 atan x. 2+1 7 3 3 t 0 In particular,

∫

t (t − 1)4 22 − π. dt = 7 t2 + 1

1 4 0

Source: [522]. Remark: π < 22/7. Fact 14.1.9. Let a > 0 and x ∈ [0, a). Then,

∫

x 0

∫

x 0

a2

t2 a+x x a − x = a atanh − x, dt = log 2 a−x a − t2

1 t a 1 a+x π 1 a 1 x π dt = atan + log − = atan + atanh − , 2a x 4a a − x 4a 2a x 2a a 4a a4 − t4 ∫ x 1 t2 a3 + x3 dt = log . 6 6 6a3 a3 − x3 0 a −t 2

Source: [1563]. √ Fact 14.1.10. Let a > 0, let n ≥ 2, assume that n is even, and let x ∈ [0, n+1 a). Then,

∫

√3

In particular, if x ∈ [0, a), then

x 0

a − xn+1 tn 1 log . dt = n+1 a tn+1 − a ∫ 0

x

t2 1 a − x3 dt = log . 3 a t3 − a

Source: [1563]. Fact 14.1.11. Let a, c > 0, b ≥ 0, and x ∈ [0, ∞). Then,

√ √ √ ac ax + a2 x2 + ac at2 + c dt = 12 x ax2 + c + log , √ 2a ac 0 √ √ √ 2 2 2 2 2 x2 + abx log( a2 x + ab + a x) √ ax(2a x + 3abx + b ) − b a b2 at2 + bt dt = + 3/2 log ab. √ 8a 4a2 ax2 + bx ∫

∫

x 0

x

√

1095

INTEGRALS

Fact 14.1.12. Let a > 0 and x ∈ [0, a). Then,

∫ x √ 2 3/2 2 3/2 [2a5/2 − (2a + 3x)(a − x)3/2 ], t a − t dt = a − t dt = [a − (a − x) ], 3 15 0 0 ∫ x √ √ 2 [8a7/2 − (8a3 + 4a2 x + 3ax2 − 15x3 ) a − x], t2 a − t dt = 105 0 ) ( ∫ x √ √ 1 √ 2 x 1 2 2 2 2 , t a2 − t2 dt = [a3 − (a2 − x2 )3/2 ], a − t dt = x a − x + a atan √ 2 3 0 a2 − x2 ∫ x √ ( ) √ 1 x x(2x2 − a2 ) a2 − x2 + a4 asin . t2 a2 − t2 dt = 8 a 0

∫

∫

x 0

x

√

Fact 14.1.13. Let a > 0 and x ∈ [0, a). Then,

∫

x

0

1

√ a−t

√ √ dt = 2( a − a − x),

∫

x

√ 0

∫

t a−t

dt =

√ 2 3/2 [2a − (x + 2a) a − x], 3

√ 2 [8a5/2 − (3x2 + 4ax + 8a2 ) a − x], 15 a−t 0 √ ∫ x ∫ x √ t x t a2 − t 2 1 , a2 − x2 , dt = asin = − atan 2 dt = a − √ √ a t − a2 0 0 a2 − t2 a2 − t2 ∫ x ∫ x ) √ √ 1( 2 1 t2 x t3 dt = dt = [2a3 − (x2 + 2a2 ) a2 − x2 ]. a asin − x a2 − x2 , √ √ 2 a 3 0 0 a2 − t2 a2 − t 2 Fact 14.1.14. Let x ∈ [1, ∞). Then, ∫ x√ √ √ t−1 dt = x2 − 1 − log(x + x2 − 1). t+1 1 x

√

t

2

dt =

Fact 14.1.15. Let x and a be real numbers, and assume that 0 < x < a. Then,

∫

x 0

a−x t+a dx = 2a log + x. t−a a

Fact 14.1.16. Let x be a real number and a be a positive number. Then,

∫

x

0

sin2 t dt = x − sin2 t + a

√

√   a  a + 1 atan  tan x . a+1 a

Source: Fact 14.4.47. Fact 14.1.17. Let x ∈ (− π2 , π2 ). Then,

∫

x 0

sec t dt = log(sec x + tan x) = log tan( 2x + π4 ).

Source: [2309]. Remark: This integral has applications to cartography. See [49, p. 175], [1027,

p. 64], and [1945, p. 176]. Fact 14.1.18. Let x ∈ [0, π2 ]. Then, ∫ x( 0

In particular,

∫

π/6 0

(

) x 1 − cot t dx = log . t sin x

) 1 π − cot t dt = log , t 3

∫ 0

π/4

(

) 1 3 − cot t dt = log π − log 2, t 2

1096

CHAPTER 14

∫

π/3

(

0

√ ) 1 2 3π − cot t dt = log , t 9

∫

π/2 0

(

) π 1 − cot t dt = log . t 2

Fact 14.1.19. Let x ∈ [−1, 1]. Then,

∫

x

asin t 1 dt = asin2 x. √ 2 0 1 − t2 Fact 14.1.20. Let x ≥ 0, a > 0, and b > 1. Then, ∫ x 1 1 b − e−at dt = log . at a b−1 0 be − 1

14.2 Facts on Definite Integrals of Rational Functions Fact 14.2.1. The following statements hold: i) Let n ≥ 1. Then, ∫ 1 4n (n!)2 (2n)!! = . (1 − x2 )n dx = (2n + 1)!! (2n + 1)! 0

ii) Let n ≥ 1, assume that n is odd, and let a > 0. Then, ∫ a n!! an+1 π (a2 − x2 )n/2 dx = . (n + 1)!! 2 0 iii) Let a, b, c be real numbers, and assume that a > 0, b > −1, and c > −2. Then, ( ) ( ) ∫ a c+2 ab+c+1 Γ b+1 2 Γ 2 b 2 2 c/2 ) ( . x (a − x ) dx = 0 2Γ b+c+3 2 iv) Let n ≥ 1 and m ≥ 0. Then,

∫

1

(1 − x1/n )m dx = (

0

v) Let a, b ∈ (−1, ∞). Then, ∫ 1 ∫ xa (1 − x)b dx = 0

∞ 0

Now, let n, m ≥ 0. Then, ∫ 1 ∫ n m x (1 − x) dx = 0

0

If, in particular, n ≤ m, then ∫

1

Γ(a + 1)Γ(b + 1) xa dx = . a+b+2 Γ(a + b + 2) (x + 1) ∞

n!m! xn . dx = (n + m + 1)! (x + 1)n+m+2

xn (1 − x)m−n dx =

0

vi) Let n ≥ 0 and α > −1. Then, ∫

1 0

1 ). n+m m

1 (m + 1)

xn (1 − x)α dx =

( ). m n

n! . (α + n + 1)n+1

1097

INTEGRALS

vii) Let p, q ∈ (−1, ∞) and a, b ∈ R, where a < b. Then, ∫ b Γ(p + 1)Γ(q + 1) . (x − a) p (b − x)q dx = (b − a) p+q+1 Γ(p + q + 2) a Now, let n, m ≥ 0. Then, ∫ b (x − a)n (b − x)m dx = (b − a)n+m+1 a

In particular, ∫ b

(x − a)n (b − x)n dx = (b − a)2n+1

a

n!m! . (n + m + 1)!

( )2n+1 (n!)2 (2n)!! b−a = . (2n + 1)! (2n + 1)!! 2

Source: i) is given in [819], [1568, p. 145], and [1675, p. 56]; ii) and iii) are given in [3024, p. 330]; iv) is given in [1610]; v) is given in [1885]; vi) is given in [1568, p. 165]. Remark: Recursions involving the integrals in vii) and their connection with series involving binomial coefficients are given in [2947]. Fact 14.2.2.

∫ 1 ∞ ∞ ∑ ∑ 1 i+1 1 x x+1 dx = (−1)i+1 ≈ 0.40303444442, x dx = (−1) i ≈ 0.78343051071, i (i + 1)i 0 0 i=1 i=1 ∫ 1 ∫ 1 ∞ ∞ ∑ ∑ 1 x 1 1 dx = ≈ 1.2912859971, dx = ≈ 0.6284737129, x i x i (i + 1)i 0 x 0 x i=1 i=1 ∫ 1 −x ∫ 1 ∞ ∞ ∑ ∑ 1 1 x −1 1 − xx dx = dx = (−1)i+1 i+1 ≈ 0.88642971056, ≈ 1.138389995. i+1 x x i i 0 0 i=1 i=1

∫

1

x

Source: [506, pp. 103, 104], [517, pp. 4, 44], and [1179]. Related: Fact 14.8.7. Fact 14.2.3. The following statements hold:

i) Let n ≥ 0. Then,

∫ 0

ii) Let n ≥ 0. Then,

∫

1

0

In particular,

∫

1 0

iii) Let n ≥ 0. Then,

1

xn dx = (−1)n (H⌊n/2⌋ − Hn + log 2). x+1

∞ ∑ x2n 1 dx = Hn − H2n + log 2 = (−1)i+1 . x+1 i i=2n+1

1 dx = log 2, x+1 ∫ 0

In particular,

∫

1

∫ 0

1

1

x2 dx = log 2 − 21 . x+1

0

x2n+1 dx = H2n+1 − Hn − log 2. x+1

x dx = 1 − log 2, x+1

0

iv) Let n ≥ 0. Then,

1

∫

∫

1 0

x3 5 dx = − log 2. x+1 6

x2n π ∑ 1 dx = (−1)n + (−1)i+1 . 2 4 i=1 2n − 2i + 1 x +1 n

1098

CHAPTER 14

In particular, ∫ 1 0

∫

1 0

∫

1 π dx = , 4 x2 + 1

0

∫

x6 13 π − , dx = 15 4 x2 + 1

v) Let n ≥ 0. Then,

∫

In particular, ∫ 1 log 2 x dx = , 2 2 0 x +1 vi) Let n ≥ 0. Then,

0

1 0

1

∫

x2 π dx = 1 − , 2 4 x +1

x4 π 2 dx = − , 4 3 +1

1 0

76 x8 π , dx = − 4 105 x2 + 1

x2

∫

0

0

∫

1

1

log 4 − 1 x5 dx = , 2 4 x +1

x10 263 π − . dx = 315 4 x2 + 1

1

1 x4n+1 dx = (Hn − H2n + log 2). 2 x2 + 1

0

∫

1

∫

1 0

12 log 2 − 7 x9 dx = . 24 +1

x2

1 x4n+3 dx = (H2n+1 − Hn − log 2). 2 2 x +1

In particular, ∫ 1 ∫ 1 11 ∫ 1 1 − log 2 5 − 6 log 2 47 − 60 log 2 x7 x x3 dx = dx = dx = , , . 2 2 2 2 12 120 0 x +1 0 x +1 0 x +1 vii) Let n ≥ 1. Then,

∫ 0

1

∑ 1 xn−1 dx = log 2 − . n (x + 1) i2i i=1 n−1

Source: ii) is given in [2013, p. 178]. vii) is given in [1100]. Fact 14.2.4.

∫

1 0

√ √ ∫ 1 ∫ 1 3π log 2 3π log 2 x x2 log 2 + , − , , dx = dx = 3 3 9 3 9 3 3 x + 1 x + 1 0 0 √ √ √ ∫ 1 2π 2 log(3 + 2 2) π 1 x dx = dx = , + , 4 4 8 8 8 0 x +1 0 x +1 √ √ √ ∫ 1 ∫ 1 3 2 log 2 x x 2π 2 log(3 + 2 2) dx = dx = − , , 4+1 4+1 8 8 4 x x 0 0 √ √ ∫ 1 ∫ 1 2 x6 (1 + 2)π x +1 2π , , dx = dx = 1 − 4 2 4 4 8 0 (x + 1)(x + 1) 0 x +1 √  √ √ ∫ 1   √ 1 1  10 + 2 5 5  .  π + + 2 log 2 dx = 5 atanh  5  10  5 3 0 x +1

1 dx = x3 + 1 ∫ 1

Fact 14.2.5. Let a and b be positive numbers. Then,

  √  x 1  a2 + 1 b dx = 2 + atan a , log 2 2 2 b+1 a a +b 0 (a x + 1)(bx + 1) [ ] x2 1 b2 b 2 dx = log(a + 1) + log(b + 1) − atan a . a (a2 x2 + 1)(bx + 1) b(a2 + b2 ) 2a2 ∫

∫

1 0

Source: [560].

1

1099

INTEGRALS

Fact 14.2.6. The following statements hold:

i) Let n ≥ 1. Then,

∫

1

0

ii) Let m, n ≥ 1. Then,

∫

1 − xn dx = 1−x

∫

1

0

1 − (1 − x)n dx = Hn . x

∑ 1 − xm m dx = . 1 − xn (ni + 1)(ni + m + 1) i=0 ∞

1

0

In particular, √ √ ∫ 1 ∫ 1 ∫ 1 1−x 3π 3π log 3 1−x 1 − x2 dx = log 2, dx = dx = , + , 2 3 3 9 18 2 1 − x 1 − x 1 − x 0 0 0 ∫ 1 ∫ 1 ∫ 1 π log 2 π π 3 log 2 1−x 1 − x2 1 − x3 dx = dx = dx = + + , , . 4 4 4 8 4 4 8 4 0 1−x 0 1−x 0 1−x iii) Let n ≥ 1. Then,

∫ 0

1

1 1 − (1 − x)n log dx = Hn,2 . x 1−x

1−xn 1−x

= 1 + x + · · · + xn−1 and [511, p. 176]; ii) is given in [2455]; iii) is Source: i) follows from given in [2013, p. 178]. Related: Fact 13.5.68 gives expressions for the infinite series in ii). Fact 14.2.7. Let p > −1. Then, ∫

0

1

1 − xp dx = H p = ψ(p + 1) + γ. 1−x

In particular,

√ ∫ 1 1 − x−1/2 1 − x−1/3 3π 3 log 3 dx = −2 log 2, dx = − , 1−x 1−x 6 2 0 0 √ ∫ 1 ∫ 1 3π 3 log 3 1 − x−1/4 π 1 − x1/6 dx = − 3 log 2, dx = 6 − − − 2 log 2, 1 − x 2 1 − x 2 2 0 0 √ ∫ 1 ∫ 1 3π 3 log 3 π 1 − x1/3 1 − x1/4 dx = 4 − − 3 log 2, dx = 3 − − , 1−x 2 1−x 6 2 0 0 √ ∫ 1 ∫ 1 3π 3 log 3 1 − x1/2 1 − x2/3 3 dx = 2 − 2 log 2, dx = + − , 1 − x 1 − x 2 6 2 0 0 ∫ 1 ∫ 1 4 π 1 − x3/2 8 1 − x3/4 dx = + − 3 log 2, dx = − 2 log 2. 1 − x 3 2 1 − x 3 0 0 ∫ 1 ∫ 1 1 − x5/2 46 1 − x7/2 352 dx = − 2 log 2, dx = − 2 log 2. 1−x 15 1−x 105 0 0 ∫

1

If, in addition, m and n are positive integers such that m < n, then ) ∫ 1 ⌊(n−1)/2⌋ ∑ ( 1 − xm/n n 2miπ iπ π mπ dx = + 2 cos log sin − cot − log 2n. 1 − x m n n 2 n 0 i=1 In particular, ∫ 1 π π 1 − x1/7 π π 3π π π dx = 7−log 14− cot +2(sin 3π 14 ) log sin 7 −2(sin 14 ) log cos 14 −2(cos 7 ) log cos 14 . 1 − x 2 7 0

1100

CHAPTER 14

Related: Fact 13.3.4. ∑ i Fact 14.2.8. Let n ≥ 1 and k ≥ 2, and define p ∈ R[s] by p(s) = k−1 i=0 s . Then,

∫

1 0

xnk p′ (x) dx = log k + Hn − Hkn . p(x)

Source: [1682]. Fact 14.2.9. Let a ∈ (−1, 1). Then,

∫

1

0

aπ xa dx = . a (1 − x) sin aπ

Fact 14.2.10. Let a ∈ (0, 1). Then,

∫

1 0

1 dx = (1 − x)1−a xa

∫

π 1 dx = . a 1−a sin aπ (1 − x) x

1 0

Fact 14.2.11. Let a and b be positive numbers, and let n ≥ 1. Then,

∫

1 0

In particular, ∫ 1 0

∑ n! xb−1 (1 − x)n ∏n . dx = a 1−x j=0 (ia + b + j) i=0 ∞

∑ (1 − x)n n! , dx = ∏n+1 1 − xa j=1 (ia + j) i=0 ∞

∫

1

1 ∑ n! . = ∏ n i=0 n+1 j=1 (i + j) ∞

(1 − x)n−1 dx =

0

Source: [113]. Remark: The last equality is given by Fact 13.5.26. Fact 14.2.12. Let a > 0 and n ≥ 1. Then,

∫

1 0

∫

1 0

∑ ∑ 1 (1 − x)n 1 n! ( ), dx = = ∏ n+1 1 − xa (n + 1)! i=0 ai+n+1 j=1 (ai + j) i=0 ai ∞

∞

∑ ∑ 1 n! 1 (1 − x)n i = dx = (−1) (−1)i (ai+n+1) . ∏ n+1 1 + xa (n + 1)! j=1 (ai + j) i=0 i=0 ai ∞

∞

Now, assume that 1/a = 2k, where k ≥ 1. Then, ∫ 1 ∫ 1 2k−1 2k−1 ∑ ∑ (1 − x)n (n − 1)! (n − 1)! (1 − x)n ∏ dx = dx = (−1)i ∏n , . n 1/(2k) 1/(2k) [i/(2k) + j] 1 + x 0 1−x 0 j=1 j=1 [i/(2k) + j] i=0 i=0 In particular, ∫ 1 ∫ 1 5 1−x dx = , 1/2 3 0 0 1−x ∫ 1 ∫ 1 1 1−x dx = , 1/2 3 0 0 1+x Source: [2481]. Fact 14.2.13.

∫ 0

(1 − x) 7 dx = , 30 1 + x1/2 2

∫

1

0

∫

0

1

1−x 319 dx = , 1/4 105 1−x 1−x 31 dx = , 105 1 + x1/4

∫

1

(1 − x)2 9217 dx = , 1/4 6930 1−x

1

(1 − x)2 1409 dx = . 6930 1 + x1/4

0

∫

0

√ √ ∫ 1 1 2 3π x 3π dx = , dx = , 2 2 9 9 0 1−x+x 0 1−x+x √ √ ∫ 1 x2 3π 1 2 4 3π dx = 1 − , dx = + , 2 2 9 3 27 1 − x + x2 0 (1 − x + x ) ∫

1

(1 − x)2 23 dx = , 1/2 30 1−x

1

1101

INTEGRALS

√ √ ∫ 1 1 2 3π 4 3π 1 x x2 dx = + dx = , − , 2 2 2 2 3 27 27 3 0 (1 − x + x ) 0 (1 − x + x ) √ √ ∫ ∫ 1 1 x4 x3 5 3π 2 2 2 3π − , , dx = dx = − 2 2 2 2 27 3 3 27 0 (1 − x + x ) 0 (1 − x + x ) √ √ ∫ 1 ∫ 1 √ √ 1 4 5 5 x 1 dx = dx = log 2 (1 + 5), log 12 (3 + 5), 2 2 5 5 1 + x − x 1 + x − x 0 0 √ √ ∫ 1 ∫ 1 2 √ √ 2 8 5 5 1 x dx = dx = + log(9 + 4 5) − 1, log 21 (1 + 5), 2 2 2 5 5 25 0 (1 + x − x ) 0 1+x−x √ √ ∫ 1 ∫ 1 2 √ √ x x 1 4 5 3 8 5 1 log 2 (1 + 5), log 12 (1 + 5), dx = + dx = − 2 2 2 2 5 25 5 25 0 (1 + x − x ) 0 (1 + x − x ) √ ∫ 1 3 √ 4 7 5 x dx = − log 21 (3 + 5), 2 )2 5 25 (1 + x − x 0 √ ∫ 1 √ x4 12 26 5 − log 12 (3 + 5), dx = 2 2 5 25 0 (1 + x − x ) √ √ ∫ 1 ∫ 1 log 3 1 x 3π 3π dx = dx = , − , 2 2 9 2 18 1 + x + x 1 + x + x 0 0 √ √ ∫ 1 ∫ 1 3π x2 log 3 1 2 3π dx = 1 − − , dx = , 2 2 2 2 18 27 0 1+x+x 0 (1 + x + x ) √ √ ∫ 1 ∫ 1 3π 1 x2 2 3π 1 x dx = − , dx = − , 2 2 2 2 3 27 27 3 0 (1 + x + x ) 0 (1 + x + x ) √ √ ∫ 1 ∫ 1 4 3 log 3 5 3π 4 3π x x dx = dx = − − , − log 3. 2 2 2 2 2 54 3 27 0 (1 + x + x ) 0 (1 + x + x ) ∫

1

Now, let n ≥ 1. Then, √ ∫ 1 n−3 1 − 3n−2 (2n − 3)!! ∑ 1−n −i 2i (n − i − 1)! 2n−1 (2n − 3)!! 3π 1 dx = n−1 + + , (3 −3 ) 2 n (2n − 2i − 1)!! (3n + 1)(n − 1)! 3 (n − 1) 2(n − 1)! i=2 0 (1 + x + x ) ∫ ∫ 1 1 1 − 3n−1 1 1 x dx = − dx. 2 )n n−1 2 (1 + x + x 2(1 − n)3 (1 + x + x 2 )n 0 0 Source: [817]. Fact 14.2.14.

∫

1 0

∑ 2(3i)! ∑ 1−x 2 dx = = = 2 (3i + 3)! (3i + 1)(3i + 2)(3i + 3) x +x+1 i=0 i=0 ∞

∞

√ 3π − 3 log 3 . 6

Source: [2577]. Fact 14.2.15. Let n > m ≥ 1. Then,

∫

0

1

∑ 1 (n − 1)xm (1 − x)n−m + 1 ( ni ) . dx = [xm (1 − x)n−m − 1]2 i=0 ∞

mi

In particular, ∫ 1 ∞ ∑ 1 mxm (1 − x) + 1 ((m+1)i) , dx = m 2 0 [x (1 − x) − 1] i=0 mi

∫ 0

1

∑ 1 (m + 1)xm (1 − x)2 + 1 ((m+2)i) , dx = [xm (1 − x)2 − 1]2 i=0 ∞

mi

1102

CHAPTER 14

√ ∞ ∑ 1 4 2 3π x(1 − x) + 1 (2i) = + dx = , 2 3 27 0 [x(1 − x) − 1] i=0 i √ √ ∞ ∑ √ 3π 2 5 16 3x2 (1 − x)2 + 1 1 1 ( ) = + − log 5). (1 + dx = 2 4i 15 27 25 [x2 (1 − x)2 − 1]2 i=0 ∫

∫

1

0

1

2i

Source: [2509, 2728]. Fact 14.2.16. Let x ∈ (−4, 4). Then,

∫

1 0

2x(1 − t) 10x − x2 dt = + 24 3 [1 − xt(1 − t)] (x − 4)2

√

√ ∞ x x ∑ xi = . asin 2 Ci (4 − x)5 i=1

Source: [343]. Related: Fact 13.4.2 and Fact 13.9.1. Fact 14.2.17. Let a > 0 and b > 0, and assume that a , b. Then,

∫

1

a x b1−x dx =

0

∞ ∏

1 1/2i 2 (a

i

+ b1/2 ) =

i=1

a−b , log a − log b

∫

0

1

log a − log b 1 dx = . ax + b(1 − x) a−b

Source: [635]. Related: Fact 2.2.63. Fact 14.2.18. Let a > 0 and b > 0. Then,

∫

1 0

1 1 dx = . ab [ax + b(1 − x)]2

Source: [2106, p. 18]. Fact 14.2.19. Let a > 0 and b > 0. Then,

∫

∞ 0

(

) 1 a 1 − dx = log , x+b x+a b

∫

∞ 0

∫

1 dx = 1, (ax + b)(x/b + 1/a)

If a , b, then ∫ ∞ ∫ ∞ 1 1 log a − log b dx = dx = , a−b 0 (ax + b)(x + 1) 0 (x + a)(x + b)

∫

∞ 0

1

1 1 . dx = ab (ax + b)2

a x b1−x dx =

0

a−b . log a − log b

Now, let c > 0 and d > 0, and assume that ad , bc. Then, ∫ ∞ log ad − log bc 1 dx = . ad − bc 0 (ax + b)(cx + d) Related: Fact 2.2.63, Fact 10.10.49, and Fact 10.11.69. Fact 14.2.20. Let a, b, c, d, e, f be positive numbers, and assume that ad , bc, be , a f, and

c f , de. Then, ∫ ∞ 0

a log ab c log dc e log ef 1 dx = + + . (ax + b)(cx + d)(ex + f ) (ad − bc)(be − a f ) (ad − bc)(c f − de) (be − a f )(c f − de)

Fact 14.2.21. The following statements hold: i) Let n ≥ 1 and a > 0. Then, ( ) ∫ ∞ π 2n − 2 (2n − 3)!!π 1 dx = = n . 2 2 n 2n−1 n−1 (2a) 2 (n − 1)!a2n−1 0 (x + a )

In particular,

∫

∞ 0

1 π dx = , 2a x2 + a2

∫ 0

∞

1 π dx = 3 . (x2 + a2 )2 4a

1103

INTEGRALS

ii) Let n ≥ 1. Then, ∫ ∞

( ) (2n − 2)!π 1 (2n − 3)!!π π 2n − 2 = 2n−1 = n dx = 2n−1 . 2 n 2 (n − 1)! n−1 (x + 1) 2 2 [(n − 1)!]2

0

iii) Let n ≥ 2 and a > 0. Then, ∫ ∞ 1 x dx = , 2 + a2 )n (x (2n − 2)a2n−2 0

∫ 0

iv) Let n ≥ 3 and a > 0. Then, ∫ ∞ x3 1 dx = , 2 2 )n 2 − 6n + 4)a2n−4 (x + a (2n 0 v) Let a > 0. Then,

∫

∫

∞

(2n − 5)!!π x2 dx = n . (x2 + a2 )n 2 (n − 1)!a2n−3

∫

∞ 0

(x2

3(2n − 7)!!π x4 . dx = n + a 2 )n 2 (n − 1)!a2n−5

∫ ∞ x 1 π x2 , dx = , dx = 2 2 2 2 2 2 2 4a 2a 0 (x + a ) 0 (x + a ) ∫ ∞ ∫ ∞ 1 3π x 1 x2 π dx = , dx = , dx = , 2 2 3 2 2 3 4 2 2 3 5 (x + a ) 4a 16a3 16a 0 (x + a ) 0 (x + a ) ∫ ∞ ∫ ∞ x3 1 x4 3π . dx = , dx = 2 2 3 2 2 2 )3 16a (x + a ) 4a (x + a 0 0

∞ 0

∞

vi) Let a > 0 and b ∈ (−1, 1) ∪ (1, 3). Then, ∫ ∞ xb (1 − b)π bπ dx = sec . 2 2 2 3−b 2 4a 0 (x + a ) In particular, ∫ 0

∞

√ x 2π dx = 5/2 , 2 2 2 (x + a ) 8a √

∫

∞ 0

√ 2π x3/2 dx = 3/2 . 2 2 2 (x + a ) 8a

vii) Let a > 0, k ≥ 0, and n ≥ 2, and assume that k + 2 ≤ n. Then, ∫ ∞ π xk dx = . n + a2 2(n−k−1)/n x na sin (k+1)π 0 n Hence, ∫ 0

∞

√ ∫ ∞ xk π xk 2 3π dx = , dx = , x2k+2 + a2 (2k + 2)a(2k+2)/n x3k+3 + a2 (9k + 9)a(4k+4)/n 0 √ ∫ ∞ 2π xk dx = . 4k+4 2 x +a (2k + 2)a(6k+6)/n 0

If, in addition, n ≥ 3, then

∫ 0

∫

∞

x π , dx = 2 2(n−2)/n +a na sin 2π 0 n ∫ ∞ ∫ ∞ x−1 1 2π π x dx = dx = ∑n−1 i dx = csc . n−1 xn + 1 x n n 0 0 i=0 x xn

∞

viii) Let n ≥ 2 and a > 0. Then,

∫

∞ 0

1 π dx = 2(n−1)/n . xn + a2 na sin πn

1104

CHAPTER 14

ix) Let n ≥ 3 and a > 0. Then,

∫

∞ 0

x) Let n ≥ 4 and a > 0. Then,

∫

xn

π x . dx = 2 2(n−2)/n +a na sin 2π n

xn

x2 π . dx = 2 2(n−3)/n +a na sin 3π n

∞ 0

xi) Let a > 0 and b ∈ (−1, 1). Then, ∫ ∞ x2

0

xii) Let n ≥ 1. Then,

∫

∞ 0

∫

∞

n ∏

0

xiii) Let n ≥ 1. Then, ∫ ∞ 0

i=1

n ∏ i=1

xb 1 bπ dx = ab−1 π sec . 2 2 + a2

π 1 , dx = 2(2n − 1)(n − 1)!n! x 2 + i2

1 π dx = 2n−1 . x2 + (2i − 1)2 2 (2n − 1)[(n − 1)!]2 √ πΓ(n + 21 ) 8x2 (1 − x2 )2n dx = 4(n + 1)! (x2 + 1)2n+3 0 ( ) (2n)!π π 2n πCn = n+1 = n+1 = n+1 . 4 n!(n + 1)! 4 (n + 1) n 4

x2 dx = (x2 + 1)n+2

∫

1

Source: i) is given in [511, p. 114]; ii) is given in [69], [1568, p. 166], and [2068, pp. 247, 248];

v) is given in [1136, p. 202]; vi) is given in [2106, p. 340]; vii) is given in [561], [645, p. 82], and [2447]; viii) and x) are given in [1136, p. 202]; ix) and xii) are given in [69]; xi) is given in [953]; xiii) is given in [822]. Remark: ii) is Wallis’s formula. See [2068, p. 247]. Fact 14.2.22. Let b > 0 and c > 0. If a ∈ (−1, b − 1), then ∫ ∞ xa c(a+1−b)/b π . dx = xb + c b sin (a+1)π 0 b If a ∈ (−1, 2b − 1), then

∫ 0

In particular, ∫ ∫

∞ 0

∞

xa (b − a − 1)c(a+1−2b)/b π dx = . (xb + c)2 b2 sin (a+1)π b

√ √ ∫ ∞ ∫ ∞ 2 3π π x 2π x x dx = dx = dx = , , , 2+1 3+1 4+1 2 9 4 x x x 0 0 0 √ √ √ ∫ ∞ ∫ ∞ x x 2 3π π x 2π , , dx = dx = dx = . 2 2 3 2 4 2 8 27 8 (x + 1) 0 (x + 1) 0 (x + 1) ∞

√

Furthermore, if d > 1, then

∫

∞ 0

xd−1 π dx = . 2d +1

x2d

1105

INTEGRALS

Fact 14.2.23. Let a ∈ (−1, 1). Then,

∫

0

In particular,

∞

√ 2 3π xa , dx = x2 + x + 1 3 + 6 cos 2aπ 3 ∫ 0

∞

√

x dx = x2 + x + 1

0

√ 4 3π cos aπ xa 3 . dx = x2 − x + 1 3 + 6 cos 2aπ 3

∫

∞

∫

√ 3π , 3

∞

0

√

x dx = π. x2 − x + 1

Source: [150, p. 13]. Fact 14.2.24. Let a and b be real numbers such that 0 < a < b. Then,

∫

∞

0

(x1/a

Γ(a + 1)Γ(b − a) 1 . dx = b Γ(b) + 1)

∫

In particular,

∞ 0

Source: [2574]. Setting b = Fact 14.2.25.

∫

∞ 0

1 2 (1

1 dx = 1. (xα + 1)α

√ √ + 5) and a = 1/b = b − 1 = 12 ( 5 − 1) yields the special case.

√  √  2 7  π 1 7   − atan  , dx = 2 7 2 7 x +x+2

∫

∞ 0

√ √ 1 log 2 3 7(π − atan 7) + . dx = 8 28 x3 + x + 2

Source: [645, p. 82]. Fact 14.2.26. The following statements hold:

i) Let a and b be positive numbers. Then, ) ( ∫ ∞ 1 a 1 bπ + log . dx = 2 2 b a2 + b2 2a 0 (x + a )(x + b) ii) Let a and b be positive numbers. Then, ) ( ∫ ∞ x 1 b dx = aπ + 2b log . 2 2 a 2(a2 + b2 ) 0 (x + a )(x + b) iii) Let a and b be positive numbers. If a , b and c ∈ (0, 4), then  π(bc−2 − ac−2 )      cπ , a , b and c , 2,    2(a2 − b2 ) sin    2    π(2 − c)   ∫ ∞  , a = b and c , 2,   xc−1  4a4−c sin cπ dx =   2 2 2 2 2   0 (x + a )(x + b )  1 a    log , a , b and c = 2,   2 2  b a −b       1    , a = b and c = 2. 2a2 In particular, ∫ ∞ ∫ ∞ 1 π x2 π dx = , dx = . 2 2 2 2 2 2 2 2 2ab(a + b) 2(a + b) 0 (x + a )(x + b ) 0 (x + a )(x + b ) iv) Let a, b, c be positive numbers. Then, ∫ ∞ 1 (a + b + c)π dx = , 2 + a2 )(x2 + b2 )(x2 + c2 ) 2abc(a + b)(b + c)(c + a) (x 0

1106

CHAPTER 14

∫ ∫

∞

+

(x2

0 ∞ 0

a2 )(x2

π x2 dx = , 2(a + b)(b + c)(c + a) + b2 )(x2 + c2 )

(ab + bc + ca)π x4 dx = . 2(a + b)(b + c)(c + a) (x2 + a2 )(x2 + b2 )(x2 + c2 )

If, in addition, a, b, c are distinct, then ∫ ∞ [ab(a2 − b2 ) + bc(b2 − c2 ) + ca(c2 − a2 )]π 1 dx = − , 2 2 2 2 2 2 2abc(a2 − b2 )(b2 − c2 )(c2 − a2 ) 0 (x + a )(x + b )(x + c ) ∫ ∞ x (a2 − b2 ) log c + (b2 − c2 ) log a + (c2 − a2 ) log b dx = , 2 2 2 2 2 2 (a2 − b2 )(b2 − c2 )(c2 − a2 ) 0 (x + a )(x + b )(x + c ) ∫ ∞ x3 (a2 − b2 )c2 log c + (b2 − c2 )a2 log a + (c2 − a2 )b2 log b dx = − . 2 2 2 2 2 2 (a2 − b2 )(b2 − c2 )(c2 − a2 ) 0 (x + a )(x + b )(x + c ) v) Let a, b, c, d be positive numbers. Then, ∫ ∞ 1 [(a + b + c + d)3 − (a3 + b3 + c3 + d3 )]π dx = 2 2 2 2 2 2 2 2 6abcd(a + b)(a + c)(a + d)(b + c)(b + d)(c + d) 0 (x + a )(x + b )(x + c )(x + d ) [a2 (b + c + d) + a(b + c + d)2 + (b + c)(b + d)(c + d)]π = , 2abcd(a + b)(a + c)(a + d)(b + c)(b + d)(c + d) ∫

∞

0

∫

∞

0

∫ 0

(a + b + c + d)π x2 , dx = 2 2 2 2 2 2 2 2 2(a + b)(a + c)(a + d)(b + c)(b + d)(c + d) (x + a )(x + b )(x + c )(x + d ) x4 (abc + bcd + cda + dab)π , dx = 2 2 2 2 2 2 2 2 2(a + b)(a + c)(a + d)(b + c)(b + d)(c + d) (x + a )(x + b )(x + c )(x + d )

∞

x6 (x2

+

a2 )(x2

+

dx + c2 )(x2 + d2 ) [a2 (b + c)(b + d)(c + d) + a(bc + bd + cd)2 + bcd(bc + bd + cd)]π = . 2(a + b)(a + c)(a + d)(b + c)(b + d)(c + d)

b2 )(x2

vi) Let z ∈ C, assume that Re z , 0. Then, ∫ ∞ 1 (sign Re z)π . dx = 2 + z2 2z x 0 In particular, if a is a nonzero real number, then ∫ ∞ a (sign a)π . dx = 2 2 2 x +a 0 vii) Let z1 , . . . , zn ∈ C, assume that z1 , . . . , zn are distinct, and assume that, for all i ∈ {1, . . . , n}, Re zi , 0. Then, ∫ ∞∏ n n n ∑ 1 (sign Re zi )π ∏ 1 dx = . 2zi x2 + z2i z2j − z2i 0 j=1 i=1 i=1 j,i

Source: i) is given in [1963]; ii) is given in [1217, p. 330]; vii) is given in [2106, p. 56]. Related:

Fact 14.10.1. Fact 14.2.27. Let a be a complex number such that a < πZ + 21 π. Then,

∫

∞ 0

1 dx = 4 2 x + 2x cos 2a + 1

∫

∞ 0

x2 π dx = . 4 2 4 cos a x + 2x cos 2a + 1

1107

INTEGRALS

√ ∫ ∞ 1 π 1 2π dx = , dx = , 4 − x2 + 1 4+1 2 4 x x 0 0 √ ∫ ∞ ∫ ∞ π 3π 1 1 dx = dx = , , 4 2 4 2 6 4 x +x +1 x + 2x + 1 0 0 √ √ ∫ ∞ √ √ 1 1 π π 4 + 2 2, 4 − 2 2. dx = dx = √ √ 4 4 0 x4 − 2x2 + 1 x4 + 2x2 + 1

In particular,

∫

∫

∞ 0

∞

Source: [2106, pp. 56–60]. Replacing x by 1/x changes the numerator from 1 to x2 . Fact 14.2.28. Let a ∈ (− π2 , π2 ). Then,

∫

∞ 0

1 dx = 2 x + 2x cos 2a + 1

∫

∞

x4

0

+

2a 2x dx = . sin 2a cos 2a + 1

2x2

√ ∫ ∞ 4 3π π 1 1 dx = dx = , , 2 2 9 2 x −x+1 x +1 0 0 √ ∫ ∞ ∫ ∞ 1 1 2 3π , dx = dx = 1, 2+x+1 2 + 2x + 1 9 x x 0 0 √ √ ∫ ∞ ∫ ∞ 3 2π 2π 1 1 dx = dx = , . √ √ 4 4 0 0 x2 − 2x + 1 x2 + 2x + 1

In particular,

∫

∞

Fact 14.2.29. Let a be a complex number such that ȷa < πZ + 12 π. Then,

∫

∞ 0

In particular, ∫ ∞ 0

x4 +

1 dx = 4 2 x + 2x cosh 2a + 1

1 17 2 4 x

+1

dx =

∫

π , 5

∞ 0

∫

∞ 0

x2 π . dx = 4 2 4 cosh a x + 2x cosh 2a + 1

1 x4 +

82 2 9 x

+1

dx =

∫

3π , 20

∞

1 x4 +

0

257 2 16 x

+1

dx =

2π . 17

Source: Replacing x by 1/x changes the numerator from 1 to x . Fact 14.2.30. Let a be a real number such that a < (π ȷ/2)Z. Then, 2

∫

∞ 0

1 dx = x2 + 2x cosh 2a + 1

In particular,

∫

∞ 0

1

16 log 2 , dx = 17 2 15 x + 4 x+1

∫ 0

∞

∫

0

∞

2x 2a . dx = sinh 2a x4 + 2x2 cosh 2a + 1

1

9 log 3 , dx = 82 2 20 x + 9 x+1

∫

∞

1 x2

0

+

257 16 x

+1

dx =

128 log 2 . 255

Fact 14.2.31. The following statements hold:

i) Let a be a complex number, let b be a positive number, and assume that 0 < Re a < b. Then, ∫ ∞ a−1 x π dx = aπ . b+1 b sin x 0 b In particular, ∫ ∞ 1 dx = π, √ x(x + 1) 0

∫

∞ 0

xa−1 π dx = , x+1 sin aπ

∫

∞ 0

1 dx = 4 x +1

√

2π . 4

1108

CHAPTER 14

ii) Let a ∈ (0, 1) and b ∈ (0, ∞). Then, ∫ ∞

1 π dx = a . + b) b sin aπ

xa (x

0

In particular, √ √ ∫ ∞ ∫ ∞ ∫ ∞ 2π 1 1 π 2 3π 1 dx = √ , dx = √3 , dx = √4 . √ √3 √4 x(x + b) x(x + b) x(x + b) 0 0 0 b 3 b b iii) Let a ∈ (−1, 1) and b ∈ (0, ∞). Then, ∫ ∞ 1 aπ . dx = a+1 a (x + b)2 x b sin aπ 0 iv) Let a ∈ (−2, 1) and b ∈ (0, ∞). Then, ∫ ∞ xa (x

0

v) Let a ∈ (−3, 1) and b ∈ (0, ∞). Then, ∫ ∞ xa (x2

0

In particular, if c ∈ (−2, 2), then ∫ ∞ 0

vi) Let a, b ∈ (0, 1). Then,

∫

∞

0

(x2

a(a + 1)π 1 . dx = a+2 3 + b) 2b sin aπ 1 (a + 1)π dx = a+3 . 2 2 +b ) 4b cos aπ 2

xc+1 cπ dx = 2−c . + b2 )2 4b sin cπ 2

xa−1 − xb−1 dx = π(cot πa − cot πb). 1−x

vii) Let n ≥ 0 and m ≥ n + 2. Then, ∫ ∞ xn n!(m − n − 2)! 1 ( ). dx = = m (x + 1) (m − 1)! 0 (m − 1 − n) m−1 n viii) Let a, b, c, d be real numbers such that a > −1, d > 0, c > (a + 1)/d, and b > 0. Then, ( ) ( ) ∫ ∞ b(a−dc+1)/d Γ a+1 Γ c − a+1 xa d d dx = . d c dΓ(c) 0 (x + b) ix) Let a, b, c be real numbers such that b > 0, a < 1, and 1 − a < c. Then, ∫ ∞ 1 Γ(1 − a)Γ(a + c − 1) dx = . a (x + b)c x ba+c−1 Γ(c) 0 In particular, ∫ ∞ 0

x 1 dx = , 2b (x + b)3

∫

∞ 0

x2 1 dx = , 3b (x + b)4

∫

∞ 0

√

1 π dx = √ . x(x + b) b

Source: i) is given in [1136, p. 208]; ii) is given in [1146, p. 69], [1568, p. 40], and [3024, p. 330, formula 589]; iii) is given in [1136, p. 206]; iv) follows from iii); v) is given in [1136, p. 208] and [1217, p. 325]; vi) is given in [3024, p. 330]; vii) is given in [1568, p. 40]. Related: Fact 14.1.6.

1109

INTEGRALS

Fact 14.2.32. Let a and b be positive numbers, and let m > n ≥ 0. Then,

∫

∞ 0

xn 1 ( ). dx = m+1 (ax + b) an+1 bm−n (m − n) mn

Remark: See [511, pp. 48–52]. Related: Fact 1.16.11. Fact 14.2.33. Let a be a real number, and let b be a positive number. Then,

( [ )]  1 a π    − atan , √ √    2 2 2  b − a b − a     ∫ ∞    √   1  a + a2 − b   1 dx =   ,  log   √ √  x2 + 2ax + b  0 2−b 2−b  2 a a − a        1    , a Source: [511, p. 32]. Remark: Extensions are given in [1217, pp. and b ≤ a2 , then allowing cancellation of singularities yields ∫ ∞ 1 dx = 0. 2 + 2ax + b x −∞

a2 < b, b < a2 and a > 0, b = a2 and a > 0. 325, 326]. Remark: If a > 0

See [2106, p. 301]. Fact 14.2.34. Let a, b, c be real numbers, assume that b ≥ 0 and b2 < ac, and let n ≥ 0. Then, ( )   ( ) ∫ ∞ n 2i−1 2 (i−1)/2  ∑  an 2n  1 b 2 (ac − b ) n acot √  . (2i) dx = n −b  2 + 2bx + c)n+1 2 )n+1/2  i 2 (ax 4 (ac − b 0 i(ac) i ac − b i=1 In particular,

∫

∞

1 b 1 acot √ . dx = √ 2 ax2 + 2bx + c ac − b ac − b2 Source: [2068, pp. 233, 234]. Example: ∫ ∞ ∫ ∞ π 1 1 1 1 1 dx = dx = − + atan . − atan 2, 2 2 2 2 5 2 2 x + 4x + 5 0 (x + 4x + 5) 0 0

Fact 14.2.35. Let a, b, c be real numbers, assume that a > 0 and b2 < 4ac, and let n ≥ 1. Then,

∫

∞

−∞

2n (2n − 3)!!an−1 π 1 dx = . (ax2 + bx + c)n (n − 1)!(4ac − b2 )n−1/2

In particular, ∫ ∞

∫ ∞ 1 2π 1 4aπ dx = , dx = . √ 2 2 2 (4ac − b2 )3/2 −∞ ax + bx + c −∞ (ax + bx + c) 4ac − b2 Now, assume that n ≥ 2. Then, ∫ ∞ 2n−1 (2n − 3)!!an−2 bπ x dx = − . 2 n (n − 1)!(4ac − b2 )n−1/2 −∞ (ax + bx + c) In particular, ∫ ∞ −∞

2bπ x dx = − , (ax2 + bx + c)2 (4ac − b2 )3/2

∫

∞

−∞

6abπ x dx = − . (ax2 + bx + c)3 (4ac − b2 )5/2

Source: [645, p. 82], [1217, pp. 325, 326], and [1524, p. 267].

1110

CHAPTER 14

Fact 14.2.36. Let a ∈ (−1, ∞) and n ≥ 1. Then,

∫

∞ 0

( )( ) n ∑ π 1 i 2n − 2i n + i dx = (a + 1)i . 2 n−i n (x4 + 2ax2 + 1)n+1 23n+3/2 (a + 1)n+1/2 i=0

Source: [509]. Fact 14.2.37. Let a, b, r be real numbers such that a > −1 and r > 21 . Then,

(

)r 2 x +1 x2 dx = dx 4 2 x + 2ax + 1 x2 0 0 0 )r ∫ ∫ ∞( (a + 1)1/2−r 1 xr−3/2 x2 x2 + 1 dx dx = = √ x4 + 2ax2 + 1 x2 (xb + 1) 2r+1/2 0 0 1−x √ (a + 1)1/2−r πΓ(r − 12 ) = . 2r+1/2 Γ(r) √ Now, let c and d be real numbers such that c ≥ 0, d > 0, and −a < cd. Then, √ √ )r ∫ ∞( (a + cd)1/2−r πΓ(r − 21 ) x2 dx = . √ dx4 + 2ax2 + c 0 2r+1/2 dΓ(r) ∫

∞

x2 x4 + 2ax2 + 1

)r

∫

∞

(

x2 x4 + 2ax2 + 1

)r

1 1 dx = 2 2 x

∫

∞

(

Now, assume that r is a positive integer. Then, )r ∫ ∞( (a + 1)1/2−r ( 21 )r−1 π x2 , dx = x4 + 2ax2 + 1 2r+1/2 (r − 1)! 0 √ )r ∫ ∞( (a + cd)1/2−r ( 21 )r−1 π x2 . dx = √ dx4 + 2ax2 + c 0 2r+1/2 d(r − 1)! In particular,

√ ∫ ∞ 3π 2 x4 x3 dx = dx = , , 4 2 3 4 2 5/2 144 243 0 (x + x + 1) 0 (x + 7x + 1) √ ∫ ∞ π3/2 x 1 Γ2 (3/4) dx = , dx = √ 4 √4 . √ 4 2 5/4 2 3/4 (x + 14x + 1) x(x + x + 1) 0 4 2π Γ2 (3/4) 12 ∫

∫

∞ 0

∞

Source: [510]. Remark: The value of the fourth integral is independent of b. Fact 14.2.38. Let a ∈ (1, ∞). Then,

∫

∞

xa − 2x + 1 dx = 0. x2a − 1

∞

xn − 2x + 1 dx = 0. x2n − 1

0

Furthermore, if n ≥ 2, then

∫ 0

Source: [156, 2447]. Fact 14.2.39. Let a ∈ (0, 1). Then,

∫

∞ 0

Source: [1317, p. xxv].

1 π dx = ∑∞ i(i+1)/2 . 2i x2 ) (1 + a 2 i=0 i=0 a

∏∞

1111

INTEGRALS

14.3 Facts on Definite Integrals of Radicals Fact 14.3.1. Let a > 0. Then,

∫ 0

In particular, ∫

1

(1 − xa )1/a dx =

Γ2 ( a1 ) 2aΓ( a2 )

.

∫ 1 ∫ 1 √3 3 √ 1 1 1 (1 − x) dx = , , , (1 − x) dx = (1 − 4 x)4 dx = 6 20 70 0 0 0 √ ∫ 1√ ∫ 1√ ∫ 1√ Γ2 ( 31 ) πΓ( 13 ) Γ2 ( 14 ) 3 4 π 4 dx = 1 − x2 dx = , 1 − x3 dx = 1 − x , = √ . √ 4 8 π 6Γ( 32 ) 3 3 4Γ( 56 ) 0 0 0 √3 √ Remark: Γ( 31 )Γ( 56 ) = 2 πΓ( 32 ). Fact 14.3.2. Let a > 0, b > −1, and k ≥ 0. Then, ∫ 1 √ Γ[(k + 1)a + 1]Γ(b + 1) . xk (1 − a x)b dx = (k + 1)Γ[(k + 1)a + b + 1] 0 1

√

2

In particular, √ 3 1 √ ∫ 1√ ∫ 1√ ∫ 1√ πΓ( 31 ) 3Γ ( 3 ) 2 π 2 3 1 − x dx = , 1 − x dx = , 1 − x dx = = , √3 11 3 4 6Γ( 6 ) 0 0 0 10 2π √ √ √ ∫ 1√ ∫ 1√ 5 πΓ( 56 ) πΓ( 14 ) 2πΓ2 ( 41 ) 4 5 1 − x dx = 1 − x dx = = , , 7 12π 8Γ( 74 ) 7Γ( 10 ) 0 0 √ ∫ 1√ ∫ 1√ ∫ 1√ πΓ( 31 ) 4Γ( 31 )Γ( 54 ) 3 3 3 3 2 4 dx = 1 − x dx = , 1 − x dx = 1 − x , , 7 4 7Γ( 12 ) 5Γ( 56 ) 0 0 0 √ ∫ 1√ ∫ 1√ ∫ 1√ Γ( 41 )Γ( 13 ) πΓ( 41 ) 4 4 4 4 3 dx = , , 1 − x dx = , 1 − x2 dx = 1 − x 7 5 6Γ( 34 ) 7Γ( 12 ) 0 0 0 ∫ 1 ∫ 1 1 1 π dx = 2, dx = , √ √ 2 2 0 0 1−x 1−x √ 2 1 √ √ ∫ 1 ∫ 1 4 πΓ( 3 ) πΓ( 45 ) 2Γ ( 4 ) 1 1 = dx = , dx = √ √ √ , 3 5 8 π Γ( 4 ) Γ( 6 ) 0 0 1 − x3 1 − x4 √ √ ∫ 1 ∫ 1 ∫ 1 3 πΓ( 23 ) 3 1 2 3π 1 1 dx = , dx = , dx = , √3 √3 √3 2 9 Γ( 61 ) 0 0 0 1−x 1 − x2 1 − x3 √ ∫ 1 ∫ 1 ∫ 1 4Γ( 34 )Γ( 31 ) 2 πΓ( 34 ) 1 4 1 1 , . dx = , dx = dx = √4 √4 √ 4 1 3 ) Γ( 41 ) Γ( 12 0 0 0 1−x 1 − x2 1 − x3 Furthermore, if n ≥ 1, then ∫ 1 ∫ 1 √ 1 8 2 , (1 − x)n dx = 2 , √ n/2 dx = 2 n − 6n + 8 n + 3n + 2 x) 0 (1 − 0 ∫ 1 ∫ 1 √3 n √ 6 24 (1 − x) dx = , (1 − 4 x)n dx = . (n + 3)(n + 2)(n + 1) (n + 4)(n + 3)(n + 2)(n + 1) 0 0

1112

CHAPTER 14

Fact 14.3.3. Let n ≥ 0 and a > 0. Then,

∫

a

0

In particular, ∫ a

√

0

√ 4n+1 n!(n + 1)!a3/2+n . xn a − x dx = (2n + 3)! ∫

2a3/2 , a − x dx = 3

∫

√ 4a5/2 , x a − x dx = 15

a 0

√ 16a7/2 . x2 a − x dx = 105

a 0

Source: [815]. Fact 14.3.4. Let n ≥ 0 and a > 0. Then,

∫

a

∫

( a )2n+2 √ x a2 − x2 dx = πCn , 2

a

2n

0

In particular, ∫ a√ a2 π a2 − x2 dx = , 4 0

∫

∫

1

√

0

x−

ax2

0

In particular, ∫ 1√

√ x−

1 2 2x

0

dx =

∫

2π , 8

0

a

Source: [816]. Fact 14.3.5. Let a ∈ (0, 1]. Then,

1

√ x2n+1 a2 − x2 dx =

√ a3 x a2 − x2 dx = , 3

∫

22n+1 a2n+3 . (2n + 3)(2n + 2)(2n + 1)Cn a

0

√ a4 π . x2 a2 − x2 dx = 16

√ √ a − a2 (2a − 1) + asin a dx = . 4a3/2 √

x−

0

3 2 4x

√ 2 3π 1 + , dx = 12 27

∫

1

√

x − x2 dx =

0

Fact 14.3.6. Let a ≥ 0 and b > 0. Then,

∫

1 0

In particular, if n ≥ 0, then ∫ 1 1 dx = 2, √ 0 1−x

∫

1

0

∫

1

0

√ ( a+1 ) πΓ b ( ). dx = √ bΓ 2a+b+2 1 − xb 2b xa

√ (4) ∫ 1 πΓ 3 1 π ( ) , dx = , dx = √ √ 2 0 0 Γ 56 1 − x2 1 − x3 √ 2 (1) √ (1) √ (5) ∫ 1 πΓ 4 πΓ 4 2Γ 4 1 ( ) = ( ) = dx = , √ √ 8 π 0 4Γ 43 Γ 34 1 − x4 √ √ ∫ 1 xn πΓ(n + 1) πΓ[(n + 1)/2] xn dx = dx = , , √ √ 2 Γ(n + 3/2) 2Γ(1 + n/2) 0 1−x 1−x √ ∫ 1 xn πΓ[(n + 1)/3] x2n+1 (2n)!! , . dx = dx = √ √ (2n + 1)!! 3Γ[(2n + 5)/6] 0 1 − x2 1 − x3 ∫

1

1

Source: [2960]. Fact 14.3.7. Let n ≥ 0 and a > 0. Then,

∫

√

a

x2n −a

a+x a2n+1 (2n)!π dx = , a−x 4n (n!)2

∫

√

a

x2n+1 −a

a+x a2n+2 (2n + 2)!π dx = n+1 . a−x 4 [(n + 1)!]2

π . 8

1113

INTEGRALS

In particular, ∫ a √ ∫ a √ ∫ a √ ∫ a√ a+x a+x a+x a+x a2 π a3 π 3a4 π dx = aπ, dx = , dx = , dx = . x x2 x3 a−x a−x 2 a−x 2 a−x 8 −a −a −a −a Source: [823]. Fact 14.3.8. Let n ≥ 0. Then,

∫ 1√ 0

1−x π dx = −1, 1+x 2

∫

√

1

x 0

1−x π dx = 1− , 1+x 4

∫

√

1

x

2

0

1−x π 2 dx = − , 1+x 4 3

∫

√

1

x

3

0

1−x 2 3π dx = − . 1+x 3 16

Source: [820]. Fact 14.3.9. Let n ≥ 0. Then,

∫ 4 √ ∫ 4−x 1 2 2n √ 1 4 − x2 dx dx = xn x Cn = 2π 0 x π 0 ∫ ∫ 22n+1 1 n−1/2 4n+1 π/2 = x (1 − x)1/2 dx = (sin2 x) cos2n x dx. π π 0 0

Source: [816, 1753, 2217]. Fact 14.3.10. Let b ≥ 0 and a > b + 1. Then,

∫

1 0

In particular, ∫ 1

1

dx =

xb π dx = . (1 − xa )(b+1)/a a sin (b+1)π a ∫

π , a sin πa

1

1

dx =

π , 2

∫

1

1

dx =

√ √3 0 1 − x2 1 − x3 Source: [352, p. 41]. Related: Fact 13.10.22. Fact 14.3.11. If a > 0, then ∫ 1 √ π 1 − x2 dx = . √ 2 2 x +a 0 2a(a + a2 + 1) 0

√a 1 − xa

If a > 1, then

∫

1

0

0

√ 2 3π . 9

√ √ aπ 1 − x2 dx = (a − a2 − 1). 2 a2 − x2

Source: [150, p. 14]. Fact 14.3.12. Let a ∈ (−1, 0) ∪ (0, 1). Then,

∫

1 0

In particular, ∫ 1√ 0

∫ 0

1

√ 4

1 − 1 dx = x

1 − 1 dx = x

∫

∫

1

√ 0

Source: [352, p. 41].

4

1

√ 0

1 1 x

−1

1 1 x

−1

(

)1/a 1 aπ −1 dx = . x sin aπ ∫

π dx = , 2

√ 2π dx = , 4

1 0

∫

1 0

√ 5

√ 3

1 − 1 dx = x

1 − 1 dx = x

∫

∫

1

√ 0

1

√ 0

5

3

1 1 x

−1

1 1 x

−1

dx = √

dx =

√ 2 3π , 9

2π 5

√

√ 1+

5 . 5

1114

CHAPTER 14

Fact 14.3.13. Let a > 0 and b > 0. Then,

∫

1 π dx = √ . x(1 − x)[ax + b(1 − x)] 0 ab Source: [1381, p. 223]. Related: Fact 10.11.68. Fact 14.3.14. Let a > 0 and b > 0. Then, ∫ ∞ ∫ ∞ 1 1 dx dx = √ √ 0 0 (x2 + a2 )(x2 + b2 ) (x2 + [( 12 (a + b)]2 )(x2 + ab) ∫ π/2 1 = dx √ 2 2 0 a cos x + b2 sin2 x ∫ π/2 2 = dx. √ 0 2 2 2 (a + b) − (a − b) sin x 1

√

In particular,

∫

∞ 0

∫

1

= √ (x2 + 1)(x2 + 2)

Source: [2960]. Related: Fact 12.18.55. Fact 14.3.15. Let a > 0 and b > 1. Then,

∫

∞

(

√

x2

+

0

a2

bab+1 − x) dx = 2 , b −1

( ) πΓ 14 ( ) . dx = √ 4Γ 43 1 − x2

1 0

∫

∞

b

0

√

1

(

√

1 x2

+

a2

+

x)b

dx =

b . ab−1 (b2 − 1)

Source: [823] and [1524, p. 267]. Fact 14.3.16. Let a be a positive number. Then,

∫

∞

1

π 1 dx = √ . √ (x + a) x − 1 a+1

Source: [2106, pp. 43–45].

14.4 Facts on Definite Integrals of Trigonometric Functions Fact 14.4.1. Let a > −1. Then,

∫

π/2

∫ sin x dx = a

0

In particular,

∫

π/2

π/2

0

√ πΓ( 38 )

( ) πΓ a+1 2 ( ). cos x dx = a 2Γ 2 + 1 √

a

∫

π/2

Γ( 14 ) 1 dx = √ √ , 2Γ( 78 ) 0 0 2 2π sin x sin x √ (5) √ (2) √ √ ∫ π/2 √ ∫ π/2 √ πΓ 8 πΓ 3 4 πΓ( 85 ) 3 πΓ( 32 ) 4 3 ( ) = ( ) = sin x dx = , sin x dx = , Γ( 81 ) Γ( 16 ) 0 0 2Γ 98 2Γ 67 √ (5) √ (3) √ ∫ π/2 ∫ π/2 √ πΓ 4 πΓ 4 Γ2 ( 1 ) 2 2 3 ( ) = ( ) = √4 . sin3/2 x dx = sin x dx = Γ ( 4 ), π 0 0 6 2π 2Γ 45 2Γ 74 √4

1

dx =

,

1115

INTEGRALS

Furthermore, if n ≥ 0, then ∫ π/2 ∫ π/2 ∫ sin2n x dx = cos2n x dx = 0 π/2

∫

0

∫

sin2n+1 x dx =

0

1

0

∫ 2n+1 cos x dx =

π/2

0

1

0

( ) (2n − 1)!!π π 2n (2n)!π 2(1 − x2 )2n , dx = = = 2(2n)!! (1 + x2 )2n+1 22n+1 (n!)2 22n+1 n 4n (n!)2 4n 2(1 − x2 )2n+1 (2n)!! ( ). = = dx = (2n + 1)!! (2n + 1)! (2n + 1) 2n (1 + x2 )2n+2 n

Remark: See [511, p. 113], [822], [1136, p. 205], and [1158, p. 153]. Related: Fact 14.4.60. Fact 14.4.2. Let a > −1 and b > −1. Then,

∫

π/2

b+1 Γ( a+1 2 )Γ( 2 )

(sina x) cosb x dx =

2Γ( a+b+2 2 )

0

In particular, ∫

π/2

0

Γ2 ( 1 ) 1 dx = √4 , √ 2 π (sin x) cos x

∫

√

π/2

(sin x) cos x dx =

0

Fact 14.4.3. Let a be a nonzero real number. Then,

∫

π

sin2 ax dx =

0

∫

π sin 2aπ − , 2 4a

π

∫

∫

π

π

sin ax dx = 2

0

0

π

cos2 ax dx =

0

∫

π , 2

π

π

Γ2 ( 34 ) 2π3/2 = √ . π Γ2 ( 14 )

(sin ax) cos ax dx =

0

cos2 ax dx =

0

If a is an integer, then ∫ π ∫ sin2 ax dx =

∫

π sin 2aπ + , 2 4a

cos2 ax dx =

0

If 2a is an integer, then

.

sin2 aπ . 2a

π . 2

(sin ax) cos ax dx = 0.

0

If 2a is an integer but a is not an integer, then ∫ π 1 (sin ax) cos ax dx = . 2a 0 Fact 14.4.4. Let a be a nonzero real number. Then,

∫

2π

sin2 ax dx = π −

0

∫

sin 4aπ , 4a

If 4a is an integer, then

2π

cos2 ax dx = π +

0

∫

∫

2π

0

2π

(sin ax) cos ax dx =

0

cos2 ax dx = π.

2π

∫

2π

cos ax dx = π, 2

0

(sin ax) cos ax dx = 0.

0

If 4a is an integer but 2a is not an integer, then ∫ 2π 1 . (sin ax) cos ax dx = 2a 0 Fact 14.4.5. Let n ≥ 0. Then,

∫

0

If 2a is an integer, then ∫ ∫ 2π 2 sin ax dx = 0

2π

sin2 ax dx =

sin 4aπ , 4a

∫ 0

2π

cos2n x dx =

2π(2n)! . 4n (n!)2

sin2 2aπ . 2a

1116

CHAPTER 14

Source: [2106, p. 328]. Fact 14.4.6. Let a and b be real numbers, and assume that a2 , b2 . Then,

∫

π

b(sin aπ) cos bπ − a(cos aπ) sin bπ , a2 − b2 0 ∫ π a(sin aπ) cos bπ − b(cos aπ) sin bπ , (cos ax) cos bx dx = a2 − b2 0 ∫ π a − a(cos aπ) cos bπ − b(sin aπ) sin bπ (sin ax) cos bx dx = . a2 − b2 0 (sin ax) sin bx dx =

Now, let n and m be integers. If n , m, then ∫ π ∫ π (sin nx) sin mx dx = (cos nx) cos mx dx = 0. 0

0

If n − m is even, then

∫

π

(sin nx) cos mx dx = 0.

0

If n − m is odd, then

∫

π

(sin nx) cos mx dx =

0

∫

Fact 14.4.7. Let n ≥ 0. Then, π/2

(sin2n x) sin 2nx dx = (−1)n+1

0

∫

2n ∑ 2i

1 22n+1

i=1

i

∫

π/2

,

n2

2n . − m2

(sin2n+1 x) sin (2n + 1)x dx = (−1)n

0

∫

π , 4n+1

n π/2 1 ∑ 2i n (cos x) sin nx dx = , , 2n+1 2n+1 i=1 i 0 0 ∫ π/2 ∫ π/2 2n+1 π 1 ∑ 2i (sin2n x) cos 2nx dx = (−1)n 2n+1 , (sin2n+1 x) cos (2n + 1)x dx = (−1)n n+1 , i 2 4 0 0 i=1 ∫ π ∫ π π (sin2n+1 x) sin (2n + 1)x dx = (−1)n 2n+1 , (sin2n x) sin 2nx dx = 0, 2 0 0 ∫ π ∫ π π (cosn x) sin nx dx = 0, (cosn x) cos nx dx = n , 2 0 0 ∫ π ∫ π π (sin2n x) cos 2nx dx = (−1)n n , (sin2n+1 x) cos (2n + 1)x dx = 0. 4 0 0 π/2

π

(cos x) cos nx dx = n

Fact 14.4.8. Let a and b be real numbers, and assume that a2 , b2 . Then,

∫

∫ ∫

0 2π

0 2π

(sin ax) sin bx dx =

b(sin 2aπ) cos 2bπ − a(cos 2aπ) sin 2bπ , a2 − b2

(cos ax) cos bx dx =

a(sin 2aπ) cos 2bπ − b(cos 2aπ) sin 2bπ , a2 − b2

2π

(sin ax) cos bx dx =

0

Now, let n and m be integers. Then, ∫ 2π ∫ (sin nx) sin mx dx = 0

0

2π

a − a(cos 2aπ) cos 2bπ − b(sin 2aπ) sin 2bπ . a2 − b2 ∫

2π

(cos nx) cos mx dx = 0

(sin nx) cos mx dx = 0.

1117

INTEGRALS

Fact 14.4.9. Let n ≥ 0. Then,

∫

π/2

x2 cos2n+2 x dx =

0

In particular,

∫

π/2 0

x2 cos2 x dx = ∫

π 2 (π − 6), 48

π/2 0

( ) π (2n + 1)!! π2 − Hn+1,2 . 4 (2n + 2)!! 6 ∫ 0

π/2

x2 cos4 x dx =

( ) 15π 2 49 x cos x dx = π − . 1152 6 2

Source: [826]. Related: Fact 13.5.44. Fact 14.4.10. ∫ π/6

( ) 3π π2 5 − , 64 3 2

6

4G π x dx = − acosh 2, sin x 3 6 0 √ √ √ ∫ π/4 ∫ π/4 2 2π (4 − π) 2 , + − 1, x cos x dx = x sin x dx = 8 2 8 0 0 ∫ π/4 3 ∫ π/4 2 x x π2 π 3Gπ π3 3π2 105 + log 2, − + − ζ(3), dx = G − dx = 16 4 4 64 32 64 sin2 x sin2 x 0 0 ∫ π/4 ∫ π/4 ∫ π/4 1 x π 1 π2 π x2 dx = 1, dx = dx = − log 2, + log 2 − G, 4 2 16 4 cos2 x cos2 x cos2 x 0 0 0 ∫ π/4 63 3Gπ π3 3π2 x3 + log 2 + ζ(3) − , dx = 2 64 32 64 4 cos x 0 ∫ π/4 ∫ π/4 ∫ π/4 1 G π Gπ π2 21 tan x dx = log 2, x tan x dx = − log 2, x2 tan x dx = − log 2− ζ(3), 2 2 8 4 32 64 0 0 0 ∫ π/4 ∫ π/4 π π2 1 π x tan2 x dx = − − log 2, tan2 x dx = 1 − , 4 4 32 2 0 0 ∫ π/4 ∫ π/4 π3 π 1 1 π2 − + log 2 − G, tan3 x dx = − log 2, x2 tan2 x dx = 16 192 4 2 2 0 0 ∫ π/2 ( ∫ π ( ∫ π( ) ) ) π π 1 π π − x tan x dx = − x tan x dx = − x tan x dx = log 2, 2 2 2 2 2 0 π/2 0 ∫ π/4 1 G π π x tan3 x dx = + log 2 − − , 4 8 2 2 0 ∫ π/4 2 2 1 π π 21 π Gπ x2 tan3 x dx = log 2 + + log 2 + ζ(3) − − , 2 16 32 64 4 4 0 ∫ π/4 ∫ π/4 G π 35 Gπ π2 xcot x dx = + log 2, x2 cot x dx = + log 2 − ζ(3), 2 8 4 32 64 0 0 ∫ π/4 3 2 π π π − + log 2, x2 cot2 x dx = G − 16 192 4 0 ∫ π/4 3 4 2 3Gπ π π 3π 105 x3 cot2 x dx = − − + log 2 − ζ(3), 4 64 1024 32 64 0 √ ∫ π/4 2 ∫ π/3 2 x tan x π2 1 π x tan x 2π2 3π dx = + log 2 − , dx = + log 2 − , 16 2 4 9 3 cos2 x cos2 x 0 0

1118

CHAPTER 14

∫

π/4

π π x2 tan2 x 1 π2 G + + − log 2 − , dx = 2x 3 48 3 12 6 cos 0 ∫ π/2 ∫ π/2 2 π 7 π x cot x dx = log 2, log 2 − ζ(3), x2 cot x dx = 2 4 8 0 0 ∫ π/2 ∫ π/2 4 3 9π 9π2 93 π π log 2 − ζ(3), log 2 − ζ(3) + ζ(5), x4 cot x dx = x3 cot x dx = 8 16 16 16 32 0 0 ∫ π/2 2 ∫ π/2 7 x x dx = 2G, dx = 2Gπ − ζ(3), sin x sin x 2 0 0 ∫ π/2 ∫ π/2 x2 3π2 x3 log 2 − 7ζ(3), dx = π log 2, dx = 4 sin2 x sin2 x 0 0 ∫ π/2 2 ∫ π/2 π π2 7 x x dx = log 2, dx = log 2 − ζ(3), tan x 2 tan x 4 8 0 0 ∫ π/2 2 ∫ π/2 3 2 2 x cos x π π 21 x cos x dx = 4G − , dx = 6Gπ − − ζ(3), 2 2 4 8 2 sin x sin x 0 0 ∫ π/2 3 ∫ π/2 4 3 2 x cos x π x cos x π4 21 3π 3π log 2 − , log 2 − − ζ(3). dx = dx = 2 16 2 32 4 sin3 x sin3 x 0 0 Fact 14.4.11. Let n ≥ 1. Then,

∫ π/2 n ∑ 2 sin (2n − 1)x π sin 2nx i+1 dx = (−1) , dx = , sin x 2i − 1 sin x 2 0 0 i=1 ∫ π/2 ∫ π n ∑ sin 2nx 1 sin 2nx 2 dx = dx = (−1)n−1 (−1)i+1 , cos x 2 0 cos x 2i −1 0 i=1 ∫ π ∫ π n ∑ sin 2nx sin (2n − 1)x x sin 2nx 4 . dx = 0, dx = π, dx = − sin x sin x sin x (2i − 1)2 0 0 i=1 ∫

∫

π

0

π/2

In particular, ∫ π/2 sin 2x dx = 2, sin x 0

∫

π/2 0

sin 4x 4 dx = , sin x 3

Source: [815] and [1217, p. 471]. Fact 14.4.12.

∫

∫ 0

π/2

sin 6x 26 dx = , sin x 15

∫

π/2 0

sin 8x 152 dx = . sin x 105

) ) ∫ π/4 ( √ 1 4 1 1 1 − dx = 1 − , − dx = log[(1 + 2)π] − 3 log 2, x sin x π x2 sin2 x 0 0 √ ) ∫ π/4 ( √ 1 1 1 8 3π 1 2 1 + 3− + − 2− + log[(1 + 2)π], dx = 3 2x x 12 2 2 2 π sin x 0 √ ) ) ∫ π/3 ( ∫ π/3 ( 1 π 1 1 1 3 3 1 − dx = log − log 3, − dx = − , 2 2 x sin x 2 2 3 π x sin x 0 0 ) ∫ π/3 ( 1 5 1 1 9 1 π 1 dx = + 3− − 2 + log − log 3, 3 2x x 12 2π 2 2 4 sin x 0 ) ) ∫ π/2 ( ∫ π/2 ( 1 π 1 1 2 1 − dx = log , − dx = − , 2 2 x sin x 4 π x sin x 0 0 π/4

(

1119

INTEGRALS

∫

π/2 0

(

) 1 2 1 1 1 1 + − dx = − + log π − log 2, 2x x3 sin3 x 12 π2 2

Fact 14.4.13. Let a and b be positive numbers. Then,

∫

∞

0

 π   , b < a,    (sin ax) cos bx  π2 dx =   4 , b = a,  x   0, a < b,

If, in addition, a , b, then

∫

∞ 0

∫ 0

∞

(sin ax) sin bx π dx = min {a, b}. 4 x2

(sin ax) sin bx 1 a+b dx = log . x 2 |a − b|

Fact 14.4.14. Let n ≥ 1 and a > 0. Then,

( ) π 2n (2n + 2)(sin2n+1 ax) cos ax , dx = 2n+1 n x 2 0 0 ( ) ∫ ∞ ( ) ∫ ∞ (sin2n+1 ax) cos2n−1 ax π 2n − 1 sin2n ax aπ 2n − 2 dx = 4n , dx = 2n−1 , x n−1 n−1 2 x2 2 0 0 ( ) ( ) ∫ ∞ ⌊n/2⌋ (cosn x) sin nx π 1 sinn ax an−1 π ∑ i n−1 n dx = 1− n , dx = n (−1) (n − 2i) , x 2 2 xn 2 (n − 1)! i=0 i 0 ( ) ∫ ∞ (cos2n x) tan x π 2n dx = 2n+1 . x n 2 0 ∫

∫

∞ 0

In particular,

∫

∞ 0

∞

sin2n+1 ax dx = x

∫

∞

∫ ∞ ∫ ∞ π sin3 ax π sin5 ax 3π sin ax dx = , dx = , dx = , x 2 x 4 x 16 0 0 0 ∫ ∞ ∫ ∞ sin2 ax sin3 ax aπ 3a2 π , , dx = dx = 2 3 2 8 x x 0 0 ∫ ∞ ∫ ∞ sin5 ax sin6 ax sin4 ax a3 π 115a4 π 11a5 π , , , dx = dx = dx = 3 384 40 x4 x5 x6 0 0 ∫ ∞ ∫ ∞ sin7 ax 5887a6 π sin8 ax a7 π dx = , dx = . 23040 630 x7 x8 0 0 ∫

∞

Source: [513, ∫ ∞ 758], [1217, pp. 462, 463, 472], [1444], [1568, p. 39], and [1889, 2050, 2691]. Remark: 0 sinxax dx is the Dirichlet integral. See [1945, p. 132]. Fact 14.4.15. Let n ≥ 1 and m ≥ 0. Then, the following statements hold:

i) If 1 ≤ m ≤ n, then

∫ 0

ii) If 0 ≤ m ≤ n, then ∫ 0

∞

∞

( ) n 2m−1 sin2n x π ∑ 2n i+m (2i) dx = n (−1) . 4 i=1 (2m − 1)! n − i x2m

( ) n 2m sin2n+1 x π ∑ 2n + 1 i+m (2i + 1) dx = 2n+1 (−1) . (2m)! n−i x2m+1 2 i=0

1120

CHAPTER 14

iii) If 2 ≤ m ≤ n, then ∫

∞ 0

( ) n 2m−2 1 ∑ 2n sin2n x i+m (2i) dx = (−1) log i. (2m − 2)! n − i x2m−1 22n−1 i=1

iv) If 1 ≤ m ≤ n, then ( ) ∫ ∞ n 2m−1 2n + 1 sin2n+1 x 1 ∑ i+m (2i + 1) log(2i + 1). (−1) dx = 4n i=0 (2m − 1)! n − i x2m 0 In particular,

∫

∞

0

∫

∞

sin3 ax 3a log 3, dx = 4 x2

∫

∞

sin5 ax 5a (3 log 3 − log 5), dx = 2 16 x 0 0 ∫ ∞ ∫ ∞ sin7 ax 7a aπ 3aπ sin4 ax sin6 ax dx = dx = dx = (9 log 3 − 5 log 5 + log 7), , , 2 2 2 64 4 16 x x x 0 0 ∫ ∞ ∫ ∞ ∫ ∞ 5aπ 5a2 π sin5 ax sin8 ax sin4 ax 2 dx = dx = a log 2, dx = , , 2 3 3 32 32 x x x 0 0 0 ∫ ∞ ∫ ∞ 3a2 5a3 sin5 ax sin6 ax dx = dx = (8 log 2 − 3 log 3), (25 log 5 − 27 log 3), 16 96 x3 x4 0 0 ∫ ∞ ∫ ∞ πa3 7a3 sin6 ax sin7 ax dx = dx = , (125 log 5 − 81 log 3 − 49 log 7), 8 384 x4 x4 0 0 ( ) ∫ ∞ ∫ ∞ 77πa3 sin7 ax sin6 ax 4 27 dx = a dx = log 3 − 2 log 2 , . 5 5 16 768 x x 0 0

Source: [1680]. Fact 14.4.16. Let a > 0 and b > 1. Then,

∫

∞

sin ax dx = b

0

In particular,

∫

∞

Γ( 1b ) ba1/b

∫

π sin , 2b

cos axb dx =

0

∫

∞

sin ax dx = 2

0

∞

0

√ cos ax dx = 2

Γ( b1 ) ba1/b

cos

π . 2b

π . 8a

Fact 14.4.17. Let a > 0 and b ≥ 0. Then,

∫

∞

(sin ax2 ) cos 2bx dx =

0

( ) ∫ ∞ ( ) π π b2 b2 b2 b2 (cos ax2 ) cos 2bx dx = cos − sin , cos + sin . 8a a a 8a a a 0

Source: [1524, p. 269]. Fact 14.4.18. Let a and b be real numbers, and assume that b , 0. If a ∈ (0, 2), then

∫

∞

0

If a ∈ (0, 1), then

sin bx aπ (sign b)|b|a−1 π aπ dx = (sign b)|b|a−1 Γ(1 − a) cos = csc . a x 2 2Γ(a) 2 ∫

∞ 0

cos bx aπ |b|a−1 π aπ a−1 dx = |b| Γ(1 − a) sin = sec . a x 2 2Γ(a) 2

1121

INTEGRALS

In particular, √ ∫ ∞ π sin bx , √ dx = (sign b) 2|b| x 0

∫

∞

0

cos bx √ dx = x

√

π , 2|b|

∫

∞ 0

√ sin bx dx = (sign b) 2π|b|. 3/2 x

Source: [116, p. 50], [968, pp. 261, 262], and [1136, p. 211]. Fact 14.4.19. Let a and b be positive numbers. Then,

∫

0

∞

sin axb π dx = . x 2b

Source: [1217, p. 483]. Fact 14.4.20. Let a, b ∈ R, and assume that c > 0 and either a > 1 and b > 1 or a ∈ (−∞, 1)

and a + b > 1. Then,

∫

∞ 0

( ) c(a−1)/b 1 − a (a + b − 1)π sin cxb dx = Γ cos . xa b b 2b

In particular, if either a > 1 or a ∈ (1/2, 1), then ( ) ( ) ∫ ∞ sin cxa c(a−1)/b 1 π c(a−1)/b 1 − a π dx = Γ cos = − Γ cos . a x a − 1 a 2a a a 2a 0 ∫

Hence,

∞ 0

1√ sin cx2 dx = cπ. 2 2 x

Source: [2106, p. 135]. Fact 14.4.21. Let a > 0. Then,

∫

∫ ∞ ∫ ∞ sin2 ax2 sin2 ax2 1√ sin2 ax2 aπ 2√ 3 , dx = aπ, dx = dx = a π, 2 3 4 2 4 3 x x x 0 0 0 √ √ ∫ ∞ ∫ ∞ sin3 ax2 sin3 ax2 3 − 3 aπ 3a , log 3, dx = dx = 2 3 4 2 8 x x 0 0 √ √ ∫ ∞ ∫ ∞ sin3 ax2 sin3 ax2 3a2 π 3 − 1 a3 π , dx = , dx = 2 2 16 x4 x5 0 0 √ √ ∫ ∞ ∫ ∞ sin3 ax2 3 3 − 1 a5 π sin ax3 π dx = , dx = , 6 5 2 x 6 x 0 0 √ 2/3 √3 ∫ ∞ ∫ ∞ 3 3 sin ax sin ax a 2 3a dx = Γ( 3 ), dx = Γ( 13 ), 2 3 2 4 x x 0 0 √ ∫ ∞ ∫ ∞ √ 1/4 3 sin ax4 1 sin ax4 1√ dx = ), dx = 2 − 2a Γ( 2aπ, 4 2 3 2 4 x x 0 0 √ ∫ ∞ ∫ ∞ √ 3/4 1 √ 3/5 2 1 sin ax5 1 sin ax4 dx = 2 + 2a Γ( ), dx = (1 + 5)a Γ( 5 ). 4 6 12 x4 x4 0 0 ∞

Fact 14.4.22. If a is a real number, then

∫

0

If a is a positive number, then

∞

sin ax − ax cos ax πa2 . dx = (sign a) 4 x3 ∫

∞ 0

cos ax + x sin ax π dx = a . e x2 + 1

1122

CHAPTER 14

Source: [1217, p. 447]. Fact 14.4.23.

∫

∞ 0

∫

π x2 − sin2 x dx = , 3 x4

∞ 0

∫

x3 − sin3 x 13π , dx = 32 x5

0

∞

x4 − sin4 x 7π . dx = 15 x6

Fact 14.4.24. Let a and b be nonnegative numbers. Then,

∫

∞ 0

π cos ax − cos bx dx = (b − a). 2 x2 ∫

In particular,

0

∞

aπ 1 − cos ax dx = . 2 2 x

Source: [1136, p. 211]. Fact 14.4.25. Let a and b be positive numbers. Then,

∫

∞

0

∫ ∞ b b sin ax − a sin bx b cos ax − cos bx dx = log , dx = ab log , 2 x a a x 0 ∫ ∞ 2 2 sin ax − sin bx 1 a dx = log . x 2 b 0

Now, let c ∈ (0, 1). Then, ∫

∞ 0

√ √ √4 √4 cos ax − cos bx 1 1 2 + 2Γ(− )( a − b), dx = 4 2 x5/4 0 √ ∫ ∞ √3 √ √ √3 √ cos ax − cos bx 3 cos ax − cos bx 1 Γ(− )( dx = a − b), dx = 2π( b − a). 3 4/3 3/2 2 x x 0

In particular, ∫

( cos ax − cos bx cπ ) Γ(−c)(ac − bc ). dx = cos 2 xc+1

∞ 0

∫

∞

Source: [1217, pp. 447, 448]. Remark: limc→1 cos(cπ/2)Γ(−c) = − π2 . Fact 14.4.26. Let a and b be positive numbers. Then,

∫

∞ 0

cos ax π , dx = 2 2 x +b 2beab

∫

∞ 0

cos ax π(ab + 1) . dx = 2 2 2 (x + b ) 4b3 eab

Source: [1136, pp. 201, 203]. Fact 14.4.27. Let a, b, and c be positive numbers. Then,

∫

∫

∞ −∞

∫ ∞ sin ax π sin ab cos ax π cos ab dx = − dx = , , ac 2 2 2 2 ce ceac −∞ (x + b) + c −∞ (x + b) + c ∫ ∞ x sin ax π(b sin ab + c cos ab) x cos ax π(c sin ab − b cos ab) dx = , dx = . ac 2 2 2 2 ce ceac (x + b) + c −∞ (x + b) + c ∞

Source: [2106, p. 340]. Fact 14.4.28. Let a, b, c be positive numbers, and assume that b , c. Then,

∫

∞ 0

(e−ac − e−ab )π x sin ax dx = , (x2 + b2 )(x2 + c2 ) 2(b2 − c2 )

∫

∞ 0

cos ax (be−ac − ce−ab )π dx = . (x2 + b2 )(x2 + c2 ) 2bc(b2 − c2 )

Fact 14.4.29. Let a and b be positive numbers. Then,

∫

0

∞

x sin ax πe−ab , dx = 2 x2 + b2

∫

0

∞

x sin ax aπe−ab , dx = 4b (x2 + b2 )2

∫

∞ 0

sin ax π(1 − e−ab ) dx = . x(x2 + b2 ) 2b2

1123

INTEGRALS

Source: [1136, p. 211]. Fact 14.4.30. Let a and b be distinct positive numbers. Then,

∫

∞ 0

x sin x (sinh a − cosh a − sinh b + cosh b)π dx = . (x2 + a2 )(x2 + b2 ) 2(a2 + b2 ) ∫

In particular,

∞ 0

(x2

(e − 1)π x sin x dx = . 2 + 1)(x + 4) 6e2

Fact 14.4.31. Let a and b be positive numbers. Then,

∫

∞

0

√ 2π cos ax dx = √ 4 2 3/2 x +b 4b ea 2b/2

√  √   a 2b a 2b   . + cos sin 2 2

Source: [1136, p. 202]. Fact 14.4.32. Let a and b be positive numbers. Then,

∫

∞

0

∫ ∞ sin2 ax π π cos2 ax −2ab dx = dx = (1 − e ), (1 + e−2ab ), 2 2 2 + b2 4b 4b x +b x 0 ∫ ∞ π (sin2 ax) cos2 ax dx = (1 − e−4ab ). 2 + b2 16b x 0

Source: [1136, p. 203] and [1524, p. 268]. Fact 14.4.33. Let a, b, and c be positive numbers. Then,

∫

0

∞

( ) 1 −2(a+b)c 1 2(b−a)c π (sin2 ax) cos2 bx −2bc −2ac 1+e −e − e − e . dx = 8c 2 2 x 2 + c2

Source: [1217, p. 461]. Fact 14.4.34. Let a, b, and c be positive numbers, and assume that b , c. Then,

∫

∞

(x2

0

∫

∫

∞ 0

(b − c + ce−2ab − be−2ac )π sin2 ax dx = , 2 2 2 + b )(x + c ) 4bc(b2 − c2 )

∞

cos2 ax (b − c − ce−2ab + be−2ac )π dx = , 2 2 2 2 4bc(b2 − c2 ) 0 (x + b )(x + c ) ∫ ∞ (2ab − 1 + e−2ab )π sin2 ax dx = , x2 (x2 + b2 ) 4b3 0 ∫ ∞ sin2 ax [1 − (2ab + 1)e−2ab ]π cos2 ax [1 + (2ab + 1)e−2ab ]π dx = , dx = . 2 2 2 3 2 2 2 (x + b ) 8b 8b3 0 (x + b )

Source: [1217, p. 462]. Fact 14.4.35. Let a ∈ (0, ∞) and m, n ≥ 1. If m is odd, then

∫

∞ 0

22a−3 sin mπx dx = , (sin πx)Γ(a + x)Γ(a − x) Γ(2a − 1)

∫

0

∞

4n (n!)2 π sin mπx dx = . ∏n 2 2(2n)! x i=1 (1 − xi2 )

If m is even, then both integrals are 0. Source: [1317, p. 216]. Remark: These integrals depend on the parity of m but not its value. Fact 14.4.36. If x ∈ (− π4 , 3π 4 ), then √ √ ( ∫ x ) √ √  2 1 2 x   dt = 2 atanh  tan − 1  + 2 atanh 2 2 2 0 sin t + cos t

1124

CHAPTER 14

√ √ tan( 2x + π8 ) sin(x − π4 ) + 1 2 2 = log √ = log . 2 2 tan π8 ( 2 − 1) cos(x − π4 ) In particular, ∫

π/6

0

√ √ √ √ √ √ 2− 3 2 1 dx = 2 atanh = log[ 21 ( 6 − 2)(4 + 3 2)] ≈ 0.435953, sin x + cos x 3 2 √ √ √ ∫ π/4 √ 1 2 2 2 dx = atanh = log(1 + 2) ≈ 0.623225, sin x + cos x 2 2 2 0 √ √ ∫ π/3 √ √ √ √ 2 1 dx = 2 atanh 2 − 3 = log( 2 + 3) ≈ 0.810497, sin x + cos x 2 0 √ ∫ π/2 √ √ √ 2 1 dx = 2 atanh = 2 log(1 + 2) ≈ 1.24645. sin x + cos x 2 0

Furthermore,

∫ 0

π/2

sin2 x dx = sin x + cos x

∫

π/2 0

√ √ 2 cos2 x dx = log(1 + 2). sin x + cos x 2

Source: The last integral is given in [2106, p. 52]. Remark: atanh

√ log(1 + 2) =

√ 1 2 log(3 + 2 2).

Fact 14.4.37.

√ 2 2

√ = 2 atanh( 2 − 1) =

√ ) ( ∫ π/4 √ √ x 1 (1 − 2)π 1 17 dx = 2 − 2, dx = + log +3 2 , sin x + 1 sin x + 1 4 4 4 0 0 ∫ π/4 ∫ π/4 √ √ x π √ 1 dx = 2 − 1, dx = log(2 + 2) − 2 log 2 + ( 2 − 1), cos x + 1 cos x + 1 4 0 0 ∫ π/3 ∫ π/3 √ √ √ 1 x π dx = 3 − 1, dx = ( 3 − 2) − 2 log( 3 − 1), sin x + 1 sin x + 1 3 0 0 √ √ ∫ π/3 ∫ π/3 1 x 4 3 3π dx = , dx = − log , cos x + 1 3 cos x + 1 9 3 0 0 ∫ π/2 ∫ π/2 ∫ π/2 2 x x 1 dx = 1, dx = log 2, dx = 2π log 2 − 4G, sin x + 1 sin x + 1 sin x+1 0 0 0 ∫ π/2 ∫ ∫ π/2 π/2 x3 3π2 63 1 x π dx = log 2− ζ(3), dx = 1, dx = −log 2, sin x + 1 2 8 cos x + 1 cos x + 1 2 0 0 0 ∫ π/2 ∫ π/2 2 3 3 2 2 x x π π 3π 63 dx = + π log 2 − 4G, dx = + log 2 + ζ(3) − 6πG, cos x + 1 4 cos x + 1 8 4 8 0 0 ∫ π ∫ π ∫ π 2 x x 1 dx = 2, dx = π, dx = π2 + 2π log 2 − 8G, sin x + 1 sin x + 1 sin x + 1 0 0 0 ∫ π ∫ π/2 x3 x2 π2 dx = π3 + 3π2 log 2 − 12πG, dx = + 2π log 2 − 8G, 2 0 sin x + 1 −π/2 cos x + 1 √   ∫ π/4 ∫ π/4 √  2  π cos x sin x   , dx = 2 − 2 + , dx = log 1 + sin x + 1 4 sin x + 1 2 0 0 ∫ π/4 ∫ π/4 √ √ sin x cos x π dx = 2 log 2 − log(2 + 2), dx = 1 − 2 + , cos x + 1 cos x + 1 4 0 0 ∫

π/4

1125

INTEGRALS

√   ∫ π/3 √  π 3  sin x cos x  , dx = 1 − 3 + , dx = log 1 + sin x + 1 3 sin x + 1 2 0 0 √ ∫ π/3 ∫ π/3 sin x 3 4 π cos x dx = log , dx = − , cos x + 1 3 cos x + 1 3 3 0 0 ∫ π/2 ∫ π/2 ∫ π/2 π cos x cos x sin x sin x dx = dx = − 1, dx = dx = log 2, sin x + 1 cos x + 1 2 sin x + 1 cos x + 1 0 0 0 ∫ π/2 ∫ π/2 π2 x cos x x sin x dx = − log 2, dx = π log 2 − 2G, sin x + 1 8 sin x + 1 0 0 ∫ π/2 ∫ π/2 π π2 π x cos x x sin x dx = 2G − log 2, dx = − + log 2, cos x + 1 2 cos x + 1 8 2 0 0 ∫ π ∫ π ∫ π 2 sin x cos x (x − π) dx = π − 2, dx = 0, dx = 4π log 2, 0 sin x + 1 0 sin x + 1 0 cos x + 1 ∫ π ∫ π x sin x π2 x cos x dx = − π, dx = π log 2 − 4G, 2 0 sin x + 1 0 sin x + 1 ∫ π ∫ π (x − π)2 cos x π2 (x − π)2 sin x dx = 2π2 log 2 − 7ζ(3), dx = − 4π log 2. cos x + 1 cos x + 1 3 0 0 ∫

∫ 0

π/2

π/3

Fact 14.4.38.

∫

π/2 0

∫

∫ π/2 (x − π2 )2 x2 π2 dx = dx = 4G + π log 2 − , 1 − sin x 1 − cos x 4 0 ∫ π/2 π 2 x(x − 2 ) 105 π2 dx = ζ(3) − 4πG − log 2, 1 − sin x 8 4 0 π/2

x3 π3 3π2 105 dx = 6πG − + log 2 − ζ(3), 1 − cos x 8 4 8 0 ∫ π ∫ π x(x − π2 )2 (x − π2 )2 π2 π3 dx = 8G + 2π log 2 − , dx = 4πG + π2 log 2 − , 2 4 0 1 − sin x 0 1 − sin x ∫ π π ∫ π/2 π ( − x) cos x ( 2 − x) cos x π 2 dx = 2G + log 2, dx = 4G + π log 2, 1 − sin x 2 1 − sin x 0 0 ∫ π ∫ π x2 x3 dx = 4π log 2, dx = 6π2 log 2 − 21ζ(3), 0 1 − cos x 0 1 − cos x ∫ π/2 2 ∫ π/2 x sin x π x sin x π2 35 dx = 2G + log 2, dx = 2πG + log 2 − ζ(3), 1 − cos x 2 1 − cos x 4 8 0 0 ∫ π ∫ π 2 x sin x x sin x dx = 2π log 2, dx = 2π2 log 2 − 7ζ(3), 0 1 − cos x 0 1 − cos x ∫ π/2 ∫ π/2 x − sin x π (x − sin x)2 π π2 dx = 2 − , dx = 1 + − , 1 − cos x 2 1 − cos x 2 4 0 0 ∫ π/2 3 1 (x − sin x) dx = (π2 − 12)(3 − π), 1 − cos x 8 0 ∫ π ∫ π ∫ π 2 (x − sin x) (x − sin x)3 3π2 x − sin x dx = 2, dx = π, dx = − 8. 1 − cos x 1 − cos x 2 0 0 0 1 − cos x

1126

CHAPTER 14

Fact 14.4.39. Let a and b be real numbers, and assume that b , 0 and |a| < b. Then,

∫

π/2

acos ba 1 dx = √ , a sin x + b b2 − a2 π + acos ba 1 dx = √ , a sin x + b b2 − a2

∫

π

2 acos ab 1 dx = √ , 0 0 a sin x + b b2 − a2 ∫ 3π/2 ∫ 2π 2π 1 dx = √ , a sin x + b 0 0 b2 − a2 ∫ π ∫ π/2 acos ab 1 π 1 , , dx = √ dx = √ 2 2 2 a cos x + b a cos x + b 0 0 b −a b − a2 ∫ 3π/2 ∫ 2π 2π − acos ba 1 2π 1 dx = √ dx = √ , . a cos x + b a cos x + b 0 0 b2 − a2 b2 − a2 Fact 14.4.40. Let a and b be real numbers, and assume that 0 < |a| < b. Then, √ √ ∫ 2π ∫ π π b + b2 − a2 2π b + b2 − a2 x sin x x sin x dx = log , dx = log . a 2(b − a) a cos x + b a 2(a + b) 0 0 a cos x + b Fact 14.4.41. Let a > 1 and n ≥ 0. Then,

√ cos nx π( a2 − 1 − a)n . dx = √ 0 cos x + a a2 − 1 Source: [1136, p. 205] and [2249, p. 326]. Fact 14.4.42. Let a and b be real numbers, and assume that a ≥ b > 0. Then, ∫ π √ π sin2 x dx = 2 (a − a2 − b2 ). b 0 b cos x + a ∫

π

Source: [1136, p. 205] and [1472, p. 161]. Fact 14.4.43. Let a > −1, and assume that a , 0. Then,

∫

∫ π/2 √ √ 1 x sin 2x π x sin 2x π log(a + 1 − log (1 + dx = a + 1), a + 1). dx = a a 2 a cos2 x + 1 a sin2 x + 1 0 0 Source: [1217, p. 453]. Fact 14.4.44. Let a be a real number. If |a| > 1, then ∫ π ∫ π √ π2 x x sin 2x 2 dx = dx = 2π log[2(1 − a + a a2 − 1)]. , √ 2 2 2 2 0 a − cos x 0 a − cos x 2a a2 − 1 If a ∈ (0, 1), then ∫ π π x sin x 1 + a . dx = log 1 − a 2 2 2a 0 a − cos x π/2

Source: [1217, pp. 453, 454]. Fact 14.4.45. Let a and b be real numbers. If |a| < |b|, then

∫

∞

π sin x . dx = √ 0 (a cos 2x + b)x 2 b2 − a2 Fact 14.4.46. Let a ∈ (−1, 1). Then, ∫ π 1 π dx = . 2 (a cos x + 1) (1 − a2 )3/2 0 Source: [2106, p. 331].

1127

INTEGRALS

Fact 14.4.47. Let a be a positive number. Then,

∫ π √ π 2 sin x 1 dx = dx = √ acoth a + 1, , √ 2 2 0 sin x + a 0 sin x + a a+1 a2 + a √ √ ) ( ∫ π ∫ π 2 3 a sin x sin x 2 a+1 a , , atanh dx = π 1 − dx = 2 − √ 2 2 a+1 a+2 0 sin x + a 0 sin x + a a+1 √ ( 3/2 ) ∫ π ∫ π 1 2 a+1 sin4 x a 4 a2 sin5 x + atanh dx = π dx = − a , − 2a + . √ √ 2 2 3 a+2 0 sin x + a 0 sin x + a a+1 2 a+1 √ √ Source: [1136, p. 205]. Remark: acoth a + 1 = 21 log a+2+2a a+1 . Related: Fact 14.1.16. Fact 14.4.48. Let a be a positive number. If n ≥ 1 is odd, then ∫ π ∫ π cosn x cosn x dx = dx = 0. 2 2 0 sin x + a 0 cos x + a Furthermore, √ ) ( ∫ π ∫ π a cos2 x π 1 , , dx = dx = π 1 − √ 2 2 a+1 0 cos x + a 0 cos x + a a2 + a ( 3/2 ( ) ) ∫ π ∫ π a a 3 1 cos4 x cos6 x a5/2 2 dx = π √ dx = π a − + − √ + −a , . 2 2 2 8 0 cos x + a 0 cos x + a a+1 2 a+1 √  ∫ π ∫ π  a + 1  cos2 x sin2 x  dx = π  dx = − 1 , 2 2 a 0 sin x + a 0 cos x + a [ ] ∫ π 3 √ 1 sin x a−1 dx = √ (a + 1)π − 4 a − 2(a + 1) atan √ . 2 2 a 2 a 0 cos x + a ∫

π

Fact 14.4.49. Let a ≥ b > 0. Then,

∫

π 0

π x sin x dx = √ atanh 2 (a + b)b a + b sin x

√

b , a+b

∫

π 0

π x sin x dx = √ atan 2 a + b cos x ab

In particular, ∫ π

√ √ √ √ x sin x 2π 2 2π dx = atanh = log( 2 + 1), 2 2 2 2 0 sin x + 1 Source: [1158, p. 150] and [2106, p. 153]. Fact 14.4.50. If n ≥ 0, then ∫ π/2 π sin (2n + 1)x dx = . sin x 2 0 If n ≥ 1, then

∫

π/2 0

If n ≥ 0, then

∑ sin 2nx 1 dx = 2 (−1)i . sin x 2i +1 i=0 n−1

∫

π 0

Source: [1158, p. 153].

∫

cos nx − 1 dx = nπ. cos x − 1

π 0

√

x sin x π2 dx = . 2 4 cos x + 1

b . a

1128

CHAPTER 14

Fact 14.4.51. Let a > 0 and n ≥ 1. Then,

∫

π 0

π sin na cos nx − cos na dx = . cos x − cos a sin a

Source: [2106, p. 62] and [2294, p. 413]. Fact 14.4.52. ∫ π/4 0

√ cos3/2 2x (4 2 − 5)π . dx = 4 cos3 x

Source: [1502]. Fact 14.4.53.

∫

π/2 0

x(π − x) dx = 72 ζ(3), sin x

∫

π/4 0

π log 2 π2 x2 − + G. dx = 4 16 sin2 x

Source: [516, p. 63] and [2278]. Fact 14.4.54. Let a be a real number. If |a| < 1, then

∫

π/2

0

∫ π/2 sin2 x π π cos2 x dx = dx = , , 2 − 2a cos 2x 4(1 + a) 4(1 − a) 1 + a2 − 2a cos 2x 1 + a 0 ∫ ∫ π π 1 2π 1 − a2 dx = π, dx = . 2 2 1 − a2 −π 1 + a − 2a cos 2x 0 1 + a − 2a cos 2x

If |a| > 1, then

∫ 0

Now, let n ≥ 1. Then, ∫

π

0

π/2

cos2 x π . dx = 4a(a − 1) 1 + a2 − 2a cos 2x

 an π    , |a| < 1,   cos 2nx  1 − a2 dx =   π  1 + a2 − 2a cos 2x   , |a| > 1.  2 (a − 1)an

Related: Fact 13.6.7. Fact 14.4.55. Let a be a real number. Then,

∫

π

0

∫

2π 0

π   log(1 + a), a ∈ (−1, 0) ∪ (0, 1],     x sin x a ( ) dx =   1 π  1 + a2 − 2a cos x     a log 1 + a , |a| > 1,

 2π    log(1 − a),    x sin x a ( ) dx =    2π 1 1 + a2 − 2a cos x    ,  log 1 − a a

Source: [1217, p. 449]. Fact 14.4.56. Let a ∈ R and n ≥ 0. Then,

∫

π 0

a ∈ [−1, 0) ∪ (0, 1), |a| > 1.

 an π    , |a| < 1,   cos nx  1 − a2 dx =   π  1 + a2 − 2a cos x   , |a| > 1.  2 (a − 1)an

1129

INTEGRALS

Fact 14.4.57. Let a be a real number, and assume that a , 0 and a , 1. Then,

∫

) ( sin x π a + 1 −1 . dx = 4a a − 1 (1 + a2 − 2a cos x)x

∞ 0

Source: [1217, p. 450]. Fact 14.4.58. Let a, b ∈ R and n ≥ 0. Then,

∫

2π

∫

(a sin x + b cos x)2n+1 = 0,

0

2π

(a sin x + b cos x)2n =

0

(2n − 1)!! 2π(a2 + b2 )n . (2n)!!

Source: [1217, p. 405]. Fact 14.4.59. Let a, b > 0 and n ≥ 1. Then,

∫

) n−1 ( )( ∑ π 2i 2n − 2i − 2 1 1 dx = . √ 2 i n−i−1 2 n 2n−1 n−i−1 ab (a sin x + b cos x) 2 ab i=0 i

π/2

0

In particular, ∫ π/2 0

∫ π/2 1 π 1 (a + b)π , dx = √ , dx = 2 2 2 2 2 4(ab)3/2 a sin x + b cos x (a sin x + b cos x) 0 2 ab ) ( ∫ π/2 1 3 2 π 3 + + . dx = √ (a sin2 x + b cos2 x)3 0 16 ab a2 b2 ab

Source: [2013, p. 182] and [2106, p. 100]. Fact 14.4.60.

∫

∫

π/2 √

sin x dx =

0

∫

√

π/2 √

cos x dx =

0

∫

π/2 √

π/2 √

0

√ 2π , cot x dx = 2

0

∫

π/2 √

csc x dx =

∫

π/2 √

tan x dx =

0

∫

2 2 3 Γ ( 4 ), π

π/2 0

sec x dx =

√

√

0

π sin x dx = . √ 4 sin x + cos x

Source: [2106, p. 49]. Related: Fact 14.4.1. Fact 14.4.61.

∫ 0

π

Γ2 ( 1 ) dx = √4 , √ 4 π 3 − cos x 1

∫

( )3/2

π/2

π 2

1

. dx = √ Γ2 ( 34 ) 2 cos2 x + sin2 x

0

Fact 14.4.62. Let n ≥ 0. Then,

∫

π/4 0

∫

log 2 ∑ 1 + (−1)i−1 , 2 2n − 2i − 2 i=1 n

tan2n+1 x dx = (−1)n π/4

0

In particular,

∫ 0 π/4

∫ 0

π ∑ 1 + (−1)i−1 . 4 i=1 2n − 2i + 1 n

tan2n x dx = (−1)n

π/4

log 2 , tan x dx = 2

1 log 2 tan3 x dx = − , 2 2

Source: [1158, p. 152] and [1217, p. 395].

∫

π/4 0

∫

π tan2 x dx = 1 − , 4

π/4 0

1 √ Γ2 ( 14 ), 2 2π

tan4 x dx =

π 2 − . 4 3

1130

CHAPTER 14

Fact 14.4.63. Let a ∈ (−1, 1). Then,

∫

π/2

tana x dx =

0

Source: [1568, p. 42]. Fact 14.4.64. Let a ≥ 0. Then,

∫

∞ 0

π . 2 cos πa 2

∫

1 π dx = , 4 (x2 + 1)(xa + 1)

π/2 0

π 1 dx = . 1 + tana x 4

Source: A proof and generalization of the first integral are given in [630]. The second integral can

be obtained from the first integral by defining√x = tan θ. See [511, p. 254]. The second integral is given in [1757, pp. 32, 33] for the case a = 2, and in [3024, p. 451] for the case where a is an integer. Remark: Both integrals are independent of a. Fact 14.4.65. Let n ≥ 1. Then, 2n+1 ∫ 1 n+1 (2π) B2n+1 (x) cot πx dx, ζ(2n + 1) = (−1) 2(2n + 1)! 0 where Bn is the nth Bernoulli polynomial. Source: [2513, p. 167]. Related: Fact 13.2.1. Fact 14.4.66.

∫

π/2 0

1 1 dx = 2 2 1 + 8 sin tan x

∫

∞ −∞

1 π(2e2 + 1) dx = . 6(2e2 − 1) (1 + 8 sin2 x)(1 + x2 )

Source: [1684].

14.5 Facts on Definite Integrals of Inverse Trigonometric Functions ∫

Fact 14.5.1. Let n ≥ 1. Then, 1

( ) π 4n (n!)2 1 − , x asin x dx = 2n + 1 2 (2n + 1)!

∫

2n

0

Source: [822]. Fact 14.5.2.

1

x 0

2n+1

( ) π 1 (2n + 2)! asin x dx = − . 2n + 2 2 22n+3 (n + 1)!

∫ 1 ∫ 1 π π 2 π x asin x dx = , x2 asin x dx = − , asin x dx = − 1, 2 8 6 9 0 0 0 √ 2 √ ∫ 1 ∫ √2/2 ∫ 1 asin x asin x 3π π G π asin x dx = dx = log 2, dx = + log 2, , 2−x+1 x 2 x 2 8 18 x 0 0 0 ∫ 1 ∫ 1 π acos x π acos x dx = log 2 + G, dx = − log 2 + G, 2 2 0 1+x 0 1−x ∫ π/2 ∫ π/2 2 1 π cos x 5π2 acos dx = , acos dx = , 1 + 2 cos x 6 1 + 2 cos x 24 0 0 ∫ 1 ∫ 1 √ √ √ 1 atan x atan x dx = G, √ dx = 2 acoth 2 + (1 − 2)π, 2 x x 0 0 ∫ 1 ∫ 1 ∫ 1 atan x π2 G π π atan x x atan x dx = dx = − log 2, dx = log 2, , 2+1 2+1 x + 1 8 32 2 8 x x 0 0 0 ∫ 1 ∫ 1 2 2 π π 1 x atan x G π2 π x atan x dx = − − log 2, dx = − − log 2, 2 4 32 2 4 16 8 x2 + 1 0 (x + 1)(x + 1) 0 ∫

1

1131

INTEGRALS

∫

1 0

∫

1 0

∫

π atan x dx = log 2 + G, 8 x3 + x

π acot x dx = − log 2 + G, 8 x2 + x

1 0

∫ 1 π2 π πG 7 atan2 x atan2 x dx = G − dx = − ζ(3), + log 2, x 2 8 16 4 x2 0 √ ∫ 1 ∫ 1 π3 atan2 x atan x2 + 2 5π2 dx = , . dx = √ 2 192 96 0 x +1 0 (x2 + 1) x2 + 2

Source: [388], [1107], [1217, pp. 600–602], and [2106, pp. 190–201, 220]. Fact 14.5.3.

∫

1 0

∫

1

π (asin x) log x dx = 1 − log 2 − , 2

(atan x) log x dx =

0

1 π π2 log 2 − + , 2 4 48

Source: [1217, p. 607]. Fact 14.5.4. Let a > 0. Then,

∫

1 0

1 atan ax dx = 2 (atan a) log(a2 + 1), a2 x + 1 2a

Source: [1217, p. 603]. Fact 14.5.5. Let n ≥ 0. Then,

∫

1

( ) 1 π 2n n! x asin x dx = − , 2n + 1 2 (2n + 1)!! ∫ 1 2n n! x2n acos x dx = , (2n + 1)(2n + 1)!! 0

∫

(acos x) log x dx = log 2 − 2,

0

∫

1 0

π π2 1 (acot x) log x dx = − log 2 − − . 2 4 48

0

) ( acot ax 1 π 1 + acot a log(a2 + 1). dx = a2 x + 1 a2 4 2

∫

1

∫

1

2n

0

1

x

2n+1

0

∫

1

) ( π (2n + 1)!! , asin x dx = 1 − n+1 4n + 4 2 (n + 1)!

x2n+1 acos x dx =

0

π(2n + 1)!! . + 1)(n + 1)!

2n+3 (n

Source: [1217, p. 601]. Fact 14.5.6. Let a ∈ (−∞, −1) ∪ (0, ∞). Then,

∫

1/a 1/(a+1)

atan ax dx = x

∫

1/a 1/(a+1)

acot(a + 1)x dx. x

Source: [109, pp. 225, 226]. Fact 14.5.7. Let a ≥ 0. Then,

∫

∫ ∞ π π atan ax atan ax dx = dx = log(a2 + 1), log(a + 1), 2 4 x3 + x x − x3 0 0 ∫ ∞ √ π atan ax dx = log(a + a2 + 1). √ 2 2 0 x 1−x Fact 14.5.8. If a ∈ [1, 2], then ∫ ∞ atan x π aπ dx = csc . a x 2(a − 1) 2 0 ∞

If a ∈ [0, 1], then

∫

∞ 0

Source: [1217, p. 602].

π aπ acot x dx = csc . a x 2(1 − a) 2

1132

CHAPTER 14

Fact 14.5.9. Let a > 0 and b > 0. Then,

∫

0

∞

) ( ∫ ∞ ) ( π acot bx atan bx abπ b b − log ab , + log ab , dx = dx = 2 (x + a)2 a2 b2 + 1 2 a2 b2 + 1 2ab 0 (x + a) ∫ ∞ ∫ ∞ x acot bx π π ab + 1 x atan bx dx = 2 log(ab + 1), dx = log , 2 2 2 2 2 ab x +a x(x + a ) 2a 0 0 ∫ ∞ ∫ ∞ x atan bx bπ π x acot bx . dx = dx = 2 , 2 2 2 2 2 2 4a(ab + 1) 4a (ab + 1) 0 (x + a ) 0 (x + a )

Source: [1217, p. 603]. Fact 14.5.10. Let a > 0 and b > 0. Then,

∫

0

∞

π a atan ax − atan bx dx = log , x 2 b

∫

∞

0

(a + b)a+b (atan ax) atan bx π log . dx = 2 x2 aa bb

Source: [701] and [1217, p. 604]. Fact 14.5.11. Let a > 0 and b > 0. Then,

∫

∞ 0

[ ( )] ( a) 1 b π 1 log 1 + + log 1 + . (acot ax) acot bx dx = 2 a b b a

Source: [1217, p. 599]. Fact 14.5.12. ∫

∫ ∞ x atan x atan x π π2 log 2, , dx = dx = 2 16 x3 + x x4 + 1 0 0 ∫ ∞ √ G atan x π 2π 3) − , dx = log 2 + log(2 + 4+x 12 9 9 x 0 ∫ ∞ ∫ ∞ atan x acot x π dx = dx = log 2 + G, 2+x x + 1 4 x 0 0 ∫ ∞ ∫ ∞ atan x acot x dx = dx = 2G. √ √ 2 0 0 x +1 x x2 + 1 Source: [1217, p. 601, 602]. ∞

14.6 Facts on Definite Integrals of Logarithmic Functions Fact 14.6.1. Let z ∈ C, and assume that Re z > 1. Then,

∫

∞

0

e−x − e−xz dx = x

∫

1 0

xz−1 − 1 dx = log z. log x

Source: [2513, p. 18]. Fact 14.6.2. The following statements hold:

i) Let a ∈ (−1, ∞) and n ≥ 0. Then, ∫ 1 xa logn x dx = (−1)n 0

In particular,

∫

1

∫

1

log x dx = (−1) n!, n

n

0

ii) Let a, b ∈ (−1, ∞). Then,

n! . (a + 1)n+1

x logn x dx = (−1)n

0

∫

1

xa logb 0

1 Γ(b + 1) dx = . x (a + 1)b+1

n! . 2n+1

1133

INTEGRALS

In particular, √ √ ∫ 1 ∫ 1 ∫ 1√ √ 3 π 1 1 π 3/2 1 , dx = , log dx = π, log dx = √ x 2 x 4 0 0 0 log 1x √ ∫ 1 ∫ 1√ √ 1 1 6π dx = 2π, x log dx = . √ x 9 0 0 x log 1x iii) Let n ≥ 0. Then,

∫

e

  n  ∑ n−i n!   e − (−1)n n!.  log x dx =  (−1) i! i=0 n

1

In particular, ∫ e ∫ e ∫ e ∫ e log x dx = 1, log2 x dx = e − 2, log3 x dx = 6 − 2e, log4 x dx = 9e − 24. 1

1

1

1

Source: [511, p. 97] and [815, 818]. Fact 14.6.3. If n ≥ 0, then

∫

1

xn log(1 − x) dx = −

0

In particular, ∫ 1 log(1 − x) dx = −1,

∫

1 0

0

If n ≥ 0, then

∫

1

3 x log(1 − x) dx = − , 4

xn log2 (1 − x) dx =

0

In particular, ∫ 1

∫

1

log2 (1 − x) dx = 2,

0

If n ≥ 0, then

∫

1

x2 log(1 − x) dx = −

0

11 . 18

2 + Hn+1,2 Hn+1 . n+1

x log2 (1 − x) dx =

0

∫

Hn+1 . n+1

7 , 4

∫

1

x2 log2 (1 − x) dx =

0

85 . 54

Hn/2 − H(n−1)/2 log 2 1 , + − 2n + 2 n + 1 (n + 1)2 0   ∫ 1 2n+1 ∑ 1  1  2n i log 4 + x log(x + 1) dx = (−1)  , 2n + 1  i 1

xn log(x + 1) dx =

0

∫

i=1

1

2n+2 ∑

1 (−1)i+1 , i 0 i=1   ∫ 1 n+1 2 4  π ∑ 1  n+1/2 n+1 i  −  . x log(x + 1) dx = log 2 + (−1) (−1) 2n + 3 2n + 3  4 i=0 2i + 1  0 x2n+1 log(x + 1) dx =

In particular, ∫ 1 log(x + 1) dx = 2 log 2 − 1, 0

∫ 0

1

1 2n + 2

x log(x + 1) dx =

1 , 4

∫ 0

1

x2 log(x + 1) dx =

2 log 2 5 − , 3 18

1134

CHAPTER 14

∫

1

∫

7 , x log(x + 1) dx = 48

1

3

0

Furthermore,

∫

1

0

∫ 0

∫

1

x2 log2 (x + 1) dx =

0

∫

1 0

∫

4 log 2 241 − , x log (x + 1) dx = 3 288 3

1

log2 (x + 1) dx = 2(log 2 − 1)2 ,

0

47 2 log 2 − . 5 300

x4 log(x + 1) dx =

1

2

5 x log2 (x + 1) dx = 2 log 2 − , 4

(6 log 2 − 11)(6 log 2 − 5) , 54 2 log2 2 92 log 2 6589 − + . 5 75 9000

x4 log2 (x + 1) dx =

0

Source: [1101, 2766]. Remark: For noninteger p, H p is defined in Fact 13.3.4. Fact 14.6.4. If a > 0, then

∫

1 0

In particular,

∫

1 0

π2 log(xa + 1) dx = , x 12a ∫

log(x + 1) π2 dx = , x 12

1 0

∫

0

1

π2 log(1 − xa ) dx = − . x 6a

log(1 − x) dx = x

∫

log x π2 dx = − . 1−x 6

1

0

Fact 14.6.5. The following statements hold:

i) Let a > 0. Then, ∫ a 1 log(ax + 1) dx = (atan a) log(1 + a2 ), 2+1 2 x 0 In particular,

∫

1

log2n x π2n+1 |E2n | dx = . 2 1+x 4n+1

∫

0

iii) Let n ≥ 1. Then,

∫ 0

In particular, ∫ 1 0

∫

π3 log2 x , dx = 2 16 x +1 1

1

log4 x 5π5 , dx = 2 64 x +1

0

∫

1

0

log6 x 61π7 . dx = 2 256 x +1

2n 2n log2n−1 x n (2 − 1)π B2n dx = (−1) . 4n 1 − x2

∫

π2 log x , dx = − 8 1 − x2

iv) Let a, b > −1. Then,

0

π log(x + a) dx = log 2a2 . 8a x 2 + a2

π log(1 + x) dx = log 2. 8 1 + x2

0

In particular, ∫ 1

a

1 0

ii) Let n ≥ 1. Then,

∫

1 0

∫

1 0

log3 x π4 , dx = − 16 1 − x2

∫ 0

1

log5 x π6 . dx = − 8 1 − x2

xa − xb a+1 dx = log . log x b+1

Source: i) is given in [511, p. 243], [2013, p. 182], and [2106, p. 55]; ii) is given in [1217, p. 550]; iii) is given in [511, p. 241]; iv) is given in [3024, p. 335].

1135

INTEGRALS

Fact 14.6.6. Let a ∈ (0, 1). Then,

∫

1 0

In particular, ∫

log x dx = (1 − x)a

∫

1

0

H1−a log(1 − x) dx = . xa a−1

√ ∫ 1 log x 16 2π 9 log 3 − 9 − 3 log x − , , dx = 4 log 2 − dx = √4 √3 9 3 4 0 0 1−x 1−x √ ∫ 1 ∫ 1 3π + 9 log 3 log x log x − 9. dx = dx = 4 log 2 − 4, √ 2/3 2 0 0 (1 − x) 1−x 1

Furthermore, √ ∫ 1 ∫ 1 √4 πΓ( 45 ) 2 x log2 x log2 x 2π2 2 , dx = 16+8 log 2−16 log 2− dx = (9π −48π+144G−64), √ √ 3 9Γ( 74 ) 0 0 1−x 1−x ∫ 1 ∫ 1 √ x log2 x 80 4π2 x log2 x π3 224 +12 log2 2− log 2− , dx = −π+2π log2 2−2π log 2, dx = √ √ 6 27 9 9 0 0 1−x 1−x √ ∫ 1 ∫ 1 log x log2 x 2π 2 1 16G + π2 2 1 Γ ( 4 ), Γ ( 4 ), dx = − dx = √ √ √ 8 0 0 8 2π x(1 − x2 ) x(1 − x2 ) √ √ ∫ 1 ∫ 1 2(π − 4)π3/2 log3 x 48G + 5π2 π 2 1 x log x Γ ( 4 ), . dx = − dx = √ √ 16 2 Γ2 ( 14 ) 0 0 x(1 − x2 ) x(1 − x2 ) Source: [1217, p. 538]. Fact 14.6.7. Let n ≥ 0. Then,

 2n   1 x2n log x (2n − 1)!! π ∑  (−1)i−1 − log 2 , dx = √ (2n)!! 2 i=1 i 0 1 − x2   ∫ 1 2n+1 2n+1  1 x log x (2n)!! π  ∑ i−1  (−1) + log 2 , dx = √ (2n + 1)!! 2 i=1 i 0 1 − x2   ∫ 1 2n √ 1 (2n − 1)!! π ∑ 1  2n i−1 2  (−1)  , x (log x) 1 − x dx = − log 2 − (2n + 2)!! 2  i=1 i 2n + 2  0 2n+1  ∫ 1 √ 1 1  (2n)!! π  ∑ 2n+1 i−1 2  (−1)  . x (log x) 1 − x dx = + log 2 − (2n + 3)!! 2  i=1 i 2n + 3  0 ∫

In particular,

∫

1

log x π dx = − log 2, √ 2 4 0 1−x ∫ 1 √ π π (log x) 1 − x2 dx = − − log 2, 8 4 0

Furthermore, ∫ 1 0

1

∫

1

x log x dx = log 2 − 1, √ 0 1 − x2 ∫ 1 √ 1 4 x(log x) 1 − x2 dx = log 2 − . 3 9 0

∫ 1 log2 x π3 π x log2 x π2 dx = + log2 2, dx = 2 − + log2 2 − 2 log 2, √ √ 24 2 12 0 1 − x2 1 − x2 ∫ 1 log3 x π3 π 3π dx = − log 2 − log3 2 − ζ(3). √ 2 8 2 4 0 1−x

1136

CHAPTER 14

Source: [1217, p. 538]. Fact 14.6.8. Let a be a real number. If |a| ≤ 1, then

∫

1 0

log(ax + 1) π2 1 − acos2 a. dx = √ 8 2 x 1 − x2

If |a| ≤ 1 and a , 0, then √ √ ∫ 1 x log(ax + 1) 1 − a2 (1 − 1 − a2 )π dx = asin a + − 1. √ a 2a 0 1 − x2 If |a| ≥ 1, then √ ∫ 1 √ π a2 − 1 x log(ax + 1) log(1 + a2 − 1) + − 1. dx = √ a 2a 0 1 − x2 Furthermore, ∫ 1 ∫ 1 π π log(1 + x) x log(1 + x) dx = − log 2 + 2G, dx = − 1, √ √ 2 2 0 0 1 − x2 1 − x2 ∫ 1 ∫ 1 log(1 − x) π x log(1 − x) π dx = − log 2 − 2G, dx = − − 1. √ √ 2 2 2 2 0 0 1−x 1−x Source: [1217, p. 558]. Fact 14.6.9. Let a ∈ [−1, 1). Then, ∫ 1 (a − x) log(1 − x) π2 (π − acos a)2 log2 (2 − 2a) dx = − − . 12 8 8 x2 − 2ax + 1 0 In particular, ∫ 1 ∫ 1 log2 2 π2 1 (x + 1) log(1 − x) (2x + 1) log(1 − x) 5π2 2 dx = dx = − , log 3 − , 4 12 4 36 x2 + 2x + 1 x2 + x + 1 0 0 ∫ 1 ∫ 1 1 5π2 (1 − 2x) log(1 − x) π2 x log(1 − x) 2 dx = log 2 − , dx = . 2 2 8 96 18 x +1 x −x+1 0 0 Source: [1682]. Fact 14.6.10. Let p > −1. Then,

∫

1 0

x p log x π2 dx = H p,2 − = −ψ′ (p + 1). 1−x 6

In particular, ∫ 1 ∫ 1 ∫ 1 log x log x π2 log x 2 dx = −π − 8G, dx = − , dx = 8G − π2 , 3/4 (1 − x) 1/2 (1 − x) 1/4 (1 − x) 2 0 x 0 x 0 x ∫ 1 ∫ 1 1/4 ∫ 1 1/2 log x π2 x log x x log x π2 2 dx = − , dx = −π − 8G + 16, dx = 4 − , 6 1−x 1−x 2 0 1−x 0 0 ∫ 1 ∫ 1 5/4 ∫ 1 3/4 2 x log x 16 x log x π x log x 416 dx = −π2 +8G+ , dx = 1− , dx = −π2 −8G+ , 1−x 9 6 1−x 25 0 1−x 0 0 ∫ 1 3/2 ∫ 1 7/4 ∫ 1 2 x log x 40 π2 x log x 928 x log x 5 π2 dx = − , dx = −π2 + 8G + , dx = − , 1−x 9 2 1−x 441 1−x 4 6 0 0 0

1137

INTEGRALS

∫ 0

1

∫

49 π2 x3 log x dx = − , 1−x 36 6

0

1

205 π2 x4 log x dx = − , 1−x 144 6

∫ 0

1

5269 π2 x5 log x dx = − . 1−x 3600 6

Remark: ψ is the digamma function. See Fact 13.3.3. Related: Fact 13.3.4 and Fact 14.2.7. Fact 14.6.11. Let p > −1 and n ≥ 1. Then,

∫

1 0

x p logn x dx = (−1)n n![ζ(n + 1) − H p,n+1 ] = −ψ(n) (p + 1). 1−x

In particular,

∫

1

4n−1 2n log2n−1 x dx = − π |B2n |, 1−x n 0 ∫ 1 ∫ 1 log2 x log2 x 3 dx = 2π + 56ζ(3), dx = 14ζ(3), 3/4 (1 − x) 1/2 (1 − x) 0 x 0 x ∫ 1 ∫ 1 1/2 2 log2 x x log x 3 dx = 14ζ(3) − 16, dx = 56ζ(3) − 2π , 1/4 (1 − x) 1−x x 0 0 ∫ 1 5/2 2 ∫ 1 3/2 2 x log x 56432 x log x 448 dx = 14ζ(3) − , dx = 14ζ(3) − . 1 − x 3375 1 − x 27 0 0

Related: Fact 13.3.4 and Fact 14.6.12. Fact 14.6.12. Let m ≥ 0 and n ≥ 1. Then,

∫

1 0

xm logn x dx = (−1)n n![ζ(n + 1) − Hm,n+1 ]. 1−x

In particular, ∫ 1 ∫ 1 ∫ 1 2 2 log2 x x log2 x x log x 9 dx = 2ζ(3), dx = 2ζ(3) − 2, dx = 2ζ(3) − , 1 − x 1 − x 1 − x 4 0 0 0 ∫ 1 ∫ 1 ∫ 1 2 3 3 3 4 4 4 log x π x log x π x log x 51 π dx = − , dx = 6 − , dx = − . 15 1−x 15 1−x 8 15 0 1−x 0 0 Furthermore, ∫ 1 ∫ 1 2n 2 n ∑ 2 log2n x 22n+1 − 1 x log x 7 ζ(3) − dx = (2n)!ζ(2n + 1), dx = , 2 2n+1 2 4 2 1−x (2i − 1)3 0 1−x 0 i=1 ∫ 1 ∫ 1 (1 − 4n+1 )π2n+2 π2n+2 log2n+1 x x log2n+1 x dx = dx = − |B |, |B2n+2 |, 2n+2 4n + 4 4n + 4 1 − x2 1 − x2 0 0 ∫ 1 ∫ 1 log2n x (4n − 1) 2n π2n+1 (1 + x2 ) log2n x dx = π |B |, dx = n+1 |E2n |. 2n 2 2 2 2 (1 − x ) 4 0 1+x 0 Related: Fact 13.3.4, Fact 14.6.11, and Fact 14.8.15. Fact 14.6.13. Let a > 0. Then,

∫

1 0

(− log x)a dx = Γ(a + 1)ζ(a + 1). 1−x

In particular, √ √ √ ∫ 1 ∫ 1 ∫ 1 (− log x)1/2 π 3 (− log x)3/2 3 π 5 (− log x)5/2 15 π 7 dx = ζ( 2 ), dx = ζ( 2 ), dx = ζ( 2 ). 1−x 2 1−x 4 1−x 8 0 0 0

1138

CHAPTER 14

Fact 14.6.14. Let n ≥ 0. Then,

∫

∫ 1 2n+1 2n 2n+1 π2 ∑ π2 ∑ 1 x2n log x x log x i+1 1 dx = − + dx = + (−1) 2 , (−1)i 2 , x + 1 12 x + 1 12 i i 0 0 i=1 i=1   ∫ 1 n 2 n ∑   3 x log x 1   dx = (−1)n  ζ(3) + 2 (−1)i 3  , x+1 2 i 0 i=1 ∫ 1 ∫ 1 2n+1 2n n (4 − 1)(2n)! (1 − 22n+1 )π2n+2 log x log x dx = ζ(2n + 1), dx = |B2n+2 |, n x+1 4 x+1 2n + 2 0 0 1

Source: [1217, pp. 540, 550]. Fact 14.6.15. ∫

(

) 1 π2 x log x dx = + − 1, 2 1 − x (1 − x) 6 0 ∫ 1 ∫ 1 ∫ 1 π2 4π2 log2 x log3 x log4 x dx = , dx = −6ζ(3), dx = , 2 2 2 3 15 0 (1 − x) 0 (1 − x) 0 (1 − x) ∫ 1 ∫ 1 ∫ 1 log3 x π2 2π4 log5 x log4 x dx = −120ζ(5), dx = −3ζ(3) − dx = 12ζ(3) + , , 2 3 3 2 15 0 (1 − x) 0 (1 − x) 0 (1 − x) ∫ 1 ∫ 1 ∫ 1 π2 7 π4 log x log2 x log3 x dx = − dx = dx = − , , ζ(3), 2 2 2 8 4 16 0 1−x 0 1−x 0 1−x ∫ 1 ∫ 1 ∫ 1 2 3 2 π 1 π4 x log x x log x x log x dx = − dx = dx = − , ζ(3), , 2 2 2 24 4 240 0 1−x 0 1−x 0 1−x ∫ 1 ∫ 1 ∫ 1 log x π2 log2 x 3 log3 x 7π4 dx = − , dx = ζ(3), dx = − , 12 2 120 0 x+1 0 x+1 0 x+1 ∫ 1 ∫ 1 ∫ 1 x log x π2 x log2 x 3 π2 x log3 x π2 31 dx = − 1, dx = − , dx = − , x+1 12 x+1 4 12 x+1 12 36 0 0 0 ∫ 1 ∫ 1 ∫ 1 log3 x log2 x π2 9 log x , dx = − log 2, dx = dx = − ζ(3), 2 2 2 6 2 0 (x + 1) 0 (x + 1) 0 (x + 1) ∫ 1 ∫ 1 ∫ 1 log x log2 x log3 x 1 log 2 π2 9ζ(3) + π2 − , + log 2, , dx = − dx = dx = − 3 3 3 4 2 12 4 0 (x + 1) 0 (x + 1) 0 (x + 1) ∫ 1 ∫ 1 7 1 π2 log x log 2 log2 x dx = − dx = + − , + log 2, 4 4 24 3 6 18 0 (x + 1) 0 (x + 1) ∫ 1 ∫ 1 3 π2 log4 x π2 7π4 log3 x dx = − ζ(3) − − log 2, dx = 9ζ(3) + + , 4 4 2 4 3 90 0 (x + 1) 0 (x + 1) ( ) √ ∫ 1 ∫ 1 (−8 + π + 2 log 2) πΓ 34 x−1 log x ( ) dx = dx = log 2, , √4 0 log x 0 2Γ 41 1 − x2 √ √ ( ) ∫ 1 ∫ 1 (−12 + 3π + 3 log 3) πΓ 23 π log 2 log x log x ( ) dx = dx = − , , √3 √ 7 2 2 2 0 0 8Γ 6 1−x 1−x ∫ 1 ∫ 1 ∫ 1 log(1 − x) log2 2 π2 1+x 1 + x2 π dx = − , log dx = 2 log 2, log dx = − log 2, 2 1+x 2 12 1−x 2 1−x 0 0 0 √ √ ∫ 1 ∫ 1 3 3 1+x 3π 3 3π 1+x log − log 3 + 2 log 2, − 1, dx = log dx = 3 2 6 2 3 1 − x 1 − x 0 0 1

1139

INTEGRALS

∫

∫ 1 π2 π2 1 + x2 ex + 1 1 dx = dx = , log , log x e −1 4 8 1 − x2 0 0 0 x ∫ 1 ∫ 1 5 13 (log x) log(2 − x) (log x) log(2 + x) dx = − ζ(3), dx = − ζ(3), 1−x 8 1+x 24 0 0 ∫ 1 ∫ 1 2 π π2 (log x) log(1 + x) dx = 2 − 2 log 2 − , (log x) log(1 − x) dx = 2 − , 12 6 0 0 ∫ 1 ∫ 1 2 2 π π log 2 log(x + 1) (log x) log(1 − x) dx = dx = log 2, − , 2 2+1 24 2 8 (1 + x) x 0 0 ∫ 1 ∫ 1 ∫ 1 3 4 2 4 π log (1 − x) log (1 − x) log (1 − x) dx = 2ζ(3), dx = − , dx = 24ζ(5), x x 15 x 0 0 0 ∫ 1 ∫ 1 log2 (1 − x) log4 (1 − x) π2 2π2 4π4 , + , dx = dx = 12ζ(3) + 2 4 3 3 45 x x 0 0 ∫ 1 ∫ 1 ∫ 1 log(x + 1) π2 log2 (x + 1) 1 log2 (x + 1) π2 dx = , dx = ζ(3), dx = − 2 log2 2, x 12 x 4 6 x2 0 0 0 ∫ 1 ∫ 1 log3 (x + 1) log3 (x + 1) 3 3 π2 3 ζ(3) − 2 log 2, − 3 log2 2 − ζ(3), dx = dx = 2 3 4 4 8 x x 0 0 ∫ 1 ∫ 1 3 3 log (x + 1) 3ζ(4) log (1 − x) dx = −6ζ(3), dx = − 2 log3 2, 2 2 6 x x 0 0 ∫ 1 ∫ 1 log3 (x + 1) π2 π2 3ζ(3) log3 (1 − x) , − − 3 log2 2, dx = −3ζ(3) − dx = 2 4 8 x3 x3 0 0 √ ∫ 1 ∫ 1 (log x) log2 (1 − x) π4 x2 log x (2 − 2)π2 dx = − , , dx = 2 4 x 180 32 0 0 (x − 1)(x + 1) √ √ ∫ 1 ∫ 1 √ 3π 1+ 5 2 2 log(1 − x + x ) dx = − 2, log(1 + x − x ) dx = 2 5 log − 2, 3 2 0 0 ∫ 1 ∫ 1 π2 log(1 − x2 ) π2 log(x2 + 1) dx = , dx = − , x 24 x 12 0 0 ∫ 1 ∫ 1 2 2 log(x + 1) log(1 − x ) π dx = − log 2, dx = −2 log 2, 2 2 x x2 0 0 √ √ ∫ 1 ∫ 1 log(1 − x3 ) log(x3 + 1) 3π 3 log 3 3π dx = − − , dx = , 3 3 12 4 6 x x 0 0 ∫ 1 ∫ 1 1 x log(x + 1) 1 log(x + 1) dx = log2 2, dx = 2 log 2 − log2 2 − 1, x+1 2 x+1 2 0 0 ∫ 1 2 ∫ 1 1 x log(x + 1) 1 5 log(x + 1) dx = (1 − log 2), dx = log2 2 + − 2 log 2, 2 x + 1 2 4 2 (x + 1) 0 0 ∫ 1 ∫ 1 2 1 1 x log(x + 1) x log(x + 1) dx = (log 2 + log2 2 − 1), dx = (3 log 2 − 2 log2 2 − 1), 2 2 2 2 (x + 1) (x + 1) 0 0 ∫ 1 ∫ 1 log(x + 1) 3 5 1 x log(x + 1) 3 dx = dx = − log 2, − log 2, 3 3 16 8 16 8 (x + 1) (x + 1) 0 0 ∫ 1 2 7 1 13 x log(x + 1) dx = log 2 + log2 2 − , 3 8 2 16 (x + 1) 0 1

1+x 1 log dx = x 1−x

∫

∞

1140

CHAPTER 14

∫

∫ 1 π log 2 3 log2 2 π2 log(x2 + 1) log(x2 + 1) dx = dx = − , − G, 2+1 + x 1 4 48 2 x 0 0 √ ∫ 1 ∫ 1 3π π2 log(x3 + 1) log(x3 + 1) dx = + 2 log 2 − 3, dx = , 3 x 36 0 0 √ √ ∫ 1 ∫ 1 3π 3π log(x3 + 1) log(x3 + 1) dx = dx = − 2 log 2, , 2 3 3 6 x x 0 0 ∫ 1 ∫ 1 ∫ 1 log x π G π 3G log x 1 log x dx = −G, dx = − dx = − − − , − , 2 2 2 2 3 8 2 8 8 16 0 (x + 1) 0 x +1 0 (x + 1) ∫ 1 ∫ 1 ∫ 1 2 π 1 1 1 x log x x log x x log x dx = − , dx = − log 2, dx = − log 2 − , 2 2 2 2 3 48 4 8 16 0 x +1 0 (x + 1) 0 (x + 1) ∫ 1 ∫ 1 ∫ 1 π3 π3 π log2 x 3π2 log2 x log2 x dx = dx = G + dx = G + , , + , 2 2 2 2 3 16 32 16 128 0 x +1 0 (x + 1) 0 (x + 1) ∫ 1 ∫ 1 π π log(x + 1) 3 log(x + 1) dx = dx = log 2, (log 2 − 1) + log 2, 2 2 2 8 16 8 x +1 0 (x + 1) 0 ∫ 1 ∫ 1 3π π2 1 3 1 x log(x + 1) log(x + 1) dx = dx = (log 2 − 1) + log 2 − , + log2 2, 2 3 2 64 8 16 96 8 x +1 0 0 (x + 1) ∫ 1 ∫ 1 x log(x + 1) 1 x log(x + 1) 1 π π − log 2, − log 2, dx = dx = 2 + 1)2 2 + 1)3 16 8 32 32 (x (x 0 0 ∫ 1 2 ∫ 1 2 x log(x + 1) π x log(x + 1) π 3 π dx = 2 log 2 − log 2 − 1, dx = − + log 2, 2 2 2 8 16 8 16 x +1 (x + 1) 0 0 ∫ 1 2 π π 1 x log(x + 1) − + log 2, dx = 2 + 1)3 16 64 64 (x 0 ∫ 1 ∫ 1 log(1 − x) π π log(1 − x) 1 1 dx = log 2 − G, dx = (log 2 − 1) + log 2 − G, 2+1 2 + 1)2 8 16 8 2 x (x 0 0 ∫ 1 ∫ 1 3π 1 log(1 − x) 1 3G x log(1 − x) 5π2 2 dx = dx = (log 2 − 1) + log 2 − , log 2 − , 2 3 64 8 8 4 96 x2 + 1 0 (x + 1) 0 ∫ 1 ∫ 1 π 1 x log(1 − x) π 1 1 x log(1 − x) dx = − − log 2, dx = − − log 2 − , 2 + 1)2 2 + 1)3 16 8 32 32 16 (x (x 0 0 ∫ 1 2 ∫ 1 2 x log(1 − x) x log(1 − x) 1 G π π (log 2 + 1) − log 2 − , dx = log 2 + G − 1, dx = 2+1 2 + 1)2 8 16 8 2 x (x 0 0 ∫ 1 2 ∫ 1 2 π G log(x + 1) π 1 x log(1 − x) dx = (log 2 − 1) − , dx = − log2 2, 2 3 2 64 8 12 2 (x + 1) x +x 0 0 ∫ 1 2 2 π 5 log (x + 1) 2 1 dx = 1 + − log 2 − log2 2 + log3 2 − ζ(3), 2+x 6 2 3 2 x 0 ∫ 1 ∫ 1 ∫ 1 2 2 2 2 π log(1 − x ) log(1 − x ) log (1 − x2 ) dx = − , dx = ζ(3), dx = −2 log 2, x 12 x x2 0 0 0 ∫ 1 ∫ 1 log(1 − x2 ) log(1 − x2 ) 1 G π π π log 2 − G, dx = dx = log 2 + log 2 − − , 2 2 2 4 8 2 2 8 x +1 (x + 1) 0 0 ∫ 1 ∫ 1 2 2 log(x + 1) π log(x + 1) 1 π π+1 G dx = log 2 − G, dx = − + log 2 − , 2+1 2 + 1)2 2 4 8 4 2 x (x 0 0 1

1141

INTEGRALS

∫

∫ 1 log2 2 log 2 + 1 (2 log 2 − 1)π G x log(x2 + 1) log(x2 + 1) dx = dx = , + − , 2+1 2 + 1)2 4 4 8 2 x (x 0 0 ∫ 1 ∫ 1 2 2 2 1 − log 2 π 1 x log(x + 1) log(x + 1) dx = dx = , − log2 2, 2 2 3 4 24 4 (x + 1) x +x 0 0 ∫ 1 ∫ 1 2 2 2 log 2 π log(x + 1) π log 2 G π4 [log(1 − x)] log2 (1 + x) dx = − + − , dx = − , 2 4 96 4 2 x 240 0 (x + 1)(x + 1) 0 ∫ 1 1 1 π2 [log(1 − x)] log(1 + x) dx = ζ(3) + log3 2 − log 2, 1+x 8 3 12 0 ∫ 1 π2 1 [log(1 − x)] log(1 + x) − (log 2)(1 + log 2), dx = 24 2 (1 + x)2 0 ∫ 1 ∫ 1 5 [log(1 − x)] log(1 + x) [log(1 − x)] log(1 + x) π2 dx = − ζ(3), dx = − − log2 2, 2 x 8 12 x 0 0 ∫ 1 ∫ 1 2 2 log(1 − x)[log(2 − x) − 1 − log 2] log 2 π log(x + 1/x) π dx = + , dx = log 2. 2 2+1 2 12 2 (2 − x) x 0 0 1

Source: [107, pp. 37, 206–209], [511, p. 97], [514], [516, p. 63], [1107], [1158, pp. 160, 161], [1217, pp. 240, 241], [1341], [1568, pp. 197, 198], [2013, p. 198], [2106, pp. 54, 55], [2483, 2765, 2946]. Fact 14.6.16. Let a ∈ [−1, 1]. Then, ∫ 1 ∫ 1 (π )2 √ log(1 − a2 x2 ) log(1 − a2 x2 ) 1 2 dx = π log 2 (1 + 1 − a ), dx = − − acos |a| , √ √ 2 0 0 1 − x2 x 1 − x2 ∫ ∫ 1 1 √ 1 log(1 + a2 x2 ) 1 + ax dx = π asin a. dx = π log 12 (1 + 1 + a2 ), log √ √ 2 2 1 − ax 0 x 1−x 0 1−x In particular, ∫ 1 ∫ 1 log(1 − a2 x2 ) π2 log(1 − x2 ) dx = −π log 2, dx = − , √ √ 4 0 0 1 − x2 x 1 − x2 ∫ ∫ 1 1 2 √ 1 π2 log(1 + x ) 1+x dx = . dx = π log 21 (1 + 2), log √ √ 1−x 2 0 x 1 − x2 0 1 − x2 Source: [1217, pp. 562, 563]. Fact 14.6.17. Let a > 0. Then,

∫

∫ 0

1

√ log(a + x) 1 dx = √ (acot a) log(a2 + a), 2 x +a 2 a 0 ∫ 1 √ log(ax + 1) 1 1 a2 x 2 + 1 dx = (atan a) log(a + 1), log dx = − atan2 a, √ 2 ax2 + 1 a2 + 1 2 a 0 1−x √ √ ∫ 1√ 1+ 1+a π1− 1+a π 2 2 + 1 − x log(ax + 1) dx = log . √ 2 2 41+ 1+a 0 1

Source: [1217, pp. 556, 557, 560, 562]. Fact 14.6.18. Let a and b be distinct positive numbers. Then,

∫

0

1

1 log(x + 1) a + b 2 log 2 dx = , log + 2 a(a − b) b (ax + b)2 b − a2

1142

CHAPTER 14

∫

1

0

1 1 log(ax + b) dx = [ (a + b) log(a + b) − b log b − a log 2]. a−b 2 (x + 1)2

Source: [1217, pp. 556, 557]. Fact 14.6.19. Let p ∈ R[s], assume that p has no roots on the unit circle, and let mroots(p) =

{λ1 , . . . , λn }ms . Then, ∫ 1 0

p′ (x) log(1 − x) 1∑ nπ2 dx = . [log2 |1 − λi | + arg2 (1 − λi )] − p(x) 2 i=1 12 n

Source: [1682]. Fact 14.6.20. Let n ≥ 0. Then,

∫

1

log

∑n i=0

x

0

xi

dx =

nπ2 . 6(n + 1)

Source: [2465]. Fact 14.6.21. Let n ≥ 0 and a ∈ [0, 2π]. Then,

∫

0

1

(2π)2n+2 (log2n x) log(x2 − 2x cos a + 1) a dx = (−1)n+1 B2n+2 ( 2π ). x (2n + 2)(2n + 1)

In particular,

∫ ∫

1 0

π2 1 log(x2 − 2x cos a + 1) dx = − (a − π)2 , x 6 2

(log2n x) log(x2 + 1) (2π)2n+2 dx = (−1)n+1 B2n+2 ( 41 ), x (2n + 2)(2n + 1) 0 ) ( ∫ 1 (logn x) log(1 + x) 1 n dx = (−1) n! 1 − n+1 ζ(n + 2), x 2 0 ∫ 1 n (log x) log(1 − x) dx = (−1)n+1 n!ζ(n + 2), x 0 ∫ 1 (logn x) log(1 − x2 ) n! dx = (−1)n+1 n+1 ζ(n + 2), x 2 0 ) ( ∫ 1 logn x 1+x 1 n log dx = (−1) n! 2 − n+1 ζ(n + 2), x 1−x 2 0 ∫ 1 ∫ 1 (log2 x) log(1 − x) π4 (log2 x) log(1 + x) 7π4 dx = − , dx = , x 45 x 360 0 0 ∫ 1 ∫ 1 8π6 (log4 x) log(1 + x) 31π6 (log4 x) log(1 − x) dx = − , dx = , x 315 x 1260 0 0 ∫ 1 ∫ 1 (log2 x) log(1 + x2 ) 7π4 (log4 x) log(1 + x2 ) 31π6 dx = , dx = . x 2880 x 40320 0 0 1

∫

Furthermore,

0

In particular,

∫ 0

1

1

log2n x 4n π2n+1 a dx = − B2n+1 ( 2π ). (2n + 1) sin a x2 − 2x cos a + 1

log x dx = −G, x2 + 1

∫

1 0

log2 x π3 dx = , 2 16 x +1

∫

1 0

5π5 log4 x dx = , 2 64 x +1

1143

INTEGRALS

∫ 0

1

√ 10 3π3 log2 x dx = , 243 x2 − x + 1

If n ≥ 1, then

∫

1 0

Finally,

∫

4

∫

1 0

π2 log2 x dx = . 3 (1 − x)2

x log2n x dx = 4n−1 π2n B2n ( 41 ). (x2 + 1)2 ∫

2

log 2x dx = 0. √ 4x − 2x − x2 Source: [705] and [2106, p. 70]. Related: Fact 13.5.93. √

log x

x2

0

Fact 14.6.22.

∫

1

log[ 21 (1 +

0

1

log log 0

Furthermore, ∫

1

1 dx = −γ, x

log log 1/x dx = x2 + 1

0

0

√ 4x + 1)] π2 dx = , x 15

Fact 14.6.23.

∫

dx =

∫

π/2 π/4

π = log 2

∫

log[ 21 (1 +

1 0

∫

1

0

√ log log 1/x dx = − π(2 log 2 + γ). √ log 1/x

log log tan x dx = √ 2πΓ( 43 ) Γ( 14 )

√ 8x + 1)] π2 1 dx = + log2 2. x 12 2

d πγ Γ(x)L(x) = − + L′ (1) dx 4 x=1

π 4π3 , log 4 Γ4 ( 14 )

=

where L is the Dirichlet L function defined in Fact 13.3.6. Furthermore, ∫ 1 ∫ 1 log log 1/x 1 1 log log 1/x π γ dx = log − , dx = − log2 2, 2 x + 1 2 2 2 2 (x + 1) 0 0 √3 √ √ ( ) ∫ 1 ∫ 1 2 2πΓ( 3 ) 3π 2 3π 5 log log 1/x log log 1/x 1 dx = dx = ) log , log 2π − log Γ( 6 . 2 2 3 3 6 Γ( 13 ) 0 x −x+1 0 x +x+1 If a > 0, then ∫ 1 γ log a xa−1 log log 1/x dx = − − , a a 0

∫

1

xa−1 log 2 log log 1/x dx = − log 2a2 . a+1 x 2a 0 √ √ △ Finally, let a ∈ (2, 6.1 · 1013 ), and define ϕ = 21 ( a + 2 − a − 2). Then, ( ) ∫ 1 log log 1/x 2 log 2π 2π 1 ȷ dx = log ϕ + arg Γ + log ϕ . √ √ 2 2 π 0 x + ax + 1 k2 − 1 k2 − 1 Source: [1217, pp. 534, 535, 570] and [1688, 2783]. Fact 14.6.24.

∫

1

0

∫

1 0

∫ 1 1−x 2 1−x 1 dx = log , dx = − log 2, 2 ) log x (1 + x) log x π 2 (1 + x)(1 + x 0 ∫ 1 x(1 − x) 3 dx = log 2 + log π − 2 log Γ( 41 ), 2 2 0 (1 + x)(1 + x ) log x √ ∫ 1 2 2 2 (1 − x)2 π x (1 − x) dx = log , dx = log , 2 2 ) log x π 4 (1 + x)(1 + x ) log x (1 + x 0

1144

∫

CHAPTER 14 1

0

∫

x(1 − x)2 dx = log 16π2 − 4 log Γ( 14 ), (1 + x2 ) log x

0

1

1 − x2 dx = log 8π2 − 4 log Γ( 41 ). (1 + x2 ) log x

Source: [1217, p. 545]. Fact 14.6.25. Let r ≥ 0, let a ≥ 0, and define △

∫

∞

f (a, r) = 1

logr x dx. xa

If a ∈ [0, 1], then f (a, r) = ∞. If a > 1, then f (a, r) ∈ R. Fact 14.6.26. Let n ≥ 1. Then, ∫ ∞ ∫ ∞ ( π )2n+1 log2n x log2n+1 x |E |, dx = dx = 0, 2n 2 2 x +1 x2 + 1 0 0 In particular,

∫

∞

π3 log2 x dx = , 2 8 x +1

∫

∞

5π5 log4 x dx = , 2 32 x +1 ∫ ∞ log2n x

∫

∞

0

∫

log2n−1 x (1 − 4n )π2n |B2n |. dx = 2 2n 1−x ∞

61π7 log6 x dx = . 2 128 x +1 0 0 0 ∫ 1 log2n x Source: [630, 650] and [1217, p. 550]. Remark: 0 x2 +1 dx = 2 0 x2 +1 dx. See Fact 14.6.21. Fact 14.6.27. Let a > 0, k ≥ 0, and n ≥ 2, and assume that k + 2 ≤ n. Then, ∫ ∞ k a(2k+2)/n π x log x dx = (2 log a − π cot (k+1)π n ). n 2 x +a a2 n2 sin (k+1)π 0 n In particular, ∫ ∞ k ∫ ∞ k √ π log a 2π x log x x log x dx = dx = (2 3 log a − π), , 2 2k+2 2 3k+3 2 4/3 2 2a(k + 1) x +a x +a 27a (k + 1) 0 0 √ ∫ ∞ k 2π x log x dx = (2 log a − π). 4k+4 + a2 3/2 (k + 1)2 x 16a 0 Source: [561] and use Fact 14.2.21. Fact 14.6.28. Let n ≥ 1 and k ≥ 2, and define Qn ∈ R[s] as in Fact 13.2.14. Then,

∫

∞ 0

[ ] ( [ ]) ( )n+1 π(k − 2) logn x π(k − 2) n π sec dx = (−1) Q tan . n k 2k 2k xk + 1

In particular, ∫

∫

∞ 0

∫ ∞ ∫ ∞ log x log2 x π3 log3 x dx = 0, dx = , dx = 0, 8 x2 + 1 x2 + 1 x2 + 1 0 0 0 √ ∫ ∞ ∫ ∞ ∫ ∞ log x 2π2 log2 x 10 3π3 log3 x 14π4 dx = − , dx = , dx = − , 27 243 243 x3 + 1 x3 + 1 x3 + 1 0 0 0 √ √ √ ∫ ∞ ∫ ∞ ∫ ∞ 2π2 log x log2 x 3 2π3 log3 x 11 2π4 dx = − , dx = , dx = − , 4 4 4 16 64 256 x +1 x +1 x +1 0 0 0 √ ( ) ∑  √ i ∫ ∞ n n+1 n √ ( π )n+1 ∑  3  logn x logn x n2 3 π n   2 dx = (−1) E , dx = (−1) En,i .  n,i  3 3 3 4 x3 + 1 x4 + 1 0 i=0 i=0 ∞

Source: [630, 650].

1145

INTEGRALS

Fact 14.6.29. Let a > 0. Then,

∫

0

∫

∞

∫

π log a log x dx = , 2 2 2a x +a

0

∫

∞

∞

π log2 a π3 log2 x dx = + , 2 8a x2 + a2

∞ log a log2 a π2 log x log2 x dx = dx = , + , 2 2 a a 3a 0 (x + a) 0 (x + a) ∫ ∞ ∫ ∞ 2 π(log a − 1) π(log a − 2) log a π3 log x log x dx = , dx = + , 2 2 2 2 2 2 4a3 4a3 16a3 0 (x + a ) 0 (x + a ) ) ) ∫ ∞( ∫ ∞( 2 log x log x 1 log x log2 x 1 π2 − dx = log2 a, − dx = log3 a + log a. x+1 x+a 2 x+1 x+a 3 3 0 0

Source: [1158, p. 151], [1568, p. 166], and [2013, pp. 197, 198]. Fact 14.6.30.

∫

∫

∞ 0

∞

0

∫

log x dx = 0, (x + 1)2

π log x dx = − , 4 (x2 + 1)2

0 ∞

∞

x log x dx = 0, (x2 + 1)2

∫

∞

0 ∞

∫ 0

∫ 0

π2 log2 x dx = , 2 3 (x + 1)

log x π dx = , 16 (x2 + 1)2

∞

2

3

x log2 x π2 , dx = 24 (x2 + 1)2

∫

∫

∞

0 ∞ 0

∫ 0

log3 x dx = 0, (x + 1)2 log3 x 3π3 dx = − , 2 2 16 (x + 1)

∞

x log3 x dx = 0, (x2 + 1)2

∫ ∞ 2 2 ∫ ∞ 2 3 π x log x π3 x log x 3π3 x2 log x dx = , dx = , dx = , 2 2 2 2 2 2 4 16 16 0 (x + 1) 0 (x + 1) 0 (x + 1) ∫ ∞ ∫ ∞ ∫ ∞ log2 x log x π2 log3 x π4 , dx = dx = 0, dx = , 2 2 2 4 8 x −1 x −1 x −1 0 0 0 √ 3 ∫ ∞ ∫ ∞ ∫ ∞ 2 3 2 log x 4π 8 3π 16π4 log x log x dx = dx = − dx = , , , 27 243 243 x3 − 1 x3 − 1 x3 − 1 0 0 0 √ ∫ ∞ ∫ ∞ ∫ ∞ x log3 x x log x x log2 x 8 3π3 16π4 , , dx = 0, dx = dx = 243 243 x3 − 1 x3 − 1 x3 − 1 0 0 0 ∫ ∞ ∫ ∞ ∫ ∞ log x log2 x log3 x 1 π2 π2 , , , dx = − dx = dx = − 3 3 3 2 6 2 0 (x + 1) 0 (x + 1) 0 (x + 1) ∫ ∞ ∫ ∞ ∫ ∞ x log x x log2 x x log3 x 1 π2 π2 , , dx = dx = dx = , 3 3 3 2 6 2 0 (x + 1) 0 (x + 1) 0 (x + 1) ∫ ∞ ∫ ∞ ∫ ∞ 2 3 3 π 3π 3π3 log x log x π log x dx = − dx = dx = − , + , , 2 3 2 3 2 3 4 64 8 16 0 (x + 1) 0 (x + 1) 0 (x + 1) ∫ ∞ ∫ ∫ ∞ ∞ x log x 1 x log2 x π2 x log3 x π2 dx = − , dx = , dx = − , 2 3 2 3 2 3 8 48 32 0 (x + 1) 0 (x + 1) 0 (x + 1) ∫ ∞ ∫ ∞ ∫ ∞ x log2 x π3 π x log x x log3 x dx = 0, dx = dx = 0. − , 2 3 2 3 2 3 64 8 0 (x + 1) 0 (x + 1) 0 (x + 1) ∫

Fact 14.6.31. The following statements hold: i) Let a > 0. Then, ∫ ∞ log x π2 + log2 a dx = . 2(a + 1) 0 (x + a)(x − 1)

1146

CHAPTER 14

ii) Let a ≥ 0. Then,

∫ 0

∞

x2

log x dx = 0. + ax + 1

iii) Let a ∈ (0, 1). Then, ∫ ∞ ∫ ∞ π2 cos aπ 2π2 log x log x dx = dx = . , a a 2 x (x + 1) x (x − 1) 1 − cos 2aπ sin aπ 0 0 iv) Let a ∈ (−1, 1). Then, ∫ ∞ log x aπ2 cos aπ − π sin aπ . dx = xa (x + 1)2 sin2 aπ 0 Source: [1136, pp. 208, 211], [1217, p. 536], and [2106, p. 68]. Fact 14.6.32. If n ≥ 0, then

∫

∞ 0

In particular, ∫ ∞ 0

log2n x dx = |E2n |π2n+1 , √ x(x + 1)

∫

∞ 0

log2n+1 x dx = 0. √ x(x + 1)

∫ ∞ ∫ ∞ log2 x log4 x log6 x 3 5 dx = π , dx = 5π , dx = 61π7 , √ √ √ x(x + 1) x(x + 1) x(x + 1) 0 0 ∫ ∞ ∫ ∞ log8 x log10 x 9 dx = 1385π , dx = 50521π11 . √ √ x(x + 1) x(x + 1) 0 0

Source: [2513, pp. 374, 375]. Fact 14.6.33. Let a ∈ (1, 2). Then,

∫

∞

0

log(x + 1) π = . xa (1 − a) sin aπ

In particular, √ ∫ ∞ ∫ ∞ ∫ ∞ √ √ √ log(x + 1) log(x + 1) log(x + 1) 4 + 2 2π, 2π, dx = 4 dx = 8 dx = 2 3π, 9/8 4/3 5/4 x x x 0 0 0 √ √ ∫ ∞ ∫ ∞ ∫ ∞ √ √ log(x + 1) log(x + 1) 8 log(x + 1) 8 4 − 2 2π, 4 − 2 2π, dx = dx = 2π, dx = 3 5 x11/8 x3/2 x13/8 0 0 0 √ √ ∫ ∞ ∫ ∞ ∫ ∞ √ √ log(x + 1) log(x + 1) log(x + 1) 4 2 8 π, dx = 3π, dx = dx = 4 + 2 2π. 7/4 5/3 15/8 3 7 x x x 0 0 0 Fact 14.6.34. Let a ≥ 0 and b ∈ (1, 3). Then,

∫

∞

log(a2 x2 + 1) ab−1 π bπ dx = sec . 1−b 2 xb

0

In particular, ∫ ∞ √ log(a2 x2 + 1) dx = 2 2aπ, 3/2 x 0

∫ 0

∞

log(a2 x2 + 1) dx = aπ, x2

Fact 14.6.35. Let a ≥ 0 and b > 1. Then,

∫

0

∞

) ( π ab log 1 + b dx = aπ csc . b x

∫

∞ 0

log(a2 x2 + 1) 2√ 3 dx = 2a . 5/2 3 x

1147

INTEGRALS

In particular, √ ) ) ) ( ( ( ∫ ∞ ∫ ∞ ∫ ∞ √ 2 3aπ a3 a4 a2 log 1 + 3 dx = , log 1 + 4 dx = 2aπ. log 1 + 2 dx = aπ, 3 x x x 0 0 0 Fact 14.6.36. Let a > 0. Then,

∫

∞ 0

∫

log(x2 + 1) 2a π + 2 log a, dx = 2 2 (x + a) a +1 a +1

0

∞

) ( a2 (log x) log 1 + 2 dx = aπ(log a − 1). x

Source: [1217, pp. 530, 560]. Fact 14.6.37. Let a > 0 and b > 0. Then,

∫

∞ 0

∫ ∞ ∫ ∞ π π π2 log ax a log x log x dx = dx = dx = log ab, log , , 2 2 2 2 2 2 2 2 2b 2ab b 4ab x +b 0 a +b x 0 a −b x ∫ ∞ ∫ ∞ aa x 2 + a2 x2 + a2 dx = (a − b)π, dx = π(b − a) + π log b , (log x) log 2 log 2 2 2 x +b x +b b 0 0 ( ) ∫ ∞ 2 b [log(a2 + x2 )] log 1 + 2 dx = 2π[(a + b) log(a + b) − a log a − b], x 0 )] ( ) ∫ ∞[ ( 2 2 b a log 1 + 2 log 1 + 2 dx = 2π[(a + b) log(a + b) − a log a − b log b], x x 0 ( ) ( ) ∫ ∞ b2 ab + 1 [log(a2 x2 + 1)] log 1 + 2 dx = 2π log(ab + 1) − b , a x 0 ) )] ( ) ( ∫ ∞[ ( 2 1 b ab + 1 2 log a + 2 log 1 + 2 dx = 2π log(ab + 1) − b log b , a x x 0 ∫ ∞ a−1 ∫ ∞ a−1 Γ( b )Γ( a+1 a x − xb−1 x − xb−1 2 ) dx = log , dx = log 2a b+1 , log x b (x + 1) log x Γ( 2 )Γ( 2 ) 0 0 ∫ ∞ log(a2 x2 + 1) − log(b2 x2 + 1) dx = π(a − b). x2 0

Source: [1217, pp. 530, 545, 560]. Fact 14.6.38. Let a and b be distinct positive numbers. Then,

∫

∫

∞ 0 ∞

0

log2 a − log2 b log x dx = , (x + a)(x + b) 2(a − b)

∫

0

∞

log x π ba dx = log , (x2 + a2 )(x2 + b2 ) 2ab(a2 − b2 ) ab

∫ ∞ x log x log2 a − log2 b x2 log x π aa , dx = , dx = log 2 2 2 2 (x2 + a2 )(x2 + b2 ) 2(a2 − b2 ) 2(a2 − b2 ) bb 0 (x + a )(x + b ) ∫ ∞ ∫ ∞ log(x + 1) 1 a log(x + a) 1 aa dx = log , dx = log , a(a − b) b b(a − b) (ax + b)2 (x + b)2 bb 0 0 ∫ ∞ ∫ ∞ log(ax + b) log(x + 1) 1 a 1 aa . dx = log , dx = log a(a − b) b a−b (ax + b)2 (x + 1)2 bb 0 0

Source: [1136, p. 208] and [1217, pp. 537, 556, 557]. Fact 14.6.39. Let a, b, c, d be positive numbers. Then,

∫

0

Source: [1217, p. 560].

∞

( ) log(a2 x2 + b2 ) π ad dx = log + b . cd c c2 x2 + d2

1148

CHAPTER 14

Fact 14.6.40. Let a, b, c, d be positive numbers. If ad , bc, then

∫

0

Furthermore, ∫

∞ 0

∫

∞

b log b log(ax + b) a(log a − log c + log d) − . dx = 2 c(ad − bc) d(ad − bc) (cx + d)

√ x log(ax + b) a2 (2 log a − log c + log d) b(a cdπ + 2bc log b) dx = + . (cx + d)2 4c(a2 d + b2 c) 4cd(a2 d + b2 c)

Fact 14.6.41. Let a ≥ 0 and b > 0. Then,

0

∞

∫ ∞ π log(ab + 1) [log(ab + 1) − ab/(ab + 1)]π log(a2 x2 + 1) log(a2 x2 + 1) dx = dx = , , 2 2 b x +b (x2 + b2 )2 2b3 0 ∫ ∞ [3(ab + 1)2 log(ab + 1) − ab(4ab + 3)]π log(a2 x2 + 1) dx = , 2 2 3 (x + b ) 8b5 (ab + 1)2 0 ( ) ∫ ∞ log x b 1 atan log(a2 + b2 ), dx = 2 2 2b a 0 (x + a) + b ( )( 2 ) ∫ ∞ 2 b π 1 1 log x 1 2 b 2 2 2 atan − atan + log (a + b ) . dx = 2 2 b a 3 3 a 4 0 (x + a) + b

Source: [511, pp. 259–261] and [2106, pp. 333–339]. Fact 14.6.42.

∫ ∞ log (1 − x2 )4 log (1 − x2 )2 dx = dx = 0, x2 x3 0 1 0 ∫ ∞ ∫ ∞ ∫ ∞ log(x + 1) log(x + 1) log(x + 1) π π2 5π2 log 2, , , dx = G + dx = dx = 2 2 3 4 6 48 x +1 x +x x +x 0 0 0 ∫ ∞ ∫ ∞ ∫ ∞ log(x2 + 1) log3 (x2 + 1) log2 (x2 + 1) π2 , dx = π, dx = dx = 3ζ(3), 2 3 6 x x x3 0 0 0 ∫ ∞ ∫ ∞ ∫ ∞ log(x2 + 1) 5π2 π2 log(x2 + 1) log(x2 + 1) dx = π log 2, dx = dx = , , 24 12 x2 + 1 x2 + x x3 + x 0 0 0 ∫ ∫ ∞ ∞ (2 log 2 − 1)π x log(x2 + 1) 1 log(x2 + 1) dx = , dx = , 2 2 2 2 4 2 (x + 1) (x + 1) 0 0 ∫ ∞ 2 ∫ ∞ x log(x2 + 1) log(x2 + 1) (2 log 2 + 1)π (12 log 2 − 7)π , , dx = dx = 4 32 (x2 + 1)2 (x2 + 1)3 0 0 ∫ ∞ ∫ ∞ 2 x log(x2 + 1) 1 x log(x2 + 1) (4 log 2 − 1)π dx = , dx = , 2 3 2 + 1)3 8 32 (x + 1) (x 0 0 ∫ ∞ 3 ∫ ∞ 4 x log(x2 + 1) 3 x log(x2 + 1) 3(4 log 2 + 3)π dx = , dx = , 2 + 1)3 2 + 1)3 8 32 (x (x 0 0 ∫ ∫ ∞ ∞ x log(x2 + 1) log(x2 + 1) (60 log 2 − 37)π 1 , , dx = dx = 2 4 2 4 192 18 (x + 1) (x + 1) 0 0 ∫ ∞ 2 ∫ ∞ 3 x log(x2 + 1) (12 log 2 − 5)π x log(x2 + 1) 5 dx = , dx = , 2 4 2 4 192 72 (x + 1) (x + 1) 0 0 ∫ ∞ 4 ∫ ∞ 5 x log(x2 + 1) (12 log 2 − 1)π x log(x2 + 1) 11 dx = , dx = , 2 4 192 36 (x + 1) (x2 + 1)4 0 0 ∫

∞

log (1 − x2 )2 dx = x2

∫

∞

1149

INTEGRALS

∫ 0

∞

∫

x6 log(x2 + 1) 5(12 log 2 + 11)π , dx = 192 (x2 + 1)4 ∞ 0

∫ 0

∫

1 π log 2 x , dx = − log 2 2 2 2 (x + 1) x +1

If n ≥ 1, then

∫

∞

0

) ( 2 1 log2 (x2 + 1) π 2 − + log 2 − log 2 π, dx = 12 2 (x2 + 1)2

∞

∞

0

1 π3 x 2 + π log2 2. dx = log 48 (x2 + 1)2 x2 + 1

1 ( x )2n+1 x Hn − H2n − 1/(2n) ( ) . log 2 dx = x2 x2 + 1 x +1 2n 2n n

Source: [510] and [511, pp. 259–261]. Fact 14.6.43. Let a > 0. Then,

 1 1    + , a , 1,   1  1 − a log a dx =   2 2  1  0 (x + a)(log x + π )   , a = 1. 2 Source: [2106, p. 184]. Remark: lima→1 [1/(1 − a) + 1/ log a] = 1/2. Fact 14.6.44. If a ∈ (1, 2), then ∫ ∞ ∫ ∞ log |x − 1| π log(x + 1) π dx = cot aπ, dx = csc aπ. a a x a−1 x 1−a 0 0 ∫

∞

Source: [1217, p. 558]. Fact 14.6.45. Let n ≥ 1. Then,

∫

∞

log2n

0

In particular, ∫ ∞

|x − 1| dx = 2|(1 − 4n )B2n |π2n . x+1

∫ ∞ ∫ ∞ |x − 1| |x − 1| |x − 1| dx = π2 , log4 dx = π4 , log6 dx = 3π6 , x+1 x+1 x+1 0 0 0 ∫ ∞ ∫ ∞ ∫ ∞ |x − 1| |x − 1| |x − 1| log8 dx = 17π8 , log10 dx = 155π10 , log12 dx = 2073π12 . x+1 x+1 x+1 0 0 0

Furthermore,

log2

∫

∞

|x − 1| dx = −21ζ(3), x+1 0 ∫ ∞ 1 |x − 1| dx = − 40005ζ(7), log7 x + 1 2 0 log3

∫

∞

|x − 1| dx = −465ζ(5), x+1 0 ∫ ∞ |x − 1| log9 dx = −1448685ζ(9). x+1 0 log5

Source: [2106, p. 275]. Remark: B2n is a Bernoulli number, and 2|(1 − 4n )B2n | is the nth unsigned Genocchi number of even index. Related: Fact 13.1.7. Fact 14.6.46. If a > 0, then

∫

a+1

∫

log Γ(x + a) dx =

a

Furthermore,

1

log Γ(x + a) dx =

0

∫ 0

1

∫

1

log Γ(x) dx = 0

1 log 2π + a log a − a. 2

log Γ(1 − x) dx =

1 log 2π. 2

1150

CHAPTER 14

If n ≥ 1, then ∫ 1 1 [log Γ(x)] sin 2πnx dx = (log 2πn + G), 2πn 0 If n ≥ 0, then ∫

1 0

∫

1

[log Γ(x)] cos 2πnx dx =

0

1 . 4n

  n  π ∑ 2 1  1  + [log Γ(x)] sin (2n + 1)πx dx = log + . (2n + 1)π  2 i=1 2i − 1 2n + 1 

Source: [1103, p. 27] and [1217, p. 656].

14.7 Facts on Definite Integrals of Logarithmic, Trigonometric, and Hyperbolic Functions Fact 14.7.1. For all n ≥ 0,

∫

π/4

log2n tan x dx =

0

In particular, ∫ π/4 0

log2 tan x dx =

π3 , 16

∫

π/4 0

π2n+1 |E2n |. 4n+1

log4 tan x dx =

5π5 , 64

∫

π/4 0

log6 tan x dx =

61π7 . 256

Furthermore, ∫ π/4 ∫ π/4 ∫ π/4 G π G π log cos x dx = − log 2, log tan x dx = −G, log sin x dx = − − log 2, 2 4 2 4 0 0 0 ∫

∫ π/4 Gπ π2 35 Gπ π2 21 x log sin x dx = − − log 2 + ζ(3), x log cos x dx = − log 2 − ζ(3), 8 32 128 8 32 128 0 0 ∫ π/4 Gπ x log tan x dx = 7ζ(3) − , 4 0 ∫ π/4 ∫ π/4 π π log(1 − tan x) dx = log 2 − G, log(1 + tan x) dx = log 2, 8 8 0 0 ∫ π/4 2 π Gπ 21 x log(1 + tan x) dx = log 2 − + ζ(3), 64 8 64 0 ∫ π/4 2 Gπ 7 π log 2 − − ζ(3), x log(1 − tan x) dx = 64 8 64 0 ∫ π/4 ∫ π/4 π π log(1 + tan2 x) dx = log 2 − G, log(1 − tan2 x) dx = log 2 − G, 2 4 0 0 ∫ π/4 2 π Gπ 21 x log(1 + tan2 x) dx = log 2 − + ζ(3), 16 4 64 0 ∫ π/4 2 Gπ 7 π log 2 − + ζ(3), x log(1 − tan2 x) dx = 32 4 32 0 ∫ π/4 ∫ π/4 π π log(cot x + 1) dx = log 2 + G, log(cot x − 1) dx = log 2, 8 8 0 0 ∫ π/4 ∫ π/4 2 2 π Gπ 7 π Gπ 35 x log(cot x+1) dx = log 2− − ζ(3), x log(cot x−1) dx = log 2+ − ζ(3), 64 8 64 64 8 64 0 0 π/4

1151

INTEGRALS

∫ ∫

π/4 0

G π log(cos x + sin x) dx = − log 2, 2 8

∫

π/4 0

log(cos x − sin x) dx = −

G π − log 2, 2 8

∫ π/4 21 π2 π2 35 x log(cos x+sin x) dx = ζ(3)− log 2, ζ(3)− log 2, x log(cos x−sin x) dx = − 128 64 128 64 0 0 ∫ π/4 ∫ π/4 2 7 π π log 2 − ζ(3), x log(tan x + cot x) dx = log(tan x + cot x) dx = log 2, 2 16 64 0 0 ∫ π/4 ∫ π/4 7 π2 π log 2 − ζ(3), x log(cot x − tan x) dx = log(cot x − tan x) dx = log 2, 4 32 32 0 0 ∫ π/4 ∫ π/4 √ √ √ √ π π G G log( tan x + cot x) dx = log 2 + , log( tan x + cot x) dx = log 2 + , 8 2 8 2 0 0 ∫ π/4 ∫ π/4 2 √ √ √ √ 21 π π log 2− ζ(3). x log ( tan x− cot x)2 dx = log ( tan x− cot x)2 dx = log 2−G, 4 32 32 0 0 π/4

Fact 14.7.2.

∫

π/3 0

∫ π/3 ( x) 7π3 x) 17π4 log 2 sin dx = , x log2 2 sin dx = , 2 108 2 6480 0 ∫ π/3 ( ( π) x) 2 x− log 2 sin dx = ζ(3). 3 2 3 0 2

(

Source: [2513, p. 581] and [3022]. Fact 14.7.3.

∫

∫ π/2 π log sin x dx = log cos x dx = − log 2, log tan x dx = 0, 2 0 0 0 ∫ π/2 ∫ π/2 7 π2 3π π3 x log sin x dx = ζ(3) − log 2, x2 log sin x dx = ζ(3) − log 2, 16 8 16 24 0 0 ∫ π/2 (sin x) log sin x dx = log 2 − 1, π/2

∫

0

π/2

∫ π/2 π2 π π3 7 log 2, x2 log cos x dx = − ζ(3) − log 2, x log cos x dx = − ζ(3) − 16 8 4 24 0 0 ∫ π/2 ∫ π/2 ∫ π/2 7 7π log sec x π2 x log tan x dx = ζ(3), x2 log tan x dx = ζ(3), dx = , 8 16 tan x 24 0 0 0 ∫ π/2 ∫ π/2 ∫ π/2 1 log3 sec x π4 log4 sec x 3 log2 sec x dx = ζ(3), dx = , dx = ζ(5), tan x 4 tan x 240 tan x 4 0 0 0 ∫ π/2 ∫ π/2 π π log(1 + sin x) dx = 2G − log 2, log(1 − sin x) dx = −2G − log 2, 2 2 0 0 ∫ π/2 ∫ π/2 2 π π Gπ 7 log(1 + tan x) dx = log 2 + G, x log(1 + tan x) dx = log 2 + + ζ(3), 4 16 4 16 0 0 ∫ π/2 ∫ π/2 π Gπ π2 log(cos x + sin x) dx = G − log 2, x log(cos x + sin x) dx = − log 2, 4 4 16 0 0 ∫ π/2 ∫ π/2 π2 log(tan x + cot x) dx = π log 2, x log(tan x + cot x) dx = log 2, 4 0 0 ∫

π/2

1152

CHAPTER 14

∫

π/2

π π3 log 2 + ζ(3), 12 16 0 ∫ π/2 ∫ π/2 π4 3π 1 − cos x x3 log(tan x + cot x) dx = log 2 + ζ(3), dx = −4G, log 32 64 1 + cos x 0 0 ∫ π/2 ∫ π/2 π2 log 2, log (tan x − cot x)2 dx = π log 2, x log (tan x − cot x)2 dx = 4 0 0 ∫ π/2 ∫ π/2 √ √ √ √ π π2 Gπ log 2 + , log( tan x + cot x) dx = log( tan x + cot x) dx = log 2 + G, 4 16 4 0 0 ∫ π/2 ∫ π/2 √ √ √ √ π π2 Gπ log 2 − . log ( tan x − cot x)2 dx = log 2 − 2G, x log ( tan x − cot x)2 dx = 2 8 4 0 0 x2 log(tan x + cot x) dx =

Furthermore, the following statements hold: i) Let n ≥ 0. Then, ∫ π/2

log2n+1 tan x dx = 0.

0

ii) Let a > 0. Then, ∫ π/2 ∫ log(1 + a sin2 x) dx = 0

π/2 0

iii) Let a > 0. Then, ∫ π/2 0

∫

√ log(1 + a cos2 x) dx = π log 21 ( a + 1 + 1). π/2

log (a sin x) dx = 2

log2 (a cos x) dx =

0

∫

π/2

[log(a sin x)] log(a cos x) dx =

0

iv) Let a ∈ [−1, 1]. Then,

π3 π 2 + log2 , 24 2 a

2 π3 π log2 − . 2 a 48

log(a2 − sin2 x)2 dx = −2π log 2.

v) Let a and b be real numbers such that 0 < b < a. Then, ∫ π/2 a + b sin x π b (csc x) log dx = asin . a − b sin x 2 a 0 vi) Let a and b be positive numbers. Then, ∫ π/2 (a2 cos2 x + b2 sin2 x) dx = π log 12 (a + b). 0

vii) Let a and b be positive numbers. Then, ∫ π/2 (a2 + b2 tan2 x) dx = π log(a + b). 0

Source: [441], [511, pp. 242, 260], [1890], [2513, pp. 570, 581], and [3022]. i) is given in [1217,

p. 533]; ii) given in [1217, p. 532] and [2013, pp. 70, 71]; iii) is given in [2106, p. 236]; iv) is given in [1217, p. 532]; v) is given in [701]; vi) is given in [1217, p. 532]; vii) is given in [1217, p. 534]. Fact 14.7.4.

∫

π 0

x log sin dx = 2

∫ 0

π

∫

π

log sin x dx = −π log 2, 0

x log sin x dx = −

π2 log 2, 2

1153

INTEGRALS

∫

π

log2 sin x dx =

0

∫

π

∫

π3 + π log2 2, 12

π

log3 sin x dx = −

0

π3 3π ζ(3) − π log3 2 − log 2, 2 4

π3 19π5 log2 2 + , 2 240 0 ∫ π ∫ π ( ( π3 x) x) dx = 0, dx = , log2 2 sin log 2 sin 2 2 12 0 0 ∫ π ∫ π ( ( x) 3π 19π5 x) log3 2 sin dx = − ζ(3), dx = , log4 2 sin 2 2 2 240 0 0 ∫ π ∫ π log(1 + sin x) dx = 4G − π log 2, log(1 − sin x) dx = −4G − π log 2, ∫ 0

∫ 0

log4 sin x dx = 6πζ(3) log 2 + π log4 2 +

0 π

0

π π π2 x log(1 + sin x) dx = 2πG − log 2, log 2, x log(1 − sin x) dx = −2πG − 2 2 0 ∫ π ∫ π log(1 + cos x) dx = log(1 − cos x) dx = −π log 2, 0

π

∫

2

0

π 7 log 2, = x log(1 + cos x) dx = − ζ(3) − 2 2 ∫ π x log cos2 x dx = −π2 log 2, 2

∫

π

x log(1 − cos x) dx =

0

∫

0

π

7 π2 ζ(3) − log 2, 2 2

x log tan2 x dx = 0,

0

∫ π ( ( x) x) 11π5 2 2 (π − x)2 log2 2 sin x log 2 cos dx = dx = , 2 2 180 0 0 ∫ π/2 ∫ π/2 log cos x dx = −π log 2, log(1 − cos x) dx = −π log 2 − 4G. ∫

π

−π/2

−π/2

Furthermore, the following statements hold: i) Let a ∈ R. If |a| ≤ 1, then ∫ π log(a2 + 1 − 2a cos x) dx = 0. 0

If |a| ≥ 1, then

∫

π

log(a2 + 1 − 2a cos x) dx = 2π log |a|.

0

ii) Let a, b be real numbers such that 0 < |b| ≤ a. Then, ∫ π √ log(a + b cos x) dx = π log 12 (a + a2 − b2 ). 0

iii) Let a, b be positive numbers. Then, ∫ π log(a2 + b2 − 2ab cos x) dx = 2π log max {a, b}. 0

iv) Let a be a real number, and assume that |a| ≥ 1. Then, ∫ π a2 π log |1 + a cos x| dx = log . 2 4 0 In particular,

∫

π 0

∫

π

log |1 ± 2 cos x| dx = 0, 0

log |1 ± 3 cos x| dx =

π 9 log . 2 4

1154

CHAPTER 14

Furthermore, ∫

2π/3

∫

π

log(1 + 2 cos x) dx = −

0

log |1 + 2 cos x| dx ≈ 0.676628.

2π/3

v) Let a ∈ [1, ∞). Then, ∫ π √ √ (1 + cos x) log(a + cos x) dx = [a − a2 − 1 + log 21 (a + a2 − 1)]π. 0

vi) Let a ∈ (−1, 1). Then,

∫

π

(sec x) log(1 + a cos x) dx = π asin a.

0

vii) Let a ∈ [−1, 1]. Then, ∫

π

0

log (1 + a cos x)2 dx = 2π log 12 (1 +

√

1 − a2 ).

viii) Let a ≥ −1. Then, ∫ π ∫ π √ 2 log (1 + a sin x) dx = log (1 + a cos2 x) dx = 2π log 21 (1 + a + 1), 0 0 ∫ π ∫ π √ x log (1 + a sin2 x) dx = x log (1 + a cos2 x) dx = π2 log 12 (1 + a + 1). 0

0

ix) Let a and b be positive numbers. Then, ∫ π (a2 cos2 x + b2 sin2 x) dx = 2π log 21 (a + b). 0

x) Let a and b be positive numbers. Then, ∫ π (a2 + b2 tan2 x) dx = 2π log(a + b). 0

Source: [441], [516, p. 58], [1101], [1568, p. 167], [2013, pp. 185, 186], and [2513, p. 193].

i)–iii) are given in [1158, pp. 155, 542, 543] and [1217, p. 531]. iv) corrects a misprint in [1217, p. 531] and [1524, p. 274]; v) is given in [2962]; vi) is given in [2106, p. 97]; vii)–x) are given in [1217, pp. 531–534]. Fact 14.7.5.

∫ 0

∞

log cos2 x dx = −π, x2

Source: [1217, p. 569] and [1685]. Fact 14.7.6.

∫

∞ 0

∫

( )2 1 1 + sin x dx = π2 , log x 1 − sin x

∞

log cos2 x 1 dx = − log2 2. 2 1 + e2x

0

∫ 0

∞

( )2 π2 1 1 + tan x dx = . log x 1 − tan x 2

Let a > 0 and b > 0. Then, ∫ ∞ ∫ ∞ log sin2 ax 1 − e−2ab log cos2 ax 1 + e−2ab π π log , , dx = dx = log 2 2 2 2 b 2 b 2 x +b x +b 0 0 ∫ ∞ ∫ ∞ π x sin x π log tan2 ax dx = tanh ab, dx = 2a+b , 2 + b2 2 + b2 ) b x (cosh 2a − cos x)(x e −1 0 0

1155

INTEGRALS

∫

∞ 0

π(e2a + e−2b ) 1 dx = . 2 b(sinh 2a)(e2a − e−2b ) (sin x + sinh a)(x2 + b2 ) 2

Source: [1217, pp. 568, 569] and [1684]. Fact 14.7.7. Let a > 0. Then,

∫

0

∞

∫ ∞ (log x) sin ax aπ π (log x) sin2 ax dx = − (γ + log 2a − 1), dx = − (γ + log a), 2 x 2 2 x 0 ∫ ∞ πγ2 π3 π (log2 x) sin ax dx = + + πγ log a + log2 a. x 2 24 2 0

Source: [1217, p. 594]. Fact 14.7.8. Let a > 0 and b > 0. Then,

∫

∫

∞ 0

∞

a (log x)(cos ax − cos bx) dx = (γ + 21 log ab) log , x b 0 (log x)(cos ax − cos bx) π dx = [(a − b)(γ − 1) + a log a − b log b]. 2 x2

Source: [1217, p. 594]. Fact 14.7.9. Let a > 0 and b > 0. Then,

∫

∫ 0

( ) b (log x)(sin bx) 1 b 2 2 a atan − bγ − log(a + b ) , dx = 2 eax a 2 a + b2 ( ) (log x)(cos bx) b 1 a 2 2 b atan + aγ + log(a + b ) . dx = − 2 eax a 2 a + b2

∞ 0 ∞

Source: [1217, p. 594]. Fact 14.7.10. Let a > 0. Then,

∫

√

π) 2π ( log 4a + γ − , a 2 0 √ ∫ ∞ 1 2π ( π) (log x) cos ax2 dx = − log 4a + γ + , 8 a 2 0 ∫ ∞ √ 1 (log x) sin ax3 dx = − √3 (6γ − 2 3π + 9 log 3 + 6 log a)Γ( 31 ), 108 a 0 ∫ ∞ √ √ 1 (log x) cos ax3 dx = − √3 [2π + 2 3(a + γ) + 3 3 log 3]Γ( 13 ), 36 a 0 √  √ √ ∫ ∞   √ 1  2 − 2  4  (log x) sin x dx = π − 2 − 2(γ + 3 log 2) Γ( 14 ),   32 2 0 √ (√ ) ∫ ∞ √ √ 1 (log x) cos x4 dx = − 2 − 2π + 2 + 2(2γ + π + 6 log 2) Γ( 41 ). 64 0 ∞

1 (log x) sin ax dx = − 8 2

Source: [1217, pp. 593]. The second equality corrects a misprint. Fact 14.7.11. Let n ≥ 1. Then,

∫

0

∞

logn tanh x dx = (−1)n n!(1 − 2−n−1 )ζ(n + 1).

1156

CHAPTER 14

In particular, ∫ ∞ π2 log tanh x dx = − , 8 0 ∫ 0

Fact 14.7.12. ∞

log x π 4π3 , dx = log cosh x 2 Γ4 ( 41 )

∫

∞

log2 tanh x dx =

0

∫ 0

∞

∫

7 ζ(3), 4

π log x dx = log − γ, 4 cosh2 x

∫ 0

∞

log3 tanh x dx = −

0 ∞

π4 . 16

π 2G 4π3 log x dx = log − . 3 4 π Γ4 ( 41 ) cosh x

Fact 14.7.13. Let p, q ∈ R[s], where p(s) = bn s + · · · + b1 s + b0 and q(s) = an sn + · · · + a1 s + a0 , n

assume that deg p ≤ deg q = n, assume that an + bn , 0 and p + q is asymptotically stable, and △ define {λ1 , . . . , λr }ms = mroots(q) ∩ ORHP . Then,   ∫ ∞ r  bn an−1 − an bn−1 ∑  q(ω ȷ)  dω = π  + λi  . log q(ω ȷ) + p(ω ȷ) 2an (an + bn ) 0 i=1 In particular, if deg p ≤ (deg q) − 1, then   ∫ ∞ r  −bn−1 ∑  q(ω ȷ)  dω = π  log + λi  , q(ω ȷ) + p(ω ȷ) 2an 0 i=1 and, if deg p ≤ (deg q) − 2, then ∫ 0

∞

log

r ∑ q(ω ȷ) dω = π λi . q(ω ȷ) + p(ω ȷ) i=1

Source: [931, pp. 99–101], [1076], and [2422, pp. 54–56]. The cases deg q − deg p ∈ {0, 1} are considered in [2422, pp. 54–56]. Example: For all α > 0, ∫ ∞ ∫ ∞ (ω ȷ + 1)2 (ω ȷ + 2)(ω ȷ − 1) log dω = 0, log (ω ȷ + 2)(ω ȷ − 1) + 2 + α dω = π, 2 + α (ω ȷ + 1) 0 0 ∫ ∞ (ω ȷ + 3)(ω ȷ − 1)2 log dω = 2π. 2 (ω ȷ + 3)(ω ȷ − 1) + (8 + α)ω ȷ 0 Remark: This integral is the Bode sensitivity integral, which shows that the presence of open-right△

half-plane poles in the loop transfer function L = p/q ∈ R(s) of an asymptotically stable servo loop △ with reference-to-error sensitivity transfer function S = 1/(1 + L) = q/(q + p) ∈ R(s) limits the achievable control-system performance. Fact 14.7.14. Let L ∈ R(s), assume that L is strictly proper, and let ω0 ∈ (0, ∞). Then, ∫ 2ω0 ∞ log |L(ω ȷ)| − log |L(ω0 ȷ)| dω arg L(ω0 ȷ) = π 0 ω2 − ω20 ∫ 2 ∞ log |L(ω0 e x ȷ)| − log |L(ω0 ȷ)| = dx π 0 sinh x ) ∫ ( 2 ∞ d x = log |L(ω0 e x ȷ)| log coth dx. π 0 dx 2 Source: [931, pp. 109–113]. Example: Let L(s) = 1/s and ω0 > 0. Then,

arg

∫ ∫ ∫ ω 1 2ω0 ∞ log ω0 2 ∞ log x 2 ∞ x = dω = − dx = − dx ω0 ȷ π 0 ω2 − ω20 π 0 x2 − 1 π 0 sinh x ) ∫ ( ∫ x 2 ∞ d x 2 ∞ π x log coth dx = − . =− log ω0 e log coth dx = − π 0 dx 2 π 0 2 2

1157

INTEGRALS

Remark: This is the Bode phase integral, which relates the phase angle of the loop transfer function

of a servo loop to the slope of the magnitude of the loop transfer function.

14.8 Facts on Definite Integrals of Exponential Functions Fact 14.8.1. Let n ≥ 0. Then,

∫

1

xn e−x dx =

0

i=0

In particular, ∫ 1 ∞ ∑ e−x dx = (−1)i 0

∫

1

1

i=0 ∞ ∑

∞ ∑ x4 e−x dx = (−1)i

0

i=0

an 1 = n! − . (i + n + 1)i! e ∫

1 1 =1− , (i + 1)i! e

1

xe−x dx =

∞ ∑ (−1)i

0

∫

1 5 x2 e−x dx = (−1)i =2− , (i + 3)i! e i=0

0

∫

∞ ∑ (−1)i

x3 e−x dx =

i=0 ∞ ∑

(−1)i

0

i=0

1

∞ ∑

∫

65 1 = 24 − , (i + 5)i! e

1

x5 e−x dx =

0

2 1 =1− , (i + 2)i! e

(−1)i

i=0

16 1 =6− , (i + 4)i! e

326 1 = 120 − . (i + 5)i! e

Source: [116, p. 7]. Remark: an is the nth arrangement number. See Fact 1.18.1. Related: Fact

14.8.9. Fact 14.8.2. Let n ≥ 1. Then,

∫

∞

[1 − (1 − e−x )n ] dx = Hn .

0

Fact 14.8.3. Let n ≥ 0. Then,

∫

π

e x sin nx dx =

0

∫

n [(−1)n+1 eπ + 1], n2 + 1

π

e x cos nx dx =

0

1 [(−1)n eπ − 1]. n2 + 1

Source: [821]. Fact 14.8.4. Let a ∈ R and n ≥ 0. Then,

∫

π

eax sin2n x dx =

0

∫

π

eax sin2n+1 x dx =

0

In particular,

∫

π

(2n)!(eaπ − 1) , ∏n−1 2 a i=0 [a + 4(n − i)2 ]

(2n + 1)!(eaπ + 1) . ∏ 2 2 (1 + a2 ) n−1 i=0 [a + (2n − 2i + 1) ]

eaπ − 1 e dx = , a 0 ∫ π 2(eaπ − 1) eax sin2 x dx = , a(a2 + 4) 0 ax

∫

π

eaπ + 1 , a2 + 1 0 ∫ π 6(eaπ + 1) eax sin3 x dx = 4 . a + 10a2 + 9 0 eax sin x dx =

Source: [821]. Fact 14.8.5. Let x ∈ Cn and y ∈ Cn . Then,

∫

π

−π

e−t ȷ ∥x + et ȷ y∥22 dt = 2πx∗ y.

Remark: This is the integral analogue of x) of Fact 11.8.3.

1158

CHAPTER 14

Fact 14.8.6. ∞ ∑

1 2e = √ 1 π Γ(i + 2 )

i=1

∫

1

e−x dx, 2

0

∞ ∑ (−1)i i=0

1 2 = √ 3 πe Γ(i + 2 )

∫

1

2

e x dx. 0

Source: [1904]. Fact 14.8.7.

∫

∫ 1 ∞ ∞ ∑ ∑ (−1)i (−1)i+1 ≈ −0.246115, ≈ 0.963281, cos(x log x) dx = (2i)2i (2i − 1)2i−1 0 0 i=1 i=1 ∫ 1 ∫ 1 ∞ ∞ ∑ ∑ 1 1 sinh(x log x) dx = − ≈ −0.253927, ≈ 1.03735. cosh(x log x) dx = 2i (2i) (2i − 1)2i−1 0 0 i=1 i=1 1

sin(x log x) dx =

Source: [1179]. Related: Fact 14.2.2. Fact 14.8.8. Let n ≥ 0 and a > 0. Then,

∫

∞ 0

In particular,

∫ 0

∞

1 1 dx = , x a log a ∫ ∞ 1 1 dx = , ax e a 0

∫

∞ 0

∫

∞ 0

∞

n −x

x e

∞

0

n! xn dx = n+1 . ax e a

x 1 dx = , x a log2 a x 1 dx = 2 , eax a

Fact 14.8.9. Let n ≥ 1. Then,

∫

∫

n! xn dx = , x a logn+1 a

∫

∞

dx = n!,

0

∫

∞ 0

∫

∞ 0

xn e−x dx =

1

2 x2 dx = , x a log3 a

2 x2 dx = 3 . eax a ⌊en!⌋ . e

Source: [1345]. Related: Fact 14.8.1. Fact 14.8.10. Let a ∈ (0, ∞) and b ∈ (−1, ∞). Then,

∫

∞

xb e−ax dx =

0

In particular,

∫

∞ 0

e−ax √ dx = x

Fact 14.8.11. Let a > 0. Then,

√

π , a

∫

∞ 0

Γ(b + 1) . ab+1 √

xe

−ax

1 dx = 2

√

π . a3

∫ ∞ x π 2π log 2 1 dx = , dx = , √ √ ax ax a a2 0 0 e −1 e −1 ∫ ∞ ∫ ∞ x2 x3 π3 + 12π log2 2 12πζ(3) + 8π log3 2 + 2π3 log 2 , . dx = dx = √ √ 3a3 a4 0 0 eax − 1 eax − 1 Fact 14.8.12. Let n ≥ 0. Then,   ∫ ∞ n ∑ (2n)!π  2  xe−nx dx = n Hn + 2 log 2 − . √ 2i − 1  4 (n!)2  0 ex − 1 i=1 ∫

∞

1159

INTEGRALS

In particular,

∫

∞

π xe−x dx = π log 2 − , √ 2 ex − 1

0

∫

In addition, ∫

∞

∞

7π xe−2x 3π log 2 − . dx = √ 4 16 ex − 1

0

x2 e−x π3 dx = − π + 2π log2 2 − 2π log 2, √ 6 ex − 1

0 ∞

∫

3 −x

x e dx = π[6ζ(3) + 4 log3 2 − 6 log2 2 − 6 log 2 − 3] + π3 (log 2 − 12 ). √ x 0 e −1 Fact 14.8.13. Let a > 0 and b ≥ −1. Then, ∫ ∞ xb (2b − 1)Γ(b + 1)ζ(b + 1) dx = . ax 2b ab+1 0 e +1 ∫

In particular,

0 ∞

∫ ∫

0 ∞

∫ 0

log 2 1 dx = , ax e +1 a

x2 3ζ(3) , dx = ax e +1 2a3

0 ∞

0 ∞

∫

∞

4

45ζ(5) x dx = , +1 2a5

eax

2835ζ(7) x6 dx = , eax + 1 4a7

x8 80325ζ(9) , dx = ax e +1 2a9

∫

∞

∫

0 ∞

π2 x , dx = eax + 1 12a2 x3 7π4 , dx = +1 120a4

eax

0

∫

∞

31π6 x5 dx = , +1 252a6

eax

0

∫

∞

127π8 x7 dx = , eax + 1 240a8

∞

x9 511π10 . dx = eax + 1 132a10

0

∫

0

Source: [511, p. 98]. Remark: limb→0 (2 − 1)Γ(b + 1)ζ(b + 1) = log 2. Fact 14.8.14. Let n ≥ 0. Then, b

∫

0

∞

  n  π2 ∑  xe−nx n i1  dx = (−1)  + (−1) 2  . x e +1 12 i=1 i

Furthermore, if n ≥ 1, then ∫ ∞ ∫ 0

∞

  n ∑  π2  xe−x 4 2  , dx = n  + Hn,2 − 2 x/(2n) 3 (2i − 1)  +1 0 e i=1  n  ∑ xe−x π2  4 1 2  dx = (n − 2 )  − Hn−1,2 −  . 3 (2i − 1)2 e x/(2n−1) + 1 i=1

In particular, ∫ ∞ ∫ ∞ ∫ ∞ π2 xe−x π2 xe−x 31 3π2 xe−x dx = 1 − , dx = − 3, dx = − , x x/2 x/3 12 3 4 4 +1 +1 0 e 0 e 0 e +1 ∫ ∞ ∫ ∞ ∫ ∞ xe−x xe−x xe−x 4π2 115 3019 25π2 2919 dx = − , − , . dx = dx = 3π2 − x/4 + 1 x/5 + 1 x/6 + 1 3 9 144 12 100 e e e 0 0 0 Furthermore,

∫ 0

∞

xe−x dx = 1 − G. e2x + 1

1160

CHAPTER 14

Source: [1217, p. 354]. Fact 14.8.15. Let a > 0 and b > 0. Then,

∫

∞

Γ(b + 1)ζ(b + 1) xb dx = . −1 ab+1

eax

0

∫

In particular,

∞

0 ∞

∫ 0

Hence, if n ≥ 1, then

x π2 dx = 2 , ax e −1 6a π x dx = , −1 15a4 4

3

eax

∫

∫ ∫

∞

x2 2ζ(3) , dx = −1 a3

eax

0 ∞ 0

24ζ(5) x4 dx = . −1 a5

eax

( )2n x2n−1 B2n n−1 2π dx = (−1) . eax − 1 a 4n

∞ 0

Furthermore, if m ≥ 1 and n ≥ 0, then ∫ ∞ m −nx x e dx = m![ζ(m + 1) − Hn,m+1 ]. ex − 1 0 In particular, ∫

∞

0

∫

∞ 0

∫

x2 dx = 2ζ(3), ex − 1 ∞ 0

If a > 0, then ∫ ∞ 0

x π2 dx = , x e −1 6

π4 x3 dx = , ex − 1 15 x

2eax

−1

dx =

∫

∞

0 ∞

∫ 0

∫

∞ 0

xe−x π2 dx = − 1, x e −1 6

x2 e−x dx = 2ζ(3) − 2, ex − 1 x3 e−x π4 dx = − 6, ex − 1 15

π2 − 6 log2 2 , 12a2

∫ 0

∞

∫

∞

0

∫ ∫ 0

0 ∞

xe−2x π2 5 dx = − , x e −1 6 4

∞

x2 e−2x 9 dx = 2ζ(3) − , ex − 1 4

x3 e−2x π4 51 dx = − . ex − 1 15 8

x2 21ζ(3) + 4 log3 2 − π2 log 4 dx = . −1 12a3

2eax

Furthermore, ∫ ∞ ∫ ∞ 2 −x ∫ ∞ xe−x x e xe−x π2 π2 G 7 − 1, + − 1, dx = dx = dx = ζ(3) − 2, 2x 4x 2x 8 16 2 4 0 e −1 0 e −1 0 e −1 √ 3 ∫ ∞ 2 −x ∫ ∞ 2 −x 3 x e 26 x e 7 4 3π π + ζ(3) − 2, + ζ(3) − 2, dx = dx = 3x − 1 4x − 1 243 27 32 8 e e 0 0 ∫ ∞ 4 −x ∫ ∞ 3 −x 4 x e π x e dx = − 6, dx = 24ζ(5) − 24, x−1 2x − 1 16 e e 0 0 √ ∫ ∞ 4 −x ∫ ∞ 4 −x x e 93 968 x e 16 3 5 dx = dx = ζ(5) − 24, ζ(5) + π − 24, 2x 3x 4 81 729 0 e −1 0 e −1 ∫ ∞ 5 −x ∫ ∞ 5 −x x e 8π6 x e π6 dx = − 120, dx = − 120. x 2x 63 8 0 e −1 0 e −1 ∫∞ 3 π4 Source: [511, p. 232]. Remark: 0 exx−1 dx = 15 is Planck’s integral. Related: Fact 14.6.11. Fact 14.8.16. Let n ≥ 1. Then, ∫ ∞ 2n−1 x Γ(2n)ζ(2n) (−1)n+1 B2n dx = = . 2πx 4n −1 (2π)2n 0 e

1161

INTEGRALS

Source: [116, p. 29] and [645, p. 99]. Fact 14.8.17. Let a, b > 0. Then,

∫

∞ 0

∫ 0

=

∞

∫

b e−ax − e−bx dx = log , x a

∞

0

b 2a (e−ax − e−bx )2 dx = 2b log + 2(a + b) log , a a+b x2

(e−ax − e−bx )3 dx x3

3 2 [3b log b − 3a2 log 3a + 4a(a + b) log(2a + b) + b2 log[27(2a + b)] − (a + 2b)2 log(a + 2b), 2 ∫ ∞ ∫ ∞ (1 − e−bx )2 27 3 (1 − e−bx )3 dx = 2b log 2, dx = b2 log . 2 3 2 16 x x 0 0

Source: [1158, p. 179]. Fact 14.8.18. Let a and b be real numbers. If either 0 < a < b or b < a < 0, then

∫

∞ −∞

e−ax aπ π dx = csc . b b 1 + e−bx

If 0 < a < 1 and b > 0, then ∫ ∞ −ax π e dx = 1−a csc aπ, −x b −∞ b + e

∫

∞

−∞

π e−ax dx = 1−a cot aπ. −x b−e b

Source: [1524, p. 270]. Fact 14.8.19. Let a ∈ (0, 1). Then,

∫

∞ −∞

eax − e(1−a)x dx = −2π cot aπ. ex − 1

Next, for all n ≥ 0, define Qn ∈ R[s] in Fact 13.2.14. If n ≥ 0, then ∫ ∞ n ax x e dx = πn+1 (csc aπ)Qn (− cot aπ). x −∞ e + 1 In particular,

∫ ∫

∞

eax dx = π csc aπ, ex + 1

−∞ ∞ 2 ax −∞

∫

∞

−∞

xeax dx = −π2 (csc aπ) cot aπ, ex + 1

x e dx = π3 (csc aπ)(2 cot2 aπ + 1) = 21 π3 (cos 2aπ + 3) csc3 aπ. ex + 1

Source: [1430]. Remark: Ignoring removable singularities, it follows that, for all n ≥ 0,

∫

∞ −∞

xn eax dx = πn+1 Pn (− cot aπ), ex − 1

where Pn ∈ R[s] is defined in Fact 13.2.14. See [1430] and [2106, p. 305]. ∫ ∞ Fact 14.8.20. e xe x dx = − 1. 2 2 (x + 1) 0 Source: [1217, p. 341] and [1524, p. 270]. Fact 14.8.21. Let a > 0 and b > −1. Then,

∫ 0

∞

xb e−ax dx = 2

Γ

(

)

b+1 2 . 2a(b+1)/2

1162

CHAPTER 14

Consequently, if n ≥ 0, then ) ( √ √ ∫ ∞ ∫ ∞ Γ n + 12 (2n)! π (2n − 1)!! π n! 2 2n −ax2 = = , x2n+1 e−ax dx = n+1 . x e dx = n+1 an 1 1 a 2 2a n+ 2 n+ 0 0 2a 22n+1 n!a 2 In particular, √ √ ∫ ∞ ∫ ∞ ∫ ∞ π 1 π 1 2 2 −ax2 e dx = , , xe−ax dx = x2 e−ax dx = 3/2 . 2 a 2a 4a 0 0 0 Fact 14.8.22. Let a ≥ 0 and b ≥ 0. Then,

∫

2 2 √ √ e−ax − e−bx dx = bπ − aπ. 2 x

∞ 0

Source: [2106, p. 90]. Fact 14.8.23. Let a > 0, b > −1, and c > 0. Then,

∫

∞

xb e−ax dx = c

0

In particular,

∫

∞ 0

)

(

Γ

∫

b+1 c , (b+1)/c ca

∞ 0

∫

e−x dx = Γ( 43 ),

∞

3

√

∫

∞

0

e−ax dx =

0

∫

1 , a

∞

∫

π , 2

xe−x dx =

Furthermore, if m ≥ 1 and n ≥ 0, then ( ) ∫ ∞ Γ n+1 m m xn e−ax dx = , (n+1)/m ma 0 Furthermore, √ ∫ ∞ −ax e π , dx = √ a x 0

∫

Γ(b + 1) , xb e−ax dx = ab+1

∫

∞

∞

4

0

1 2a

√

π , a

∫

∞

e

1 = √ π

∫

∞

−∞

Source: [116, p. 278]. Fact 14.8.25. Let a > 0 and b ≥ 0. Then,

∫

∞

e

−ax2 −b/x2

0

In particular,

∫

1 dx = 2

∞

√

π −2 √ab e , a

0

e−ax dx = 2

0

1 2

√

π , a

e−x e2zx ȷ dx. 2

∫

∞

e−ax−b/x dx = √ x

∞

e−ax √ dx = x

0

∫ 0

√

√

π −2 √ab e . a

π . a

Source: [2013, p. 200] and [2504, p. 113]. Fact 14.8.26. Let a and b be real numbers, and assume that a > 0. Then,

∫

√

∞

e −∞

−ax2 +bx

dx =

( 2) b π exp . a 4a

) .

xe−ax dx =

Source: [352, p. 41], [1158, p. 178], and [2106, pp. 119, 125]. Fact 14.8.24. Let z be a complex number. Then, −z2

c+1 c 1/c a

n! . an+1

xn e−ax dx =

xe−ax dx =

(

1 5 Γ( ). 4 8

x3/2 e−x dx =

0

∞

Γ

0

0

√

e−ax dx = c

1 . a2

1163

INTEGRALS

∫

Furthermore,

∞

xe

−x2 −x

−∞

1 dx = − 2

√

√

∫

π e,

∞

x2 e−x

2

−x

−∞

dx =

3 4

√

√ π e.

Source: [1524, p. 271] and [2106, pp. 115, 116]. Fact 14.8.27. Let a, b, c be real numbers, and assume that a > 0. Then,

∫

∞

−ax

e 0

b cos c + a sin c , sin(bx + c) dx = a2 + b2 ∫

In particular,

∞

e−ax sin bx dx =

0

Furthermore,

∫

∞

xe−ax sin bx dx =

0

b , 2 a + b2

2ab , (a2 + b2 )2

∫

∞

e−ax cos(bx + c) dx =

0

∫

∞

e−ax cos bx dx =

0

∫

∞

a2

xe−ax cos bx dx =

0

a cos c − b sin c . a2 + b2

a . + b2

a2 − b2 . (a2 + b2 )2

Fact 14.8.28. Let a and b be real numbers, and assume that a > 0. Then,

∫

∞

−ax2

e 0

√ √ ( 2) ( 2) ∫ ∞ b b π b 1 π −ax2 exp − , xe sin bx dx = exp − , cos bx dx = 3 2 a 4a 4a a 4 0 √ ( 2) ∫ ∞ b π 2a − b2 2 −ax2 x e cos bx dx = exp − , 5 8 4a a 0 √ ( 2) ∫ ∞ π 3(6ab − b3 ) b 2 x3 e−ax sin bx dx = exp − . 16 4a a7 0

Fact 14.8.29. Let a > 0 and b > 0. Then,

∫

∫

∞

0

∞

e−ax sin bx b dx = atan , x a 0 √ √ √ √ ∫ ∞ −ax −ax 2 2 ( a + b − a)π ( a2 + b2 + a)π e sin bx e cos bx dx = dx = , . √ √ 2 2 2(a + b ) 2(a2 + b2 ) x x 0

Source: [1158, p. 178] and [1890]. Fact 14.8.30. Let a, b, c be real numbers, and assume that a ≥ 0 and a2 + b2 > 0. Then,

(

) 1 b c2 b e sin atan − , e (sin bx ) cos cx dx = √4 2 a 4(a2 + b2 ) 0 2 a2 + b2 ( ) √ ∫ ∞ 1 b c2 b π 2 2 2 2 e−ax (cos bx2 ) cos cx dx = √4 e−c a/[4(a +b )] cos atan − . 2 a 4(a2 + b2 ) 0 2 a2 + b2 In particular, √ √ ∫ ∞ ∫ ∞ ∫ ∞ π π −c2 /4 1 2 2 −x2 sin |b|x dx = , e cos cx dx = e cos bx dx = . 2 2|b| 2 0 0 0 ∫

∞

−ax2

√ π

2

Source: [352, p. 166] and [2013, p. 184]. Fact 14.8.31. Let a ≥ 1 and b > 0. Then,

∫

∞ 0

−c2 a/[4(a2 +b2 )]

( ) e−bx − cos x 1 ( √a π ) a−1 dx = b − b sin Γ . √a b 2a a x

1164

CHAPTER 14

In particular, ∫ ∞ −x ∫ ∞ −x √ √ √ 1 1 e − cos x e − cos x dx = (4 + 2 − 6)Γ( 56 ), dx = (5 − 5)Γ( 54 ), √6 √5 4 4 x x 0 0 √ ) ( ∫ ∞ −x ∫ ∞ −x √ e − cos x 1 1 e − cos x 2 − 2 − 2 Γ( 43 ), dx = dx = Γ( 23 ), √4 √3 2 2 x x 0 0 ∫ ∫ ∞ −x ∞ −x √ √ e − cos x e − cos x 1 dx = 0. dx = (2 − 2) π, √ 2 x x 0 0 Fact 14.8.32. Let a ≥ 0, b ≥ 0, and c, d ∈ R, and assume that a2 + c2 > 0 and b2 + d2 > 0.

∫

Then,

∞ 0

e−ax cos cx − e−bx cos dx 1 b2 + d2 . dx = log 2 x 2 a + c2

Fact 14.8.33. Let a be a positive number. Then,

∫

∞

0

∫ ∫ ∞ x 1 ∞ x π2 dx = dx = , log coth ax dx = eax − e−ax 2 0 sinh ax 8a 0 ∫ ∞ ∫ ∞ x 1 x π dx = dx = . ax + e−ax e 2 cosh ax 2a 0 0

Fact 14.8.34. Let a > 0, and define △

∫

∞

ϕ(a) = 0

Then,

∫

∞ 0

sin ax dx = ϕ(a) + √ 2π e x−1

√

cos ax dx. −1

√ e2π x

( ) 1 2π3 π2 ϕ − , 3 a 2a a

lim ϕ(a) = 0.

a→∞

In particular, √ √ √ ∫ ∞ ∫ ∞ ∫ ∞ cos π5 x cos 2π 1 1 3 1 3 5 5 5 10 5 x , − , , dx = dx = + dx = − √ √ √ 2π x − 1 2π x − 1 2π x − 1 12 2 4 8 2 16 0 e 0 e 0 e √ √ √ ∫ ∞ ∫ ∞ ∫ ∞ cos 2π cos π2 x 1 1 3 3 3 1 2 cos πx 3 x dx = dx = + dx = − , − , , √ √ √ x x x 2π 2π 2π 4π 3 8π 16 4 8 −1 −1 −1 0 e 0 e 0 e √ √ ∫ ∞ ∫ ∞ ∫ ∞ 1 3− 2 13 − 4 3 cos 4πx cos 6πx cos 2πx dx = dx = dx = , , , √ √ √ 2π x − 1 2π x − 1 2π x − 1 16 32 144 0 e 0 e 0 e √ ∫ ∞ ∫ ∞ ∫ ∞ sin π2 x 1 2 1 3 sin πx 1 sin 2πx 1 dx = − , dx = dx = − , − . √ √ √ 2π x 2π x 2π x 4 4π 8 2π 16 8π −1 −1 −1 0 e 0 e 0 e Furthermore, ( ) ∫ ∞ ∫ ∞ x cos π2 x 13 1 x cos 2πx 1 1 3 5 dx = 2 − , dx = − + 2 , √ √ 2π x − 1 2π x − 1 2π 64 2 π π 8π 0 e 0 e ( ) ∫ ∞ 2 x cos 2πx 1 5 5 dx = 1− + 2 . √ 2π x 256 π π e −1 0 Source: [1317, pp. xxv, 66, 67]. Fact 14.8.35. Let a, b ∈ R. If a is nonzero and b is positive, then

∫

∞ 0

sin ax π aπ 1 dx = coth − . 2b b 2a ebx − 1

1165

INTEGRALS

If a and b are positive, then

∫

∞

π aπ 1 sin ax − csch . dx = bx 2a 2b b e +1

0

If a is positive, then

∫

∞

e x/2 sin ax π dx = tanh aπ. ex − 1 2

0

Source: [1217, p. 489] and [1317, p. 56]. Fact 14.8.36. Let a > 0 and b > 0. Then,

∫

∞ 0

( 2 ) ∫ ∞ π b a a (1 − e−ax ) sin bx (1 − e−ax ) sin bx b aπ dx = − atan , dx = log 2 + 1 + − a atan , 2 x 2 a 2 2 b x b 0 ( 2 ) ∫ ∞ −ax 1 a (1 − e ) cos bx dx = log 2 + 1 . x 2 b 0

Source: [1217, p. 501]. Fact 14.8.37. Let a ∈ (0, ∞). Then,

∫

( ) atan ax a 1 a 1 1 log Γ(a) + + − log a − log 2π. dx = 2 2 4 2 4 e2πx − 1

∞ 0

Furthermore,

∫

0 ∞

∫ 0

√ √ 3π 1 atan x3 1 dx = log 2π − − log(1 + e− 3π ), 2πx 4 12 2 e −1 √ 3 √ 1 3π 1 atan x −2 3π dx = log 12π − − log(1 − e ). 8 12 4 e4πx − 1

∞

Source: [1317, p. 51]. Fact 14.8.38. Let a and b be real numbers, and assume that |b| < a. Then,

∫

∞

b , a2 − b2

e−ax sinh bx dx =

0

∫

∞

e−ax cosh bx dx =

0

a . a2 − b2

Fact 14.8.39. Let z ∈ C, and assume that Im z ∈ (− 21 , 12 ). Then,

∫

∞

−∞

Now, let a ∈ (−1, 1). Then,

∫

∞ −∞

In particular, ∫ ∞ −∞

√ e−x/2 dx = 2π, cosh x

Furthermore, if n ≥ 0, then

∫

∞ −∞

∫ 0

In particular,

∫ 0

∞

1 π dx = , cosh x 2

e−2πzt ȷ 1 dt = . cosh πt cosh πz e−ax π dx = . cosh x cos aπ/2

√ e−x/3 2 3π dx = , cosh x 3

∞

∫

∞

−∞

e−x/4 dx = cosh x

√

√ 4 − 2 2π.

( π )2n+1 x2n dx = |E2n |. cosh x 2

∫ 0

∞

x2 π3 dx = , cosh x 8

∫

∞ 0

x4 5π5 dx = . cosh x 32

1166

CHAPTER 14

∫

Finally,

∞ 0

x dx = 2G. cosh x

Source: [629] and [2513, p. 115]. Fact 14.8.40. Let a > 0 and b > 0. Then,

∫

∞

0

Furthermore, if n ≥ 1, then

∫

2a+1 − 1 xa dx = a a+1 Γ(a + 1)ζ(a + 1). sinh bx 2b ∞

0

∫

In particular,

∫

xn (2n+1 − 1)n! dx = ζ(n + 1). sinh bx 2n bn+1

∞

π2 x dx = 2 , sinh bx 4b

∞

π x dx = 4 , sinh bx 8b

0

0

3

4

∫

∞

x2 7 dx = 3 ζ(3), sinh bx 2b

∞

93 x4 dx = 5 ζ(5). sinh bx 2b

0

∫

0

Fact 14.8.41. Let a and b be real numbers, and assume that b > 0. Then,

∫

∫

∞ 0

∫ ∞ sin ax π aπ cos ax π aπ dx = tanh , dx = sech , sinh bx 2b 2b cosh bx 2b 2b 0 0 ∫ ∞ 2 2 aπ (a + b )π cos ax cos ax aπ aπ dx = 2 csch , dx = sech . 2 3 3 2b 2b 2b 4b cosh bx 0 cosh bx ∞

If, in addition, |a| < b, then ∫ ∞ sinh ax π aπ dx = tan , 2b 2b 0 sinh bx

∫

∞ 0

cosh ax π aπ dx = sec . cosh bx 2b 2b

Source: [1524, p. 273] and [1217, pp. 509, 510]. Fact 14.8.42. Let a ∈ R. Then,

∫

∫ ∫

0 ∞

0 ∞

(x2

∞

√ cos 2ax 3 dx = , 2(1 + 2 cosh 2a) 1 + 2 cosh 2π x 3

cos ax dx = (cosh a) log(2 cosh a) − a sinh a, + 1) cosh π2 x

cos 2ax dx = 2 cosh a − e2a atan e−a − e−2a atan ea , 2 + 1) cosh πx (x 0 √ √ ∫ ∞ ∫ ∞ (cos πx2 ) cos 2πax −1 + 2 sin πa2 1 + 2 sin πa2 (sin πx2 ) cos 2πax , , dx = dx = √ √ cosh πx cosh πx 0 0 2 2 cosh πa 2 2 cosh πa ∫ ∞ ∫ ∞ (sin πx2 ) sin 2πax sin πa2 (cos πx2 ) sin 2πax cosh πa − cos πa2 dx = , dx = , sinh πx 2 sinh πa sinh πx 2 sinh πa 0 0 ∫ ∞ (sin πx2 ) sin 2πax dx = 12 (tanh πa) sin( π4 + πa2 ), tanh πx 0 ∫ ∞ (cos πx2 ) sin 2πax dx = 21 (tanh πa)[1 − cos( π4 + πa2 )], tanh πx 0

1167

INTEGRALS

∫ 0

∞

(sin πx2 ) cos πax 1 + 2 cosh 2

Furthermore, ∫ 0

∞

√

3π 3 x

− 2 cos π−3πa 12 2

dx =

8 cosh

√ 3π 3 a

√

3

−4

∫

∞

,

(cos πx2 ) cos πax 1 + 2 cosh 2

0

√

3π 3 x

dx =

1 − 2 sin π−3πa 12 √

8 cosh

3π 3 a

∫ ∞ 1 π x x dx = log 2 − , dx = − 1, 2 + 1) sinh π x 2 2 (x2 + 1) sinh πx (x 0 2 √ ∫ ∞ √ 2π √ x + 2 log(1 + 2) − 2, π dx = 2 2 0 (x + 1) sinh 4 x √ ∫ ∞ √ x 2π √ dx = − 2 log(1 + 2). π 2 2 0 (x + 1) cosh 4 x

Source: [1217, pp. 375, 376] and [1317, pp. 55, 61–66]. Fact 14.8.43. ∫ ∞ ∫ ∞ 2

1 1 x cos πx2 x sin πx dx = , dx = , sinh πx 4π sinh πx 8 0 0 √ √ ∫ ∞ 2 ∫ ∞ 2 2 2 x sin πx 1 x cos πx 1 2 2 dx = − , dx = − , cosh πx 8 16 cosh πx 16 4π 0 0 ∫ ∞ 3 ∫ ∞ 3 x sin πx2 1 x cos πx2 1 3 . dx = , dx = − sinh πx 16π sinh πx 64 16π2 0 0

Source: [1317, pp. 63, 65]. Fact 14.8.44. If a > 0 and b > 1, then

∫

∞ 0

4 xb dx = Γ(b + 1)ζ(b). 2 (2a)b+1 sinh ax

If a > 0 and n ≥ 1, then ∫ ∞ ∫ ∞ 2n x2n π2n (4n − 1) ( π )2n x cosh ax dx = dx = |B2n |. |B |, 2n 2 a a a2n+1 sinh2 ax 0 sinh ax 0 If a , 0 and n ≥ 1, then ∫ ∞ 2n+1 x cosh ax (22n+1 − 1)(2n + 1)! ζ(2n + 1). dx = 2 4n a2n+2 sinh ax 0 Source: [1217, pp. 379, 380]. Fact 14.8.45. If a > 0, b > −1, and b , 1, then ∫ ∞ xb 4(1 − 21−b ) dx = Γ(b + 1)ζ(b). 2 (2a)b+1 0 cosh ax In particular, ∫ ∞ √ 0

∫

√ √ 1 dx = 2( 2 − 4) πζ(− 21 ), 2 x cosh ax

If a , 0, then

∫

∞ 0

If a > 0 and n ≥ 1, then

1 1 dx = , 2 a cosh ax ∫

∞ 0

∞ 0

∫

∞ 0

√   2  √ dx =  − 1 πζ( 12 ). 2 2 cosh ax √

x

x log 2 dx = 2 . 2 a cosh ax

x2n (4n − 2)π2n dx = |B2n |. 2 4n a2n+1 cosh ax

2

−4

.

1168

CHAPTER 14

If a > 0 and n ≥ 0, then

∫ 0

If a > 0 and b > 0, then

∞

2n + 1 ( π )2n+1 x2n+1 sinh ax dx = |E2n |. a 2a cosh2 ax ∫

∞ 0

√ πΓ(b) dx = . 2b+1 2 4a bΓ(b + 12 ) cosh ax x sinh ax

Source: [1217, pp. 379, 380]. Fact 14.8.46. Let a, b, c be real numbers, and assume that |b| < c and a2 + b2 > 0. Then,

∫

∫

∞

0

0

∞

aπ π(sin bπ b (cos ax) sinh bx c ) cosh c , dx = − 2 + b2 ) 2aπ 2bπ ecx + 1 2(a c(cosh c − cos c )

π(sin 2bπ (cos ax) sinh bx b c ) − dx = . cx 2 2 2aπ e −1 2(a + b ) 2c(cosh c − cos 2bπ c )

In particular, if 0 < |b| < c, then ∫ ∞ π aπ 1 sinh bx dx = csc − , cx e +1 2c c 2b 0

∫

∞ 0

sinh bx 1 π bπ dx = − cot . cx e −1 2b 2c c

Source: [1217, p. 523] and [3024, p. 336]. Fact 14.8.47. Let a and b be nonzero real numbers such that division by zero does not occur in

the expressions below. Then, ∫ ∞

∫

∞ 0

√ 1 a + b + a2 + b2 1 log , dx = √ √ 0 a + b sinh x a2 + b2 a + b − a2 + b2 √   1 a + b + a2 + b2    log , |b| < |a|, √ √     a + b − a2 + b2 1  a2 + b2 dx =  √   a + b cosh x   b2 − a2 2    atan , |a| < |b|, √ a+b b2 − a2

√   1 a + b + a2 − b2    log , |b| < |a|, √ √  ∫ ∞    a + b − a2 − b2 1  a2 − b2 dx =  √    0 a sinh x + b cosh x  b2 − a2 2    atan , |a| < |b|. √ a+b b2 − a2 Source: [1524, p. 273]. Fact 14.8.48. Let a > 0. Then, √ ∫ ∞ ∫ ∞ 2 π log x 1 (log 2)(2 log a + log 2) x log x (2 − log 4a − γ). dx = dx = − , 2 ax ax 2a 8 a3 e 0 e +1 0 Source: [1217, p. 574]. Fact 14.8.49. Let a and b be positive numbers. Then,

∫

∞ 0

( ) log bx 1 b dx = log − γ , eax a a

∫

∞ 0

log2 bx π2 γ2 1 ( a) ( a) dx = + + log 2γ + log , eax 6a a a b b

1169

INTEGRALS

∫

∞ 0

( 1 [ 2 a) ( a )] ( a) 2 log3 bx 2 dx = − π + 2γ + 2 log 2γ + log γ + log − ζ(3), eax 2a b b b a √ ∫ ∞ log bx 1 π (2 log b − γ − log 4a), dx = 2 ax 4 a e 0 √ ∫ ∞ 1 log2 bx π 2 dx = [π + 2(γ + log 4a − 2 log b)2 ], ax2 16 a e 0  √ ∫ ∞   3π γ 1 1 log bx 1 4  + + log 3 + log a − log b , dx = − √3 Γ( 3 )  3 ax 18 3 2 3 e a 0 ) ( ∫ ∞ log bx 1 1 π γ 3 5 + + log 2 + log a − log b . Γ( dx = − ) √ 4 4 eax4 a 4 8 4 4 0

In particular, ∫ ∞ log x dx = −γ, ex 0

∫

∞ 0

∫

∫

π2 log2 x dx = + γ2 , x e 6

∞ 0

1 log3 x dx = −γ3 − γπ2 − 2ζ(3), ex 2

∞

log4 x 3π4 dx = + γ4 + π2 γ2 + 8γζ(3), ex 20 0 √ √ ∫ ∞ ∫ ∞ π π 2 log x log2 x dx = − dx = (γ + 2 log 2), [π + 2(γ + 2 log 2)2 ]. 2 2 x x 4 16 e e 0 0 Fact 14.8.50. Let a > 0 and n ≥ 0. Then, ∫ ∞ n x log x n! dx = n+1 (Hn − log a − γ), ax e a 0  2n+1 √ ∫ ∞ n+1/2  π(2n + 1)!!  ∑ 2 x log x  dx = − log 4a − γ . ax n+1 n+3/2 e 2i + 1 2 a 0 i=1 Source: [1217, p. 573]. Fact 14.8.51. Let a > 0. Then,

∫

∞

log(1 + e−ax ) dx =

0

If n ≥ 1, then,

∫

π2 , 12a

∞

log(1 + e−ax ) dx = −

0

∫

∞

logn (1 − e−ax ) dx = (−1)n

0

In particular, ∫ ∞ 2 log2 (1 − e−ax ) dx = ζ(3), a 0

π2 , 6a

∫

∞

∞

log2 (1 + e−ax ) dx =

0

π4 , 15a

∫

∞

log4 (1 − e−ax ) dx =

0

Source: [1217, p. 530].

14.9 Facts on Integral Representations of G and γ Fact 14.9.1.

G=

1 2

∫

π/2

0

∫ =2 0

π/4

x dx = sin x

∫ 0

1

atan x dx = − x ∫

π/4

log 2 cos x dx = −2 0

∫

1 0

1 ζ(3). 4a

n! ζ(n + 1). a

log3 (1 − e−ax ) dx = −

0

∫

log x π2 dx = 4 x2 + 1 ∫

log 2 sin x dx = − 0

π/2

∫

−x dx cos πx

1 1 2 0

log 2 sin 2x dx

24 ζ(5). a

1170

CHAPTER 14

∫

π/4

=

1 log cot x dx = 2

0

=

1 4

∫

∞

0

∫

∞ 0

3 x dx = cosh x 4

x2 sinh x π2 π − log 2 + dx = 16 4 cosh2 x

∫

∫

π/4 0

π/2

log |1 + 2 cos x| dx

0

∑1 i x2 atan . dx = lim 2 n→∞ i n sin x i=1 n

Furthermore, π π π π3 π π π3 − log + < G < − log 2 − ≈ 0.91874. 2 2 2 576 2 4 288 Source: [11], [1217, p. 380], and [2411]. The inequalities are given in [2952]. Related: Fact 13.5.97 and Fact 13.6.2. Fact 14.9.2. Let a > 0 and b > 0. Then, ) ∫ 1( ∫ 1 ∫ 1 1 1 1 1 + dx = (1 − e−x − e−1/x ) dx γ=− log log dx = x 1 − x log x 0 0 x 0 ) ∫ 1 ∫ ∞ ∫ ∞ ( 1 1 log x 1 1 = − dx = − dx = − cos x dx x(log x)(1 − log x) ex x 2 0 log x 0 0 ) ) ∫ ∞ ( ∫ ∞ ∫ ∞( 1 2 1 2 1 1 − dx = dx = (cos x2 − cos x) dx = − x−1 xx x x2 e e x e x e 0 0 0 ) ) ) ∫ ∞( ∫ ∞ ( ∫ ∞ ( 1 1 b 1 1 b 1 b = 1 − ex − e−x dx = − dx = − cos x dx e e x x a + 1 e xb x xa + 1 0 0 0 ) ∫ ∞ ( ∫ 1 1 2 1 2 ∞1 = log + acot x − e−ax dx = log − (sin ax) log x dx a x π a π 0 x 0 ∫ ∞ ∫ ∞ 2 1 1 x 2 = 1 − log 2a − (sin ax) log x dx = dx + 2 2 2πx − 1) aπ 0 x2 2 0 (x + 1)(e ) ) ∫ ∞ ( ∫ ∞ ( a a e−x − 1 sin xa 1 a 1 =1+ + − dx = 1 + dx x xb + 1 xa x xb + 1 xa 0 0 ) ( ) ( ∫ ∞ ∫ ∞ log x sin x 3 2a 1 cos xa − 1 dx. =1+ cos x − dx = + + x x 2 x 2(xb + 1) x2a 0 0 0.91528 ≈

If a , b, then

∫ ∞ −xb ∫ ∞ a e − e−x ab cos xb − cos xa ab dx = dx b−a 0 x b−a 0 x ) ∫ ∞ ∫ ∞ ( a ab cos xa − e−x ab 1 sin xb sin xa = dx = 1 + − dx b−a 0 x b−a 0 x xa xb ∫ ∞ bb 2 1 1 log a + (cos ax − cos bx) log x dx. =1+ a−b a π(a − b) 0 x2

γ=

Furthermore,

∫ 0

∞

( ) ∫ ∞ 1 1 1 xb−1 − xa−1 − dx = dx = 0. a b x xa + 1 x b + 1 0 (x + 1)(x + 1)

Source: [208], [511, pp. 176–179], [1217, p. 906], [2013, p. 198], [2256], [2504, p. 113], and

[2513, pp. 15–22].

1171

INTEGRALS

14.10 Facts on Definite Integrals of the Gamma Function Fact 14.10.1. Let a, b ∈ (0, ∞). Then,

∫

∞

|Γ(a + x ȷ)|2 cos 2bx dx =

0

∫

∞

0

If a +

1 2

< b, then

1 2

√

πΓ(a)Γ(a + 12 ) sech2a b,

√ π|Γ(a + b ȷ)|2 (sech x) cos 2bx dx = . 2Γ(a)Γ(a + 12 ) 2a

√ 2 1 1 Γ(a + x ȷ) dx = πΓ(a)Γ(a + 2 )Γ(b − a − 2 ) , Γ(b + x ȷ) 2Γ(b)Γ(b − 12 )Γ(b − a) 0 2 √ ∫ ∞∏ ∞ 1+ x πΓ(b)Γ(a + 12 )Γ(b − a − 21 ) (b+i)2 dx = . x2 2Γ(a)Γ(b − 12 )Γ(b − a) 0 i=0 1 + (a+i)2 ∫

∞

Remark: If b − a is an integer, then the infinite product in the last integral is a finite product. For

example, setting a = 11/10 and b = 61/10 yields ∫ ∞ 1 dx 2 2 2 2 2 2 2 2 2 2 0 (x + 11 )(x + 21 )(x + 31 )(x + 41 )(x + 51 ) 5π 5π = = . 12 · 13 · 16 · 17 · 18 · 22 · 23 · 24 · 31 · 32 · 41 377244828499968 Source: [1317, pp. 20, 21, 54]. Related: Fact 14.2.26. Fact 14.10.2. Let a, b ∈ (0, ∞). If a > 12 , then ∫ ∞ 1 cos πx dx = . 2 2 4Γ(2a − 1)Γ2 (a) 0 Γ (a + x)Γ (a − x) If a + b > 32 , then ∫

∞

0

∫

∞ 0

∫

∞

1 Γ(2a + 2b − 3) dx = 4 , 2 (a + x)Γ2 (b − x) Γ Γ (a + b − 1) 0 Γ(2a + 2b − 3) 1 dx = , Γ(a + x)Γ(a − x)Γ(b + x)Γ(b − x) 2Γ(2a − 1)Γ(2b − 1)Γ2 (a + b − 1) 4a+b−3 Γ(a + b − 23 ) cos πx dx = √ . Γ(a + x)Γ(a − x)Γ(b + 2x)Γ(b − 2x) πΓ(a)Γ(2b − 1)Γ(2a + b − 2)

Source: [1317, pp. 226–229].

14.11 Facts on Integral Inequalities Fact 14.11.1. The following statements hold:

i) If n ≥ 1, then

2n+1 − 1 ≤ n+1

ii) If x ∈ (0, ∞), then

∫

∞ x

iii) If x ∈ (0, 1), then

∫

1 x

∫

e−t

π/2

(cos x + 1)n dx.

0 2

/2

dt ≤

1 −x2 /2 e . x

log(t + 1) 1−x dt < (2 log 2) . t 1+x

1172

CHAPTER 14

Source: [2527, pp. 117, 118].

14.12 Facts on the Gaussian Density Fact 14.12.1. Let A ∈ Rn×n, assume that A is positive definite, let b ∈ Rn , and let a ∈ R. Then,

∫

e

−xTAx+2bTx+a

Rn

∫

πn/2 bTA−1 b+a dx = √ e , det A

In particular,

πn/2 −bTA−1 b+a T T e . e−x Ax+2b x ȷ+a dx = √ Rn det A

∫

πn/2 T . e−x Ax dx = √ Rn det A Source: [2013, pp. 103, 104]. Related: Fact 15.14.12. Fact 14.12.2. Let A ∈ Rn×n, assume that A is positive definite, and define f : Rn 7→ R by 1 T −1

f (x) = Then,

∫

e− 2 x A x . √ (2π)n/2 det A

∫ Rn

f (x) dx = 1,

∫ Rn

f (x)xxT dx = A,

Rn

f (x) log f (x) dx = − 12 log[(2πe)n det A].

Source: ∫ [788] and Fact 14.12.5. Remark: f is the multivariate Gaussian density. The entropy of

f is −

Rn

f (x) log f (x) dx.

Fact 14.12.3. Let A, B ∈ Rn×n, assume that A and B are positive definite, and, for all k ∈

{0, 1, 2, 3}, define

△

Ik = Then,

I0 = 1,

1 √

(2π)n/2

∫

det A

I1 = tr AB,

1 T −1

Rn

(xTBx)k e− 2 x A

x

dx.

I2 = (tr AB)2 + 2 tr (AB)2 ,

I3 = (tr AB)3 + 6(tr AB) tr (AB)2 + 8 tr (AB)3 . I4 = (tr AB)4 + 12(tr AB)2 tr (AB)2 + 32(tr AB) tr (AB)3 + 12[tr (AB)2 ]2 + 48 tr (AB)4 . Source: [2043, p. 80], which uses a moment generating function. A Kronecker product approach is used in [2402]. Remark: These are Lancaster’s formulas. Fact 14.12.4. Let A, B, C, D ∈ Rn×n, assume that A is positive definite, assume that B, C, and D

are symmetric, and let µ ∈ Rn. Then, ∫ 1 1 T −1 xTBxe− 2 (x−µ) A (x−µ) dx = tr AB + µTBµ, √ n/2 n (2π) det A R ∫ 1 1 T −1 xTBxxTCxe− 2 (x−µ) A (x−µ) dx = (tr AB) tr AC + 2 tr ACAB + µTCµ tr AB √ (2π)n/2 det A Rn + 4µTBACµ + µTBµ tr CA + µTBµµTCµ, 1 √ n/2 (2π) det A

∫

1 T −1

xTBxxTCxxTDxe− 2 x A

Rn

x

dx

= (tr AB)(tr AC) tr AD + 2[(tr AB) tr ACAD + (tr AC) tr ADAB + (tr AD) tr ABAC] + 4(tr ABACAD + tr ABADAC).

1173

INTEGRALS

Source: [2043, p. 83] and [2403, pp. 414–419]. Remark: These are the mean and covariance of quadratic forms in multivariate Gaussian random variables. Setting µ = 0 and C = B yields I2 of Fact 14.12.3. Fact 14.12.5. Let A ∈ Rn×n, assume that A is positive definite, let B ∈ Rn×n, let a, b ∈ Rn, and let α, β ∈ R. Then, ∫ 1 T −1 πn/2 T T (xTBx + bTx + β)e−(x Ax+a x+α) dx = √ (2β + tr A−1B − bTA−1a + 21 aTA−1BA−1a)e 4 a A a−α . n R 2 det A Source: [1343, p. 322]. Fact 14.12.6. Let A1 , A2 ∈ Rn×n, assume that A1 and A2 are positive definite, and let µ1 , µ2 ∈ Rn. Then, 1 1 1 T −1 T −1 T −1 e 2 (x−µ1 ) A1 (x−µ1 ) e 2 (x−µ2 ) A2 (x−µ2 ) = αe 2 (x−µ3 ) A3 (x−µ3 ) ,

where △

−1 A3 = (A1−1 + A−1 2 ) ,

△

µ3 = A3 (A1−1 µ1 + A−1 2 µ2 ),

△

1

T −1

α = e 2 (µ1 A1

T −1 µ1+µT2 A−1 2 µ2 −µ3 A3 µ3 )

.

Remark: A product of Gaussian densities is a weighted Gaussian density.

14.13 Facts on Multiple Integrals Fact 14.13.1. Let n ≥ 1. Then,

∫ 1∫ 0

Fact 14.13.2.

∫ 1∫

∫ 1∫ 0

1 0

1

|x − y|n dx dy =

0

1 π2 dx dy = , 1 − xy 6

2 . (n + 1)(n + 2)

∫ 1∫ 0

1 0

1 π2 dx dy = , 1 + xy 12

∫ 1∫ 1 1 1 π dx dy = 2 log 2, dx dy = − log 2, 2 2 2 0 0 1 − xy 0 0 1 + xy √ √ ∫ 1∫ 1 ∫ 1∫ 1 3π 3 3π 1 1 dx dy = dx dy = + log 3, , 3 3 12 4 6 1 − xy 1 + xy 0 0 0 0 ∫ 1∫ 1 ∫ ∫ ∫ 1∫ 1 1 1 π2 1 π/2 π/2 1 dx dy = G, dx dy = , dx dy = 2 2 2 2 8 2 0 1 + (cos x) cos y 0 0 1+x y 0 0 0 1−x y √ ∫ 1∫ 1 ∫ 1 1 atanh x3/2 3π 3 dx dy = dx = + log 3 − 2 log 2, 2 y3 3/2 6 2 1 − x x 0 0 0 ∫ 1∫ 1 ∫ 1∫ 1 2 π xy π2 xy dx dy = − 1, dx dy = 1 − , 6 12 0 0 1 + xy 0 0 1 − xy ∫ 1∫ 1 ∫ 1∫ 1 xy xy 1 1 dx dy = , dx dy = log 2 − , 2 2 2 2 0 0 1 + xy 0 0 1 − xy √ √ ∫ 1∫ 1 ∫ 1∫ 1 1 xy xy 3π 3 3π 1 dx dy = − + log 3, dx dy = − , 3 3 4 24 8 12 4 1 − xy 1 + xy 0 0 0 0 ∫ 1∫ 1 ∫ 1∫ 1 2 2 xy xy π π , , dx dy = dx dy = 2 2 2 2 24 48 0 0 1−x y 0 0 1+x y √ √  ∫ 1∫ 1 ∫ 1∫ 1  3 1  1 xy 3   dx dy =  −  π + log 2, dx dy = − log 2, 2 y3 2 y3 3 2 6 1 + x 1 + x 0 0 0 0 1

1174

CHAPTER 14

√ ∫ 1∫ 1 2 2 ∞ ∑ (i!)3 3π x x y[x y(1 − x)(1 − y) + 1] dx dy = dx dy = , ≈ 1.178403258, 3 3 9 (3i)! [1 − x2 y(1 − x)(1 − y)]3 0 0 1−x y 0 0 i=0 ∫ 1∫ 1 ∫ 1∫ 1 π2 1 1 x dx dy = − log2 2, dx dy = 1 − log 2, 12 2 0 0 2 − xy 0 0 2 − xy ∫ 1∫ 1 ∫ 1∫ 1 π2 xy xy 1 dx dy = − 1 − log2 2, dx dy = (1 − log 2), 2 2 − xy 6 2 2 − xy 0 0 0 0 ∫ 1∫ 1 ∞ ∑ √ (1 − y)[2 − xy2 (1 − x) + x2 y4 (1 − x)2 ] (i!)2 8G π dx dy = = − log(2 + 3), 2 3 (2i + 1)(2i + 1)! 3 3 [1 − xy (1 − x)] 0 0 i=0 ∫ 1∫ 1 ∞ ∑ x2 y2 π log 2 7 log2 2 13π2 G dx dy = (−1)i+1 (H2i −Hi −log 2)2 = + − − , 2 2 8 8 192 2 0 0 (1 + x y )(1 + x)(1 + y) i=1 ∫ 1∫ 1 log2 tan π8 π2 1 dx dy = − . 2 π 2 2 16 tan π8 4 tan π8 0 0 1 − x y tan 8 ∫ 1∫

1

Source: [511, pp. 226, 234] and [1107, 1266, 1491, 2483]. Fact 14.13.3. The following statements hold:

i) If α > −1 and β > −1, where α , β, then ∫ 1∫ 1 α β 1 1+α x y dx dy = log . β−α 1+β 0 0 log xy In particular,

∫ 1∫

√

1

∫ 1∫

2 x dx dy = 2 log , 3 0 0 log xy ∫ 1∫ 1 √ 3 xy dx dy = 2 log , log xy 4 0 0

ii) If α > −1, then

∫ 1∫ 0

In particular,

∫ 1∫

0

1

1

∫ 1∫ 0

In particular, ∫ 1∫ 1 0

∫ 1∫ 0

0

0 1

1 0

x dx dy = − log 2, log xy

0

∫ 1∫ 0

1 0

xy2 2 dx dy = log . log xy 3

(xy)α 1 dx dy = − . log xy 1+α

1 dx dy = −1, 0 0 log xy ∫ 1∫ 1 √ xy 2 dx dy = − , log xy 3 0 0

iii) If α > −1, then

0

1

∫ 1∫ 0

1

√

0

∫ 1∫ 0

1 dx dy = −2, xy log xy

1 0

xy 1 dx dy = − . log xy 2

1 log(1 + α) dx dy = − . (1 + αxy) log xy α

1 dx dy = − log 2, (2 − xy) log xy

1 dx dy = − log 2, (1 + xy) log xy

∫ 1∫ 0

0

∫ 1∫ 0

1

1 0

1 2 dx dy = log , (2 + xy) log xy 3

1 1 dx dy = − log 3. (1 + 2xy) log xy 2

1175

INTEGRALS

iv) If k ≥ 1, then

∫ 1∫ 0

0

1

(1 − y)k y log(1 − x) 1 dx dy = − 2 . (1 − xy)2 k

Source: [1266]. Fact 14.13.4. Let n ≥ 2. Then,

∫ 1∫ 0

In particular,

∫ 1∫ 0

1 0

Source: [2106, p. 164]. Fact 14.13.5.

∫ 1∫

1 0

logn−2 xy dx dy = (−1)n (n − 1)!ζ(n). 1 − xy

π2 1 dx dy = , 1 − xy 6

∫ 1∫ 0

1 0

log xy dx dy = −2ζ(3). 1 − xy

∫ 1∫ 1 log x 3 log xy dx dy = − ζ(3), dx dy = −2ζ(3), 1 + xy 4 1 − xy 0 0 0 0 ∫ 1∫ 1 ∫ 1∫ 1 log(1 − x) log(1 − xy) dx dy = 2 dx dy = −2ζ(3), 1 − xy 1 − xy 0 0 0 0 ∫ 1∫ 1 ∫ 1∫ 1 ∫ 1∫ 1 log(2 − xy) log(1 + x) 5 x−1 dx dy = γ, dx dy = dx dy = ζ(3), 1 − xy 1 − xy 8 0 0 0 0 0 0 (1 − xy) log xy ∫ 1∫ 1 ∫ 1∫ 1 2 log(2 − x) log(1 + xy) π log 2 dx dy = dx dy = − ζ(3), 1 − xy 1 − xy 4 0 0 0 0 ∫ 1∫ 1 ∫ 1∫ 1 log xy π2 1 7 1 dx dy = − log 2, dx dy = log 2 − log3 2 − ζ(3), 6 3 4 0 0 2 − xy 0 0 (2 − xy) log xy ∫ 1∫ 1 ∫ 1 ∞ ∞ ∑ ∑ x x−1 1 i 1 1 dx dy = dx = log = log ≈ −0.507834, i i (2 − xy) log xy (x − 2) log x 2 i 2 i + 1 0 0 0 i=1 i=1 ∫ 1∫ 1 ∫ 1∫ 1 1 log xy π3 π dx dy = − , dx dy = − , 2 y2 2 y2 ) log xy 16 4 1 + x (1 + x 0 0 0 0 ∫ 1∫ 1 ∫ 1∫ 1 ∫ 1∫ 1 3 2 log x π π π2 x log xy x log xy dx dy = − dx dy = dx dy = − , −G, , 2 2 2 2 2 2 32 48 12 0 0 1+x y 0 0 1+x y 0 0 1−x y ∫ 1∫ 1 ∫ 1∫ 1 1+x 1−x π dx dy = − log π, dx dy = log , (1 + xy) log xy (1 + xy) log xy 4 0 0 0 0 ∫ 1∫ 1 ∫ 1∫ 1 2 2 x x Γ (3/4) dx dy = log , , dx dy = log √ 2 2 π 0 0 (1 + x y ) log xy 0 0 (1 + xy) log xy 2π ∫ 1∫ 1 ∫ 1∫ 1 (1 + 21 xy)[log(1 − x)] log(1 − y) √ 1 π2 2 dx dy = 2 log 2 + , dx dy = π, 1 3/2 3 3 (− log xy) (1 − 2 xy) 0 0 0 0 ∫ 1∫ 1 ∫ 1∫ 1 √ √ x x dx dy = 2( 2 − 1) π, dx dy = Γ( 34 ), 3/2 (− log xy) (− log xy)5/4 0 0 0 0 √ ∫ 1∫ 1 ∫ 1∫ 1 Γ( 41 ) 2G 1 1−x π √ 1 − x2 dx dy = (8 2−10), dx dy = log + − , 2 5/2 2 2 3 2Γ( 34 ) π 2 0 0 (− log xy) 0 0 (1 + x y ) log xy ∫ 1∫ 1 ∫ 1∫ 1 log4 xy 225 x log2 xy π2 π3 dx dy = ζ(5), dx dy = G − + , 2 2 2 2 2 48 32 0 0 (1 + xy) 0 0 (1 + x y ) 1

1176

CHAPTER 14

∫ 1∫ 0

∫ 1∫ 0

0

1

1 0

1 1 dx dy = − (π + 2), 8 (1 + x2 y2 )2 log xy

∑ 1 dx dy = (−1)i (1 + x)(1 + y)(1 + xy) i=0 ∞

∫ 1∫ 0

(∫ ∞ ∑ = (−1)i i=0

If z ∈ C\[1, ∞), then

∫ 1∫ 0

If n ≥ 1, then

0

∫ 1∫ 0

If n ≥ 0, then

1

∫ 1∫ 0

1 0

1 0

1 0

∫ 1∫ 0

1 0

G x2 dx dy = − , π (1 + x2 y2 )2 log xy

1

xi yi dx dy 0 (1 + x)(1 + y) ∞ )2 ∑ ∞ ∑ xi 1 i (−1)  (−1) j−1 dx = x+1 i+ j=1 i=0

2 2   = π .  j 24

log(1 − z) 1 dx dy = . (1 − xyz) log xy z ∑2 (xy)n log xy dx dy = − 2ζ(3). 1 − xy i3 i=1 n

(n + 3)!ζ(n + 4) (log x)(log y) logn xy dx dy = (−1)n . 1 − xy 6

If m ≥ 2 and n ≥ max {0, m − 3}, then ] ∫ 1∫ 1 m−1 [ ∑ (log x)(log y) logn xy m−1 n (n + 3)! dx dy = (−1) ζ(n + 4 − i). i (1 − xy)m 6(m − 1)! i=1 0 0 If n ≥ 1, then

∫ 1∫ 0

0

1

)2 ∞ ∫ ∞( ∑ n! [(x(1 − x)y(1 − y)]n dx dy = − dx. (1 − xy) log xy xn+1 i=n+1 i

Source: [116, p. 392] and [1105, 1106, 1266, 2483, 2484, 2492]. Remark:

[

m−1 i

]

is a Stirling number of the first kind. √ Fact 14.13.6. Let α = 21 (1 + 5). Then, ∫ 1∫ 1 ∫ 1∫ 1 1 π2 π2 1 2 dx dy = dx dy = − log α, − log2 α, 2 10 15 0 0 α − xy 0 0 α − xy ∫ 1∫ 1 ∫ 1∫ 1 1 π2 1 1 π2 1 dx dy = + log2 α, dx dy = − log2 α, 10α α 15 2 0 0 1 + αxy 0 0 α + xy ∫ 1∫ 1 ∫ 1∫ 1 1 1 dx dy = −2 log α, dx dy = − log α, 2 − xy) log xy (α − xy) log xy (α 0 0 0 0 ∫ 1∫ 1 ∫ 1∫ 1 2 1 1 dx dy = − log α, dx dy = − log α, α 0 0 (α + xy) log xy 0 0 (1 + αxy) log xy ∫ 1∫ 1 log xy 4π2 4 8 dx dy = log α − log3 α − ζ(3), 2 15 3 5 α − xy 0 0 ∫ 1∫ 1 2 π 1 3 dx dy = − 3 log2 α. 3 6 − x2 y2 24α 4α α 0 0 Source: [1266].

1177

INTEGRALS

Fact 14.13.7.

∫

∫

1

0

1

∫

0

1 0

( ) 1 2 1 1 dx dy dz = Γ . 1 − (cos x)(cos y) cos z 4 4

Source: [2106, p. 212]. Fact 14.13.8. Let n ≥ 2. Then,

∫

∫

1

0

Source: [2513, p. 402]. Fact 14.13.9. Let n ≥ 2. Then,

∫

1

∫

···

0

0

1

1 ∑n

xi

i=1

1

0

1 dx1 · · · dxn = ζ(n). 1 − x1 · · · xn

∫

∞

···

dx1 · · · dxn = 0

(

1 − e−x x

)n dx =

( ) n ∑ n n−1 log i. (−1)n−i in−2 (n − 1)! i=2 i−1

Source: [1103, pp. 48–50] and [1104]. Fact 14.13.10. Let n ≥ 1. Then,

∫

1

∫

···

0

In particular,

0

∫

1 0

1

π(n+1)/2 1 ( ∑n 2 )(n+1)/2 dx1 · · · dxn = 2n (n + 1)Γ ( n+1 ) . 1 + i=1 xi 2

1 π dx = , 2 4 1+x

∫

1

∫

0

△

1 0

(1 +

x2

π 1 dx dy = . 2 3/2 6 +y )

Fact 14.13.11. Let n ≥ 1, S = {x ∈ (0, ∞) : ∥x∥1 ≤ 1}, and α1 , . . . , αn ∈ ORHP . Then,

∫

···

n

∫ ∏ n S i=1

xiαi −1 dx1 · · · dxn =

∏n

Γ(αi ) ∑n . i=1 αi )

i=1

Γ(1 +

In particular, if k and l are nonnegative integers, then ∫ 1 ∫ 1−y ∫ 1 1 k!l! xk yl dx dy = yl (1 − y)k+1 dy = . k+1 0 (k + l + 2)! 0 0 Source: [116, pp. 32, 33]. △ Fact 14.13.12. Let n ≥ 1, S = {x ∈ (0, ∞)n : ∥x∥1 = 1}, and α1 , . . . , αn ∈ ORHP . Then,

∫

···

∫ ∏ n S i=1

xiαi −1

dx1 · · · dxn−1

∏n Γ(αi ) ∑ . = i=1 Γ( ni=1 αi )

Source: [116, p. 35]. Fact 14.13.13. Let n ≥ 1, for all i ∈ {1, . . . , n}, let αi ∈ ORHP, ai > 0, and pi > 0, and let △

S = {x ∈ (0, ∞)n :

∑n

i=1 (x(i) /ai )

∫

≤ 1}. Then, ∏n ∫ ∏ αi n i=1 (ai /pi )Γ(αi /pi ) αi −1 ∑ ··· xi dx1 · · · dxn = . Γ(1 + ni=1 αi /pi ) S i=1 pi

△

In particular, the volume of the hyperellipsoidal solid E = {x ∈ Rn : ∏ πn/2 ni=1 ai ) . ( vol(E) = Γ n2 + 1 Source: [116, p. 33]. Related: Fact 5.5.14 and Fact 5.5.15.

∑n

i=1 (x(i) /ai )

2

≤ 1} is

1178

CHAPTER 14

Fact 14.13.14. Let n ≥ 2 and a < −1/n. Then,

∫

π

−π

∫

···

π −π

∏

|e ȷxi − e ȷx j |2a dx1 · · · dxn =

(2π)n Γ(na + 1) , Γn (a + 1)

where the product is taken over all i, j ∈ {1, . . . , n} such that i < j. In particular, ∫ π∫ π ∫ π∫ π∫ π |e ȷx − e ȷy |2 dx dy = 2π2 , |(e ȷx − e ȷy )(e ȷx − e ȷz )(e ȷy − e ȷz )|2 dx dy dz = 48π3 . −π

−π

−π

−π

−π

Source: [116, p. 426] and [179, p. 144]. Remark: This is Dyson’s integral. Fact 14.13.15. Let n ≥ 2 and a > −1/n. Then,

∫

  n n ∏  1 ∑ ∏ Γ(ia + 1) 2  ··· exp − |xi − x j |2a dx1 · · · dxn = (2π)n/2 , xi  2 Γ(a + 1) −∞ −∞ i=1 i=1 ∞

∫

∞

where the product under the integral is taken over all i, j ∈ {1, . . . , n} such that i < j. Source: [116, p. 407] and [179, p. 143].

14.14 Notes Historical notes on the accuracy of integral tables are given in [2585]. Proofs of many of the formulas in [1217] are given in [2069, 2070]. An extensive collection of formulas is given in [2268]. Integrals with matrix arguments are considered in [2253, pp. 230–237].

Chapter Fifteen The Matrix Exponential and Stability Theory The matrix exponential function is fundamental to the study of linear ordinary differential equations. This chapter focuses on the properties of the matrix exponential as well as on stability theory.

15.1 Definition of the Matrix Exponential The scalar differential equation x˙(t) = ax(t), x(0) = x0 ,

(15.1.1) (15.1.2)

where t ∈ [0, ∞), a ∈ R, and x(t) ∈ R, has the solution x(t) = eatx0 ,

(15.1.3)

where t ∈ [0, ∞). We are interested in systems of linear differential equations of the form x˙(t) = Ax(t),

(15.1.4)

x(0) = x0 ,

(15.1.5)

where the derivative is one-sided where t ∈ [0, ∞), x(t) ∈ Rn, and A ∈ Rn×n. Here x˙(t) denotes for t = 0 and two-sided for t > 0. The solution of (15.1.4), (15.1.5) is given by dx(t) dt ,

x(t) = etA x0 ,

(15.1.6)

where t ∈ [0, ∞) and etA is the matrix exponential. The following definition is based on (12.8.2). Definition 15.1.1. Let A ∈ Fn×n. Then, the matrix exponential, which is written as either eA ∈ n×n F or exp(A) ∈ Fn×n , is the matrix △

eA =

∞ ∑ 1 i A. i! i=0

(15.1.7)

△

Note that 0! = 1, A0 = In , and e0n×n = In. Proposition 15.1.2. Let A ∈ Fn×n. Then, the following statements hold: i) The series (15.1.7) converges absolutely. ii) The series (15.1.7) converges to eA. iii) Let ∥ · ∥ be a normalized submultiplicative norm on Fn×n. Then, e−∥A∥ ≤ ∥eA ∥ ≤ e∥A∥.

(15.1.8)

Proof. To prove i), let ∥ · ∥ be a normalized submultiplicative norm on Fn×n. Then, for all k ≥ 1, k ∑ i=0

i 1 i! ∥A ∥

≤

k ∑ i=0

i 1 i! ∥A∥

≤ e∥A∥.

1180

CHAPTER 15

)∞ (∑ of partial sums is increasing and bounded, there exists α > 0 Since the sequence ki=0 i!1 ∥Ai ∥ ∑∞ 1 i k=0 ∑ 1 i such that the series i=0 i! ∥A ∥ converges to α. Hence, the series ∞ i=0 i! A converges absolutely. ii) follows from i) using Proposition 12.3.4. Next, to prove the second inequality in (15.1.8), note that ∞ ∞ ∞ ∑ ∑ ∑ A i i i ∥A∥ 1 1 1 ∥e ∥ = ≤ A ∥A ∥ ≤ i! ∥A∥ = e . i=0 i! i=0 i! i=0 Finally, 1 ≤ ∥eA ∥∥e−A ∥ ≤ ∥eA ∥e∥A∥ , and thus e−∥A∥ ≤ ∥eA ∥. Proposition 15.1.3. Let A ∈ Fn×n. Then, )k ( eA = lim I + 1k A .

 (15.1.9)

k→∞

Proof. It follows from the binomial theorem that

(

I + 1k A

)k

k ∑

=

αi (k)Ai,

i=0

( ) 1 k! △ 1 k αi (k) = i = i . k i k i!(k − i)!

where

For all i ∈ P, it follows that αi (k) → 1/i! as k → ∞. Hence, k ∞ ∑ ∑ )k ( A 1 i αi (k)Ai = lim I + 1k A = lim i! A = e .

k→∞

k→∞

i=0



i=0

Proposition 15.1.4. Let A ∈ Fn×n. Then, for all t ∈ R,

∫

etA − I =

t

AeτA dτ,

(15.1.10)

0

d tA e = AetA. dt Proof. Note that

∫

t

AeτA dτ =

0

(15.1.11)

∫ t∑ ∞ ∞ ∑ 1 k k+1 1 tk+1 k+1 τ A dτ = A = etA − I, k! k + 1 0 k=0 k! k=0 

which yields (15.1.10), while differentiating (15.1.10) with respect to t yields (15.1.11). Proposition 15.1.5. Let A, B ∈ Fn×n. Then, AB = BA if and only if, for all t ∈ [0, ∞), etAetB = et(A+B).

(15.1.12)

Proof. Suppose that AB = BA. By expanding etA, etB, and et(A+B), it can be seen that the

expansions of etAetB and et(A+B) are identical. Conversely, differentiating (15.1.12) twice with respect to t and setting t = 0 yields AB = BA.  Corollary 15.1.6. Let A, B ∈ Fn×n, and assume that AB = BA. Then, eAeB = eBeA = eA+B. △

The converse of Corollary 15.1.6 is false. Let A = e = −I and e B

A+B

[

0 π −π 0

]

(15.1.13) △

and B =

[

√ ] 0 (7+4 3)π √ . Then, eA (−7+4 3)π 0

=

= e e = e e = I. However, AB , BA. A partial converse is given by Fact 15.15.1. A B

B A

1181

THE MATRIX EXPONENTIAL AND STABILITY THEORY

Proposition 15.1.7. Let A ∈ Fn×n and B ∈ Fm×m. Then,

eA⊗Im = eA ⊗ Im , e e

(15.1.14)

In⊗B

= In ⊗ e ,

(15.1.15)

A⊕B

= e ⊗e .

(15.1.16)

B

A

B

Proof. Note that

eA⊗Im = Inm + A ⊗ Im + 2!1 (A ⊗ Im )2 + · · · = In ⊗ Im + A ⊗ Im + 2!1 (A2 ⊗ Im ) + · · · = (In + A + 2!1 A2 + · · · ) ⊗ Im = eA ⊗ Im and similarly for (15.1.15). To prove (15.1.16), note that (A⊗Im )(In ⊗B) = A⊗B and (In ⊗B)(A⊗Im ) = A ⊗ B, which shows that A ⊗ Im and In ⊗ B commute. Thus, by Corollary 15.1.6, eA⊕B = eA⊗Im +In⊗B = eA⊗ImeIn⊗B = (eA ⊗ Im )(In ⊗ eB ) = eA ⊗ eB.



15.2 Structure of the Matrix Exponential To elucidate the structure of the matrix exponential, recall that, by Theorem 6.6.1, each term Ak △ in (15.1.7), where k ≥ r = deg µA , can be expressed as a linear combination of I, A, . . . , Ar−1. The following result provides an expression for etA in terms of I, A, . . . , Ar−1. Proposition 15.2.1. Let A ∈ Fn×n. Then, for all t ∈ R, I n−1 ∑ 1 (zI − A)−1 etz dz = ψi (t)Ai, (15.2.1) etA = 2πȷ C i=0 where, for all i ∈ {0, . . . , n − 1}, ψi (t) is given by I [i+1] χA (z) tz △ 1 ψi (t) = e dz, 2πȷ C χA (z)

(15.2.2)

where C is a simple, closed contour in the complex plane enclosing spec(A), χA (s) = sn + βn−1 sn−1 + · · · + β1 s + β0 ,

(15.2.3)

[n] and the polynomials χ[1] A , . . . , χA are defined by the recursion [i] sχ[i+1] A (s) = χA (s) − βi ,

where

△ χ[0] A =

χA and

χ[n] A (s)

i = 0, . . . , n − 1,

= 1. Furthermore, for all i ∈ {0, . . . , n − 1} and t ≥ 0, ψi (t) satisfies

ψ(n) i (t)

′

+ βn−1 ψ(n−1) (t) + · · · + β1 ψi (t) + β0 ψi (t) = 0, i

(15.2.4)

where, for all i, j ∈ {0, . . . , n − 1}, ψ(i j) (0) = δi, j .

(15.2.5)

 The coefficient ψi(t) of A in (15.2.1) can be further characterized in terms of the Laplace transform. Define ∫ ∞ △ △ xˆ(s) = L{x(t)} = e−stx(t) dt. (15.2.6) Proof. See [1785, 1881], [2912, p. 31], and Fact 6.9.7. i

0

Then, L{ x˙(t)} = s xˆ(s) − x(0),

(15.2.7)

1182

CHAPTER 15

L{ x¨(t)} = s2 xˆ(s) − sx(0) − x˙(0).

(15.2.8)

The following result shows that the resolvent of A is the Laplace transform of the exponential of A. See (6.4.24). Proposition 15.2.2. Let A ∈ Fn×n, and define ψ0 , . . . , ψn−1 as in Proposition 15.2.1. Then, for all s ∈ C\spec(A), ∫ ∞ L{etA } = e−st etA dt = (sI − A)−1. (15.2.9) 0

Furthermore, for all i ∈ {0, . . . , n − 1}, the Laplace transform ψˆ i (s) of ψi (t) is given by χ[i+1] (s) A , χA (s) n−1 ∑ = ψˆ i (s)Ai.

ψˆ i (s) = (sI − A)−1

(15.2.10) (15.2.11)

i=0

Proof. Let s ∈ C satisfy Re s > αmax (A) so that A − sI is asymptotically stable. It thus follows from Lemma 15.10.2 that ∫ ∞ ∫ ∞ tA −st tA L{e } = e e dt = et(A−sI) dt = (sI − A)−1. 0

0

By analytic continuation, the expression L{e } is given by (15.2.9) for all s ∈ C\ spec(A).  Comparing (15.2.11) with the expression for (sI − A)−1 given by (6.4.24) shows that there exist B0 , . . . , Bn−2 ∈ Fn×n such that tA

n−1 ∑ i=0

sn−1 sn−2 s 1 ψˆ i (s)Ai = I+ Bn−2 + · · · + B1 + B0 . χA (s) χA (s) χA (s) χA (s)

(15.2.12)

To further illustrate the structure of etA , where A ∈ Fn×n, let A = SBS −1, where B = diag(B1 , . . . , Bk ) is the Jordan form of A. Hence, by Proposition 15.2.8, etA = SetBS −1,

(15.2.13)

where etB = diag(etB1 , . . . , etBk ). The structure of e can thus be determined by considering the block Bi ∈ F i ∈ {1, . . . , k} has the form tB

(15.2.14) αi ×αi

Bi = λi Iαi + Nαi .

, which, for all (15.2.15)

Since λi Iαi and Nαi commute, it follows from Proposition 15.1.5 that etBi = et(λi Iαi +Nαi ) = eλi tIαi etNαi = eλi t etNαi . Since

Nααii

= 0, it follows that e

tNαi

(15.2.16)

is a finite sum of powers of tNαi . Specifically,

etNαi = Iαi + tNαi + 12 t2 Nα2i + · · · +

1 tαi −1Nααii −1, (αi −1)!

(15.2.17)

1183

THE MATRIX EXPONENTIAL AND STABILITY THEORY

and thus

etNαi

  1     0    =  0  ..  .    0   0

tαi −2 (αi −2)!

t

t2 2

···

1

t

..

.

tαi −3 (αi −3)!

0 .. .

1 .. .

..

.

tαi −4 (αi −4)!

0

0

..

0

0

..

.

..

.

1

···

0

.

     tαi −2   (αi −2)!    tαi −3   (αi −3)!  , ..  .    t    1 tαi −1 (αi −1)!

(15.2.18)

which is upper triangular and Toeplitz (see Fact 15.14.1). Alternatively, (15.2.18) follows from (12.8.5) with f (s) = e st. Note that (15.2.16) follows from (12.8.5) with f (λ) = eλt . Furthermore, every entry of etBi is of the form r!1 treλi t , where r ∈ {0, αi − 1} and λi is an eigenvalue of A. Reconstructing A by means of A = SBS −1 shows that every entry of A is a linear combination of the entries of the blocks etBi . If A is real, then etA is also real. Thus, the term eλi t for complex λi = νi + ωi ȷ ∈ spec(A), where νi and ωi are real, yields terms of the form eνi t cos ωi t and eνi t sin ωi t. The following result follows from either (15.2.18) or Corollary 12.8.4. Proposition 15.2.3. Let A ∈ Fn×n. Then, mspec(eA ) = {eλ : λ ∈ mspec(A)}ms .

(15.2.19)

Proof. Note that every diagonal entry of the Jordan form of eA is of the form eλ , where λ ∈



spec(A). Corollary 15.2.4. Let A ∈ Fn×n. Then,

det eA = etr A.

(15.2.20)

Corollary 15.2.5. Let A ∈ F , and assume that tr A = 0. Then, det e = 1. Corollary 15.2.6. Let A ∈ Fn×n. Then, the following statements hold: n×n

A

i) If eA is unitary, then spec(A) ⊂ ȷR. ii) spec(eA ) ⊂ R if and only if Im spec(A) ⊂ πZ. Proposition 15.2.7. Let A ∈ Fn×n. Then, the following statements hold: i) A and eA have the same number of Jordan blocks of corresponding sizes. ii) eA is semisimple if and only if A is semisimple. iii) If µ ∈ spec(eA ), then ∑ amult exp(A)(µ) = amultA(λ),

(15.2.21)

{λ∈spec(A): eλ =µ}

gmult exp(A)(µ) =

∑

gmultA(λ).

(15.2.22)

{λ∈spec(A): eλ =µ}

iv) If eA is simple, then A is simple. v) If eA is cyclic, then A is cyclic. vi) eA is a scalar multiple of the identity matrix if and only if A is semisimple and every pair of eigenvalues of A differs by an integer multiple of 2πȷ.

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CHAPTER 15

vii) eA is a real scalar multiple of the identity matrix if and only if A is semisimple, every pair of eigenvalues of A differs by an integer multiple of 2πȷ, and the imaginary part of every eigenvalue of A is an integer multiple of πȷ. Proof. To prove i), note that, for all t , 0, def(etNαi −Iαi ) = 1, and thus the geometric multiplicity of (15.2.18) is 1. Since (15.2.18) has one distinct eigenvalue, it follows that (15.2.18) is cyclic. Hence, by Proposition 7.7.15, (15.2.18) is similar to a single Jordan block. Now, i) follows by setting t = 1 and applying this argument to each Jordan block of A. ii)–v) follow by similar arguments. To prove vi), note that, for all λi , λ j ∈ spec(A), it follows that eλi = eλ j . Furthermore, since A is semisimple, it follows from ii) that eA is also semisimple. Since all of the eigenvalues of eA are equal, it follows that eA is a scalar multiple of the identity matrix. Finally, vii) is an immediate consequence of vii).  Proposition 15.2.8. Let A ∈ Fn×n. Then, the following statements hold: T i) (eA )T = eA . ii) (eA ) = eA. ∗ iii) (eA )∗ = eA . iv) eA is nonsingular, and (eA )−1 = e−A. −1 v) If S ∈ Fn×n is nonsingular, then eSAS = SeAS −1. vi) If A = diag(A1 , . . . , Ak ), where Ai ∈ Fni ×ni for all i ∈ {1, . . . , k}, then eA = diag(eA1 , . . . , eAk ). vii) If A is Hermitian, then eA is positive definite. Furthermore, the following statements are equivalent: viii) A is normal. ∗ ∗ ix) tr eA eA = tr eA +A. ∗ ∗ x) eA eA = eA +A. ∗ ∗ ∗ xi) eAeA = eA eA = eA +A. Finally, the following statements hold: xii) If A is normal, then eA is normal. xiii) eA is normal if and only if A is unitarily similar to a block-diagonal matrix diag(A1 , . . . , Ak ) such that, for all i ∈ {1, . . . , k}, Ai is semisimple and each pair of eigenvalues of Ai differ by an integer multiple of 2πȷ, and, for all distinct i, j ∈ {1, . . . , k}, spec(eAi ) , spec(eAj ). xiv) If eA is normal and no pair of eigenvalues of A differ by an integer multiple of 2πȷ, then A is normal. xv) A is skew Hermitian if and only if A is normal and eA is unitary. xvi) If F = R and A is skew symmetric, then eA is orthogonal and det eA = 1. xvii) If eA is unitary, then either A is skew Hermitian or at least two eigenvalues of A differ by a nonzero integer multiple of 2πȷ. xviii) eA is unitary if and only if A is unitarily similar to a block-diagonal matrix diag(A1 , . . . , Ak ) such that, for all i ∈ {1, . . . , k}, Ai is semisimple, every eigenvalue of Ai is imaginary, and each pair of eigenvalues of Ai differ by an integer multiple of 2πȷ, and, for all distinct i, j ∈ {1, . . . , k}, spec(eAi ) , spec(eAj ). xix) eA is Hermitian if and only if A is unitarily similar to a block-diagonal matrix diag(A1 , . . . , Ak ) such that, for all i ∈ {1, . . . , k}, Ai is semisimple, the imaginary part of every eigenvalue of Ai is an integer multiple of πȷ, and each pair of eigenvalues of Ai differ by an integer multiple of 2πȷ, and, for all distinct i, j ∈ {1, . . . , k}, spec(eAi ) , spec(eAj ). xx) eA is positive definite if and only if A is unitarily similar to a block-diagonal matrix diag(A1 ,

THE MATRIX EXPONENTIAL AND STABILITY THEORY

1185

. . . , Ak ) such that, for all i ∈ {1, . . . , k}, Ai is semisimple, the imaginary part of every eigenvalue of Ai is an integer multiple of 2πȷ, and each pair of eigenvalues of Ai differ by an integer multiple of 2πȷ, and, for all distinct i, j ∈ {1, . . . , k}, spec(eAi ) , spec(eAj ). Proof. The equivalence of viii) and ix) is given in [987, 2476], while the equivalence of viii) and xi) is given in [2395]. xi) =⇒ x) =⇒ ix) is immediate. xii) follows from the fact that viii) =⇒ xi). xiii) is given in [2934]. xiv) is a consequence of xiii). To prove sufficiency in xv), note that ∗ ∗ eA+A = eAeA = eA (eA )∗ = I = e0. Since A + A∗ is Hermitian, it follows from iii) of Proposition T 15.4.3 that A + A∗ = 0. The converse is immediate. To prove xvi), note that eA (eA )T = eA eA = eA e−A = eA (eA )−1 = I, and, using Corollary 15.2.5, det eA = etr A = e0 = 1. To prove xvii), note that it follows from xiii) that, if every block Ai is scalar, then A is skew Hermitian, while, if at least one block Ai is not scalar, then A has at least two eigenvalues that differ by an integer multiple of 2πȷ. Finally, xviii)–xx) are analogous to xiii).  [ ] △ −2π 4π A The converse of xii) is false. Let A = −2π 2π , which satisfies e = I but is not normal. Likewise, [ 1] ∗ ∗ ∗ A = πȷ0 −πȷ satisfies eA = −I but is not normal. For both matrices, eA eA = eAeA = I, but eA eA , ∗ eA +A , which confirms that xi) does not hold. Both matrices have eigenvalues ±πȷ. Proposition 15.2.9. The following statements hold: i) If A, B ∈ Fn×n are similar, then eA and eB are similar. ii) If A, B ∈ Fn×n are unitarily similar, then eA and eB are unitarily similar. [ ] [ ] △ △ 0 2π , which satisfy eA = e B = I but are not The converse of i) is false. Let A = 00 00 and B = −2π 0 similar.

15.3 Explicit Expressions This section presents explicit expressions for the exponential of 2×2 matrices. These expressions are given in terms of either the entries of A or the eigenvalues of A. [ ] △ Lemma 15.3.1. Let A = a0 db ∈ C2×2. Then,      1 b    a  a = d, e    ,    0 1      eA =  (15.3.1)    ea b ea −ed    a−d       , a , d.     0 ed  The following result gives an expression for eA in terms of the eigenvalues of A. Proposition 15.3.2. Let A ∈ C2×2 and mspec(A) = {λ, µ}ms . Then,  λ   e [(1 − λ)I + A], λ = µ, A e =   µeλ −λeµ I + eµ −eλ A, λ , µ. µ−λ µ−λ

(15.3.2)

Proof. This result follows from Theorem 12.8.2. Alternatively, suppose that λ = µ. Then, [ ] there exists a nonsingular matrix S ∈ C2×2 such that A = S λ0 αλ S −1, where α ∈ C. Hence, eA = [ ] eλS 10 α1 S −1 = eλ [(1 − λ)I + A]. Now, suppose that λ , µ. Then, there exists a nonsingular [ ] [λ ] [λ ] matrix S ∈ C2×2 such that A = S λ0 µ0 S −1. Hence, eA = S e0 e0µ S −1. Then, the equality e0 e0µ = [ ] µeλ −λeµ eµ −eλ λ 0  µ−λ I + µ−λ 0 µ yields the desired result.

Next, we give an expression for eA in terms of the entries of A ∈ R2×2.

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Corollary 15.3.3. Let A =

[

]

△

△

∈ R2×2, and define γ = (a − d)2 + 4bc and δ = 21 |γ|1/2. Then,    b   cos δ + a−d sin δ  sin δ  a+d   2δ δ  2   ,   γ < 0, e   c a−d  sin δ cos δ − sin δ   δ 2δ          1 + a−d   b  a+d A 2 γ = 0, e 2   , (15.3.3) e =  a−d  c 1− 2           b    cosh δ + a−d sinh δ  sinh δ  a+d  2δ δ   , γ > 0.    e 2    c a−d sinh δ cosh δ − sinh δ δ 2δ a b c d

√ √ △ γ) and µ = 12 (a + d + γ). Hence, λ = µ if and only if γ = 0. The result now follows from Proposition 15.3.2.  △ [ ν ω] 2×2 Example 15.3.4. Let A = −ω ∈ R . Then, ν [ ] cos ωt sin ωt . (15.3.4) etA = eνt − sin ωt cos ωt △ [ ω] On the other hand, if A = ων −ν , then [ ] ω cosh δt + δν sinh δt δ sinh δt etA = , (15.3.5) ω cosh δt − δν sinh δt δ sinh δt √ △ where δ = ω2 + ν2. ^ [ ] △ Example 15.3.5. Let α ∈ F, and define A = 00 α1 . Then,     1 α−1 (eαt − 1)      , α , 0,       0 eαt   ^ etA =         1 t     α = 0.   0 1  , △

Proof. The eigenvalues of A are λ = 12 (a + d −

] . Then, [ ] cos θ sin θ A e = . − sin θ cos θ △

Example 15.3.6. Let θ ∈ R, and define A =

△

Furthermore, define B =

[

0 −π 2 +θ

π 2 −θ

0

]

[

0 θ −θ 0

. Then, [ eB =

sin θ − cos θ

] cos θ . sin θ

^

Example 15.3.7. Consider the second-order mechanical vibration equation

mq¨ + cq˙ + kq = 0,

(15.3.6)

where m is positive and c and k are nonnegative. Here m, c, and k denote mass, damping, and stiffness parameters, respectively. Equation (15.3.6) can be written in companion form as x˙ = Ax, where △

x=

[ ] q , q˙

△

A=

[

] 0 1 . −k/m −c/m

(15.3.7)

(15.3.8)

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THE MATRIX EXPONENTIAL AND STABILITY THEORY

In the inelastic case, k = 0, and thus A is upper triangular. In this case,    1 t      ,   k = c = 0,     0 1    tA e =      1 m (1 − e−ct/m )     c   , k = 0, c > 0,      0 e−ct/m

(15.3.9)

where c = 0 and c > 0 correspond to a rigid body and a damped rigid body, respectively. Next, we consider the elastic case where c ≥ 0 and k > 0. In this case, we define √ k c △ △ (15.3.10) ωn = , ζ= √ , m 2 mk where ωn > 0 denotes the (undamped) natural frequency of vibration and ζ ≥ 0 denotes the damping ratio. Now, A can be written as [ ] 0 1 A= , (15.3.11) −ω2n −2ζωn and Corollary 15.3.3 yields   1   cos ωn t  sin ωn t   ω  n  ,      −ωn sin ωn t cos ωn t           1   cos ωd t + √ ζ sin ωd t   sin ωd t  ω  2 d    1−ζ  −ζωn t     ,  e     −ωd ζ     √ sin ω t cos ω t − sin ω t  d d d 2  1−ζ  1−ζ 2 tA e =        1 + ωnt  t   −ωnt   ,   e     −ω2n t 1 − ωnt           1    cosh ωd t + √ ζ sinh ωd t  sinh ω t d  ω  d  ζ 2 −1  −ζωnt     e  ,  −ωd ζ    √ sinh ω t sinh ω t cosh ω t − d  d d  ζ 2 −1 2 ζ −1

ζ = 0,

0 < ζ < 1,

ζ = 1,

ζ > 1,

where ζ = 0, 0 < ζ < 1, ζ = 1, and ζ > 1 correspond to undamped, underdamped, critically damped, and overdamped oscillators, and where the damped natural frequency ωd is the positive number  √   ω 1 − ζ 2 , 0 < ζ < 1, △  n ωd =  (15.3.12) √   ω ζ 2 − 1, ζ > 1. n

15.4 Matrix Logarithms Definition 15.4.1. Let A ∈ Cn×n. Then, B ∈ Cn×n is a logarithm of A if eB = A.

The following result shows that every complex, nonsingular matrix has a complex logarithm. Proposition 15.4.2. Let A ∈ Cn×n. Then, there exists B ∈ Cn×n such that A = eB if and only if A is nonsingular. Proof. See [1299, pp. 35, 60] and [1450, p. 474].  Proposition 15.4.3. The following statements hold: i) B ∈ Cn×n is positive definite if and only if there exists a Hermitian matrix A ∈ Fn×n such that eA = B. If these conditions hold, then A is unique.

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ii) B ∈ Cn×n is Hermitian and nonsingular if and only if there exists a normal matrix A ∈ Cn×n such that, for all λ ∈ spec(A), Im λ is an integer multiple of πȷ and eA = B. iii) B ∈ Cn×n is normal and nonsingular if and only if there exists a normal matrix A ∈ Cn×n such that eA = B. iv) B ∈ Cn×n is unitary if and only if there exists a skew-Hermitian matrix A ∈ Cn×n such that eA = B. Proof. To prove i), let B = S diag(b1 , . . . , bn )S −1, where S ∈ Fn×n is unitary and b1 , . . . , bn are △ positive. Then, define A = S diag(log b1 , . . . , log bn )S −1. Next, to prove uniqueness, let A and Aˆ be ˆ ˆ Hermitian matrices such that B = eA = eA. Then, for all t ≥ 0, it follows that etA = (eA )t = (eA )t = ˆ ˆ ˆ t A , while setting t = 0 implies that A = A. ˆ As another proof, x) et A. Differentiating yields AetA = Ae A −Aˆ A−Aˆ ˆ of Fact 15.15.1 implies that A and A commute. Therefore, I = e e = e , which, since A − Aˆ is Hermitian, implies that A − Aˆ = 0. The result also follows from xiii) of Fact 15.15.1. iv) is given by v) of Proposition 15.6.7.  Corollary 15.4.4. Let A, B ∈ Cn×n, assume that A and B are normal, and assume that eA = eB . Then, A + A∗ = B + B∗. ∗ ∗ Proof. Note that eA+A = eB+B , which, by vii) of Proposition 15.2.8, is positive definite. The result now follows from i) of Proposition 15.4.3.  [ ] 0 π Although the real number −1 does not have a real logarithm, the real matrix B = −π 0 satisfies [ ] B 0 e = −1 0 −1 . The following result characterizes the real matrices that have a real logarithm. Proposition 15.4.5. Let A ∈ Rn×n. Then, there exists B ∈ Rn×n such that A = eB if and only

if A is nonsingular and, for every negative eigenvalue λ of A and for every positive integer k, the Jordan form of A has an even number of k × k blocks associated with λ. Proof. See [1450, p. 475].  n×n n×n Corollary 15.4.6. Let A ∈ R be nonsingular. Then, there exists C ∈ R such that A = C 2 n×n if and only if there exists B ∈ R such that A = eB . 1

Proof. To prove sufficiency, let C = e 2 B . To prove necessity, let λ ∈ spec(A), and assume that

λ < 0. Hence, there exists µ ∈ spec(C) ∩ IA such that µ2 = λ. Thus, µ ∈ spec(C), µ2 = λ, and amultC (µ) = amultC (µ). Therefore, for every positive integer k, the Jordan form of A has an even number of k × k blocks associated with λ.  n×n n×n Corollary 15.4.7. Let A ∈ R be nonsingular, and let k ≥ 1. Then, there exists B ∈ R such that A2k = eB . Proposition 15.4.8. Let B ∈ Rn×n . Then, the following statements hold: i) B is positive definite if and only if there exists a symmetric matrix A ∈ Rn×n such that eA = B. If these conditions hold, then A is unique. ii) B is symmetric and I ≤ B if and only if there exists a positive-semidefinite matrix A ∈ Rn×n such that eA = B. iii) B is symmetric and I < B if and only if there exists a positive-definite matrix A ∈ Rn×n such that eA = B. iv) B is normal and nonsingular and, for all λ ∈ spec(B), amultB (λ) is even if and only if there exists a normal matrix A ∈ Rn×n such that eA = B. v) B is normal, nonsingular, and, for all λ ∈ spec(B), amultB (λ) is even and | log λ| = 1 if and only if there exists an orthogonal matrix A ∈ Rn×n such that eA = B. vi) B is orthogonal and det B = 1 if and only if there exists a skew-symmetric matrix A ∈ Rn×n such that eA = B. Proof. See [2442]. 

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THE MATRIX EXPONENTIAL AND STABILITY THEORY

Replacing A and B in Proposition 15.4.5 by eA and A, respectively, yields the following result. Corollary 15.4.9. Let A ∈ Rn×n. Then, for every negative eigenvalue λ of eA and for every positive integer k, the Jordan form of eA has an even number of k × k blocks associated with λ. [ ] △ A 4π Since the matrix A = −2π −2π 2π satisfies e = I, it follows that a positive-definite matrix can have a logarithm that is not normal. However, the following result shows that every positive-definite matrix has exactly one Hermitian logarithm. Proposition 15.4.10. The function exp: Hn 7→ Pn is one-to-one and onto. Proof. The result follows from vii) of Proposition 15.2.8 and iii) of Proposition 15.4.3.  Let A ∈ Rn×n. If there exists B ∈ Rn×n such that A = eB, then Corollary 15.2.4 implies that [ ] △ 0 det A = det eB = etr B > 0. However, the converse is not true. Consider, for example, A = −1 0 −2 , which satisfies det A > 0. However, Proposition 15.4.5 implies that there does [ ]not exist△ a[ matrix ] △ 0 0 0 π and C = B ∈ R2×2 such that A = eB. On the other hand, note that A = eBeC, where B = −π 0 log 2 . 0 While the product of two exponentials of real matrices has positive determinant, the following result shows that the converse is also true. Proposition 15.4.11. Let A ∈ Rn×n. Then, there exist B, C ∈ Rn×n such that A = eBeC if and only if det A > 0. Proof. Suppose that there exist B, C ∈ Rn×n such that A = eBeC. Then, it follows from Corollary 15.2.4 that det A = (det eB )(det eC ) = etr B etr C > 0. Conversely, suppose that det A > 0. In the case where A has no negative eigenvalues, Proposition 15.4.5 implies that there exists B ∈ Rn×n such that A = eB. Hence, A = eB e0n×n . Now, suppose that A has at least one negative eigenvalue. Then, Theorem 7.4.6 on the real Jordan form implies that there[exist]a nonsingular matrix S ∈ Rn×n and matrices A1 ∈ Rn1 ×n1 and A2 ∈ Rn2 ×n2 such that A = S A01 A02 S −1, where every eigenvalue of A1 is negative and none of the eigenvalues of A2 are negative. Since det A and det A are positive, it [ −In 0 ] [ −A1 20 ] 1 follows that det A1 is positive, and thus n1 is even. Now, write A = S 0 In S −1, and note 0 A2 2 [ −In 0 ] ˆ that 0 1 In = eB , where 2 [ ]    In1 /2 ⊗ π0 −π 0n1 ×n2  △  0 ˆ   . B= 0n2 ×n1 0n2 ×n2 [ ] Furthermore, since −A0 1 A02 has no negative eigenvalues, it follows from Proposition 15.4.5 that there [ ] ˆ ˆ ˆ ˆ −1 ˆ −1 exists Cˆ ∈ Rn×n such that −A0 1 A0 = eC. Hence, eA = SeBeC S −1 = eSBS eSCS . See [2091].  2

Although eA eB may be different from eA+B, the following result, known as the Baker-CampbellHausdorff series, provides an expansion of a matrix function C(t) that satisfies eC(t) = etA etB. Proposition 15.4.12. Let A1 , . . . , Al ∈ Fn×n. Then, there exists ε > 0 such that, for all t ∈ (−ε, ε), etA1 · · · etAl = eC(t),

(15.4.1)

where, as t → 0, △

C(t) =

l ∑ i=1

∑

tAi +

1 2 2 t [Ai , Aj ]

+ O(t3 ).

(15.4.2)

1≤i< j≤l



Proof. See [1299, Chapter 3], [2379, p. 35], and [2781, p. 97].

To illustrate (15.4.1), let l = 2, A = A1 , and B = A2 . Then, the first few terms of the series are etAetB = etA+tB+(t /2)[A,B]+(t /12)[[B,A],A+B]+··· . 2

3

The radius of convergence of this series is discussed in [490, 508, 842, 994, 2126, 2304]. The following result is the Lie-Trotter product formula.

(15.4.3)

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Corollary 15.4.13. Let A, B ∈ Fn×n. Then,

( 1 1 )p eA+B = lim e p Ae p B . p→∞

(15.4.4)

Proof. Setting l = 2 and t = 1/p in (15.4.1) yields, as p → ∞,

(

1

1

e p Ae p B

)p

( 1 )p 2 = e p (A+B)+O(1/p ) = eA+B+O(1/p) → eA+B.



15.5 Principal Logarithm Let A ∈ Fn×n be positive definite so that A = SBS ∗ ∈ Fn×n, where S ∈ Fn×n is unitary and B ∈ Rn×n is diagonal with positive diagonal entries. In Section 8.5, log A is defined as log A = S (log B)S ∗ ∈ △ Hn, where (log B)(i,i) = log B(i,i) . Since log A satisfies A = elog A, it follows that log A is a logarithm of A. The following result extends the definition of log A to a larger class of matrices A ∈ Cn×n. A logarithm function defined on the set of nonsingular matrices can be based on the principal branch of the log function given by Fact 2.21.29 along with Definition 12.8.1; however, this function is not continuous, and thus an alternative approach to defining the principal logarithm is taken. Definition 15.5.1. Let A ∈ Cn×n, and assume that A has no eigenvalues in (−∞, 0]. Then, the principal logarithm of A is the unique logarithm of A whose eigenvalues are elements of {z ∈ C : | Im z| < π}. The notation log A denotes the principal logarithm of A. Theorem 15.5.2. Let A ∈ Cn×n. Then, the following statements hold: i) If A is nonsingular, then log A is a logarithm of A; that is, elog A = A. ii) log eA = A if and only if, for all λ ∈ spec(A), it follows that | Im λ| < π. iii) If A is nonsingular and ρmax (A − I) < 1, then log A is given by the series log A =

∞ ∑ (−1)i+1 i=1

i

(A − I)i,

(15.5.1)

which converges absolutely for every submultiplicative norm ∥ · ∥ such that ∥A − I∥ < 1. iv) If spec(A) ⊂ ORHP, then log A is given by the series log A =

∞ ∑

2 [(A − I)(A + I)−1 ]2i+1. 2i + 1

i=0

v) If A has no eigenvalues in (−∞, 0], then ∫ 1 log A = (A − I)[t(A − I) + I]−1 dt.

(15.5.2)

(15.5.3)

0

vi) If A has no eigenvalues in (−∞, 0] and α ∈ [−1, 1], then log Aα = α log A.

(15.5.4)

In particular, log A−1 = − log A,

log A1/2 = 12 log A.

(15.5.5)

vii) If A is real and spec(A) ⊂ ORHP, then log A is real. viii) If A is real and nonsingular, then A has a real logarithm if and only if A is nonsingular and, for every negative eigenvalue λ of A and for every positive integer k, the Jordan form of A has an even number of k × k blocks associated with λ. Now, let ∥ · ∥ be a submultiplicative norm on Cn×n. Then, the following statements hold: ix) The function log is continuous on {X ∈ Cn×n : ∥X − I∥ < 1}.

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THE MATRIX EXPONENTIAL AND STABILITY THEORY

x) If B ∈ Cn×n and ∥B∥ < log 2, then ∥eB − I∥ < 1 and log eB = B. xi) exp: Blog 2 (0) 7→ Fn×n is one-to-one. xii) If ∥A − I∥ < 1, then ∥ log A∥ ≤ −log(1 − ∥A − I∥) ≤ xiii) If ∥A − I∥ < 2/3, then

( ∥A − I∥ 1 −

∥A − I∥ . 1 − ∥A − I∥

) ∥A − I∥ ≤ ∥ log A∥. 2(1 − ∥A − I∥)

(15.5.6)

(15.5.7)

xiv) Assume that A is nonsingular, and let mspec(A) = {λ1 , . . . , λn }ms . Then, mspec(log A) = {log λ1 , . . . , log λn }ms .

(15.5.8)

log det A = tr log A.

(15.5.9)

Hence,

Proof. i) follows from the discussion in [1450, p. 420]; ii) is given in [1391, p. 32]; iii) and iv) are given by Fact 13.4.15, see [1299, pp. 34–35] and [1391, p. 273]; v) is given in [1391, p. 269]; vi) is given in [1391, p. 270]; vii) is immediate; viii) follows from Proposition 15.4.5 and the discussion in [1450, pp. 474–475]. ix) and x) are proved in [1299, pp. 34–35]. To prove the inequality in x), let ∥B∥ < 2, so that e∥B∥ < 2, and thus

∥eB − I∥ ≤

∞ ∑ (i!)−1 ∥B∥i = e∥B∥ − 1 < 1. i=1

To prove xi), let B1 , B2 ∈ Blog 2 (0), and assume that eB1 = eB2. Then, ii) implies that B1 = log eB1 = △ log eB2 = B2 . Finally, to prove xii), let α = ∥A − I∥ < 1. Then, it follows from (15.5.1) and iv) of Fact ∑∞ i  2.21.29 that ∥ log A∥ ≤ i=1 α /i = −log(1 − α). For xiii), see [1391, p. 647].

15.6 Lie Groups Definition 15.6.1. Let S ⊂ Fn×n, and assume that S is a multiplication group. Then, S is a Lie

group if S is closed relative to GLF (n). Proposition 15.6.2. Let S ⊂ Fn×n, and assume that S is a multiplication group. Then, S is a Lie group if and only if the limit of every convergent sequence in S is either an element of S or is singular. The multiplication groups SLF (n), U(n), O(n), SU(n), SO(n), U(n, m), O(n, m), SU(n, m), SO(n, m), SympF (2n), Aff F (n), SEF (n), and TransF (n) defined in Proposition 4.6.6 are closed sets, and thus are Lie groups. Although the multiplication groups GLF (n), PLF (n), and UT(n) (see Fact 4.31.11) are not closed sets, they are closed relative to GLF (n), and thus they are Lie groups. Finally, the multiplication group S ⊂ C2×2 defined by {[ ȷt ] } e 0 △ S= : t ∈ R (15.6.1) 0 eπȷt is not closed relative to GLC (2), and thus is not a Lie group. For details, see [1299, p. 4]. Proposition 15.6.3. Let S ⊂ Fn×n, and assume that S is a Lie group. Furthermore, define { } △ S0 = A ∈ Fn×n : etA ∈ S for all t ∈ R . (15.6.2) Then, S0 is a Lie algebra. Proof. See [1299, pp. 39, 43, 44].



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The Lie algebra S0 defined by (15.6.2) is the Lie algebra of S. Proposition 15.6.4. Let S ⊂ Fn×n, assume that S is a Lie group, and let S0 ⊆ Fn×n be the Lie algebra of S. Furthermore, let S ∈ S and A ∈ S0 . Then, SAS −1 ∈ S0 . −1 Proof. For all t ∈ R, etA ∈ S, and thus etSAS = SetAS −1 ∈ S. Hence, SAS −1 ∈ S0 .  Proposition 15.6.5. The following statements hold: i) glF (n) is the Lie algebra of GLF (n). ii) glR (n) = plR (n) is the Lie algebra of PLR (n). iii) plC (n) is the Lie algebra of PLC (n). iv) slF (n) is the Lie algebra of SLF (n). v) u(n) is the Lie algebra of U(n). vi) so(n) is the Lie algebra of O(n). vii) su(n) is the Lie algebra of SU(n). viii) so(n) is the Lie algebra of SO(n). ix) su(n, m) is the Lie algebra of U(n, m). x) so(n, m) is the Lie algebra of O(n, m). xi) su(n, m) is the Lie algebra of SU(n, m). xii) so(n, m) is the Lie algebra of SO(n, m). xiii) sympF (2n) is the Lie algebra of SympF (2n). xiv) osympF (2n) is the Lie algebra of OSympF (2n). xv) affF (n) is the Lie algebra of Aff F (n). xvi) seC (n) is the Lie algebra of SEC (n). xvii) seR (n) is the Lie algebra of SER (n). xviii) transF (n) is the Lie algebra of TransF (n). Proof. See [1299, pp. 38–41].  Proposition 15.6.6. Let S ⊂ Fn×n, assume that S is a Lie group, and let S0 ⊆ Fn×n be the Lie algebra of S. Then, exp: S0 7→ S. Furthermore, if exp is onto, then S is pathwise connected. Proof. Let A ∈ S0 so that etA ∈ S for all t ∈ R. Hence, setting t = 1 implies that exp : S0 7→ S. △ Now, suppose that exp is onto, let B ∈ S, and let A ∈ S0 satisfy eA = B. Then, f (t) = etA satisfies f (0) = I and f (1) = B, which implies that S is pathwise connected.  A Lie group can consist of multiple pathwise-connected components. Proposition 15.6.7. Let n ≥ 1. Then, the following functions are onto: i) exp: glC (n) 7→ GLC (n). ii) exp: glR (1) 7→ PLR (1). iii) exp: plC (n) 7→ PLC (n). iv) exp: slC (n) 7→ SLC (n). v) exp: u(n) 7→ U(n). vi) exp: su(n) 7→ SU(n). vii) exp: so(n) 7→ SO(n). Furthermore, the following functions are not onto: viii) exp: glR (n) 7→ PLR (n), where n ≥ 2. ix) exp: slR (n) 7→ SLR (n). x) exp: so(n) 7→ O(n).

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xi) exp: sympR (2n) 7→ SympR (2n). Proof. i) follows from Proposition 15.4.2; ii) is immediate; iii)–vii) can be by construc[ verified ] △ 0 and Proposition tion; see [2263, pp. 199, 212] for the proof of v) and vii). The example A = −1 0 −2 15.4.5 ] show that viii) is not onto. For λ < 0, where λ , −1, Proposition 15.4.5 and the example [ λ 0 0 1/λ given in [2379, p. 39] show that ix) is not onto; see also [218, pp. 84, 85]. viii) shows that x) is not onto. For xi), see [896].  Proposition 15.6.8. The Lie groups GLC (n), SLF (n), U(n), SU(n), and SO(n) are pathwise connected. The Lie groups GLR (n), O(n), O(n, 1), and SO(n, 1) are not pathwise connected. Proof. See [1299, p. 15].  Proposition 15.6.8 and ix) of Proposition 15.6.7 show that the converse of Proposition 15.6.6 does not hold; that is, pathwise connectedness does not imply that exp is onto. See [2379, p. 39].

15.7 Linear Time-Varying Differential Equations Let A: [0, ∞) 7→ Fn×n be continuous, let t ∈ [0, ∞), let x0 ∈ Fn , and consider x˙(t) = A(t)x(t), x(0) = x0 .

(15.7.1) (15.7.2)

Theorem 15.7.1. There exists a unique C1 solution x: [0, ∞) 7→ Fn of (15.7.1) and (15.7.2).

Furthermore, for all t ∈ [0, ∞), x(t) is given by the series ∫ tn−1 ∞ ∫ t ∑ x(t) = x0 + ··· A (t1 ) · · · A (tn ) dtn · · · dt1 x0 , n=1

0

(15.7.3)

0

which, for all T > 0, converges absolutely and uniformly on [0, T ]. △ Proof. To prove existence, let T > 0 and define c = maxt∈[0,T ] σmax [A(t)]. For all t ∈ [0, T ], △ define ψ0 : [0, T ] 7→ Fn by ψ0 (t) = x0 , and, for all k ≥ 1 and t ∈ [0, T ], define ψk : [0, T ] 7→ Fn by ∫ t ∫ tk−1 △ ψk (t) = ··· A(t1 ) · · · A(tk )x0 dtk · · · dt1 . (15.7.4) 0

Then, for all k ≥ 1,

∫ tk−1 ··· σmax [A(t1 )] · · · σmax [A(tk )] dtk · · · dt1 ∥x0 ∥2 0 0 ∫ t ∫ t 1 = ··· σmax [A(t1 )] · · · σmax [A(tk )]∥ dtk · · · dt1 ∥x0 ∥2 k! 0 0 ck T k ≤ ∥x0 ∥2 . k!

max ∥ψk (t)∥2 ≤

t∈[0,T ]

∫

0

t

(15.7.5)

Since ∞ ∑ ck T k k=0

k!

∥x0 ∥2 = exp(cT )∥x0 ∥2 ,

(15.7.6)

∑ Fact 12.16.15 implies that ∞ k=0 ψk (t) converges absolutely and uniformly on [0, T ]. Furthermore, Fact 12.16.16 implies that x : [0, T ] 7→ Fn defined by △

x(t) =

∞ ∑ k=0

ψk (t)

(15.7.7)

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is continuous. Note that, for all t ∈ [0, T ], x(t) = lim ϕn (t), n→∞

(15.7.8)

where, for all n ≥ 0 and all t ∈ [0, T ], ϕn : [0, T ] 7→ Fn is the partial sum △

ϕn (t) =

n ∑

ψk (t).

(15.7.9)

k=0

Next, for all n ≥ 0, ϕn is differentiable and, furthermore, for all n ≥ 2 and t ∈ [0, T ], ∫ t ∫ tk−1 n ∑ ˙ϕn (t) = A(t)x0 + A(t) ··· A(t2 ) · · · A(tk )dtk · · · dt2 x0 0

k=2

 ∫ n−1 ∫ t ∑  = A(t)  x0 + ··· k=1

0

0 tk−1

0

  A(t1 ) · · · A(tk )dtk · · · dt1 x0  = A(t)ϕn−1 (t).

(15.7.10)

Since A is continuous on [0, T ] and (ϕn )∞ n=1 converges to x on [0, T ], it follows from (15.7.10) that, for all t ∈ [0, T ], lim ϕ˙ n (t) = lim A(t)ϕn−1 (t) = A(t)x(t).

n→∞

n→∞

(15.7.11)

˙ ∞ Since (ϕn )∞ n=1 converges uniformly to x on [0, T ], it follows from (15.7.11) that (ϕn )n=1 converges ∞ uniformly on [0, T ]. Since, for all n ≥ 1, ϕn is continuous on [0, T ] and (ϕn )n=1 converges uniformly on [0, T ], it follows from Fact 12.16.16 that x is differentiable on [0, T ] and satisfies ∞ ∑ x˙(t) = lim ϕ˙ n (t) = ψ˙ k (t) = A(t)x(t). (15.7.12) n→∞

k=0

Hence, x satisfies (15.7.1) and (15.7.2). To prove uniqueness of the solution (15.7.3), let x satisfy (15.7.1), and let y satisfy y˙ (t) = A(t)y(t),

(15.7.13)

y(0) = x0 .

(15.7.14)

z˙(t) = A(t)z(t),

(15.7.15)

z(0) = 0.

(15.7.16)

△

Then, for all t ≥ 0, z(t) = x(t) − y(t) satisfies

Integrating (15.7.15) yields

∫ z(t) =

t

A(s)z(s) ds.

(15.7.17)

0

Hence, for all t ≥ 0, ∥z(t)∥2 ≤ f (t), where △

∫

f (t) =

t

σmax [A(s)]∥z(s)∥2 ds.

(15.7.18)

(15.7.19)

0

Differentiating (15.7.19) with respect to t, and using (15.7.18), yields f˙(t) = σmax [A(t)]∥z(t)∥2 ≤ σmax [A(t)] f (t).

(15.7.20)

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Since, for all t ≥ 0, σmax [A(t)] ≥ 0, it follows from (15.7.20) that, for all t ≥ 0, ∫ t ∫q ∫t d [ f (q) exp− 0 σmax [A(s)]ds ] dq 0 ≤ f (t) exp− 0 σmax [A(s)] ds = 0 dq ∫ t ∫q ˙[ f (q) − σmax [A(q)] f (q)] exp− 0 σmax [A(s]ds dq ≤ 0. = 0

Hence, for all t ≥ 0, f (t) = 0, and thus (15.7.18) implies that, for all t ≥ 0, z(t) = 0. Therefore, (15.7.3) is the unique solution of (15.7.1), (15.7.2). 

15.8 Lyapunov Stability Theory Let D ⊆ Rn , assume that D is open, let f : D → Rn , and assume that f is locally Lipschitz. Consider the ordinary differential equation x˙(t) = f [x(t)],

(15.8.1)

where t ≥ 0. Throughout this section, let ∥ · ∥ be a norm on Rn. We first consider the case n = 1. Theorem 15.8.1. Assume that n = 1. Then, for all x0 ∈ D, exactly one of the following statements holds: i) There exists T x0 > 0 and a unique C1 solution x: [0, T x0 ) 7→ D satisfying (15.8.1) and x(0) = x0 and such that either limt→T x0 x(t) ∈ bd D or limt→T x0 |x(t)| = ∞. ii) There exists a unique C1 solution x: [0, ∞) 7→ D satisfying (15.8.1) and x(0) = x0 . Furthermore, the following statements hold: iii) If i) holds and f is bounded, then limt→T x0 x(t) ∈ bd D. iv) If D = R and f is either bounded or globally Lipschitz, then ii) holds. Now, we consider the more general case where n ≥ 1. Theorem 15.8.2. For all x0 ∈ D, exactly one of the following statements holds: i) There exists T x0 > 0 and a unique C1 solution x: [0, T x0 ) 7→ D satisfying (15.8.1) and x(0) = x0 and such that, for every compact set D′ ⊂ D, there exists T ∈ [0, T x0 ) such that, for all t ∈ (T, T x0 ), x(t) < D′ . ii) There exists a unique C1 solution x: [0, ∞) 7→ D satisfying (15.8.1) and x(0) = x0 . Furthermore, the following statements hold: iii) If i) holds and f is bounded, then limt→T x0 x(t) ∈ bd D. iv) If D = Rn and f is either bounded or globally Lipschitz, then ii) holds. Proof. The result follows from [1297, Theorem 2.1, Theorem 3.1].  Consider the example r˙ =

1 , 1−r

θ˙ =

1 , (1 − r)2

(15.8.2)

which represents (15.8.1) with n = 2 and D = B1 (0)\{0}, where x is expressed in polar coordinates. For all r0 ∈ (0, 1), it follows that √ r(t) = 1 − (1 − r0 )2 − 2t, (15.8.3) △

where r(0) = r0 . Therefore, r(t) → 1 as t → T x0 = 12 (1 − r0 )2 . Furthermore, θ(t) → ∞ as t → T x0 . Since r(t) and θ(t) are increasing, it follows that x(t) spirals counterclockwise toward the unit circle with increasing speed, encircling the origin an infinite number of times in a finite interval of time. Consequently, x(t) does not converge as t approaches T x0 . Since, for all ε ∈ (0, 1), x(t) leaves Bε (0) in finite time, it follows that i) of Theorem 15.8.2 holds. The fact that x(t) does not converge as t

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approaches T x0 is consistent with the fact that f is not bounded on D. If xe ∈ D satisfies f (xe ) = 0, then x(t) ≡ xe is an equilibrium of (15.8.1). The following definition concerns the stability of an equilibrium of (15.8.1). Definition 15.8.3. Let xe ∈ D be an equilibrium of (15.8.1). i) xe is Lyapunov stable if, for all ε > 0 such that cl Bε (xe ) ⊂ D, there exists δ ∈ (0, ε) such that, for all x0 ∈ Bδ (xe ), there exists a unique C1 solution x : [0, ∞) 7→ Bε (xe ) satisfying (15.8.1) and x(0) = x0 . ii) xe is asymptotically stable if it is Lyapunov stable and there exists δ > 0 such that Bδ (xe ) ⊆ D and such that, for all x0 ∈ Bδ (xe ), there exists a unique C1 solution x : [0, ∞) 7→ Bδ (xe ) satisfying (15.8.1), x(0) = x0 , and limt→∞ x(t) = xe . iii) xe is globally asymptotically stable if it is Lyapunov stable, D = Rn, and, for all x0 ∈ Rn, there exists a unique C1 solution x : [0, ∞) 7→ Rn satisfying (15.8.1), x(0) = x0 , and limt→∞ x(t) = xe . iv) xe is unstable if it is not Lyapunov stable. Note that, if xe ∈ Rn is a globally asymptotically stable equilibrium of (15.8.1), then xe is the unique equilibrium of (15.8.1). Lyapunov’s direct method gives sufficient conditions for Lyapunov stability and asymptotic stability of an equilibrium of (15.8.1). The following definition is needed. Definition 15.8.4. The function V: Rn 7→ R is radially unbounded if, for every sequence ∞ (xi )i=1 ⊂ Rn such that limi→∞ ∥xi ∥ = ∞, it follows that limi→∞ V(xi ) = ∞. Theorem 15.8.5. Let xe ∈ D be an equilibrium of (15.8.1), and let V: D 7→ R be a C1 function such that V(xe ) = 0 and such that, for all x ∈ D\{xe }, V(x) > 0. Then, the following statements hold: i) Assume that, for all x ∈ D, V ′(x)f (x) ≤ 0.

(15.8.4)

Then, xe is Lyapunov stable. ii) Assume that, for all x ∈ D\{xe }, V ′(x)f (x) < 0.

(15.8.5)

Then, xe is asymptotically stable. If, in addition, D = R and V is radially unbounded, then xe is globally asymptotically stable. Proof. To prove i), let ε > 0 satisfy cl Bε (xe ) ⊂ D. Since Sε (xe ) is a compact subset of D and V is continuous on Sε (xe ), Corollary 12.4.12 implies that there exists x1 ∈ Sε (xe ) such that, for all x ∈ Sε (xe ), V(x1 ) ≤ V(x). Since x1 ∈ Sε (xe ) and V(x) > 0 for all x ∈ Sε (xe ), it follows that V(x1 ) > 0. △ Next, since V is continuous, it follows from Theorem 12.4.9 that D1 = {x ∈ Bε (xe ) : V(x) < V(x1 )} is open. Since D1 is open and contains xe , it follows that there exists δ ∈ (0, ε) such that Bδ (xe ) ⊆ D1 . Therefore, for all x ∈ Bδ (xe ), V(x) < V(x1 ). Now, let x0 ∈ Bδ (xe ), and suppose that i) of Theorem 15.8.2 holds. Then, there exists T x0 > 0 and a unique C1 solution x: [0, T x0 ) 7→ D satisfying (15.8.1) and x(0) = x0 . It therefore follows from (15.8.4) that, for all t ∈ [0, T x0 ), ∫ t V[x(t)] − V[x(0)] = V ′ [x(τ)] f [x(τ)] dτ ≤ 0. (15.8.6) n

0

Since x(0) ∈ Bδ (xe ), it follows that, for all t ∈ [0, T x0 ), V[x(t)] ≤ V[x(0)] < V(x1 ). Since, for all x ∈ Sε (xe ), V(x) ≥ V(x1 ), it follows that, for all t ∈ [0, T x0 ), ∥x(t) − xe ∥ , ε. Next, let t1 ∈ [0, T x0 ) satisfy ∥x(t1 ) − xe ∥ > ε. Then, since ∥x0 − xe ∥ < ε, it follows from the intermediate value theorem that there exists t2 ∈ (0, t1 ) such that ∥x(t2 ) − xe ∥ = ε, which is a contradiction. Therefore, i) of

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Theorem 15.8.2 does not hold. Since ii) of Theorem 15.8.2 holds, it follows that there exists a unique C1 solution x: [0, ∞) 7→ D satisfying (15.8.1) and x(0) = x0 . It therefore follows from (15.8.4) that, for all t ∈ [0, ∞), (15.8.6) holds. Since x(0) ∈ Bδ (xe ), it follows that, for all t ∈ [0, ∞), V[x(t)] ≤ V[x(0)] < V(x1 ). Since, for all x ∈ Sε (xe ), V(x) ≥ V(x1 ), it follows that, for all t ∈ [0, ∞), ∥x(t) − xe ∥ , ε. Next, suppose there exists t1 > 0 such that ∥x(t1 ) − xe ∥ > ε. Then, since ∥x0 − xe ∥ < ε, it follows from the intermediate value theorem that there exists t2 ∈ (0, t1 ) such that ∥x(t2 ) − xe ∥ = ε, which is a contradiction. Hence, for all t ∈ [0, ∞), x(t) ∈ Bε (xe ), and thus xe is Lyapunov stable. To prove ii), note that, since (15.8.5) holds for all x ∈ D\{xe } and since f (xe ) = 0, it follows that (15.8.4) holds for all x ∈ D. Therefore, xe = 0 is Lyapunov stable. Let ε > 0 satisfy cl Bε (xe ) ⊆ D, and let δ ∈ (0, ε) be such that Bδ (xe ) ⊆ D and such that, for all x0 ∈ Bδ (xe ), there exists a unique solution x : [0, ∞) 7→ Bε (xe ) satisfying (15.8.1) and x(0) = x0 . Let x0 ∈ Bδ (xe ). Then, for all t ≥ 0, dtd V[x(t)] = V ′ [x(t)] f [x(t)] < 0, and thus V[x(t)] is a decreasing function on [0, ∞) that is bounded from below by zero. Now, suppose that V[x(t)] does not converge to zero. Thus, there exists L > 0 such that, for all t ≥ 0, V[x(t)] ≥ L. Since △ D1 = {x ∈ Bδ (xe ) : V(x) < L} is open and contains xe , there exists η > 0 such that Bη (xe ) ⊆ D1 . △ Therefore, for all t ≥ 0, ∥x(t) − xe ∥ ≥ η. Next, define γ < 0 by γ = max x∈{x∈Rn : η≤∥x−xe ∥≤ε} V ′(x)f (x). Since, for all t ≥ 0, η ≤ ∥x(t) − xe ∥ < ε, it follows that ∫ t V[x(t)] − V[x(0)] = V ′ [x(τ)] f [x(τ)] dτ ≤ γt. 0

Therefore, for all t ≥ 0, V(x(t)) ≤ V[x(0)] + γt. However, for all t > −V[x(0)]/γ, V[x(t)] < 0, which is a contradiction. Therefore, limt→∞ V[x(t)] = 0. Now, suppose that either limt→∞ x(t) does not exist or limt→∞ x(t) exists but is not xe . In both cases, there exists ρ > 0 such that, for every positive integer i, there exists ti > i such that ρ < ∥x(ti ) − xe ∥ < ε. Let (ti j )∞j=1 be an increasing subsequence of (ti )∞ i=1 . Since, for all j ≥ 1, ∥x(ti j ) − xe ∥ > ρ and since {V(xi j )}∞j=1 is decreasing and nonnegative, it follows that lim j→∞ V[x(ti j )] ≥ min x∈{x∈R : ρ≤∥x−xe ∥≤ε V(x) > 0. However, lim j→∞ V[x(ti j )] = 0, which is a contradiction. Hence, limt→∞ x(t) = xe , which proves that xe is asymptotically stable. To prove the last statement, let x0 ∈ Rn. Suppose that i) of Theorem 15.8.2 holds. Let (ti )∞ i=1 ⊂ (0, T x0 ) satisfy limi→∞ ti = T x0 and limi→∞ ∥x(ti )∥ = ∞. Since V is radially unbounded, it follows that limi→∞ V[x(ti )] = ∞. However, it follows from (15.8.4) that, for all t ∈ [0, T x0 ), ∫ t V[x(t)] − V[x(0)] = V ′ [x(τ)] f [x(τ)] dτ ≤ 0, 0

and thus, for all t ∈ [0, T x0 ), V[x(t)] ≤ V[x(0)]. Therefore, limi→∞ V[x(ti )] ≤ V(x0 ), which is a contradiction. Hence, ii) of Theorem 15.8.2 holds. Since ii) of Theorem 15.8.2 holds, there exists a unique C1 solution x: [0, ∞) 7→ Rn satisfying △ (15.8.1) and x(0) = x0 . Since V[x(t)] is decreasing and nonnegative on [0, ∞), it follows that η = limt→∞ V[x(t)] exists and is nonnegative. Suppose that η > 0. Therefore, for all t ≥ 0, x(t) ∈ △ D1 = {x ∈ Rn : η ≤ V(x) ≤ V(x0 )}. Since V is radially unbounded, it follows that D1 is compact. △ Therefore, define γ = max x∈D1 V ′(x)f (x). Since η > 0, it follows that xe < D1 , and thus γ < 0. Since, for all t ≥ 0, x(t) ∈ D1 , it follows that ∫ t V[x(t)] − V[x(0)] = V ′ [x(τ)] f [x(τ)] dτ ≤ γt. 0

Therefore, for all t ≥ 0, V(x(t)) ≤ V[x(0)] + γt. However, for all t > −V[x(0)]/γ it follows that V[x(t)] < 0, which is a contradiction. Therefore, η = 0, and thus limt→∞ V[x(t)] = 0. As in the proof of asymptotic stability, it can be shown that limt→∞ x(t) = xe . 

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15.9 Linear Stability Theory We now specialize Definition 15.8.3 to the linear system x˙(t) = Ax(t),

(15.9.1)

where t ≥ 0, x(t) ∈ Rn, and A ∈ Rn×n. Note that xe = 0 is an equilibrium of (15.9.1), and that xe ∈ Rn is an equilibrium of (15.9.1) if and only if xe ∈ N(A). Hence, if xe is the globally asymptotically stable equilibrium of (15.9.1), then A is nonsingular and xe = 0. We consider three types of stability for the linear system (15.9.1). Unlike Definition 15.8.3, these definitions are stated in terms of the dynamics matrix rather than the equilibrium. Definition 15.9.1. For A ∈ Cn×n, define the following classes of matrices: i) A is Lyapunov stable if spec(A) ⊂ CLHP and, if λ ∈ spec(A) ∩ IA, then λ is semisimple. ii) A is semistable if spec(A) ⊂ OLHP ∪ {0} and, if 0 ∈ spec(A), then 0 is semisimple. iii) A is asymptotically stable if spec(A) ⊂ OLHP. iv) A is unstable if A is not Lyapunov stable. The following result concerns Lyapunov stability, semistability, and asymptotic stability. Proposition 15.9.2. Let A ∈ Rn×n. Then, the following statements are equivalent: i) xe = 0 is a Lyapunov-stable equilibrium of (15.9.1). ii) At least one equilibrium of (15.9.1) is Lyapunov stable. iii) Every equilibrium of (15.9.1) is Lyapunov stable. iv) A is Lyapunov stable. v) For every initial condition x(0) ∈ Rn, {x(t) : t ≥ 0} is bounded. vi) {etA : t ≥ 0} is bounded. vii) For every initial condition x(0) ∈ Rn, {etAx(0) : t ≥ 0} is bounded. The following statements are equivalent: viii) A is semistable. ix) limt→∞ etA exists. x) For every initial condition x(0), limt→∞ x(t) exists. If these conditions hold, then lim etA = I − AA#.

t→∞

(15.9.2)

The following statements are equivalent: xi) xe = 0 is an asymptotically stable equilibrium of (15.9.1). xii) A is asymptotically stable. xiii) αmax (A) < 0. xiv) For every initial condition x(0) ∈ Rn, limt→∞ x(t) = 0. xv) For every initial condition x(0) ∈ Rn, etA x(0) → 0 as t → ∞. xvi) etA → 0 as t → ∞. The following definition concerns the stability of a polynomial. Definition 15.9.3. Let p ∈ R[s]. Then, define the following terminology: i) p is Lyapunov stable if roots(p) ⊂ CLHP and, for all λ ∈ roots(p) ∩ IA, mult p (λ) = 1. ii) p is semistable if roots(p) ⊂ OLHP ∪ {0} and, if 0 ∈ roots(p), then mult p (0) = 1. iii) p is asymptotically stable if roots(p) ⊂ OLHP. iv) p is unstable if p is not Lyapunov stable.

1199

THE MATRIX EXPONENTIAL AND STABILITY THEORY

For the following result, recall Definition 15.9.1. Proposition 15.9.4. Let A ∈ Rn×n. Then, the following statements hold: i) A is Lyapunov stable if and only if µA is Lyapunov stable. ii) A is semistable if and only if µA is semistable. Furthermore, the following statements are equivalent: iii) A is asymptotically stable iv) µA is asymptotically stable. v) χA is asymptotically stable. Next, consider the factorization of the minimal polynomial µA of A given by µA = µAs µAu ,

(15.9.3)

where µAs and µAu are monic polynomials such that roots(µAs ) ⊂ OLHP,

roots(µAu ) ⊂ CRHP.

(15.9.4)

Proposition 15.9.5. Let A ∈ Rn×n, and let S ∈ Rn×n be a nonsingular matrix such that

[

A=S

A1 0

] A12 −1 S , A2

(15.9.5)

where A1 ∈ Rr×r is asymptotically stable, A12 ∈ Rr×(n−r), and A2 ∈ R(n−r)×(n−r) satisfies spec(A2 ) ⊂ CRHP. Then, [ ] 0 C12s µAs (A) = S S −1, (15.9.6) 0 µAs (A2 ) where C12s ∈ Rr×(n−r) and µAs (A2 ) is nonsingular, and [ u ] µA (A1 ) C12u −1 u µA (A) = S S , 0 0 where C12u ∈ Rr×(n−r) and µAu (A1 ) is nonsingular. Consequently, ( [ ]) I N[µAs (A)] = R[µAu (A)] = R S r . 0 If, in addition, A12 = 0, then µAs (A) Consequently,

[

] 0 0 =S S −1 , 0 µAs (A2 )

(15.9.8)

[

µAu (A)

] µAu (A1 ) 0 −1 =S S . 0 0

(15.9.7)

( [ ]) 0 R[µAs (A)] = N[µAu (A)] = R S . In−r

(15.9.9)

(15.9.10)

Corollary 15.9.6. Let A ∈ Rn×n. Then,

N[µAs (A)] = R[µAu (A)], N[µAu (A)] = R[µAs (A)].

(15.9.11) (15.9.12)

Proof. Theorem 7.4.6 implies that there exists a nonsingular matrix S ∈ Rn×n such that (15.9.5)

holds, where A1 ∈ Rr×r is asymptotically stable, A12 = 0, and A2 ∈ R(n−r)×(n−r) satisfies spec(A2 ) ⊂ CRHP. The result now follows from Proposition 15.9.5.  In view of Corollary 15.9.6, we define the asymptotically stable subspace Ss (A) of A by

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CHAPTER 15 △

Ss (A) = N[µAs (A)] = R[µAu (A)]

(15.9.13)

and the unstable subspace Su (A) of A by △

Su (A) = N[µAu (A)] = R[µAs (A)]. Note that

∑

dim Ss (A) = def µAs (A) = rank µAu (A) =

(15.9.14)

amultA(λ),

(15.9.15)

λ∈spec(A) Re λ 0 for △ all nonzero x ∈ D = Rn. Furthermore, Theorem 15.8.5 with f (x) = Ax implies that V ′(x)f (x) = 2xTPAx = xT (ATP + PA)x = −xTRx, which satisfies (15.8.4) for all x ∈ Rn. Thus, Theorem 15.8.5 implies that A is Lyapunov stable. If, in addition, R is positive definite, then, for all x , 0, (15.8.5) holds, and thus A is asymptotically stable. Alternatively, we now prove the first and third statements without using Theorem 15.8.5. Letting λ ∈ spec(A), and letting x ∈ Cn be an associated eigenvector, it follows that 0 ≥ −x∗Rx = x∗ (ATP + PA)x = (λ + λ)x∗Px. Therefore, spec(A) ⊂ CLHP. Now, suppose that ω ȷ ∈ spec(A), where ω ∈ R, △ and let x ∈ N[(ω ȷI − A)2 ]. Defining y = (ω ȷI − A)x, it follows that (ω ȷI − A)y = 0, and thus ∗ ∗ T Ay = ω ȷy. Therefore, −y Ry = y (A P + PA)y = −ω ȷy∗Py + ω ȷy∗Py = 0, and thus Ry = 0. Hence, 0 = x∗Ry = −x∗ (ATP + PA)y = −x∗ (AT + ω ȷI)Py = y∗Py. Since P is positive definite, it follows that y = 0, and thus (ω ȷI − A)x = 0. Therefore, x ∈ N(ω ȷI − A). Now, Proposition 7.7.8 implies that ω ȷ is semisimple. Therefore, A is Lyapunov stable. Next, to prove that A is asymptotically stable, let λ ∈ spec(A), and let x ∈ Cn be an associated eigenvector. Thus, 0 > −x∗Rx = (λ + λ)x∗Px, which implies that A is asymptotically stable. Finally, to prove that A is semistable, let ω ȷ ∈ spec(A), where ω ∈ R is nonzero, and let x ∈ Cn be an associated eigenvector. Then, −x∗Rx = x∗ (ATP + PA)x = −x∗ [(ω ȷI − A)∗P + P(ω ȷI − A]x = 0. Therefore, Rx = 0, and thus

[

] ω ȷI − A x = 0, R

which implies that x = 0, which contradicts x , 0. Consequently, ω ȷ < spec(A) for all nonzero ω ∈ R, and thus A is semistable.  Equation (15.10.1) is a Lyapunov equation. Converse results for Corollary 15.10.1 are given by Corollary 15.10.4, Proposition 15.10.6, Proposition 15.10.5, and Proposition 15.10.6. The following lemma is useful for analyzing (15.10.1). Lemma 15.10.2. Assume that A ∈ Fn×n is asymptotically stable. Then, ∫ ∞ etA dt = −A−1. (15.10.3) ∫t

0

eτA dτ = A−1 (etA − I). Letting t → ∞ yields (15.10.3).  The following result concerns Sylvester’s equation. See Fact 7.11.26 and Proposition 9.2.5. Proposition 15.10.3. Let A, B, C ∈ Rn×n. Then, there exists a unique matrix X ∈ Rn×n satisfying

Proof. Proposition 15.1.4 implies

0

AX + XB + C = 0

(15.10.4)

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CHAPTER 15

if and only if BT ⊕ A is nonsingular. If these conditions hold, then X is given by X = − vec−1 [(BT ⊕ A)−1 vec C].

(15.10.5)

If, in addition, B ⊕ A is asymptotically stable, then X is given by ∫ ∞ X= etACetB dt.

(15.10.6)

T

0

Proof. The first two statements follow from Proposition 9.2.5. Now, assume that BT ⊕ A is

asymptotically stable. Then, it follows from (15.10.5), Lemma 15.10.2, and Proposition 15.1.7 that ∫ ∞ ∫ ∞ ( T ) ( T ) X= vec−1 et(B ⊕A) vec C dt = vec−1 etB ⊗ etA vec C dt 0 ∫0 ∞ ∫ ∞ ( ) = vec−1 vec etACetB dt = etACetB dt.  0

0

The following result provides a converse to Corollary 15.10.1 for the case of asymptotic stability. Corollary 15.10.4. Let A ∈ Rn×n and R ∈ Rn×n. Then, there exists a unique matrix P ∈ Rn×n satisfying (15.10.1) if and only if A ⊕ A is nonsingular. If these conditions hold and R is symmetric, then P is symmetric. Now, assume that A is asymptotically stable. Then, P is given by ∫ ∞ T etA RetA dt. (15.10.7) P= 0

Finally, if R is (positive semidefinite, positive definite), then so is P. Proof. First note that A ⊕ A is nonsingular if and only if (A ⊕ A)T = AT ⊕ AT is nonsingular. Now, the first statement follows from Proposition 15.10.3. To prove the second statement, note that AT (P−PT )+(P−PT )A = 0, which implies that P is symmetric. Now, suppose that A is asymptotically stable. Then, Fact 15.19.33 implies that A ⊕ A is asymptotically stable. Consequently, (15.10.7) follows from (15.10.6).  The following results include converse statements. We first consider asymptotic stability. Proposition 15.10.5. Let A ∈ Rn×n. Then, the following statements are equivalent: i) A is asymptotically stable. ii) For every positive-definite matrix R ∈ Rn×n there exists a positive-definite matrix P ∈ Rn×n that satisfies (15.10.1). iii) There exist a positive-definite matrix R ∈ Rn×n and a positive-definite matrix P ∈ Rn×n that satisfy (15.10.1). Proof. It follows from Corollary 15.10.4 that i) implies ii). ii) immediately implies iii). Finally, it follows from Corollary 15.10.1 that iii) implies i).  Next, we consider the case of Lyapunov stability. Proposition 15.10.6. Let A ∈ Rn×n. Then, the following statements hold: i) If A is Lyapunov stable, then there exist a positive-definite matrix P ∈ Rn×n and a positivesemidefinite matrix R ∈ Rn×n such that rank R = ν− (A) and such that (15.10.1) is satisfied. ii) If there exist a positive-definite matrix P ∈ Rn×n and a positive-semidefinite matrix R ∈ Rn×n that satisfy (15.10.1), then A is Lyapunov stable. Proof. To prove i), suppose that A is Lyapunov stable. Then, it follows from Theorem 7.4.6 [ ] and Definition 15.9.1 that there exists a nonsingular matrix S ∈ Rn×n such that A = S A01 A02 S −1 is in real Jordan form, where A1 ∈ Rn1 ×n1 , A2 ∈ Rn2 ×n2 , spec(A1 ) ⊂ IA, A1 is semisimple, and spec(A2 ) ⊂ OLHP. Next, Fact 7.10.5 implies that there exists a nonsingular matrix S 1 ∈ Rn1 ×n1 such that A1 = S 1 J1S 1−1, where J1 ∈ Rn1 ×n1 is skew symmetric. Then, it follows that A1TP1 + P1 A1 =

THE MATRIX EXPONENTIAL AND STABILITY THEORY

1203

△

S 1−T (J1 + J1T )S 1−1 = 0, where P1 = S 1−TS 1−1 is positive definite. Next, let R2 ∈ Rn2 ×n2 be positive T definite, and let P2 ∈ Rn2 ×n2 be the [positive-definite solution[ of A 2 P2 + P2 A2 + R2 = 0. Hence, ] ] △ △ −T P1 0 −1 −T 0 0 (15.10.1) is satisfied with P = S and R = S 0 R2 S −1. Finally, Corollary 15.10.1 0 P2 S implies ii).  Corollary 15.10.7. Let A ∈ Rn×n. Then, the following statements hold: i) A is Lyapunov stable if and only if there exists a positive-definite matrix P ∈ Rn×n such that ATP + PA is negative semidefinite. ii) A is asymptotically stable if and only if there exists a positive-definite matrix P ∈ Rn×n such that ATP + PA is negative definite.

15.11 Discrete-Time Stability Theory The theory of difference equations is concerned with solutions of discrete-time dynamical systems of the form x(k + 1) = f [x(k)],

(15.11.1)

where f : Rn → Rn, k ∈ N, x(k) ∈ Rn, and x(0) = x0 is the initial condition. The solution x(k) ≡ xe is an equilibrium of (15.11.1) if xe = f (xe ). A linear discrete-time system has the form x(k + 1) = Ax(k),

(15.11.2)

where A ∈ Rn×n. For k ∈ N, x(k) is given by x(k) = Akx0 .

(15.11.3)

The asymptotic behavior of the sequence (x(k))∞ k=0 is determined by the stability of A. Definition 15.11.1. For A ∈ Cn×n, define the following classes of matrices: i) A is discrete-time Lyapunov stable if spec(A) ⊂ CIUD and, if λ ∈ spec(A) and |λ| = 1, then λ is semisimple. ii) A is discrete-time semistable if spec(A) ⊂ OIUD ∪ {1} and, if 1 ∈ spec(A), then 1 is semisimple. iii) A is discrete-time asymptotically stable if spec(A) ⊂ OIUD. iv) A is discrete-time unstable if A is not discrete-time Lyapunov stable. Proposition 15.11.2. Let A ∈ Rn×n and consider the linear discrete-time system (15.11.2). Then, the following statements are equivalent: i) A is discrete-time Lyapunov stable. ii) For every initial condition x0 ∈ Rn, (x(k))∞ k=1 is bounded. n k iii) For every initial condition x0 ∈ R , (A x0 )∞ k=1 is bounded. iv) (Ak )∞ is bounded. k=1 The following statements are equivalent: v) A is discrete-time semistable. vi) limk→∞ Ak exists. In fact, limk→∞ Ak = I − (I − A)(I − A)#. vii) For every initial condition x0 ∈ Rn, limk→∞ x(k) exists. The following statements are equivalent: viii) A is discrete-time asymptotically stable. ix) ρmax (A) < 1. x) For every initial condition x0 ∈ Rn, limk→∞ x(k) = 0. xi) For every initial condition x0 ∈ Rn, Akx0 → 0 as k → ∞.

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CHAPTER 15

xii) Ak → 0 as k → ∞. The following definition concerns discrete-time stability of a polynomial. Definition 15.11.3. For p ∈ R[s], define the following terminology: i) p is discrete-time Lyapunov stable if roots(p) ⊂ CIUD and, if λ is an imaginary root of p, then mult p (λ) = 1. ii) p is discrete-time semistable if roots(p) ⊂ OIUD∪{1} and, if 1 ∈ roots(p), then mult p (1) = 1. iii) p is discrete-time asymptotically stable if roots(p) ⊂ OIUD. iv) p is discrete-time unstable if p is not discrete-time Lyapunov stable. Proposition 15.11.4. Let A ∈ Rn×n. Then, the following statements hold: i) A is discrete-time Lyapunov stable if and only if µA is discrete-time Lyapunov stable. ii) A is discrete-time semistable if and only if µA is discrete-time semistable. Furthermore, the following statements are equivalent: iii) A is discrete-time asymptotically stable. iv) µA is discrete-time asymptotically stable. v) χA is discrete-time asymptotically stable. We now consider the discrete-time Lyapunov equation P = ATPA + R.

(15.11.4)

Proposition 15.11.5. Let A ∈ Rn×n. Then, the following statements are equivalent:

i) A is discrete-time asymptotically stable. ii) For every positive-definite matrix R ∈ Rn×n there exists a positive-definite matrix P ∈ Rn×n that satisfies (15.11.4). iii) There exist a positive-definite matrix R ∈ Rn×n and a positive-definite matrix P ∈ Rn×n that satisfy (15.11.4). Proposition 15.11.6. Let A ∈ Rn×n. Then, A is discrete-time Lyapunov-stable if and only if there exist a positive-definite matrix P ∈ Rn×n and a positive-semidefinite matrix R ∈ Rn×n that satisfy (15.11.4).

15.12 Facts on Matrix Exponential Formulas Fact 15.12.1. Let A ∈ Fn×n and t ∈ R. Then, the following statements hold:

i) ii) iii) iv) v) vi) vii) viii)

If A2 = 0, then etA = I + tA. If A2 = I, then etA = (cosh t)I + (sinh t)A. If A2 = −I, then etA = (cos t)I + (sin t)A. If A2 = A, then etA = I + (et − 1)A. If A2 = −A, then etA = I + (1 − e−t )A. If rank A = 1 and tr A = 0, then etA = I + tA. (tr A)t If rank A = 1 and tr A , 0, then etA = I + e tr A−1 A. If there exists nonzero r ∈ F such that A2 = r2I, then etA = (cosh rt)I +

sinh rt A. r

ix) If there exists nonzero r ∈ F such that A3 = r2A, then etA = I +

(cosh rt) − 1 2 sinh rt A+ A. r r2

1205

THE MATRIX EXPONENTIAL AND STABILITY THEORY

x) If there exist k ≥ 1 and nonzero r ∈ F such that Ak+1 = rAk , then ∑k−1 1 k−1 ∑ 1 i i ert − i=0 i! ri ti k tA tA + e = A. i! rk i=0 xi) If ρmax (A) < 1, then e

∑∞

1 i=1 i

tr Ai

1 . det(I − A)

=

xii) e1n×n t = I + e n−1 1n×n . Source: viii)–x) are given in [2913]. xi) follows from the power series for log(I − A) given by Fact 13.4.15 along with Corollary 15.2.4. xii) is given in [904]. Remark: Explicit expressions for the cases where A satisfies Ak+2 = r2 Ak and Ak+3 = r3Ak are given in [2913]. See also [2234]. Related: Setting r = θ ȷ in ix) yields Fact 15.12.6. [ ] △ Fact 15.12.2. Let A = I0n I0n . Then, nt

etA = (cosh t)I2n + (sinh t)A,

etJ2n = (cos t)I2n + (sin t)J2n .

Fact 15.12.3. Let A ∈ Rn×n, and assume that A is skew symmetric. Then, {eθA : θ ∈ R} ⊆ SO(n)

is a multiplication group. If, in addition, n = 2, then [

]

{eθJ2 : θ ∈ R} = SO(2).

θ sin θ Remark: Note that eθJ2 = −cos sin θ cos θ . See Fact 4.14.1. Fact 15.12.4. Let n ≥ 2 and A ∈ Rn×n, where

  0  0   0  △  A =  ...    0  0

Then,

1 0 0 2 0 0 .. .. . .

0 0 3 .. .

··· ··· ··· .. .

0

0

0

..

.

0

0

0

···

 (0) (1) (2)  0  (01) (02)  0 1  (12)  0 0  2 eA =  . .. ..  .. . .   0 0  0  0 0 0

Furthermore, if k ≥ n, then k ∑ i=1

in−1 = [1n−1 2n−1

(3) (03) (13) 2

··· ···

0

··· .. . .. .

0

···

..

.

       .   n − 1   0 0 0 0 .. .

(n−1)   0  (n−1 )   1  (n−1 )   2  ..  . .  (n−1)    n−2  (n−1)  n−1

 (k )   1    n−1 −A  · · · n ]e  ...  .  (k)  n

Source: [168]. Remark: A is called the creation matrix in [9]. Related: Fact 7.18.5.

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CHAPTER 15

Fact 15.12.5. Let A ∈ F3×3. If spec(A) = {λ}, then

etA = eλt [I + t(A − λI) + 21 t2 (A − λI)2 ]. If mspec(A) = {λ, λ, µ}ms , where µ , λ, then etA = eλt [I + t(A − λI)] +

[

] teλt eµt − eλt (A − λI)2. − (µ − λ)2 µ − λ

If spec(A) = {λ, µ, ν}, then etA =

eλt eµt (A − µI)(A − νI) + (A − λI)(A − νI) (λ − µ)(λ − ν) (µ − λ)(µ − ν) eνt (A − λI)(A − µI). + (ν − λ)(ν − µ)

Source: [153]. Remark: Additional expressions are given in [4, 361, 436, 717, 1326, 2234, 2246]. △ Fact 15.12.6. Let x ∈ R3, assume that x is nonzero, and define θ = ∥x∥2 . Then,

sin θ 1 − cos θ 2 K(x) + K (x) θ θ2 2 sin2 2θ 2 sin θ K (x) =I+ K(x) + θ θ2 sin θ 1 − cos θ T xx , = (cos θ)I + K(x) + θ θ2

eK(x) = I +

eK(x) x = x,

spec[eK(x) ] = {1, e∥x∥2 ȷ , e−∥x∥2 ȷ },

tr eK(x) = 1 + 2cos θ = 1 + 2cos ∥x∥2 .

Source: The Cayley-Hamilton theorem and Fact 4.12.1 imply that K 3 (x) + θ2K(x) = 0. Then, every

term K k (x) in the expansion of eK(x) can be expressed in terms of either K(x) or K 2 (x). Finally, Fact 4.12.1 implies that θ2I + K 2 (x) = xxT. Remark: Fact 15.12.7 shows that, for all z ∈ R3, eK(x)z is the counterclockwise (right-hand-rule) rotation of z about the vector x through the angle θ, which is given by the Euclidean norm of x. In Fact 4.14.9, the cross product is used to construct the pivot vector x for a given pair of vectors having the same length. Remark: K(x)y = x×y. See Fact 4.12.1. Related: Fact 15.12.1. Fact 15.12.7. Let x, y ∈ R3, and assume that x and y are nonzero. Then, there exists a skewsymmetric matrix A ∈ R3×3 such that y = eA x if and only if xTx = yTy. If x , −y, then one such matrix is A = θK(z), where △

z=

1 x × y, ∥x × y∥2

△

θ = acos △

xTy . ∥x∥2 ∥y∥2

⊥ T If x = −y, then one such matrix is A = πK(z), where z = ∥y∥−1 2 ν × y and ν ∈ {y} satisfies ν ν = 1. Source: This result follows from Fact 4.14.9 and Fact 15.12.6, which provide equivalent expressions for an orthogonal matrix that transforms a given vector into another given vector having the same length. This result thus provides a geometric interpretation for Fact 15.12.6. Remark: Note that z is the unit vector perpendicular to the plane containing x and y, where the direction of z is determined by the right-hand rule. An intuitive proof is to let x be the initial condition w(0) = x to the differential equation w(t) ˙ = K(z)w(t), where t ∈ [0, θ]. Then, the derivative w(t) ˙ lies in the x, y plane and is perpendicular to w(t) for all t ∈ [0, θ]. Hence, y = w(θ). Remark: Since det eA = etr A = 1, it follows that every pair of vectors in R3 having the same Euclidean length are related by a proper rotation. See Fact 4.11.5 and Fact 4.19.4. This is a linear interpolation problem. See Fact 4.11.5, Fact 4.14.9, and [1549]. Remark: Parameterizations of SO(3) are considered in [2448, 2562]. Related: Fact 4.14.9. Problem: Extend this result to Rn. See [285, 2383].

1207

THE MATRIX EXPONENTIAL AND STABILITY THEORY

Fact 15.12.8. Let A ∈ SO(3), let z ∈ R3 be an eigenvector of A corresponding to the eigenvalue

1, assume that ∥z∥2 = 1, assume that tr A > −1, and let θ ∈ (−π, π) satisfy tr A = 1 + 2cos θ. Then, A = eθK(z).

Related: Fact 7.12.2. Fact 15.12.9. Let x, y ∈ R3, and assume that x and y are nonzero. Then, xTx = yTy if and only if θ

y = e ∥x×y∥2 where △

θ = acos

(yxT −xyT )

x,

xTy . ∥x∥2 ∥y∥2

Source: Fact 15.12.7. Remark: Note that K(x × y) = yxT − xyT. Fact 15.12.10. Let A ∈ R3×3, assume that A ∈ SO(3) and tr A > −1, and let θ ∈ (−π, π) satisfy

tr A = 1 + 2cos θ. Then,

   θ = 0, 0, log A =    θ (A − AT ), θ , 0. 2sin θ

Source: [1506, p. 364] and [2064]. Related: Fact 15.16.11. △ Fact 15.12.11. Let x ∈ R3, assume that x is nonzero, and define θ = ∥x∥2 . Then,

K(x) =

K(x) θ 2sin θ [e

− e−K(x) ].

Source: Fact 15.12.10. Related: Fact 4.12.1. Fact 15.12.12. Let A ∈ SO(3), let x, y ∈ R3, and assume that xTx = yTy. Then, Ax = y if and

only if, for all t ∈ R,

AetK(x)A−1 = etK(y).

Source: [1784]. Fact 15.12.13. Let x, y, z ∈ R3. Then, the following statements are equivalent:

i) For every A ∈ SO(3), there exist α, β, γ ∈ R such that A = eαK(x)eβK(y)eγK(z). ii) yTx = 0 and yTz = 0. Source: [1784]. Credit: P. Davenport. Problem: Given A ∈ SO(3), determine α, β, γ. Fact 15.12.14. Let x, y, z ∈ R3, assume that xT (y × z) , 0, assume that ∥x∥2 = ∥y∥2 = ∥z∥2 = 1, and let α, β, γ ∈ [0, 2π) satisfy tan α2 = Then,

zT (x × y) , (x × y)T (x × z)

tan β2 =

xT (y × z) , (y × z)T (y × x)

tan γ2 =

yT (z × x) . (z × x)T (z × y)

eαK(x) eβK(y) eγK(z) = I.

Source: [2077, p. 5]. Remark: This is the Rodrigues-Hamilton theorem, which concerns a closed

rotation sequence. As stated in [929], “Successive rotations about three concurrent lines fixed in space, through twice the dihedral angles of the planes formed by them, restore a body to its original position.” An equivalent result is given by Donkin’s theorem. See [446] and [2077, p. 6]. Remark: A converse result given in [446] shows that, if a sequence of three nonzero rotations about linearly independent rotation axes is closed, then the rotation angles are uniquely determined. Remark: See [2334, 2902].

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CHAPTER 15

Fact 15.12.15. Let A ∈ R4×4, and assume that A is skew symmetric with mspec(A) =

{ωȷ, −ωȷ, µ ȷ, −µ ȷ}ms . If ω , µ, then eA = a3 A3 + a2 A2 + a1 A + a0 I, where ) ( a3 = (ω2 − µ2 )−1 µ1 sin µ − ω1 sin ω , a2 = (ω2 − µ2 )−1 (cos µ − cos ω), ( ) 2 2 a1 = (ω2 − µ2 )−1 ωµ sin µ − µω sin ω , a0 = (ω2 − µ2 )−1 (ω2 cos µ − µ2 cos ω). If ω = µ, then

sin ω A. ω Source: [1277, p. 18] and [2246]. Remark: There are misprints in [1277, p. 18] and [2246]. Related: Fact 6.9.19 and Fact 6.10.8. Fact 15.12.16. Let a, b, c ∈ R, define the skew-symmetric matrix A ∈ R4×4 by either     a b c  a b c   0  0    −a  −a 0 c −b  0 −c b  △  △   ,  , A = A =   −b −c 0 a  c 0 −a   −b  −c b −a 0 −c −b a 0 √ △ and define θ = a2 + b2 + c2 . Then, eA = (cos ω)I +

mspec(A) = {θ ȷ, −θ ȷ, θ ȷ, −θ ȷ}ms , eA = (cos θ)I +  k/2 k    (−1) θ I, k even, Ak =    (−1)(k−1)/2 θk−1A, k odd.

sin θ A, θ

Source: [2771]. Remark: (sin 0)/0 = 1. Remark: The skew-symmetric matrix A arises in the

kinematic relationship between the angular velocity vector and quaternion (Euler-parameter) rates. See [308, p. 385]. Remark: The two matrices A are similar. To show this, note that Fact 7.10.10 implies that A and −A are similar. Then, apply the similarity transformation S = diag(−1, 1, 1, 1). Related: Fact 6.9.19 and Fact 6.10.8. Fact 15.12.17. Let x ∈ R3, and define the skew-symmetric matrix A ∈ R4×4 by    0 −xT  A =   . x −K(x) Then, for all t ∈ R, 1

e 2 tA = (cos 21 ∥x∥t)I4 + Source: [1486, p. 34]. Remark:

1 2A

sin 21 ∥x∥t A. ∥x∥

expresses quaternion rates in terms of the angular velocity

vector. Fact 15.12.18. Let a, b ∈ R3, define the skew-symmetric matrix A ∈ R4×4 by

△

assume that aTb = 0, and define α =

√

  K(a) A =  T −b

 b   , 0

aTa + bTb. Then,

1 − cos α 2 sin α A+ A. α α2 Source: [2723]. Related: Fact 6.9.19 and Fact 6.10.8. e A = I4 +

1209

THE MATRIX EXPONENTIAL AND STABILITY THEORY

Fact 15.12.19. Let a, b ∈ Rn−1, define A ∈ Rn×n by △

[

A=

] 0 aT , b 0(n−1)×(n−1)

√ △ and define α = |aTb|. Then, the following statements hold: i) If aTb < 0, then 2 sin2 sin α eαA = I + A+ α α2 ii) If aTb = 0, then, for all t ∈ R,

α 2

A2.

etA = I + A + 21 A2.

iii) If aTb > 0, then eαA = I +

2 sinh2 sinh α A+ α α2

α 2

A2.

Source: [2967].

15.13 Facts on the Matrix Sine and Cosine Fact 15.13.1. Let A ∈ Cn×n, and define △

sin A =

ȷA 1 2 ȷ (e

△

− e− ȷA ),

cos A = 21 (e ȷA + e− ȷA ).

Then, the following statements hold: i) sin2 A + cos2 A = I, sin 2A = 2(sin A) cos A, cos 2A = 2(cos2 A) − I. ii) If A is real, then sin A = Im e ȷA and cos A = Re e ȷA . iii) sin(A ⊕ B) = (sin A) ⊗ cos B + (cos A) ⊗ sin B. iv) cos(A ⊕ B) = (cos A) ⊗ cos B − (sin A) ⊗ sin B. v) If A is involutory and k is an integer, then cos kπA = (−1)k I. Furthermore, the following statements are equivalent: vi) For all t ∈ R, sin t(A + B) = (sin tA) cos tB + (cos tA) sin tB. vii) For all t ∈ R, cos t(A + B) = (cos tA) cos tB − (sin tA) sin tB. viii) AB = BA. Source: [1391, pp. 287, 288, 300].

15.14 Facts on the Matrix Exponential for One Matrix Fact 15.14.1. Let A ∈ Fn×n, and assume that A is (lower triangular, upper triangular). Then, so is eA. If, in addition, A is Toeplitz, then so is eA. Related: Fact 4.23.7. Fact 15.14.2. Let A ∈ Fn×n. Then, ρmax (eA ) = eαmax (A). Fact 15.14.3. Let S : [t0 , t1 ] → Rn×n be differentiable. Then, for all t ∈ [t0 , t1 ], d 2 S (t) = S˙ (t)S (t) + S (t)S˙ (t). dt

Let S 1 : [t0 , t1 ] → Rn×m and S 2 : [t0 , t1 ] → Rm×l be differentiable. Then, for all t ∈ [t0 , t1 ], d S 1 (t)S 2 (t) = S˙1 (t)S 2 (t) + S 1 (t)S˙ 2 (t). dt △

△

Fact 15.14.4. Let A ∈ Fn×n, and define AH = 12 (A + A∗ ) and AS = 21 (A − A∗ ). Then, [AH , AS ] =

1 ∗ 2 [A , A].

Hence, AH AS = AS AH if and only if A is normal. In this case, eAH eAS is the polar decomposition of eA. Related: Fact 7.20.2 and Fact 7.20.3. Problem: Find a decomposition of nonnormal

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CHAPTER 15

A that yields the polar decomposition of eA . Fact 15.14.5. Let A ∈ Fn×n, and assume that either spec(A) ⊂ OLHP or spec(A) ⊂ OLHP . Then, ∫ 2 ∞ 2 (x I + A2 )−1 dx. A−1 = π 0 Source: [953]. Fact 15.14.6. Let A ∈ Fn×m, and assume that rank A = m. Then,

A+ =

∫

∞

∗

e−tA AA∗ dt.

0

Fact 15.14.7. Let A ∈ F

n×n

, and assume that A is nonsingular. Then, ∫ ∞ ∗ −1 A = e−tA A dt A∗.

Fact 15.14.8. Let A ∈ F

n×n

0 △

, and define k = ind A. Then, ∫ ∞ k (2k+1)∗ k+1 A AD = e−tA A dt AkA(2k+1)∗Ak. 0

Source: [1198]. Fact 15.14.9. Let A ∈ Fn×n, and assume that ind A = 1. Then,

∫

A# =

∞

e−tAA

3∗ 2

A

dt AA3∗A.

0

Source: Fact 15.14.8. △ Fact 15.14.10. Let A ∈ Fn×n, and define k = ind A. Then,

∫

t

( eτA dτ = AD (etA − I) + (I − AAD ) tI +

0

1 2 2! t A

+ ··· +

1 k k−1 k! t A

)

.

If, in particular, A is group invertible, then ∫ t eτA dτ = A# (etA − I) + t(I − AA# ). 0

Fact 15.14.11. Let A ∈ Fn×n, let mspec(A) = {λ1 , . . . , λr , 0, . . . , 0}ms , where λ1 , . . . , λr are non-

zero, and let t > 0. Then,

∫ det

t

τA

e dτ = t

0

∫t

n−r

r ∏

λi t λ−1 i (e − 1).

i=1

eτA dτ , 0 if and only if, for every nonzero integer k, 2kπȷ/t < spec(A). Finally, ∫t ∫t det(etA − I) , 0 if and only if det A , 0 and det 0 eτA dτ , 0. Remark: etA − I = A 0 eτA dτ. Fact 15.14.12. Let A ∈ Rn×n and a ∈ (0, ∞). Then, ∫ ∞ √ 2 2 e−τ /a eτA dτ = σπe(a/4)A .

Hence, det

0

−∞

Credit: S. Bhat. Related: Fact 14.12.1. Fact 15.14.13. Let A ∈ Fn×n, and assume that there exists α ∈ R such that spec(A) ⊂ {z ∈

C: α ≤ Im z < 2π + α}. Then, eA is (diagonal, upper triangular, lower triangular) if and only if A is. Source: [1887]. Fact 15.14.14. Let A ∈ Fn×n. Then, the following statements hold: i) If A is unipotent, then (15.5.1) is a finite series, log A exists and is nilpotent, and elog A = A.

1211

THE MATRIX EXPONENTIAL AND STABILITY THEORY

ii) If A is nilpotent, then eA is unipotent and log eA = A. Source: [1299, p. 60]. Remark: A is unipotent if and only if A − I is nilpotent. Fact 15.14.15. Let A ∈ Rn×n, assume that det A > 0, and assume that A is either symmetric, 1

skew symmetric, or orthogonal. Then, there exists B ∈ Rn×n such that eA = e 2 (B −B) eB . Source: [576]. Remark: A has a twisted logarithm. Fact 15.14.16. Let A ∈ Cn×n , let λ ∈ spec(A), assume that λ is simple, assume that, for all µ ∈ spec(A)\{λ}, Re µ < Re λ, and let x, y ∈ Cn be nonzero vectors that satisfy Ax = λx and ATy = λy. Then, yTx , 0 and 1 1 lim λt etA = T xyT . t→∞ e yx T

Source: [1742]. Remark: λ is a CT-dominant eigenvalue. Related: Fact 6.11.5 and Fact 15.22.24.

15.15 Facts on the Matrix Exponential for Two or More Matrices Fact 15.15.1. Let A, B ∈ Fn×n, and consider the following statements:

i) A = B. ii) eA = eB. iii) AB = BA. iv) AeB = eBA. v) eAeB = eBeA. vi) eAeB = eA+B. vii) eAeB = eBeA = eA+B. viii) For all i, j ∈ Z, eiA+ jB = eiA e jB . Then, the following statements hold: ix) iii) =⇒ iv) =⇒ v). x) iii) =⇒ vii) =⇒ vi). xi) If spec(A) is 2πȷ congruence free, then ii) =⇒ iii) =⇒ iv) ⇐⇒ v). xii) If spec(A) and spec(B) are 2πȷ congruence free, then ii) =⇒ iii) ⇐⇒ iv) ⇐⇒ v). xiii) If spec(A + B) is 2πȷ congruence free, then iii) ⇐⇒ vii). xiv) If, for all λ ∈ spec(A) and µ ∈ spec(B), it follows that (λ − µ)/(2πȷ) is not a nonzero integer, then ii) =⇒ i). xv) iii) ⇐⇒ viii). xvi) If A and B are Hermitian, then i) ⇐⇒ ii) =⇒ iii) ⇐⇒ iv) ⇐⇒ v) ⇐⇒ vi). Remark: The set S ⊂ C is 2πȷ congruence free if no two elements of S differ by a nonzero integer multiple of 2πȷ. Source: [856, 1073], [1304, pp. 88, 89, 270–272], and [2190, 2392, 2393, 2394, 2476, 2866, 2867]. The assumption of normality in operator versions of several of these statements in [2190, 2394] is not needed in the matrix[ case. the first ] xiii) and[ 2πȷ ] implication in x) are given in 0 1 [1391, p. 32]. Remark: The matrices A = 00 2πȷ and B = 0 −2πȷ satisfy eA = eB = eA+B = I, but AB , BA. Therefore, vii) =⇒ iii) does not hold. Furthermore, since vii) holds and iii) does not hold, it follows from xii) that spec(A + B) is not 2πȷ congruence free. In fact, spec(A + B) = {0, 2πȷ}. The same observation holds for the real matrices √ √      0  0 3/2  0 − 3/2  0     0 −1/2  , B = 2π  √ 0 0 −1/2  . A = 2π  √0     − 3/2 1/2 0 3/2 1/2 0 [ ] [ ] 0 Remark: The matrices A = log 2+πȷ and B = 2 log 02+2πȷ 01 satisfy eAeB = eA+B , eBeA . Therefore, 0 0

1212

CHAPTER 15

vi) =⇒ vii) does not hold. Furthermore, spec(A + B) = {3 log 2 + 3π ȷ, 0}, which is 2πȷ congruence free. Therefore, since ] not hold, it follows from xii) that iii) does not hold; that is, AB , BA. [ vii) does . Consequently, under the additional statement that spec(A + B) is 2πȷ In fact, AB − BA = 00 log 2+πȷ 0 congruence free, vi) =⇒ iii) does not hold. However, xiv) implies that vi) =⇒ iii) holds under the additional assumption that A and B are Hermitian, in which case spec(A + B) is 2πȷ congruence free. This example is due to G. Bourgeois. Fact 15.15.2. Let A ∈ Fn×n, B ∈ Fn×m, and C ∈ Fm×m. Then,  tA ∫ t (t−τ)A τC  ∫ t [ ] [ ][ ]  e e Be dτ  A I 0 t A B 0  , . eτA dτ = [I 0]et 0 0 e 0 C =  I 0 0 etC Remark: This result can be extended to arbitrary upper block-triangular matrices. See [2774]. For an application to sampled-data control, see [2171]. Fact 15.15.3. Let I be an interval, let A : I 7→ Fn×n, and assume that A is differentiable. Then, for all t ∈ I, ( ) ∫ 1 d A(t) τA(t) d e = A(t) e(1−τ)A(t) dτ. e dt dt 0 Source: [2586, pp. 256, 257]. Remark: This is the Duhamel formula. Related: Fact 15.15.4. Fact 15.15.4. Let A, B ∈ Fn×n. Then,

d A+tB e = dt Hence,

Furthermore,

Hence,

∫

1

eτ(A+tB)Be(1−τ)(A+tB) dτ.

0

∫ 1 d A+tB eτABe(1−τ)A dτ. Dexp(A;B) = e = dt 0 t=0 d tr eA+tB = tr eA+tBB. dt d tr eA+tB = tr eAB. dt t=0

Source: [353, p. 175], [962, p. 371], and [1774, 1986, 2108]. Fact 15.15.5. Let A, B ∈ Fn×n. Then,

) ( ) ( adA ∞ ∑ I − e−adA e −I d A+tB 1 = e (B)eA = eA (B) = adAk (B)eA. (k+1)! dt adA adA t=0 k=0

Source: The second and fourth expressions are given in [218, p. 49] and [1506, p. 248], while the third expression appears in [2754]. See also [2781, pp. 107–110]. Related: Fact 3.23.5. Fact 15.15.6. Let A, B ∈ Fn×n, and assume that eA = eB. Then, the following statements hold:

If |λ| < π for all λ ∈ spec(A) ∪ spec(B), then A = B. If λ − µ , 2kπȷ for all λ ∈ spec(A), µ ∈ spec(B), and k ∈ Z, then [A, B] = 0. If A is normal and σmax (A) < π, then [A, B] = 0. If A is normal and σmax (A) = π, then [A2, B] = 0. Source: [2396, 2476] and [2781, p. 111]. Remark: If [A, B] = 0, then [A2, B] = 0. Fact 15.15.7. Let A, B ∈ Fn×n, and assume that A and B are skew Hermitian. Then, etAetB is unitary, and there exists a skew-Hermitian matrix C(t) such that etAetB = eC(t). Remark: Converi) ii) iii) iv)

1213

THE MATRIX EXPONENTIAL AND STABILITY THEORY

gence of (15.4.1) is discussed in [490, 508, 842, 994, 2126, 2304], and closed-form expressions are given in [2770, 1068]. Fact 15.15.8. Let A, B ∈ Fn×n, and assume that A and B are Hermitian. Then, ( p p )1/p = eA+B. lim e 2 Ae pB e 2 A p→0

Source: [101]. Remark: This result is related to the Lie-Trotter formula given by Corollary 15.4.13. For extensions, see [19, 1127]. Fact 15.15.9. Let A, B ∈ Fn×n, and assume that A and B are Hermitian. Then, [ ( )]1/p 1 lim 21 e pA + e pB = e 2 (A+B). p→∞

Source: [443]. Fact 15.15.10. Let A, B ∈ Fn×n. Then,

2 [ 1 1 1 1 ]k lim e k Ae k Be− k Ae− k B = e[A,B] .

k→∞

Fact 15.15.11. Let A ∈ F

, X ∈ Fm×l, and B ∈ Fl×n. Then,

n×m

d tr eAXB = BeAXBA. dX Fact 15.15.12. Let A, B ∈ Fn×n. Then, d tA tB −tA −tB d √tA √tB − √tA − √tB = AB − BA. e e e e e = 0, e e e dt dt t=0 t=0 Fact 15.15.13. Let A, B, C ∈ Fn×n, assume there exists β ∈ F such that [A, B] = βB + C, and

assume that [A, C] = [B, C] = 0. Then,

eA+B = eAeϕ(β)Beψ(β)C, where

    1 (1 − e−β ), β , 0, △ β ϕ(β) =    1, β = 0,

    1 (1 − β − e−β ), β , 0, △  β2 ψ(β) =    − 1 , β = 0. 2

Source: [1181, 2599]. Fact 15.15.14. Let A, B ∈ Fn×n, and assume that there exist α, β ∈ F such that [A, B] = αA + βB.

Then, where

et(A+B) = eϕ(t)Aeψ(t)B,   t, α = β = 0,        △ α−1 log(1 + αt), α = β , 0, 1 + αt > 0, ϕ(t) =     ∫ t α−β     0 (α−β)τ dτ, α , β, αe

△

∫

ψ(t) =

t

e−βϕ(τ) dτ.

0

−β

Source: [2600]. Fact 15.15.15. Let A, B ∈ Fn×n, and assume that there exists nonzero β ∈ F such that [A, B] =

αB. Then, for all t > 0,

et(A+B) = etAe[(1−e

−αt

.

)/α]B

Source: Apply Fact 15.15.13 with [tA, tB] = αt(tB) and β = αt.

1214

CHAPTER 15

Fact 15.15.16. Let A, B ∈ Fn×n, and assume that [[A, B], A] = 0 and [[A, B], B] = 0. Then, for

all t ∈ R,

etAetB = etA+tB+(t /2)[A,B]. 2

In particular, 1

1

1

eAeB = eA+B+ 2 [A,B] = eA+Be 2 [A,B] = e 2 [A,B] eA+B ,

eBe2AeB = e2A+2B.

Source: [1299, pp. 64–66] and [2878]. Fact 15.15.17. Let A, B ∈ Fn×n , and assume that [A, B] = B2. Then,

eA+B = eA(I + B). Fact 15.15.18. Let A, B ∈ Fn×n. Then, for all t ∈ [0, ∞),

et(A+B) = etAetB +

∞ ∑

C k t k,

k=2

where, for all k ∈ N,

△

Ck+1 =

1 ([A k+1 △

Dk+1 =

△

+ B]Ck + [B, Dk ]),

1 (ADk k+1

+ Dk B),

C0 = 0, △

D0 = I.

Source: [2308]. Fact 15.15.19. Let A, B ∈ Fn×n. Then, for all t ∈ [0, ∞),

et(A+B) = etA etBetC2 etC3 · · · , where

△

C2 = − 21 [A, B],

△

C3 = 13 [B, [A, B]] + 16 [A, [A, B]].

Remark: This is the Zassenhaus product formula. See [1391, p. 236] and [2401]. Remark: Higher order terms are given in [2401]. Remark: Conditions for convergence appear to be unknown. Fact 15.15.20. Let A ∈ R2n×2n, and assume that A is symplectic and discrete-time Lyapunov stable. Then, spec(A) ⊂ UC, amultA (1) and amultA (−1) are even, A is semisimple, and there exists a Hamiltonian matrix B ∈ R2n×2n such that A = eB. Source: Since A is symplectic and discrete-time Lyapunov stable, it follows that the spectrum of A is a subset of the unit circle and A is semisimple. Therefore, the only negative eigenvalue that A can have is −1. Since all nonreal eigenvalues appear in complex conjugate pairs and A has even size, and since, by Fact 4.28.10, det A = 1, it follows that the eigenvalues −1 and 1 (if present) have even algebraic multiplicity. Theorem 2.6 of [896] implies that A has a Hamiltonian logarithm. Related: xiii) of Proposition 15.6.5. Fact 15.15.21. Let A, B ∈ Fn×n, assume that A is positive definite, and assume that B is positive semidefinite. Then, −1/2 −1/2 A + B ≤ A1/2eA BA A1/2.

Hence,

det(A + B) −1 ≤ etr A B. det A Furthermore, for each inequality, equality holds if and only if B = 0. Source: If C is positive semidefinite, then I + C ≤ eC. Fact 15.15.22. Let A, B ∈ Fn×n, and assume that A and B are Hermitian. Then, I ⊙ (A + B) ≤ log(eA ⊙ eB ). Source: [88, 2977]. Related: Fact 10.25.58.

1215

THE MATRIX EXPONENTIAL AND STABILITY THEORY

Fact 15.15.23. Let A, B ∈ Fn×n, assume that A and B are Hermitian, assume that A ≤ B, let

α, β ∈ R, assume that either αI ≤ A ≤ βI or αI ≤ B ≤ βI, and let t > 0. Then, etA ≤ S (t, eβ−α )etB, where, for all t > 0 and h > 0,

 t t  (h − 1)ht/(h −1)    , △  etlog h S (t, h) =     1,

h , 1, h = 1.

Source: [1094]. Related: S (t, h) is Specht’s ratio. See Fact 2.2.54 and Fact 2.11.95. Fact 15.15.24. Let A, B ∈ Fn×n, assume that A and B are Hermitian, let α, β ∈ R, assume that

αI ≤ A ≤ βI and αI ≤ B ≤ βI, let t > 0, and define S (t, h) as in Fact 15.15.23. Then, 1 S (1, eβ−α )S 1/t (t, eβ−α )

[αetA + (1 − α)etB ]1/t ≤ eαA+(1−α)B ≤ S (1, eβ−α )[αetA + (1 − α)etB ]1/t .

Source: [1094]. Fact 15.15.25. Let A, B ∈ Fn×n, and assume that A and B are positive definite. Then,

log det A = tr log A,

log det AB = tr(log A + log B).

Fact 15.15.26. Let S ⊂ R, assume that S is an open set, let f : S 7→ R, assume that f is convex and differentiable, let A, B ∈ Hn , and assume that all of the eigenvalues of A and B are elements of S. Then, tr[(A − B) f ′ (B)] ≤ tr[ f (A) − f (B)]. Source: [454, pp. 115, 116]. Fact 15.15.27. Let A, B ∈ Fn×n, and assume that A and B are positive definite. Then,

tr(A − B) ≤ tr A(log A − log B), (log tr A − log tr B) tr A ≤ tr A(log A − log B). Source: The first inequality follows from Fact 15.15.26 with S = (0, ∞) and f (x) = x log x. See [337] and [449, p. 281]. Remark: The first inequality is Klein’s inequality. See [454, p. 118]. Remark: tr A(log A − log B) is the relative entropy of Umegaki. Related: The second inequality is equivalent to the thermodynamic inequality. See Fact 15.15.35. Fact 15.15.28. Let A, B ∈ Fn×n, assume that A and B are positive definite, and define △

µ(A, B) = elog A+log B. Then, the following statements hold: i) µ(A, A−1 ) = I. ii) µ(A, B) = µ(B, A). iii) If AB = BA, then µ(A, B) = AB. Source: [169]. Remark: With multiplication defined by µ, Pn is a commutative Lie group. Fact 15.15.29. Let A, B ∈ Fn×n, assume that A and B are positive definite, and let p > 0. Then, 1 p tr

Alog(B p/2A pB p/2 ) ≤ tr A(log A + log B) ≤ 1p tr Alog(A p/2B pA p/2 ).

Furthermore, lim 1p tr Alog(B p/2A pB p/2 ) = tr A(log A + log B) = lim 1p tr A log(A p/2B pA p/2 ). p↓0

p↓0

Source: [101, 338, 1127, 1380]. Remark: This result arises in quantum information theory.

1216

△

CHAPTER 15

Fact 15.15.30. Let A, B ∈ Fn×n, assume that A and B are Hermitian, let q ≥ p > 0, and define

h = λmax (eA )/λmin (eB ) and

(h − 1)h1/(h−1) . e log h

△

S (1, h) =

Then, there exist unitary matrices U, V ∈ Fn×n such that A+B ∗ 1 U S (1,h) Ue

≤ e 2 AeBe 2 A ≤ S (1, h)VeA+B V ∗. 1

1

Furthermore, tr eA+B ≤ tr eAeB ≤ S (1, h) tr eA+B, p

q

p

q

tr (e pA #e pB )2/p ≤ tr eA+B ≤ tr (e 2 Be pAe 2 B )1/p ≤ tr (e 2 BeqAe 2 B )1/q , p

p

tr eA+B = lim tr (e 2 Be pAe 2 B )1/p ,

eA+B = lim (e pA #e pB )2/p .

p↓0

p↓0

Moreover, tr eA+B = tr eAeB if and only if AB = BA. Furthermore, for all i ∈ {1, . . . , n}, A+B 1 ) S (1,h) λi (e

≤ λi (eAeB ) ≤ S (1, h)λi (eA+B ).

Finally, let α ∈ [0, 1]. Then, lim (e pA #α e pB )1/p = e(1−α)A+αB ,

tr (e pA #α e pB )1/p ≤ tr e(1−α)A+αB.

p↓0

Source: [550] and [2586, pp. 254–256]. Remark: The left-hand inequality in the second string of inequalities is the Golden-Thompson inequality. See Fact 15.17.4. Remark: Since S (1, h) > 1 for all h > 1, the left-hand inequality in the first string of inequalities does not imply the GoldenThompson inequality. Remark: For i = 1, the stronger eigenvalue inequality λmax (eA+B ) ≤ λmax (eAeB ) holds. See Fact 15.17.4. Remark: S (1, h) is Specht’s ratio given by Fact 15.15.23. Remark: The generalized geometric mean is defined in Fact 10.11.72. Related: Fact 15.15.31. Fact 15.15.31. Let A, B ∈ Fn×n, and assume that A and B are normal. Then, slog

slog

σ(eA+B ) ≺ λ(⟨eA ⟩⟨eB ⟩),

λ(eA+B ) ≺ λ(⟨eA ⟩⟨eB ⟩),

| tr eA+B | ≤ tr ⟨eA ⟩⟨eB ⟩.

Furthermore, if ∥ · ∥ is a unitarily invariant norm on Fn×n , then ∥⟨eA+B ⟩∥ ≤ ∥⟨⟨eA ⟩1/2 ⟨eB ⟩⟨eA ⟩1/2 ⟩∥. Source: [1810]. Related: Fact 15.15.30. Fact 15.15.32. Let A, B ∈ Fn×n, and assume that A and B are Hermitian. Then,

(tr eA )e(tr e

A

B)/tr eA

≤ tr eA+B.

Source: [337]. Remark: This is the Peierls-Bogoliubov inequality. Related: Fact 15.15.35. Fact 15.15.33. Let A, B, C ∈ Fn×n, and assume that A, B, and C are positive definite. Then,

∫ tr elog A−log B+log C ≤ tr

∞

A(B + xI)−1C(B + xI)−1 dx.

0

Source: [1817, 1891]. Remark: −log B is correct. Remark: tr eA+B+C ≤ | tr eAeBeC | is false. Fact 15.15.34. Let A, B, C ∈ Fn×n, and assume that A, B, and C are Hermitian. Then,

∫

∞

tr eA+B+C ≤ 0

Source: [1381, pp. 180, 181].

tr eA (xI + e−C )−1 eB (xI + e−C )−1 dx.

1217

THE MATRIX EXPONENTIAL AND STABILITY THEORY

Fact 15.15.35. Let A, B ∈ Fn×n, and assume that A is positive definite, tr A = 1, and B is

Hermitian. Then,

tr AB ≤ tr Alog A + log tr eB.

Furthermore, equality holds if and only if (tr eB )A = eB. Source: [337]. Remark: This is the thermodynamic inequality. Equivalent forms are given by Fact 15.15.27 and Fact 15.15.32. Fact 15.15.36. Let A, B ∈ Fn×n, and assume that A and B are Hermitian. Then, ∥A − B∥F ≤ ∥ log(e− 2 AeBe 2 A )∥F . 1

1

Source: [454, p. 203]. Remark: This result has a distance interpretation in terms of geodesics.

See [454, p. 203] and [464, 2064, 2065]. Fact 15.15.37. Let A, B ∈ Fn×n, and assume that A and B are skew Hermitian. Then, there exist unitary matrices S 1 , S 2 ∈ Fn×n such that −1

eAeB = eS1 AS1

+S 2 BS 2−1

.

Source: [2478, 2614, 2615]. Related: Fact 10.11.70. Fact 15.15.38. Let A, B ∈ Fn×n, and assume that A and B are Hermitian. Then, there exist

unitary matrices S 1, S 2 ∈ Fn×n such that 1

1

−1

e 2 AeB e 2 A = eS1 AS1

+S 2 BS 2−1

.

Source: [2477, 2478, 2614, 2615]. Problem: Relate this result to Fact 15.15.37. Fact 15.15.39. Let A, B ∈ Fn×n, assume that A and B are positive semidefinite, and assume that

B ≤ A. Furthermore, let p, q, r, t ∈ R, and assume that r ≥ t ≥ 0, p ≥ 0, p + q ≥ 0, and p + q + r > 0. Then, [ r r ]t/(p+q+r) e 2 AeqA+pBe 2 A ≤ etA.

Source: [2757]. Fact 15.15.40. Let A ∈ Fn×n and B ∈ Fm×m. Then,

tr eA⊕B = (tr eA ) tr eB . Fact 15.15.41. Let A ∈ Fn×n, B ∈ Fm×m, and C ∈ Fl×l. Then,

eA⊕B⊕C = eA ⊗ eB ⊗ eC. Fact 15.15.42. Let A ∈ Fn×n, B ∈ Fm×m, C ∈ Fk×k, and D ∈ Fl×l. Then,

tr eA⊗I⊗B⊗I+I⊗C⊗I⊗D = (tr eA⊗B ) tr eC⊗D. Source: By Fact 9.4.38, a similarity transformation involving the Kronecker permutation matrix

can be used to reorder the inner two terms. See [2525]. Fact 15.15.43. Let A, B ∈ Rn×n, and assume that A and B are positive definite. Then, A#B is the unique positive-definite solution X of the matrix equation log A−1X + log B−1X = 0. Source: [2065].

15.16 Facts on the Matrix Exponential and Eigenvalues, Singular Values, and Norms for One Matrix Fact 15.16.1. Let A ∈ Fn×n, assume that eA is positive definite, and assume that σmax (A) < 2π. Then, A is Hermitian. Source: [1712, 2395].

1218

CHAPTER 15 △

Fact 15.16.2. Let A ∈ Fn×n, and define f : [0, ∞) 7→ (0, ∞) by f (t) = σmax (eAt ). Then,

f ′(0) = 21 λmax (A + A∗ ). △

Hence, there exists ε > 0 such that f (t) = σmax (etA ) is decreasing on [0, ε) if and only if A is dissipative. Source: iii) of Fact 15.16.7 and [2834]. Remark: The derivative is one-sided. Fact 15.16.3. Let A ∈ Fn×n. Then, for all t ≥ 0, d tA 2 ∗ ∥e ∥F = tr etA(A + A∗ )etA . dt △

Hence, if A is dissipative, then f (t) = ∥etA ∥F is decreasing on [0, ∞). Source: [2834]. Fact 15.16.4. Let A ∈ Fn×n. Then, ∗

∗

∗

| tr e2A | ≤ tr eAeA ≤ tr eA+A ≤ [ntr e2(A+A ) ]1/2 ≤ ∗

n 2

∗

+ 21 tr e2(A+A ).

∗

In addition, tr eAeA = tr eA+A if and only if A is normal. Source: [426], [1450, p. 515], and [2476]. ∗ ∗ Remark: tr eAeA ≤ tr eA+A is Bernstein’s inequality. See [93]. Related: Fact 4.10.12. Fact 15.16.5. Let A ∈ Fn×n. Then, 1

slog

∗

1

∗

∗

∗

λ⊙1/2 (eA eA ) = λ(⟨eA ⟩) = σ(eA ) ≺ σ(e 2 (A+A ) ) = λ(e 2 (A+A ) ) = λ⊙1/2 (eA+A ). That is, for all k ∈ {1, . . . , n}, k ∏

σi (eA ) ≤

i=1

k ∏

k k ∏ ( 1 ) ∏ 1 ∗ ∗ λi e 2 (A+A ) = eλi [ 2 (A+A )] ≤ eσi (A)

i=1

i=1

i=1

with equality for k = n. Furthermore, for all k ∈ {1, . . . , n}, k ∑ i=1

In particular, Equivalently,

σi (eA ) ≤

k k k ∑ ∑ ) ∑ ( 1 1 ∗ ∗ eλi [ 2 (A+A )] ≤ eσi (A). λi e 2 (A+A ) = i=1

i=1

i=1

) ( 1 1 ∗ ∗ σmax (eA ) ≤ λmax e 2 (A+A ) = e 2 λmax (A+A ) ≤ eσmax (A) . ∗

∗

∗

λmax (eAeA ) ≤ λmax (eA+A ) = eλmax (A+A ) ≤ e2σmax (A).

Furthermore, | det eA | = |etr A | ≤ e| tr A| ≤ etr ⟨A⟩ ,

tr ⟨eA ⟩ ≤

n ∑

eσi (A).

i=1

Source: [767, 1878, 2479], and use Fact 3.25.15, Fact 10.21.9, and Fact 10.21.10. Fact 15.16.6. Let A ∈ Fn×n, and let ∥ · ∥ be a unitarily invariant norm on Fn×n. Then, ∗

∗

∥eAeA ∥ ≤ ∥eA+A ∥. In particular,

∗

∗

λmax (eAeA ) ≤ λmax (eA+A ),

∗

∗

tr eAeA ≤ tr eA+A .

Source: [767]. Fact 15.16.7. Let A, B ∈ Fn×n, let ∥ · ∥ be the norm on Fn×n induced by the norm ∥ · ∥′ on Fn, let

mspec(A) = {λ1 , . . . , λn }ms , and define

△

∥I + εA∥ − 1 . ε↓0 ε

µ(A) = lim Then, the following statements hold:

1219

THE MATRIX EXPONENTIAL AND STABILITY THEORY

i) ii) iii) iv) v) vi) vii) viii) ix) x) xi) xii) xiii) xiv)

△

µ(A) = D+ f (A; I), where f : Fn×n 7→ R is defined by f (A) = ∥A∥. µ(A) = limt↓0 t−1 log ∥e tA ∥ = supt>0 t−1 log ∥e tA ∥. + + µ(A) = ddt ∥etA ∥ t=0 = ddt log ∥etA ∥ t=0 . µ(I) = 1, µ(−I) = −1, and µ(0) = 0. αmax (A) = limt→∞ t−1 log ∥e tA ∥ = inf t>0 t−1 log ∥e tA ∥. For all i ∈ {1, . . . , n}, −∥A∥ ≤ −µ(−A) ≤ Re λi ≤ αmax (A) ≤ µ(A) ≤ ∥A∥. For all α ∈ R, µ(αA) = |α|µ[(sign α)A]. For all α ∈ F, µ(A + αI) = µ(A) + Re α. max {µ(A) − µ(−B), −µ(−A) + µ(B)} ≤ µ(A + B) ≤ µ(A) + µ(B). µ: Fn×n 7→ R is convex. |µ(A) − µ(B)| ≤ max {|µ(A − B)|, |µ(B − A)|} ≤ ∥A − B∥. For all x ∈ Fn, max {−µ(−A), −µ(A)}∥x∥′ ≤ ∥Ax∥′ . If A is nonsingular, then max {−µ(−A), −µ(A)} ≤ 1/∥A−1∥. For all t ≥ 0 and i ∈ {1, . . . , n}, e−∥A∥t ≤ e−µ(−A)t ≤ e(Re λi )t ≤ eαmax (A)t ≤ ∥etA ∥ ≤ eµ(A)t ≤ e∥A∥t.

xv) µ(A) = min{β ∈ R: ∥etA ∥ ≤ eβt for all t ≥ 0}. xvi) If ∥ · ∥′ = ∥ · ∥1 , and thus ∥ · ∥ = ∥ · ∥col , then     n ∑    |A(i, j) | . µ(A) = max Re A( j, j) + j∈{1,...,n}    i=1 i, j

xvii) If ∥ · ∥′ = ∥ · ∥2 and thus ∥ · ∥ = σmax (·), then µ(A) = λmax [ 21 (A + A∗ )]. xviii) If ∥ · ∥′ = ∥ · ∥∞ , and thus ∥ · ∥ = ∥ · ∥row , then     n ∑   |A(i, j) | . µ(A) = max Re A(i,i) +  i∈{1,...,n}   j=1 j,i

Source: [879, 883, 2194, 2561], [1399, pp. 653–655], and [2695, p. 150]. Remark: µ is called either the matrix measure, logarithmic derivative, or initial growth rate. For applications, see [1399, 2798]. See Fact 15.19.11 for the logarithmic derivative of an asymptotically stable matrix. Remark: The directional differential D+ f (A; I) is defined in (12.5.2). Remark: vi) and xvii) yield Fact 7.12.27. Remark: Higher order logarithmic derivatives are studied in [460]. Fact 15.16.8. Let A ∈ Fn×n, let β > αmax (A), let γ ≥ 1, and let ∥ · ∥ be a normalized, submultiplicative norm on Fn×n. Then, for all t ≥ 0, ∥etA ∥ ≤ γeβt if and only if, for all k ≥ 1 and α > β, γ . ∥(αI − A)−k ∥ ≤ (α − β)k Remark: This is a consequence of the Hille-Yosida theorem. See [802, p. 26] and [1399, p. 672]. Related: Fact 15.16.9. Fact 15.16.9. Let A ∈ Fn×n, let β > αmax (A), and let ∥ · ∥ be a submultiplicative norm on Fn×n. Then, there exists γ ≥ 1 such that, for all t ≥ 0, ∥etA ∥ ≤ γeβt. Source: [1184, pp. 201–206], [1399, pp. 649–651], and [1572]. Note that γ ≥ supt≥0 ∥e(A−βI)t ∥. Remark: If A is asymptotically stable, then β can be chosen such that αmax (A) < β < 0 so that the bound converges to zero. Related: Fact

1220

CHAPTER 15

11.13.11 and Fact 15.16.8. Fact 15.16.10. Let A ∈ Rn×n, let β ∈ R, and assume that there exists a positive-definite matrix P ∈ Rn×n such that ATP + PA ≤ 2βP. Then, for all t ≥ 0,

σmax (etA ) ≤

√ σmax (P)/σmin (P)eβt.

Remark: See [1399, p. 665]. Related: Fact 15.19.9. √ △ Fact 15.16.11. Let A ∈ SO(3), and define θ = 2 acos( 21 1 + tr A). Then,

1 θ = σmax (log A) = √ ∥ log A∥F . 2 Remark: θ is a Riemannian metric giving the length of the shortest geodesic curve on SO(3) between A and I. See [2064]. Related: Fact 4.14.6 and Fact 15.12.10. △ Fact 15.16.12. Let A ∈ Cn×n, for all i ∈ {1, . . . , n}, define fi : [0, ∞) 7→ R by fi (t) = log σi (etA ), △ n tA and, for all x ∈ C , define g x (t): R 7→ R by g x (t) = log σmax (e x). Then, the following statements are equivalent: i) A is normal. ii) For all i ∈ {1, . . . , n}, fi is convex. iii) For all x ∈ Cn , g x is convex Source: [199, 987, 1077]. Remark: The [ ]statement in [199, 987] that convexity holds on R is △ 0 for which log σ (etA ) = |t| and log σ (etA ) = −|t|. erroneous. A counterexample is A = 10 −1 1 2 Credit: iii) is due to S. Friedland.

15.17 Facts on the Matrix Exponential and Eigenvalues, Singular Values, and Norms for Two or More Matrices Fact 15.17.1. Let A, B ∈ Fn×n. Then, ∗

∗

∗

∗

∗

| tr eA+B | ≤ tr e 2 (A+B)e 2 (A+B) ≤ tr e 2 (A+A +B+B ) ≤ tr e 2 (A+A )e 2 (B+B ) 1

1

∗

1

∗

1

∗

1

∗

≤ (tr eA+A )1/2 (tr eB+B )1/2 ≤ 21 tr(eA+A + eB+B ),  tr eAeB   ∗ ∗ ∗ ∗  1 ≤ 2 tr(eA eA + eB eB ) ≤ 21 tr(eA+A + eB+B ).  1 2A 2B   2 tr(e + e ) Now, assume that A and B are Hermitian. Then, tr eA+B ≤ tr eAeB ≤ (tr e2A )1/2 (tr e2B )1/2 ≤ 12 tr(e2A + e2B ). Source: [426, 768], [1450, p. 514], and [2206, 2523, 2524]. Remark: The following conjecture is

given in [2523]. Assume that A and B are Hermitian, let p, q ∈ (1, ∞), and assume that 1/p+1/q = 1. Then, (tr e pA )1/p (tr eqB )1/q ≤ 21 tr(e2A + e2B ). Fact 15.17.2. Let A, B ∈ Fn×n and p ∈ (0, ∞). Then,

[ ( 1 1 )p ] σmax eA+B − e p Ae p B ≤

σmax (A)+σmax (B) 1 . 2p σmax ([A, B])e

Source: [1391, p. 237] and [2066]. Related: Corollary 12.11.15 and Fact 15.17.3.

1221

THE MATRIX EXPONENTIAL AND STABILITY THEORY △

Fact 15.17.3. Let A ∈ Fn×n, define AH =

Then,

1 2 (A

)p ] [ ( 1 1 ≤ σmax eA − e p AHe p AS

△

+ A ∗ ) and AS = 21 (A − A ∗ ), and let p ∈ (0, ∞).

1 ∗ ∗ 1 2 λmax (A+A ). 4p σmax ([A , A])e

Source: [2066]. Related: Fact 12.11.15. Fact 15.17.4. Let A, B ∈ Fn×n, assume that A and B are Hermitian, and let ∥ · ∥ be a unitarily

invariant norm on Fn×n. Then,



1 1 ∥eA+B ∥ ≤

e 2 AeBe 2 A

≤ ∥eAeB ∥.

If, in addition, p > 0, then

p

p 1/p ∥eA+B ∥ ≤

e 2 AeBe 2 A

,

p

p 1/p ∥eA+B ∥ = lim

e 2 AeBe 2 A

. p↓0

Furthermore, for all k ∈ {1, . . . , n},

 k ∏     λi (eA )λi (eB )   k k k  ∏ ∏ ∏   i=1 A B A+B A B λi (e )λn−i+1 (e ) ≤ λi (e ) ≤ λi (e e ) ≤  k  ∏    i=1 i=1 i=1   σi (eAeB )   i=1

with equality for k = n; that is, n ∏

λi (eA )λn−i+1 (eB ) =

n ∏

i=1

λi (eA+B ) =

i=1

n ∏

λi (eAeB ) =

i=1

n ∏

σi (eAeB ) = det eAeB .

i=1

slog

Hence, λ(eA+B ) ≺ λ(eA eB ). In fact, n n ∏ ∏ det eA+B = λi (eA+B ) = eλi (A+B) = etr(A+B) = e(tr A)+(tr B) i=1

i=1

= etr A etr B = (det eA ) det eB = det eA eB =

n ∏

σi (eAeB ).

i=1

Furthermore, for all k ∈ {1, . . . , n}, k ∑ i=1

λi (eA+B ) ≤

k ∑

λi (eAeB ) ≤

k ∑

i=1

σi (eAeB ).

i=1

In particular, λmax (eA+B ) ≤ λmax (eAeB ) ≤ σmax (eAeB ), and, for all p > 0,

p

tr eA+B ≤ tr eAeB ≤ tr ⟨eAeB ⟩, p

tr eA+B ≤ tr e 2 AeBe 2 A .

Finally, tr eA+B = tr eAeB if and only if A and B commute. Source: [101], [449, p. 261], [2607, 2750], Fact 3.25.15, Fact 7.12.32, and Fact 10.13.2. For the last statement, see [2476]. Remark: Note that det eA+B = (det eA ) det eB despite the fact that eA+B and eAeB may not be equal. See [1391, p. 265] and [1450, p. 442]. Remark: tr eA+B ≤ tr eAeB is the Golden-Thompson inequality. See 1 1 Fact 15.15.30. Remark: ∥eA+B ∥ ≤ ∥e 2 AeBe 2 A ∥ is Segal’s inequality. See [93]. Problem: Compare the upper bound tr ⟨eAeB ⟩ for tr eAeB with the upper bound S (1, h) tr eA+B given by Fact 15.15.30.

1222

CHAPTER 15

Fact 15.17.5. Let A, B ∈ Fn×n . Then, slog

∗

∗

wlog

∗

1

∗

λ(⟨eA+B ⟩) ≺ λ(e 2 (A+A +B+B ) ≺ λ(e 2 (A+A ) e 2 (B+B ) ). 1

1

If, in addition, A and B are normal, then slog

λ(⟨eA+B ⟩) ≺ λ(⟨eA ⟩⟨eB ⟩). Source: [1878]. Fact 15.17.6. Let A, B ∈ Fn×n . Then,

   σ[e 21 (A+B) ]  slog [ σ(eA ) ]   ≺ . 1  (A+B)  σ(eB ) 2 σ[e ]

Source: [2750]. Related: Fact 10.22.4. Fact 15.17.7. Let A, B ∈ Fn×n, assume that A and B are Hermitian, let q, p > 0, where q ≤ p,

and let ∥ · ∥ be a unitarily invariant norm on Fn×n. Then,



(

e 2q AeqBe q2 A )1/q

≤

(e 2p Ae pBe 2p A )1/p

.



Source: [101]. Fact 15.17.8. Let A, B ∈ Fn×n, and assume that A and B are positive semidefinite. Then, 1/2

1/2 eσmax (AB) − 1 ≤ σmax [(eA − I)(eB − I)],

1/3

1/3 eσmax (BAB) − 1 ≤ σmax [(eB − I)(eA − I)(eB − I)].

Source: [2756]. Related: Fact 10.22.42. Fact 15.17.9. Let A, B ∈ Fn×n and t ≥ 0. Then,

∫

t

et(A+B) = etA +

e(t−τ)ABeτ(A+B) dτ.

0

Source: [1391, p. 238]. Fact 15.17.10. Let A, B ∈ Fn×n, and let ∥ · ∥ be a submultiplicative norm on Fn×n. Then, for all

t ≥ 0,

∥etA − etB ∥ ≤ e∥A∥t (e∥A−B∥t − 1).

Fact 15.17.11. Let A, B ∈ Fn×n, let ∥ · ∥ be a normalized submultiplicative norm on Fn×n, and let

t ≥ 0. Then,

∥etA − etB ∥ ≤ t∥A − B∥et max {∥A∥,∥B∥}.

Source: [1391, p. 265]. Fact 15.17.12. Let A, B ∈ Rn×n, and assume that A is normal. Then, for all t ≥ 0,

σmax (etA − etB ) ≤ σmax (etA )[eσmax (A−B)t − 1]. Source: [2866]. Fact 15.17.13. Let A ∈ Fn×n, let ∥ · ∥ be an induced norm on Fn×n, and let α > 0 and β ∈ R be

such that, for all t ≥ 0,

Then, for all B ∈ Fn×n and t ≥ 0, Source: [1399, p. 406].

∥etA ∥ ≤ αeβt. ∥et(A+B) ∥ ≤ αe(β+α∥B∥)t.

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THE MATRIX EXPONENTIAL AND STABILITY THEORY

Fact 15.17.14. Let A, B ∈ Cn×n, assume that A and B are idempotent, assume that A , B, and

let ∥ · ∥ be a norm on Cn×n. Then,

∥e ȷA − e ȷB ∥ = |e ȷ − 1|∥A − B∥ < ∥A − B∥. Source: [2109]. Remark: |e ȷ − 1| ≈ 0.96. Fact 15.17.15. Let A, B ∈ Cn×n, assume that A and B are Hermitian, let X ∈ Cn×n, and let ∥ · ∥

be a unitarily invariant norm on Cn×n. Then,

∥e ȷAX − Xe ȷB ∥ ≤ ∥AX − XB∥. Source: [2109]. Remark: This is a matrix version of xi) of Fact 2.21.27. △ ∑ Fact 15.17.16. Let A, B ∈ Hn, let k ≥ 1, and define f : (0, ∞) 7→ R by f (t) = ki=1 1t log λi (etA etB ).

Then, f is nondecreasing. Furthermore, lim f (t) =

k ∑

t↓0

λi (A + B) ≤

k ∑

i=1

log λi (eA eB ).

i=1

Source: [1079, pp. 378–380]. Remark: Setting k = n yields the Golden-Thompson inequality. △

Fact 15.17.17. Let A, B ∈ Hn , for all t ∈ (0, ∞), define C(t) =

1

1

log λi (e 2 tA etB e 2 tA ), let k ∈ △ ∑k {1, . . . , n}, and define f : (0, ∞) → 7 (0, ∞) by f (t) = i=1 λi [C(t)]. Then, f is nondecreasing. Furw

1 t

△

thermore, for all t ∈ (0, ∞), λ[C(t)] ≺ λ(A) + λ(B). Finally, C0 = limt→∞ C(t) exists, C0 ∈ Hn , and [A, C0 ] = 0. Source: [1079, pp. 380–387]. Fact 15.17.18. Let A, B ∈ Fn×n, assume that A and B are positive semidefinite, and let ∥ · ∥ be a unitarily invariant norm on Fn×n. Then, ∥e⟨A−B⟩ − I∥ ≤ ∥eA − eB ∥,

∥eA + eB ∥ ≤ ∥eA+B + I∥.

Source: [106] and [449, p. 294]. Related: Fact 11.10.86. Fact 15.17.19. Let A, X, B ∈ Fn×n, assume that A and B are Hermitian, and let ∥ · ∥ be a unitarily

invariant norm on Fn×n. Then,

∥AX − XB∥ ≤ ∥e 2 AXe− 2 B − e− 2 BXe 2 A ∥. 1

1

1

1

Source: [477]. Related: Fact 11.10.87. n×n Fact 15.17.20. Let A ∈ Fn×n, B ∈ Fn×m, C ∈ Fm×n, and D ∈ Fm×m, let (Mi )∞ , and assume i=1 ⊂ F

that limi→∞ λmin (Mi + Mi∗ ) = ∞. Then,

lim exp

i→∞

([

A − Mi C

B D

])

[ =

] 0 0 . 0 eD

Source: [2729].

15.18 Facts on Stable Polynomials Fact 15.18.1. Let p ∈ C[s], and assume that p is (asymptotically stable, discrete-time asymptotically stable). Then, so is p′ . Related: Fact 12.16.5. △ Fact 15.18.2. Let a1 , . . . , an be real numbers, let ∆ = {i ∈ {1, . . . , n − 1} : aai+1i < 0}, let b1 , . . . , bn be real numbers such that b1 < · · · < bn , define f : (0, ∞) 7→ R by f (x) = an xbn + · · · + a1 xb1 , △ and define S = {x ∈ (0, ∞) : f (x) = 0}. Furthermore, for all x ∈ S, define the multiplicity of x to be the positive integer m such that f (x) = f ′ (x) = · · · = f (m−1) = 0 and f (m) (x) , 0, and let S′ denote the multiset consisting of all elements of S counting multiplicity as roots of f. Then, card(S′ ) ≤ card(∆). If, in addition, b1 , . . . , bn are nonnegative integers, then card(∆) − card(S′ ) is even. Source: [414, 1666, 2831]. Remark: This is the Descartes rule of signs.

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CHAPTER 15

Fact 15.18.3. Let p ∈ R[s], where p(s) = sn + an−1 sn−1 + · · · + a0 . If p is asymptotically stable,

then a0 , . . . , an−1 are positive. Now, assume that a0 , . . . , an−1 are positive. Then, the following statements hold: i) If either n = 1 or n = 2, then p is asymptotically stable. ii) If n = 3, then p is asymptotically stable if and only if a0 < a1a2 . iii) If n = 4, then p is asymptotically stable if and only if a21 + a0 a23 < a1a2 a3 . iv) If n = 5, then p is asymptotically stable if and only if a2 < a3 a4 ,

a22 + a1 a24 < a0 a4 + a2 a3 a4 ,

a20 + a1 a22 + a21 a24 + a0 a23 a4 < a0 a2 a3 + 2a0 a1 a4 + a1 a2 a3 a4 . Remark: These are special cases of the Routh criterion, which provides necessary and sufficient conditions for asymptotic stability of a polynomial. See [676]. Remark: The Jury criterion for discrete-time asymptotic stability of a polynomial is given by Fact 15.21.2. Related: Fact 10.21.2

and Fact 15.18.15. Fact 15.18.4. Let p ∈ R[s], where p(s) = s3 + a2 s2 + a1 s + a0 , assume that a1 > 0, and △ △ define α = min {−a0 /a1 , −a2 } and β = max {−a0 /a1 , −a2 }. Then, there exists λ ∈ roots(p) such that λ ∈ (α, β). Source: [290]. Related: Fact 2.21.2, Fact 15.21.27, and Fact 15.21.29. Fact 15.18.5. Let ε ∈ [0, 1], let n ∈ {2, 3, 4}, let pε ∈ R[s], where pε (s) = sn + an−1 sn−1 + · · · + a1 s + εa0 , and assume that p1 is asymptotically stable. Then, for all ε ∈ (0, 1], pε is asymptotically stable. Furthermore, p0 (s)/s is asymptotically stable. Remark: This result does not hold for n = 5. A counterexample is p(s) = s5 + 2s4 + 3s3 + 5s2 + 2s + 2.5ε, which is asymptotically stable if and only if ε ∈ (4/5, 1]. This is another instance of the quartic barrier. See [787], Fact 10.17.8, and Fact 10.19.24. △ △ Fact 15.18.6. Let p ∈ R[s] be monic, and define q(s) = sn p(1/s), where n = deg p. Then, p is asymptotically stable if and only if q is asymptotically stable. Related: Fact 6.8.3 and Fact 15.18.7. △ Fact 15.18.7. Let p ∈ R[s] be monic, and assume that p is semistable. Then, q(s) = p(s)/s and △ n q(s) ˆ = s p(1/s) are asymptotically stable. Related: Fact 6.8.3 and Fact 15.18.6. Fact 15.18.8. Let p, q ∈ R[s], and assume that p is even, q is odd, and every coefficient of p + q is positive. Then, p + q is asymptotically stable if and only if every root of p and every root of q is imaginary, and the roots of p and the roots of q are interlaced on the imaginary axis. Source: [482, 676, 1438]. Remark: This is called either the Hermite-Biehler or interlacing theorem. Example: s2 + 2s + 5 = (s2 + 5) + 2s. Fact 15.18.9. Let p ∈ R[s] be asymptotically stable, and let p(s) = βn sn +βn−1 sn−1 +· · ·+β1 s+β0 , where βn > 0. Then, for all i ∈ {1, . . . , n − 2}, βi−1 βi+2 < βi βi+1 . Remark: This is a necessary condition for asymptotic stability. Remark: See [482, p. 68]. Credit: X. Xie. See [2949]. △ Fact 15.18.10. Let n ≥ 2 be even, let m = n/2, let p ∈ R[s], where p(s) = βn sn + βn−1 sn−1 + · · · + β1 s + β0 and βn > 0, and assume that p is asymptotically stable. Then, for all i ∈ {1, . . . , m − 1}, (m) (m−i)/m βi/m n ≤ β2i . i β0 Remark: This is a necessary condition for asymptotic stability. See [2949, 2950] for extensions to polynomials of odd degree, and [328] for inequalities involving either four or more terms. Credit:

A. Borobia and S. Dormido. Fact 15.18.11. Let p ∈ R[s], where p(s) = αn sn + αn−1 sn−1 + · · · + α1 s + α0 , assume that △ α0 , . . . , αn are positive, let r ∈ [1, ∞), and define p⊙r (s) = αrn sn + αrn−1 sn−1 + · · · + αr1 s + αr0 . Then, the following statements hold: i) Assume that n = 3 and all of the roots of p are real. Then, all of the roots of p⊙r are real.

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THE MATRIX EXPONENTIAL AND STABILITY THEORY

ii) Assume that n is even, all of the roots of p are real, and r > 1/[log2 (n + 2) − log2 n]. Then, all of the roots of p⊙r are real. iii) Assume that n is odd, all of the roots of p are real, and r > 2/[log2 (n + 3) − log2 (n − 1)]. Then, all of the roots of p⊙r are real. iv) If p is asymptotically stable, then so is p⊙r . Source: [2832]. Remark: p⊙r is a Schur power of p. Remark: i) is false for n = 4. Fact 15.18.12. Let p, q ∈ R[s], where p(s) = αn sn + αn−1 sn−1 + · · · + α1 s + α0 and q(s) = △ m βm s + βm−1 sm−1 + · · · + β1 s + β0 , and define (p ⊙ q)(s) = αl βl sl + αl−1 βl−1 sl−1 + · · · + α1 β1 s + α0 β0 , △ where l = min{m, n}. If p and q are (Lyapunov stable, asymptotically stable), then so is p ⊙ q. Source: [1147]. Remark: p ⊙ q is the Schur product of p and q. See [184, 1539]. Fact 15.18.13. Let p, q ∈ R[s], where p(s) = αn sn + αn−1 sn−1 + · · · + α1 s + α0 and q(s) = △ m βm s + βm−1 sm−1 + · · · + β1 s + β0 , assume that αn and βn are nonzero, define (p ⊙ q)(s) = αn βn sn + · · · + α1 β1 s + α0 β0 , assume that all of the roots of p and q are real, and assume that all of the roots of either p or q have the same sign. Then, all of the roots of p ⊙ q are real. Source: [2832]. Fact 15.18.14. Let A ∈ Rn×n, and assume that A is diagonalizable over R. Then, χA has all positive coefficients if and only if A is asymptotically stable. Source: Sufficiency follows from Fact 15.18.3. For necessity, note that all of the roots of χA are real and that χA (λ) > 0 for all λ ≥ 0. Hence, roots(χA ) ⊂ (−∞, 0). Fact 15.18.15. Let A ∈ Rn×n. Then, the following statements are equivalent: i) χA⊕A has all positive coefficients. ii) χA⊕A is asymptotically stable. iii) A ⊕ A is asymptotically stable. iv) A is asymptotically stable. Source: If A is not asymptotically stable, then Fact 15.19.32 implies that A ⊕ A has a nonnegative eigenvalue λ. Since χA⊕A (λ) = 0, it follows that χA⊕A cannot have all positive coefficients. See [1097, Theorem 5]. Remark: A similar proof is used in Proposition 10.2.8. Related: Fact 15.18.3. Fact 15.18.16. Let A ∈ Rn×n. Then, the following statements are equivalent: i) χA and χA(2,1) have all positive coefficients. ii) A is asymptotically stable. Source: [2559]. Remark: The additive compound A(2,1) is defined in Fact 9.5.18. Fact 15.18.17. For all i ∈ {1, . . . , n − 1}, let ai , bi ∈ R satisfy 0 < ai ≤ bi , define ϕ1, ϕ2 , ψ1, ψ2 ∈ R[s] by ϕ1 (s) = bn sn + an−2 sn−2 + bn−4 sn−4 + · · · , ψ1 (s) = bn−1 sn−1 + an−3 sn−3 + bn−5 sn−5 + · · · ,

ϕ2 (s) = an sn + bn−2 sn−2 + an−4 sn−4 + · · · , ψ2 (s) = an−1 sn−1 + bn−3 sn−3 + an−5 sn−5 + · · · ,

assume that ϕ1 + ψ1, ϕ1 + ψ2 , ϕ2 + ψ1, and ϕ2 + ψ2 are asymptotically stable, let p ∈ R[s], where p(s) = βn sn + βn−1 sn−1 + · · · + β1 s + β0 , and assume that, for all i ∈ {1, . . . , n}, ai ≤ βi ≤ bi . Then, p is asymptotically stable. Source: [970, pp. 466, 467]. Remark: This is Kharitonov’s theorem. Fact 15.18.18. Let p ∈ R[s], where p(s) = βn sn + βn−1 sn−1 + · · · + β1 s + β0 , and assume that β0 , . . . , βn−1 ≥ 0. Then, the following statements hold: i) αmax (p) ∈ roots(p). ii) If p(1) ≤ 2, then αmax (p) ≤ [p(1) − 1]1/n . iii) If p(1) ≥ 2, then αmax (p) ≤ p(1) − 1. √ √ iv) αmax (p) ≤ 2 max {βn−1 , 2 βn−2 , . . . , n β0 }. Source: [1172, pp. 6, 7].

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CHAPTER 15

Fact 15.18.19. Let p ∈ R[s], where p(s) = βn sn + βn−1 sn−1 √ + · · · + β1 s + β0 , and assume that, for all i ∈ {0, 1, . . . , n − 1}, βi ∈ [0, 1]. Then, αmax (p) < 21 (1 + 5). Source: [107, pp. 53, 288].

15.19 Facts on Stable Matrices Fact 15.19.1. Let A ∈ Fn×n . Then, the following statements are equivalent:

i) A is semistable. ii) For all nonzero ω ∈ R, rank(ω ȷI − A) = n, and there exist a positive-semidefinite matrix R ∈ Fn×n and a positive-definite matrix P ∈ Fn×n such that ATP + PA + R = 0.

(15.19.1)

n×n n×n iii) There exist a positive-semidefinite matrix [ R]∈ F and a positive-definite matrix P ∈ F ω ȷI−A such that, for all nonzero ω ∈ R, rank R = n and (15.19.1) holds. iv) There exist a positive-semidefinite matrix R ∈ Fn×n and a positive-definite matrix P ∈ Fn×n ∩ satisfying nk=1 N(RAk−1 ) = N(A) and (15.19.1). ∩ v) For each positive-semidefinite matrix R ∈ Fn×n satisfying nk=1 N(RAk−1 ) = N(A), there exists a positive-definite matrix P ∈ Fn×n satisfying (15.19.1). ∩ △ vi) For each positive-semidefinite matrix R ∈ Fn×n such that k = min {l ≥ 0 : ni=1 N(RAi+l−1 ) = N(A)} exists, there exists a positive-semidefinite matrix P ∈ Fn×n satisfying

AkT (ATP + PA + R)Ak = 0. △

vii) For each positive-semidefinite matrix R ∈ Fn×n such that k = min {l ≥ 0 : = N(A)} exists, there exists a positive-definite matrix P ∈ Fn×n such that

∩n i=1

N(RAi+l−1 )

ATP + PA + AkTRAk = 0. Credit: Q. Hui. Fact 15.19.2. Fact 15.19.3. Fact 15.19.4. Fact 15.19.5.

Let A ∈ Fn×n, and assume that A is semistable. Then, A is Lyapunov stable. Let A ∈ Fn×n, and assume that A is Lyapunov stable. Then, A is group invertible. Let A ∈ Fn×n, and assume that A is semistable. Then, A is group invertible. Let A, B ∈ Fn×n, and assume that A and B are similar. Then, A is (Lyapunov stable, semistable, asymptotically stable, discrete-time Lyapunov stable, discrete-time semistable, discrete-time asymptotically stable) if and only if B is. Fact 15.19.6. Let A ∈ Rn×n, and assume that A is Lyapunov stable. Then, ∫ 1 t τA e dτ = I − AA#. lim t→∞ t 0

Related: Fact 15.19.3. Fact 15.19.7. Let A ∈ Fn×n, and assume that A is semistable. Then,

lim etA = I − AA#,

t→∞

and thus

1 t→∞ t

∫

t

lim

eτA dτ = I − AA#.

0

Related: Fact 12.13.12, Fact 15.19.2, and Fact 15.19.3. Fact 15.19.8. Let A, B ∈ Fn×n. Then, limα→∞ eA+αB exists if and only if B is semistable. If these

conditions hold, then #

#

lim eA+αB = e(I−BB )A (I − BB# ) = (I − BB# )eA(I−BB ).

α→∞

1227

THE MATRIX EXPONENTIAL AND STABILITY THEORY

Source: [625]. Fact 15.19.9. Let A ∈ Rn×n, assume that A is asymptotically stable, let β ∈ (αmax (A), 0), let

P ∈ Rn×n be positive definite and satisfy

ATP + PA ≤ 2βP, and let ∥ · ∥ be a normalized, submultiplicative norm on Rn×n. Then, for all t ≥ 0, √ ∥etA ∥ ≤ ∥P∥∥P−1∥eβt. Source: [1398] and Fact 15.16.10. Fact 15.19.10. Let A ∈ Fn×n, assume that A is asymptotically stable, let R ∈ Fn×n, assume that

R is positive definite, and let P ∈ Fn×n be the positive-definite solution of A∗P + PA + R = 0. Then, for all t ≥ 0, √ √ σmax (P) −tλmin (RP−1 )/2 −1 tA σmax (e ) ≤ e , ∥etA ∥F ≤ ∥P∥F ∥P−1∥F e−tλmin (RP )/2. σmin (P)

If, in addition, A + A∗ is negative definite, then, for all t ≥ 0, ∗

∥etA∥F ≤ e−tλmin (−A−A )/2. Source: [1936]. Fact 15.19.11. Let A ∈ Rn×n, assume that A is asymptotically stable, let R ∈ Rn×n, assume

that R is positive definite, and let P ∈ Rn×n √ be the positive-definite solution of ATP + PA + R = 0. ′ △ Furthermore, define the vector norm ∥x∥ = xTPx on Rn, let ∥ · ∥ denote the induced norm on Rn×n, and let µ(·) denote the corresponding logarithmic derivative. Then, µ(A) = −λmin (RP−1 )/2. Consequently, for all t ≥ 0,

−1

∥etA∥ ≤ e−tλmin (RP

.

)/2

Source: [1475] and xiv) of Fact 15.16.7. Related: Fact 15.16.7 for the logarithmic derivative. Fact 15.19.12. Let A ∈ Fn×n. Then, A is similar to a skew-Hermitian matrix if and only if there exists a positive-definite matrix P ∈ Fn×n such that A∗P + PA = 0. Related: Fact 7.10.5. 2 Fact 15.19.13. Let A ∈ Rn×n. Then, ( A)and( A are) asymptotically stable if and only if, for all

5π 3π ȷθ λ ∈ spec(A), there exist r > 0 and θ ∈ π2 , 3π 4 ∪ 4 , 2 such that λ = re . n×n Fact 15.19.14. Let A ∈ R . Then, A is group invertible and 2kπȷ < spec(A) for all k ≥ 1 if and only if AA# = (eA − I)(eA − I)#.

In particular, if A is semistable, then this equality holds. Source: ii) of Fact 15.22.13 and ix) of Proposition 15.9.2. Fact 15.19.15. Let A ∈ Fn×n. Then, A is asymptotically stable if and only if A−1 is asymptoti−1 cally stable. Hence, etA → 0 as t → ∞ if and only if etA → 0 as t → ∞. Fact 15.19.16. Let A, B ∈ Rn×n, assume that A is asymptotically stable, and assume that σmax (B ⊕ B) < σmin (A ⊕ A). Then, A + B is asymptotically stable. Source: Since A ⊕ A is nonsingular, Fact 11.16.22 implies that A ⊕ A + α(B ⊕ B) = (A + αB) ⊕ (A + αB) is nonsingular for all 0 ≤ α ≤ 1. Now, suppose that A + B is not asymptotically stable. Then, there exists α0 ∈ (0, 1] such that A + α0 B has an imaginary eigenvalue, and thus (A + α0 B) ⊕ (A + α0 B) = A ⊕ A + α0 (B ⊕ B) is singular, which is a contradiction. Remark: This is a suboptimal solution of a nearness problem. See [1387, Sect. 7] and Fact 11.16.22.

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CHAPTER 15

Fact 15.19.17. Let A ∈ Cn×n, assume that A is asymptotically stable, let ∥ · ∥ denote either

σmax (·) or ∥ · ∥F , and define △

β(A) = {∥B∥: B ∈ Cn×n and A + B is not asymptotically stable}. Then, 1 2 σmin (A ⊗ A)

≤ β(A) = min σmin (A + γ ȷI) ≤ min{αmax (A), σmin (A), 21 σmax(A + A∗ )}. γ∈R

Furthermore, let R ∈ F , assume that R is positive definite, and let P ∈ Fn×n be the positive-definite solution of A∗P + PA + R = 0. Then, n×n

1 2 σmin (R)/∥P∥

≤ β(A).

∗

If, in addition, A + A is negative definite, then − 21 λmin (A + A∗ ) ≤ β(A). Source: [1387, 2775]. Remark: Real matrices and real perturbations are discussed in [2283]. Fact 15.19.18. Let A ∈ Fn×n, assume that A is asymptotically stable, let V ∈ Fn×n, assume that

V is positive definite, and let Q ∈ Fn×n be the positive-definite solution of AQ + QA∗ + V = 0. Then, for all t ≥ 0, ∗ −1 −∗ ∥etA ∥2F = tr etAetA ≤ κ(Q)tr e−tS VS ≤ κ(Q)tr e−[t/σmax (Q)]V, △

where S ∈ Fn×n satisfies Q = SS ∗ and κ(Q) = σmax (Q)/σmin (Q). If, in particular, AQ + QA∗ + I = 0, then ∥etA ∥2F ≤ nκ(Q)e−t/σmax (Q). ∗

∗

Source: [2934]. Remark: Fact 15.16.4 yields etAetA ≤ et(A+A ). However, this bound is poor in the case where A + A∗ is not asymptotically stable. See [427]. Related: Fact 15.19.19. Fact 15.19.19. Let A ∈ Fn×n, assume that A is asymptotically stable, let Q ∈ Fn×n be the △

positive-definite solution of AQ + QA∗ + I = 0, and define κ(Q) = σmax (Q)/σmin (Q). Then, for all t ≥ 0, 2 σmax (etA ) ≤ κ(Q)e−t/σmax (Q). √ Source: [2795, 2796] and references therein. Remark: Since ∥etA ∥F ≤ nσmax (etA ), it follows that this inequality implies the last inequality in Fact 15.19.18. Fact 15.19.20. Let A ∈ Rn×n, and assume that A >> 0. Then, A is unstable. Source: Fact 6.11.5. Fact 15.19.21. Let A ∈ Rn×n. Then, A is asymptotically stable if and only if there exist B, C ∈ n×n R such that B is positive definite, C is dissipative, and A = BC. Source: A = P−1 (−ATP − R). Remark: To reverse the ordering of the factors, consider AT. Fact 15.19.22. Let A ∈ Fn×n. Then, the following statements hold: i) All of the real eigenvalues of A are positive if and only if A is the product of two dissipative matrices. ii) A is nonsingular and A , αI for all α < 0 if and only if A is the product of two asymptotically stable matrices. iii) A is nonsingular if and only if A is the product of three or fewer asymptotically stable matrices. Source: [272, 2919]. Fact 15.19.23. Let p ∈ R[s], where p(s) = sn + βn−1 sn−1 + · · · + β1 s + β0 and β0 , . . . , βn > 0.

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THE MATRIX EXPONENTIAL AND STABILITY THEORY

Furthermore, define A ∈ Rn×n by   βn−1  1   0  △  A =  0  ..  .  0  0

βn−3 βn−2 βn−1 1 .. .

βn−5 βn−4 βn−3 βn−2 .. .

0 0

0 0

βn−7 βn−6 βn−5 βn−4 .. . ··· ···

··· ··· ··· ··· .. . ··· ···

       .    0   β0

··· ··· ··· ··· .. .

0 0 0 0 .. .

β1 β2

If p is Lyapunov stable, then every subdeterminant of A is nonnegative. Source: [184]. Remark: A is totally nonnegative. Furthermore, p is asymptotically stable if and only if every leading principal subdeterminant of A is positive. Remark: The diagonal entries of A are βn−1 , . . . , β0 . Credit: The second statement is due to A. Hurwitz. Problem: Show that this stability condition is equivalent to the condition given in [1040, p. 183]. Fact 15.19.24. Let A ∈ Rn×n, assume that A is tridiagonal, assume that A(i,i) > 0 for all i ∈ {1, . . . , n}, and assume that A(i,i+1) A(i+1,i) > 0 for all i ∈ {1, . . . , n − 1}. Then, A is asymptotically stable. Source: [636]. Credit: S. Barnett and C. Storey. Fact 15.19.25. Let A ∈ Rn×n, and assume that A is cyclic. Then, there exists a nonsingular △ matrix S ∈ Rn×n such that AS = SAS −1 is given by the tridiagonal matrix   1 0 ··· 0 0   0     −αn 0 1 ··· 0 0    ..   . −αn−1 0 0 0   0  AS =  . .. ..  , .. .. ..  .. . . . . .    ..  . 0 0 0 1   0   0 0 0 · · · −α2 −α1 where α1 , . . . , αn are real numbers. If α1α2 · · · αn , 0, then the number of eigenvalues of A in the OLHP is equal to the number of positive elements in {α1 , α1α2 , . . . , α1α2 · · · αn }ms . Furthermore, ATSP + PAS + R = 0, where △

P = diag(α1α2 · · · αn , α1α2 · · · αn−1 , . . . , α1α2 , α1 ),

△

R = diag(0, . . . , 0, 2α21 ).

Finally, AS is asymptotically stable if and only if α1 , . . . , αn are positive. Source: [302, pp. 52, 95]. Remark: AS is a Schwarz matrix. See Fact 15.19.26 and Fact 15.19.27. Fact 15.19.26. Let α1 , . . . , αn be real numbers, and define A ∈ Rn×n by   1 0 ··· 0 0   0     −α 0 1 ··· 0 0  n    ..  . −αn−1 0 0 0   0   A =  . .. ..  . .. .. ..  .. . . . . .    ..  . 0 0 0 1   0   0 0 0 · · · −α2 α1 Then, spec(A) ⊂ ORHP if and only if α1 , . . . , αn are positive. Source: [1450, p. 111]. Remark: Note the absence of the minus sign in the (n, n) entry compared to the matrix in Fact 15.19.25. This

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CHAPTER 15

minus sign changes the signs of all of the eigenvalues of A. Fact 15.19.27. Let α1 , α2 , α3 > 0, and define AR , P, R ∈ R3×3 by the tridiagonal matrix   α1/2 0   −α1 2    △   AR =  −α1/2 0 α1/2 2 3    1/2 0 −α3 0 and the diagonal matrices

△

△

P = I,

R = diag(2α1 , 0, 0).

Then, ATRP + PAR + R = 0. Remark: The matrix AR is a Routh matrix. The Routh matrix AR and the Schwarz matrix AS are related by AR = S RS AS S R−1S , where     0 0 α1/2 1   △  1/2  S RS =  0 −(α1α2 ) 0  .   (α1α2 α3 )1/2 0 0 Related: Fact 15.19.25. Fact 15.19.28. Let α1 , α2 , α3 > 0, and define AC , P, R ∈ R3×3 by the tridiagonal matrix

  0  △  AC =  −1/a2  0

 0   1/a2   −1/a1

1/a3 0 −1/a1

and the diagonal matrices △

△

P = diag(a3 , a2 , a1 ), △

△

R = diag(0, 0, 2),

△

where a1 = 1/α1 , a2 = α1/α2 , and a3 = α2 /(α1α3 ). Then, ATC P + PAC + R = 0. Source: [696, p. 346]. Remark: The matrix AC is a Chen matrix. The Schwarz matrix AS and the Chen matrix AC −1 are related by AS = S SC AC S SC , where   0 0   1/(α1α3 )   △  0 1/α2 0  . S SC =    0 0 1/α1 Remark: Schwarz, Routh, and Chen matrices are used to prove the Routh criterion. See [74, 590, 696, 2204]. Remark: A circuit interpretation of the Chen matrix is given in [1955]. Fact 15.19.29. Let α1 , . . . , αn > 0 and β1 , . . . , βn > 0, and define A ∈ Rn×n by

  −α1   β  2  . △  A =  ..    0  0

0

···

0

−α2

···

0

..

..

.

.. .

.

−αn−1

Then, χA (s) =

.

0

..

0

···

n ∏ i=1

βn

(s + αi ) +

n ∏ i=1

 −β1    0   ..  .  .   0   −αn

βi .

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THE MATRIX EXPONENTIAL AND STABILITY THEORY

Furthermore, if

π ∏ αi < , n β i=1 i n

cosn

then A is asymptotically stable. Source: [2497]. Remark: For n = 2, A is asymptotically stable for all positive α1 , β1 , α2 , β2 . Remark: This is the secant condition. Fact 15.19.30. Let A ∈ Fn×n. Then, the following statements are equivalent: i) A is asymptotically stable. ii) There exist a negative-definite matrix B ∈ Fn×n, a skew-Hermitian matrix C ∈ Fn×n, and a nonsingular matrix S ∈ Fn×n such that A = B + SCS −1. iii) There exist a negative-definite matrix B ∈ Fn×n, a skew-Hermitian matrix C ∈ Fn×n, and a nonsingular matrix S ∈ Fn×n such that A = S (B + C)S −1. Source: [831]. Fact 15.19.31. Let A ∈ Rn×n and k ≥ 2. Then, there exist asymptotically stable matrices ∑ A1 , . . . , Ak ∈ Rn×n such that A = ki=1 Ai if and only if tr A < 0. Source: [1508]. Fact 15.19.32. Let A ∈ Cn×n. Then, A is (Lyapunov stable, semistable, asymptotically stable) ∗ if and only if A ⊕ A is. Source: Use Fact 9.5.8 and vec(etAVetA ) = et(A⊕A) vec V. Fact 15.19.33. Let A ∈ Rn×n and B ∈ Rm×m. Then, the following statements hold: i) If A and B are (Lyapunov stable, semistable, asymptotically stable), then so is A ⊕ B. ii) If A ⊕ B is (Lyapunov stable, semistable, asymptotically stable), then so is either A or B. Source: Fact 9.5.8. Fact 15.19.34. Let A ∈ Rn×n, and assume that A is asymptotically stable. Then, ∫ ∞ ∫ ∞ ( ȷωI − A)−1 dω = πI, (ω2I + A2 )−1 dω = −πA−1, −∞ −∞ ∫ −1 ∞ (A ⊕ A)−1 = (− ȷωI − A)−1 ⊗ ( ȷωI − A)−1 dω. 2π −∞ Source: Use ( ȷωI − A)−1 + (− ȷωI − A)−1 = −2A(ω2I + A2 )−1 and Corollary 16.11.5. Fact 15.19.35. Let A ∈ R2×2. Then, A is asymptotically stable if and only if tr A < 0 and det A > 0. Fact 15.19.36. Let A ∈ Cn×n. Then, there exists a unique asymptotically stable matrix B ∈ Cn×n such that B2 = −A. Source: [2540]. Remark: The uniqueness of the square root of a complex matrix that has no eigenvalues in (−∞, 0] is implicitly assumed in [2541]. Related: Fact 7.17.21. Fact 15.19.37. Let A ∈ Rn×n. Then, the following statements hold:

i) ii) iii) iv) v)

If A is semidissipative, then A is Lyapunov stable. If A is dissipative, then A is asymptotically stable. If A is Lyapunov stable and normal, then A is semidissipative. If A is asymptotically stable and normal, then A is dissipative. If A is discrete-time Lyapunov stable and normal, then A is semicontractive. Fact 15.19.38. Let M ∈ Rr×r, assume that M is positive definite, let C, K ∈ Rr×r, assume that C and K are positive semidefinite, and consider the equation M¨q + Cq˙ + Kq = 0. Furthermore, define

  A =  △

0

I

−M−1K

−M−1C

   .

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CHAPTER 15

Then, the following statements hold: i) A is Lyapunov stable if and only if C + K is positive definite. [ ] ii) A is Lyapunov stable if and only if rank CK = r. iii) A is semistable if and only if (M −1K, C) is observable. iv) A is asymptotically stable if and only if A is semistable and K is positive definite. Source: [431]. Related: Fact 7.13.31.

15.20 Facts on Almost Nonnegative Matrices Fact 15.20.1. Let A ∈ Rn×n. Then, etA is nonnegative for all t ≥ 0 if and only if A is almost nonnegative. Source: Let α > 0 satisfy αI + A ≥≥ 0, and consider et(αI+A). See [422, p. 74], [423, p. 146], [434, 812], and [2451, p. 37]. Fact 15.20.2. Let A ∈ Rn×n, and assume that A is almost nonnegative. Then, etA is positive for all t > 0 if and only if A is irreducible. Source: [2418, p. 208]. Fact 15.20.3. Let A ∈ Rn×n, where n ≥ 2, and assume that A is almost nonnegative. Then, the following statements are equivalent: i) There exist α ∈ (0, ∞) and B ∈ Rn×n such that A = B − αI, B ≥≥ 0, and ρmax (B) ≤ α. ii) spec(A) ⊂ OLHP ∪ {0}. iii) spec(A) ⊂ CLHP. iv) If λ ∈ spec(A) is real, then λ ≤ 0. v) Every principal subdeterminant of −A is nonnegative. vi) For every diagonal, positive-definite matrix B ∈ Rn×n, it follows that A − B is nonsingular. Remark: A is an N-matrix if A is almost [ ] nonnegative and i)–vi) hold. Related: This result follows from Fact 6.11.13. Example: A = 00 10 . Fact 15.20.4. Let A ∈ Rn×n, where n ≥ 2, and assume that A is almost nonnegative. Then, the following statements are equivalent: i) A is a group-invertible N-matrix. ii) A is a Lyapunov-stable N-matrix. iii) A is a semistable N-matrix. iv) A is Lyapunov stable. v) A is semistable. vi) A is an N-matrix, and there exist α ∈ (0, ∞) and B ∈ Rn×n such that A = B − αI, B ≥≥ 0, and α−1B is discrete-time semistable. vii) There exists a positive-definite matrix P ∈ Rn×n such that ATP + PA is negative semidefinite. Furthermore, consider the following statements: viii) There exists a positive vector p ∈ Rn such that −Ap is nonnegative. ix) There exists a nonzero nonnegative vector p ∈ Rn such that −Ap is nonnegative. Then, viii) =⇒ [i)–vii)] =⇒ ix). Source: [423, pp. 152–155] and [424]. The statement [i)–vii)] =⇒ ix) by Fact 6.11.15. Remark: The converse of viii) =⇒ [i)–vii)] is false. For example, [ is given ] 0 1 A = 0 −1 is almost negative and semistable, but there does not exist a positive vector p ∈ R2 such that −Ap is nonnegative. However, note that viii) holds for AT, but not for either diag(A, AT ) or its transpose. Remark: A discrete-time semistable matrix is called semiconvergent in [423, p. 152]. Remark: The last statement follows from the fact that the function V(x) = pTx is a Lyapunov ˙ function for the system x˙ = −Ax for x ∈ [0, ∞)n with Lyapunov derivative V(x) = −ATp. See [683, 1286].

THE MATRIX EXPONENTIAL AND STABILITY THEORY

1233

Fact 15.20.5. Let A ∈ Rn×n, where n ≥ 2, and assume that A is almost nonnegative. Then, the

following statements are equivalent: i) A is a nonsingular N-matrix. ii) A is asymptotically stable. iii) A is an asymptotically stable N-matrix. iv) There exist α ∈ (0, ∞) and B ∈ Rn×n such that A = B − αI, B ≥≥ 0, and ρmax (B) < α. v) If λ ∈ spec(A) is real, then λ < 0. vi) If B ∈ Rn×n is nonnegative and diagonal, then A − B is nonsingular. vii) Every principal subdeterminant of −A is positive. viii) Every leading principal subdeterminant of −A is positive. ix) For all i ∈ {1, . . . , n}, the sign of the ith leading principal subdeterminant of A is (−1)i. x) For all k ∈ {1, . . . , n}, the sum of all k × k principal subdeterminants of −A is positive. xi) There exists a positive-definite matrix P ∈ Rn×n such that ATP + PA is negative definite. xii) There exists a positive vector p ∈ Rn such that −Ap is positive. xiii) There exists a nonnegative vector p ∈ Rn such that −Ap is positive. xiv) If p ∈ Rn and −Ap is nonnegative, then p ≥≥ 0 is nonnegative. xv) For every nonnegative vector y ∈ Rn , there exists a unique nonnegative vector x ∈ Rn such that Ax = −y. xvi) A is nonsingular and −A−1 is nonnegative. Source: [422, pp. 134–140] and [1450, pp. 114–116]. Remark: −A is a nonsingular M-matrix. See Fact 6.11.13. △ Fact 15.20.6. For all i, j ∈ {1, . . . , n}, let σi j ∈ [0, ∞), and define A ∈ Rn×n by A(i, j) = σi j for all ∑n △ i , j and A(i,i) = − j=1 σi j . Then, the following statements hold: i) A is almost nonnegative. ii) −A1n×1 = [σ11 · · · σnn ]T is nonnegative. iii) spec(A) ⊂ OLHP ∪ {0}. iv) A is an N-matrix. v) A is a group-invertible N-matrix. vi) A is a Lyapunov-stable N-matrix. vii) A is a semistable N-matrix. If, in addition, σ11 , . . . , σnn are positive, then A is a nonsingular N-matrix. Source: It follows from the Gershgorin circle theorem given by Fact 6.10.22 that every eigenvalue λ of A is an element of ∑ ∑ a disk in C centered at − nj=1 σi j ≤ 0 and with radius nj=1, j,i σi j . Hence, if σii = 0, then either λ = 0 or Re λ < 0, whereas, if σii > 0, then Re λ ≤ σii < 0. Thus, iii) holds. iv)–vii) follow from ii) and Fact 15.20.4. The last statement follows from the Gershgorin circle theorem. Remark: AT is a compartmental matrix. See [434, 1288, 2809]. Problem: Determine necessary and sufficient conditions on the parameters σi j such that A is a nonsingular N-matrix. Fact 15.20.7. Let G = (X, R) be a directed graph, where X = {x1 , . . . , xn }, and let L ∈ Rn×n denote either the inbound Laplacian or the outbound Laplacian of G. Then, the following statements hold: i) −L is semistable. ii) limt→∞ e−Lt exists. Remark: Fact 15.20.6. Remark: The spectrum of the Laplacian is discussed in [16].

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Fact 15.20.8. Let A ∈ Rn×n, and assume that A is asymptotically stable. Then, at least one of

the following statements holds: i) All of the diagonal entries of A are negative. ii) At least one diagonal entry of A is negative and at least one off-diagonal entry of A is negative. Source: [1080]. Remark: sign stability is discussed in [1525].

15.21 Facts on Discrete-Time-Stable Polynomials Fact 15.21.1. Let p ∈ R[s], where p(s) = sn + an−1 sn−1 + · · · + a0 , and assume that p is

discrete-time asymptotically stable. Then, the following statements hold: i) p(1) > 0. ii) If n is even, then p(−1) > 0. iii) If n is odd, then p(−1) < 0. Fact 15.21.2. Let p ∈ R[s], where p(s) = sn +an−1 sn−1 +· · ·+a0 . Then, the following statements hold: i) If n = 1, then p is discrete-time asymptotically stable if and only if |a0 | < 1. ii) If n = 2, then p is discrete-time asymptotically stable if and only if |a0 | < 1 and |a1 | < 1+a0 . iii) If n = 3, then the following conditions are equivalent: a) p is discrete-time asymptotically stable. b) |a0 + a2 | < |1 + a1 |, and |a1 − a0 a2 | < 1 − a20 . c) |a0 + a2 | < 1 + a1 , and |a1 − a0 a2 | < 1 − a20 . d) |a0 + a2 | < 1 + a1 , |3a0 − a2 | < 3 − a1 , and a20 + a1 − a0 a2 < 1. iv) If n = 4, then the following conditions are equivalent: a) p is discrete-time asymptotically stable. b) |a0 | < 1, |a1 + a3 | < 1 + a0 + a2 , (a0 a3 − a1 )(a1 + a3 ) + (1 − a20 )(1 + a0 + a2 ) > 0, and (a0 a3 − a1 )(a1 − a3 ) + (1 − a0 )2 (1 + a0 − a2 ) > 0. c) |a1 − a0 a3 | < 1 − a20 , [(a0 a3 − a1 )(a1 + a3 ) + (1 − a20 )(1 + a0 + a2 )][(a0 a3 − a1 )(a1 − a3 ) + (1 − a0 )2 (1 + a0 − a2 )] > 0, |(a3 − a0 a1 )(1 + a0 ) − a2 (a1 − a0 a3 )| < |(a0 a3 − a1 )(a1 + a3 ) + (1 − a20 )(1 + a0 + a2 )|. Source: [288, p. 185], [622, p. 6], [1399, p. 355], and [1562, pp. 34, 35]. Remark: These conditions are the Schur-Cohn criterion and Jury criterion. Remark: The Routh criterion for the asymptotic stability of a polynomial is given by Fact 15.18.3. Problem: Show that the inequalities b), c), and d) for n = 3 equivalent and that the inequalities b) and c) for n = 4 equivalent. Fact 15.21.3. Let p ∈ C[s], where p(s) = sn + an−1 sn−1 + · · · + a0 , and define pˆ ∈ C[s] by △

p(s) ˆ = sn−1 +

a1 − a0 an−1 an−1 − a0 a1 n−1 an−2 − a0 a2 n−2 s + s + ··· + . 1 − |a0 |2 1 − |a0 |2 1 − |a0 |2

Then, p is discrete-time asymptotically stable if and only if |a0 | < 1 and pˆ is discrete-time asymptotically stable. Source: [1399, p. 354]. Fact 15.21.4. Let p ∈ R[s], where p(s) = sn +an−1 sn−1 +· · ·+a0 . Then, the following statements hold: i) If a0 ≤ · · · ≤ an−1 ≤ 1, then roots(p) ⊂ {z ∈ C : |z| ≤ 1 + |a0 | − a0 }. ii) If 0 < a0 ≤ · · · ≤ an−1 ≤ 1, then roots(p) ⊂ CIUD. iii) If 0 < a0 < · · · < an−1 < 1, then p is discrete-time asymptotically stable. Source: For i), see [2427]. For ii), see [2046, p. 272]. For iii), use Fact 15.21.3. See [1399, p.

1235

THE MATRIX EXPONENTIAL AND STABILITY THEORY

355]. Remark: If there exists r > 0 such that 0 < ra0 < · · · < rn−1 an−1 < rn, then roots(p) ⊂ {z ∈ C : |z| ≤ r}. Remark: ii) is the Enestrom-Kakeya theorem. Fact 15.21.5. Let p ∈ C[s], where p(s) = sn + an−1 sn−1 + · · · + a0 , and assume that a0 , . . . , an−1 are nonzero. Then, ρmax (p) ≤ max {2|an−1 |, 2|an−2 /an−1 |, . . . , 2|a1/a2 |, |a0 /a1 |}. Credit: N. Bourbaki. See [2047]. This is Kojima’s bound. See [2979, pp. 222, 223]. Fact 15.21.6. Let p ∈ R[s], where p(s) = an sn + an−1 sn−1 + · · · + a0 , assume that a0 , . . . , an are

positive, and let λ ∈ roots(p). Then,

ai−1 ai−1 ≤ |λ| ≤ max . 1≤i≤n ai 1≤i≤n ai min

Source: [2979, p. 225]. Related: This result implies ii) of Fact 15.21.4. Fact 15.21.7. Let p ∈ R[s], where p(s) = an sn + an−1 sn−1 + · · · + a0 , assume that a0 , . . . , an are

nonzero, and let λ ∈ roots(p). Then,  n (n)  2 i Fi 3 min  2 1≤i≤n F4i

 1/i  4i a0  ≤ |λ| ≤ 2 max  F  () ai  3 1≤i≤n  2n n Fi i

1/i an−i  . an 

Source: [885]. Remark: Extensions are given in [485]. Remark: Fi is a Fibonacci number. Fact 15.21.8. Let p ∈ C[s], where p(s) = sn + an−1 sn−1 + · · · + a0 , assume that a0 , . . . , an−1 are

nonzero, and let λ ∈ roots(p). Then, |λ| ≤

n−1 ∑

|ai |1/(n−i) ,

|λ + 21 an−1 | ≤ 21 |an−1 | +

i=1

n−2 ∑

|ai |1/(n−i).

i=0

Credit: J. L. Walsh. See [2047]. Fact 15.21.9. Let p ∈ C[s], where p(s) = sn + an−1 sn−1 + · · · + a0 , and let λ ∈ roots(p). Then,

|a0 | < |λ| ≤ max {|a0 |, 1 + |a1 |, . . . , 1 + |an−1 |}. |a0 | + max {|a1 |, . . . , |an−1 |, 1} Source: The lower bound is given in [2047], while the upper bound is given in [882] and [2979, p. 222]. Remark: The upper bound is Cauchy’s estimate. Remark: The weaker upper bound |λ| < 1 + maxi=0,...,n−1 |ai | is given in [288, p. 184] and [2047]. △ Fact 15.21.10. Let p ∈ C[s], where p(s) = sn +an−1 sn−1 +· · ·+a0 , and define α = maxi=0,...,n−1 |ai |. Then, √  α(1 − nα)  n   , α ≤ 1/n,     1 − (nα)1/n ρmax (p) ≤  ) } { (   2(nα − 1) α    , 1 + , 1/n ≤ α. min (1 + α) 1 −   n+1 (1 + α)n+1 − nα Source: [500]. Remark: Extensions are given in [1513]. Fact 15.21.11. Let p ∈ C[s], where p(s) = sn + an−1 sn−1 + · · · + a0 . Then,

ρmax (p) ≤

√

2 + max |ai |2 + |an−1 |2 ,

ρmax (p) ≤ 21 (1 + |an−1 |) +

i=0,...,n−2

√

max |ai | + 14 (1 − |an−1 |)2 ,

i=0,...,n−2

ρmax (p) ≤ max {2, |a0 | + |an−1 |, |a1 | + |an−1 |, . . . , |an−2 | + |an−1 |}.

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CHAPTER 15

Source: [882]. Credit: The second inequality is due to A. Joyal, G. Labelle, and Q. I. Rahman.

See [2047]. Fact 15.21.12. Let p ∈ C[s], where p(s) = sn + an−1 sn−1 + · · · + a0 , assume that a0 , . . . , an−1 are

nonzero, define

{ } a0 a1 an−2 △ , α = max , , . . . , a1 a2 an−1

{ } a1 a2 an−2 △ , β = max , , . . . , a2 a3 an−1

and let λ ∈ roots(p). Then,

√ |λ| ≤ 21 (β + |an−1 |) + α|an−1 | + 14 (β − |an−1 |)2 , |λ| ≤ |an−1 | + α, } { √ a0 |λ| ≤ max , 2β, 2|an−1 | , |λ| ≤ 2 max |ai |1/(n−i) , |λ| ≤ 2|an−1 |2 + α2 + β2 . i=0,...,n−1 a1

Source: [882, 1857]. Remark: The third inequality is Kojima’s bound, while the fourth inequality

is Fujiwara’s bound. ∑ △ 2 Fact 15.21.13. Let p ∈ C[s], where p(s) = sn + an−1 sn−1 + · · · + a0 , define α = 1 + n−1 i=0 |ai | , and let λ ∈ roots(p). Then, v u   t n−1 ∑  1 n  1 n − 1 + |λ| ≤ |an−1 | + |ai |2 − |an−1 |2  , n n−1 n i=0 v   u v u u u t n−2   t ∑  1  2 2 |λ| ≤ |an−1 | + 1 + (|an−1 | − 1) + 4 |ai |  , 2   i=0 v   u t(   n−3 ) ∑ 2 π π 1  + (|an−2 | + 1)2 + |ai |2  , |an−1 | − cos |λ| ≤ |an−1 | + cos + 2 n n  i=0 v   u t n−1   ∑ π 1  |λ| ≤ cos + |an−1 | + |ai |2  , n+1 2  i=0 √ ( √ √ ) ) α2 − 4|a0 |2 ≤ |λ| ≤ 12 α + α2 − 4|a0 |2 , v   u t   n−3 ) ( ∑ 2 π 1 π  + (|an−2 | − 1)2 + |ai |2  , | Re λ| ≤ | Re an−1 | + cos + | Re an−1 | − cos 2 n n  i=0 √ ( 1 2

α−

v   u t(   n−3 ) ∑ 2 1 π π  | Im λ| ≤ | Im an−1 | + cos + | Im an−1 | − cos + (|an−2 | + 1)2 + |ai |2  . 2 n n  i=0 Source: [1090, 1633, 1638, 1857]. Remark: The Parodi bound refines the Carmichael-Mason bound. See Fact 15.21.14. Credit: The first bound is due to H. Linden (see [1638]), the fourth bound is due to M. Fujii and F. Kubo, and the upper bound in the fifth string, which follows from Fact 7.12.24 and Fact 7.12.34, is due to M. Parodi, see [2047]. See also [1601, 1628]. Fact 15.21.14. Let p ∈ C[s], where p(s) = sn + an−1 sn−1 + · · · + a0 , let r, q ∈ (1, ∞), assume △ ∑ r 1/r that 1/r + 1/q = 1, and define α = ( n−1 i=0 |ai | ) . Then,

ρmax (p) ≤ (1 + αq )1/q.

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THE MATRIX EXPONENTIAL AND STABILITY THEORY

In particular, if r = q = 2, then ρmax (p) ≤

√ 1 + |an−1 |2 + · · · + |a0 |2 .

Furthermore, for r → 1 and q → ∞, ρmax (p) ≤ max {1, |a0 | + · · · + |an−1 |}. Source: [1857, 2047]. For the last inequality, see [2979, p. 222]. Credit: The case r = q = 2 is due to R. D. Carmichael and T. E. Mason. The last inequality is Montel’s bound. Fact 15.21.15. Let p ∈ C[s], where p(s) = sn + an−1 sn−1 + · · · + a0 , let m ∈ {1, . . . , n}, and define △ Sm = {z ∈ C : |z| < 1 + maxi∈{1,...,m−1} |ai |1/(n−m+1) }. Then, at least m roots of p are contained in Sm . In particular, ρmax (p) < 1 + max {|a0 |, . . . , |an−1 |}. Source: [1954, pp. 123, 151]. Related: Fact 15.21.14. Fact 15.21.16. Let p ∈ C[s], where p(s) = sn + an−1 sn−1 + · · · + a0 , let m ∈ {1, . . . , n}, let r ≥ 1,

∑m−1 r 1/r △ and define Sm,r = {z ∈ C : |z| ≤ 1 + ( i=0 |ai | ) }. Then, at least m roots of p are contained in Sm,r . In particular, √ ρmax (p) ≤ 1 + |an−1 |2 + · · · + |a0 |2 .

Source: [1954, p. 151] and [1978]. Related: Fact 15.21.14. Fact 15.21.17. Let p ∈ C[s], where p(s) = sn + an−1 sn−1 + · · · + a0 . Then,

ρmax (p) ≤

√ 1 + |1 − an−1 |2 + |an−1 − an−2 |2 + · · · + |a1 − a0 |2 + |a0 |2 .

Source: [1978, 2965]. Credit: K. P. Williams. Fact 15.21.18. Let p ∈ C[s], where p(s) = sn +an−1 sn−1 +· · ·+a0 , let mroots(p) = {λ1 , . . . , λn }ms , △

and let r > 0 be the unique positive root of p(s) ˆ = sn − |an−1 |sn−1 − · · · − |a0 |. Then, √n ( 2 − 1)r ≤ max |λi | ≤ r, i∈{1,...,n}

n √n 1∑ |λi | < r. ( 2 − 1)r ≤ n i=1

Finally, the third inequality is an equality if and only if λ1 = · · · = λn . Credit: The first inequality is due to A. Cohn, the second inequality is due to A. L. Cauchy, and the third and fourth inequalities are due to L. Berwald. See [2046, p. 245] and [2047]. ∑ △ 2 Fact 15.21.19. Let p ∈ C[s], where p(s) = sn + an−1 sn−1 + · · · + a0 , define α = 1 + n−1 i=0 |ai | , and let λ ∈ roots(p). Then, √ ( √ ( √ √ ) ) 1 1 2 − 4|a |2 ≤ |λ| ≤ α − α α2 − 4|a0 |2 . 0 2 2 α+ Source: Fact 7.12.33, Fact 7.12.34, and [1634]. Fact 15.21.20. Let p ∈ C[s], where p(s) = an sn + an−1 sn−1 + · · · + a0 , and assume that a1 , . . . , an

are nonzero. Then, for all i ∈ {1, . . . , n},  [( ) ]1/i     n a0     , ∅. z ∈ roots(p) : |z| ≤      i ai

Source: [205]. Fact 15.21.21. Let n ≥ 2, let p ∈ C[s], where p(s) = an sn + a1 s + a0 , and assume that a1 and

an are nonzero. Then, { z ∈ roots(p) :

} z + a0 ≤ a0 , ∅, a1 a1

{

} a0 a0 z ∈ roots(p) : ≤ z + , ∅. a1 a1

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Consequently, {z ∈ roots(p) : |z| ≤ 2|a0 /a1 |} , ∅. Source: [205]. Remark: p is a trinomial. Credit: L. Fej´er. Fact 15.21.22. Let n ≥ 1, 0 ≤ m < n, p ∈ C[s], where p(s) = an sn + am sm + a1 s + a0 , and assume that a1 , am , and an are nonzero. Then, { } 17 a0 , ∅. z ∈ roots(p) : |z| ≤ 3 a1 Source: [205]. Remark: p is a quadrinomial. Fact 15.21.23. Let n ≥ 3, 2 ≤ m < n, p ∈ C[s], where p(s) = an sn + am sm + a1 s + a0 , and

assume that a1 , am , and an are nonzero. Then, { } 2n a0 , ∅. z ∈ roots(p) : |z| ≤ n − 1 a1

Source: [205]. Remark: 2n/(n − 1) ≤ 3. Fact 15.21.24. Let n ≥ 3, 1 ≤ m ≤ n − 2, p ∈ C[s], where p(s) = sn + an−1 sn−1 + am sm + a0 ,

and assume that a0 , am , and an−1 are nonzero. Then, } { n−1 |an−1 | ≤ |z| , ∅. z ∈ roots(p) : 2n

Source: [205]. Fact 15.21.25. Let n ≥ 3, let p ∈ C[s], where p(s) = an sn + a2 s2 + a1 s + a0 , and assume that

a2 and an are nonzero. Then,  ( )1/2     n a0     z ∈ roots(p) : |z| ≤ , ∅,      n − 2 a2

{

} a1 a ≤ z + 1 , ∅. z ∈ roots(p) : 2a2 2a2

Source: [205]. Fact 15.21.26. Let p ∈ R[s], where p(s) = sn + an−1 sn−1 + · · · + a0 , assume that a0 , . . . , an−1 are

nonnegative, and let x1 , . . . , xm ∈ [0, ∞). Then, √ √ m p( m x1 · · · xm ) ≤ p(x1 ) · · · p(xm ).

Source: [2129]. Remark: This result extends a result of C. Huygens for the case p(x) = x + 1. Credit: D. Mihet. Fact 15.21.27. Let p ∈ R[s], where p(s) = sn + an−1 sn−1 + · · · + a0 , and assume that mroots(p) =

{λ1 , . . . , λn } ⊂ R, where λn ≤ · · · ≤ λ1 . Then, 2nan−2 ≤ (n − 1)a2n−1 , and

2(n − 1)a2n−1 − 4nan−2 2(n − 1)a2n−1 − 4nan−2 ≤ (λn − λ1 )2 ≤ . n(n − 1) n Source: [400]. Related: Fact 15.21.28. Fact 15.21.28. Let p ∈ R[s], where p(s) = sn + an−1 sn−1 + · · · + a0 , and assume that at least one

of the following statements holds: i) (n − 1)a2n−1 < 2nan−2 . ii) a2n−1 < 2an−2 + na2/n 0 . iii) a0 , 0 and (n − 1)a21 < 2na2 a0 . iv) a0 , 0 and a21 < 2a2 a0 + na0(2n−2)/n . Then, p has at least two roots that are not real. Source: [2680]. Related: Fact 15.18.4, Fact 15.21.27, and Fact 15.21.29.

THE MATRIX EXPONENTIAL AND STABILITY THEORY

1239

Fact 15.21.29. Let p ∈ R[s], where p(s) = as3 + bs2 + cs + d, and assume that a , 0, and

assume that at least one of the following conditions is satisfied: i) b2 < 3ac. √3 ii) b2 < 2ac + 3 a4 d2 . iii) d , 0 and c2 < 3bd. √3 iv) d , 0 and c2 < 2bd + 3 a2 d4 . v) d , 0 and b2 c2 + 18abcd < 4b3 d + 4ac3 + 27a2 d2 . Then, p has exactly one real root. Source: [2680]. Related: Fact 2.21.2, Fact 6.10.7, Fact 15.18.4, and Fact 15.21.28. Fact 15.21.30. Let p ∈ C[s], where p(s) = sn + an−1 sn−1 + · · · + a0 , and let α and β be the largest nonnegative roots, respectively, of the polynomials △

q(s) = sn − |an−1 |sn−1 − |an−2 |sn−2 − · · · − |a1 |s − |a0 |, △

r(s) = sn + |an−1 |sn−1 − |an−2 |sn−2 − · · · − |a1 |s − |a0 |. Then,

roots(p) ⊂ Bα (0) ∩ Bβ+|an−1 | (−an−1 ).

Source: [2009]. Remark: This result extends a result due to A. L. Cauchy. n n−1 Fact 15.21.31. Let p ∈ C[s], where ( ) p(s) = an s + an−1 s + · · · + a0 , and assume that there

exists m ∈ {0, . . . , n − 1} such that |a0 | mn < |an−m |. Then, there exists λ ∈ roots(p) such that |λ| < 1. Source: [108, pp. 49, 280]. Fact 15.21.32. Let p, q ∈ C[s], assume that p and q are monic, deg p = deg q = n, ρmax (p) ≤ 1, π . Source: [1029]. and ρmax (q) ≤ 1, and let α ∈ [0, 1]. Then, ρmax [αp + (1 − α)q] ≤ cot 2n Fact 15.21.33. Let p, q ∈ R[s], assume that the leading coefficients of p and q are positive, △ assume that p and q have no roots in ORHP, and define θ = π/max {2, deg p, deg q}. Then, p + q has no roots in {z ∈ C : | arg z| < θ}. Now, let n ≥ 2, let p1 , . . . , pn ∈ R[s], assume that, for all i ∈ {1, . . . , n}, the leading coefficient of pi is positive, pi has no roots in ORHP, and deg pi ≤ n. ∑ Then, ni=1 pi has no roots in {z ∈ C : | arg z| < π/n}. Source: [1440].

15.22 Facts on Discrete-Time-Stable Matrices Fact 15.22.1. Let A ∈ R2×2. Then, the following statements are equivalent:

i) A is discrete-time asymptotically stable. ii) | tr A| < 1 + det A and | det A| < 1. Fact 15.22.2. Let A ∈ Rn×n, and assume that A is nonnegative. Then, the following statements are equivalent: i) A is discrete-time asymptotically stable. ii) Every principal subdeterminant of I − A is positive. Source: [353, p. 303]. Fact 15.22.3. Let A ∈ Fn×n. Then, A is discrete-time (Lyapunov stable, semistable, asymptotically stable) if and only if A2 is. Fact 15.22.4. Let A ∈ Rn×n, and let χA (s) = sn + an−1 sn−1 + · · · + a1 s + a0 . Then, for all k ≥ 0, Ak = x1 (k)I + x2 (k)A + · · · + xn (k)An−1, where, for all i ∈ {1, . . . , n} and k ≥ 0, xi : N 7→ R satisfies xi (k + n) + an−1 xi (k + n − 1) + · · · + a1 xi (k + 1) + a0 xi (k) = 0, with, for all i, j ∈ {1, . . . , n}, the initial conditions xi ( j − 1) = δi, j . Source: [1715].

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Fact 15.22.5. Let A ∈ Rn×n. Then, the following statements hold:

i) ii) iii) iv)

If A is semicontractive, then A is discrete-time Lyapunov stable. If A is contractive, then A is discrete-time asymptotically stable. If A is discrete-time Lyapunov stable and normal, then A is semicontractive. If A is discrete-time asymptotically stable and normal, then A is contractive. Problem: Prove these results using Fact 15.16.6. Fact 15.22.6. Let x ∈ Fn , let A ∈ Fn×n, and assume that A is discrete-time semistable. Then, ∑∞ i i=0 A x exists if and only if x ∈ R(A − I). If these conditions hold, then ∞ ∑

Ai x = −(A − I)# x.

i=0

Source: [1487]. Fact 15.22.7. Let A ∈ Fn×n. Then, A is discrete-time (Lyapunov stable, semistable, asymptotically stable) if and only if A ⊗ A is. Source: Fact 9.4.21. Remark: See [1487]. Fact 15.22.8. Let A ∈ Rn×n and B ∈ Rm×m. Then, the following statements hold:

i) If A and B are discrete-time (Lyapunov stable, semistable, asymptotically stable), then A⊗B is discrete-time (Lyapunov stable, semistable, asymptotically stable). ii) If A ⊗ B is discrete-time (Lyapunov stable, semistable, asymptotically stable), then either A or B is discrete-time (Lyapunov stable, semistable, asymptotically stable). Source: Fact 9.4.21. Fact 15.22.9. Let A ∈ Rn×n, and assume that A is (Lyapunov stable, semistable, asymptotically stable). Then, eA is discrete-time (Lyapunov stable, semistable, asymptotically stable). Problem: If B ∈ Rn×n is discrete-time (Lyapunov stable, semistable, asymptotically stable), under what conditions does there exist a (Lyapunov-stable, semistable, asymptotically stable) matrix A ∈ Rn×n such that B = eA ? See Proposition 15.4.5. Fact 15.22.10. The following statements hold: △ i) If A ∈ Rn×n is discrete-time asymptotically stable, then B = (A+I)−1(A−I) is asymptotically stable. △ ii) If B ∈ Rn×n is asymptotically stable, then A = (I+B)(I−B)−1 is discrete-time asymptotically stable. iii) If A ∈ Rn×n is discrete-time asymptotically stable, then there exists a unique asymptotically stable matrix B ∈ Rn×n such that A = (I + B)(I − B)−1. In fact, B = (A + I)−1 (A − I). iv) If B ∈ Rn×n is asymptotically stable, then there exists a unique discrete-time asymptotically stable matrix A ∈ Rn×n such that B = (A + I)−1 (A − I). In fact, A = (I + B)(I − B)−1. Source: [1354]. Related: Results on the Cayley transform are given in Fact 4.13.24, Fact 4.13.25, Fact 4.13.26, Fact 4.28.12, and Fact 10.10.35. Problem: Obtain analogous results for Lyapunovstable and semistable matrices. [ ] Fact 15.22.11. Let △

P1 P12

PT12 P2

∈ R2n×2n be positive definite, where P1, P12 , P2 ∈ Rn×n. If P1 ≥ △

T −1 P2 , then A = P−1 1 P12 is discrete-time asymptotically stable, while, if P2 ≥ P1, then A = P2 P12 −1 −1 T is discrete-time asymptotically stable. Source: If P1 ≥ P2 , then P1 − P12 P1 P1P1 P12 ≥ P1 − T P12 P−2 2 P12 > 0. See [744]. Fact 15.22.12. Let A ∈ Rn×n, where n ≥ 2, and assume that A is row stochastic. Then, the following statements hold: i) A is discrete-time Lyapunov stable.

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THE MATRIX EXPONENTIAL AND STABILITY THEORY

ii) If A is primitive, then A is discrete-time semistable. Source: For all k ≥ 1, Ak 1n×1 = 1n×1 . Since Ak is nonnegative, it follows that every entry of A is bounded. If A is primitive, then the result follows from Fact 6.11.11, which implies that ρmax (A) = 1, and viii) and xv) of Fact 6.11.5, which imply that 1 is a simple eigenvalue of A as well as the unique eigenvalue of A on the unit circle. Fact 15.22.13. Let A ∈ Rn×n, and let ∥ · ∥ be a norm on Rn×n. Then, the following statements hold: i) A is discrete-time Lyapunov stable if and only if {∥Ak ∥}∞ k=0 is bounded. △

ii) A is discrete-time semistable if and only if A∞ = limk→∞ Ak exists. △ iii) Assume that A is discrete-time semistable. Then, A∞ = I − (A − I)(A − I)# is idempotent and rank A∞ = amultA (1). If, in addition, rank A = 1, then, for each eigenvector x of A associated with the eigenvalue 1, there exists y ∈ Fn such that y∗ x = 1 and A∞ = xy∗. iv) A is discrete-time asymptotically stable if and only if limk→∞ Ak = 0. Source: For ii), see [2036, p. 640]. Related: Fact 15.22.18. Fact 15.22.14. Let A ∈ Fn×n. Then, A is discrete-time Lyapunov stable if and only if k−1 1∑ i △ A∞ = lim A k→∞ k i=0 exists. If these conditions hold, then A∞ = I − (A − I)(A − I)#. Source: [2036, p. 633]. Remark: A is Cesaro summable. Related: Fact 8.3.37. Fact 15.22.15. Let A ∈ Fn×n. Then, A is discrete-time asymptotically stable if and only if limk→∞ Ak = 0. If these conditions hold, then ∞ ∑ (I − A)−1 = Ai , i=1

where the series converges absolutely. Fact 15.22.16. Let A ∈ Fn×n, assume that A is discrete-time asymptotically stable, let ∥ · ∥ be a ∑ norm on Fn, let x0 ∈ Cn , and, for all k ≥ 0, let xk ∈ Cn satisfy xk+1 = Axk . Then, ∞ k=1 ∥xk ∥ is finite. Source: Fact 11.13.11. Fact 15.22.17. Let A ∈ Fn×n, where A is unitary. Then, A is discrete-time Lyapunov stable. △ Fact 15.22.18. Let A, B ∈ Rn×n, assume that A is discrete-time semistable, and let A∞ = k limk→∞ A . Then, lim (A + 1k B)k = A∞ eA∞ BA∞ . k→∞

Source: [499, 2876]. Remark: If A is idempotent, then A∞ = A. The existence of A∞ is guaranteed

by Fact 15.22.13, which also implies that A∞ is idempotent. Fact 15.22.19. Let A ∈ Rn×n. Then, the following statements hold: i) A is discrete-time Lyapunov stable if and only if there exists a positive-definite matrix P ∈ Rn×n such that P − ATPA is positive semidefinite. ii) A is discrete-time asymptotically stable if and only if there exists a positive-definite matrix P ∈ Rn×n such that P − ATPA is positive definite. Remark: The discrete-time Lyapunov equation (also called the Stein equation) is P = ATPA + R. Fact 15.22.20. Let A, B, C ∈ Fn×n, and assume that ρmax (A⊗ B) < 1. Then, there exists a unique

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CHAPTER 15

matrix X ∈ Fn×n such that X = AXB + C. In particular, X=

∞ ∑

AiCBi .

i=0

Source: [2991, p. 324]. Fact 15.22.21. Let A ∈ Rn×n, assume that A is discrete-time asymptotically stable, let P ∈ Rn×n

be positive definite, and assume that P satisfies P = ATPA + I. Then, P is given by ∫ 1 π T (A − e−θ ȷI)−1 (A − eθ ȷI) dθ. P= 2π −π

Furthermore, 1 ≤ λn (P) ≤ λ1 (P) ≤

[σmax (A) + 1]2n−2 . [1 − ρmax (A)]2n

Source: [1328, pp. 167–169]. n×n Fact 15.22.22. Let (Ak )∞ and, for k ∈ N, consider the discrete-time, time-varying k=0 ⊂ R

system

x(k + 1) = Ak x(k).

Furthermore, assume that there exist real numbers β ∈ (0, 1), γ > 0, and ε > 0 such that ρ(β2 + △ ρε2 )2 < 1, where ρ = (γ + 1)2n−2 /(1 − β)2n and such that, for all k ∈ N, ρmax (Ak ) < β,

∥Ak ∥ < γ,

∥Ak+1 − Ak ∥ < ε,

where ∥ · ∥ is an induced norm on Rn×n. Then, x(k) → 0 as k → ∞. Source: [1328, pp. 170–173]. Remark: This result arises from the theory of infinite matrix products. See [171, 494, 495, 836, 1278, 1437, 1725]. Fact 15.22.23. Let A ∈ Fn×n, and define △

r(A) = Then,

|z| − 1 . σ {z∈C : |z|>1} min (zI − A) sup

r(A) ≤ sup σmax (Ak ) ≤ ner(A). k≥0

Hence, if A is discrete-time Lyapunov stable, then r(A) is finite. Source: [2848]. Remark: This is the Kreiss matrix theorem. Remark: The constant ne is the best possible. See [2848]. Fact 15.22.24. Let A ∈ Cn×n , let λ ∈ spec(A), assume that λ is simple, assume that, for all µ ∈ spec(A)\{λ}, |µ| < |λ|, and let x, y ∈ Cn be nonzero vectors such that Ax = λx and ATy = λy. Then, yTx , 0 and 1 1 lim Ai = T xyT . i→∞ λi yx Source: [1742]. Remark: λ is a DT-dominant eigenvalue. Related: Fact 6.11.5 and Fact 15.14.16. Fact 15.22.25. Let p ∈ R[s], and assume that p is discrete-time semistable. Then, C(p) is

discrete-time semistable, and there exists v ∈ Rn such that

lim C k (p) = 1n×1 vT.

k→∞

Source: Since C(p) is a companion form matrix, it follows from Proposition 15.11.4 that its min-

imal polynomial is p. Hence, C(p) is discrete-time semistable. Now, it follows from Proposition 15.11.2 that limk→∞ C k (p) exists, and thus the state x(k) of the difference equation x(k + 1) = C(p)x(k) converges for all initial conditions x(0) = x0 . The structure of C(p) shows that all components of x(k) converge to the same value as k → ∞. Hence, all rows of limk→∞ C k (p) are equal.

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THE MATRIX EXPONENTIAL AND STABILITY THEORY

15.23 Facts on Lie Groups Fact 15.23.1. The groups UT(n), UT+ (n), UT±1 (n), SUT(n), and {In } are Lie groups. Furthermore, ut(n) is the Lie algebra of UT(n), sut(n) is the Lie algebra of SUT(n), and {0n×n } is the Lie algebra of {In }. Related: Fact 4.31.10 and Fact 4.31.11. Problem: Determine the Lie algebras of UT+ (n) and UT±1 (n).

15.24 Facts on Subspace Decomposition Fact 15.24.1. Let A ∈ Rn×n, and let S ∈ Rn×n be a nonsingular matrix such that

[

A=S

A1 0

] A12 −1 S , A2

where A1 ∈ Rr×r is asymptotically stable, A12 ∈ Rr×(n−r), and A2 ∈ R(n−r)×(n−r). Then, [ ] 0 B12s µAs (A) = S S −1, 0 µAs (A2 ) where B12s ∈ Rr×(n−r), and

[ µAu (A) = S

] µAu (A1 ) B12u S −1, 0 µAu (A2 )

where B12u ∈ Rr×(n−r) and µAu (A1 ) is nonsingular. Consequently, ( [ ]) I R S r ⊆ Ss (A). 0 If, in addition, A12 = 0, then [ ] 0 0 µAs (A) = S S −1, 0 µAs (A2 )

[ µAu (A) = S

µAu (A1 ) 0

] 0 S −1, µAu (A2 )

( [ ]) 0 Su (A) ⊆ R S . In−r

Source: Fact 6.10.17. Fact 15.24.2. Let A ∈ Rn×n, and let S ∈ Rn×n be a nonsingular matrix such that

[

A A=S 1 0

] A12 −1 S , A2

where A1 ∈ Rr×r, A12 ∈ Rr×(n−r), and A2 ∈ R(n−r)×(n−r) satisfies spec(A2 ) ⊂ CRHP. Then, [ s ] µA (A1 ) C12s s µA (A) = S S −1, 0 µAs (A2 ) where C12s ∈ Rr×(n−r) and µAs (A2 ) is nonsingular, and [ u ] µ (A ) C12u −1 µAu (A) = S A 1 S , 0 0 where C12u ∈ Rr×(n−r). Consequently,

If, in addition, A12 = 0, then [ s ] µA (A1 ) 0 s S −1, µA (A) = S 0 µAs (A2 )

( [ ]) I Ss (A) ⊆ R S r . 0 [

µAu (A)

] µAu (A1 ) 0 −1 S , =S 0 0

( [ ]) 0 R S ⊆ Su (A). In−r

1244

CHAPTER 15

Fact 15.24.3. Let A ∈ Rn×n, and let S ∈ Rn×n be a nonsingular matrix such that

[

A=S

A1 0

] A12 −1 S , A2

where A1 ∈ Rr×r satisfies spec(A1 ) ⊂ CRHP, A12 ∈ Rr×(n−r), and A2 ∈ R(n−r)×(n−r). Then, [ s ] µ (A ) B12s µAs (A) = S A 1 S −1, 0 µAs (A2 ) where µAs (A1 ) is nonsingular and B12s ∈ Rr×(n−r), and [ ] 0 B12u u µA (A) = S S −1, 0 µAu (A2 ) where B12u ∈ Rr×(n−r). Consequently,

If, in addition, A12 = 0, then [ s ] µ (A ) 0 µAs (A) = S A 1 S −1, 0 µAs (A2 )

( [ ]) I R S r ⊆ Su (A). 0 [ µAu (A) = S

0 0

] 0 S −1, µAu (A2 )

( [ ]) 0 Ss (A) ⊆ R S . In−r

Fact 15.24.4. Let A ∈ Rn×n, and let S ∈ Rn×n be a nonsingular matrix such that

[

A=S

A1 0

] A12 −1 S , A2

where A1 ∈ Rr×r, A12 ∈ Rr×(n−r), and A2 ∈ R(n−r)×(n−r) is asymptotically stable. Then, [ s ] µ (A ) C12s −1 µAs (A) = S A 1 S , 0 0 where C12s ∈ Rr×(n−r), and

[ µAu (A) = S

] µAu (A1 ) C12u S −1, 0 µAu (A2 )

where µAu (A2 ) is nonsingular and C12u ∈ Rr×(n−r). Consequently, ( [ ]) I Su (A) ⊆ R S r . 0 If, in addition, A12 = 0, then ] [ s µ (A ) 0 −1 S , µAs (A) = S A 1 0 0

[ µAu (A) = S

] µAu (A1 ) 0 S −1, 0 µAu (A2 )

( [ ]) 0 R S ⊆ Ss (A). In−r

Fact 15.24.5. Let A ∈ Rn×n, and let S ∈ Rn×n be a nonsingular matrix such that

[

A=S

A1 0

] A12 −1 S , A2

where A1 ∈ Rr×r satisfies spec(A1 ) ⊂ CRHP, A12 ∈ Rr×(n−r), and A2 ∈ R(n−r)×(n−r) is asymptotically stable. Then, [ s ] µA (A1 ) C12s −1 s µA (A) = S S , 0 0

1245

THE MATRIX EXPONENTIAL AND STABILITY THEORY

where C12s ∈ Rr×(n−r) and µAs (A1 ) is nonsingular, and [ ] 0 C12u µAu (A) = S S −1, 0 µAu (A2 ) where C12u ∈ Rr×(n−r) and µAu (A2 ) is nonsingular. Consequently, ( [ ]) I Su (A) = R S r . 0 If, in addition, A12 = 0, then [ s ] µ (A ) 0 −1 µAs (A) = S A 1 S , 0 0

[ µAu (A) = S

] 0 0 S −1, 0 µAu (A2 )

( [ ]) 0 Ss (A) = R S . In−r

Fact 15.24.6. Let A ∈ Rn×n, and let S ∈ Rn×n be a nonsingular matrix such that

[

A=S

A1 A21

] 0 S −1, A2

where A1 ∈ Rr×r is asymptotically stable, A21 ∈ R(n−r)×r, and A2 ∈ R(n−r)×(n−r). Then, [ ] 0 0 µAs (A) = S S −1, B21s µAs (A2 ) where B21s ∈ R(n−r)×r, and

[ µAu (A) = S

] µAu (A1 ) 0 S −1, B21u µAu (A2 )

where B21u ∈ R(n−r)×r and µAu (A1 ) is nonsingular. Consequently, ( [ ]) 0 Su (A) ⊆ R S . In−r If, in addition, A21 = 0, then [ ] 0 0 µAs (A) = S S −1, 0 µAs (A2 )

[ µAu (A) = S

] µAu (A1 ) 0 S −1, 0 µAu (A2 )

( [ ]) I R S r ⊆ Ss (A). 0

Fact 15.24.7. Let A ∈ Rn×n, and let S ∈ Rn×n be a nonsingular matrix such that

[

A=S

A1 A21

] 0 S −1, A2

where A1 ∈ Rr×r, A21 ∈ R(n−r)×r, and A2 ∈ R(n−r)×(n−r) satisfies]spec(A2 ) ⊂ CRHP. Then, [ s µ (A ) 0 µAs (A) = S A 1 S −1, C21s µAs (A2 ) where C21s ∈ R(n−r)×r and µAs (A2 ) is nonsingular, and [ u ] µ (A ) 0 −1 S , µAu (A) = S A 1 C21u 0 where C21u ∈ R(n−r)×r. Consequently,

( [ ]) 0 R S ⊆ Su (A). In−r

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CHAPTER 15

If, in addition, A21 = 0, then [ s ] µA (A1 ) 0 s µA (A) = S S −1, 0 µAs (A2 )

[

µAu (A)

] µAu (A1 ) 0 −1 =S S , 0 0

( [ ]) I Ss (A) ⊆ R S r . 0

Fact 15.24.8. Let A ∈ Rn×n, and let S ∈ Rn×n be a nonsingular matrix such that

[

] 0 S −1, A2

A A=S 1 A21

where A1 ∈ Rr×r is asymptotically stable, A21 ∈ R(n−r)×r, and A2 ∈ R(n−r)×(n−r) satisfies spec(A2 ) ⊂ CRHP. Then, [ ] 0 0 s µA (A) = S S −1, C21s µAs (A2 ) where C21s ∈ R(n−r)×r and µAs (A2 ) is nonsingular, and [ u ] µ (A ) 0 −1 µAu (A) = S A 1 S , C21u 0 where C21u ∈ R(n−r)×r and µAu (A1 ) is nonsingular. Consequently, ( [ ]) 0 Su (A) = R S . In−r If, in addition, A21 = 0, then [ ] 0 0 µAs (A) = S S −1 , 0 µAs (A2 )

[ µAu (A) = S

] µAu (A1 ) 0 −1 S , 0 0

( [ ]) I Ss (A) = R S r . 0

Fact 15.24.9. Let A ∈ Rn×n, and let S ∈ Rn×n be a nonsingular matrix such that

[

A A=S 1 A21

] 0 S −1, A2

where A1 ∈ Rr×r, A21 ∈ R(n−r)×r, and A2 ∈ R(n−r)×(n−r) is asymptotically stable. Then, [ s ] µA (A1 ) 0 −1 s µA (A) = S S , B21s 0 where B21s ∈ R(n−r)×r, and

[

µAu (A)

] µAu (A1 ) 0 =S S −1, B21u µAu (A2 )

where B21u ∈ R(n−r)×r and µAu (A2 ) is nonsingular. Consequently, ( [ ]) 0 ⊆ S(A). R S In−r If, in addition, A21 = 0, then ] [ s µA (A1 ) 0 −1 s S , µA (A) = S 0 0

[

µAu (A)

] µAu (A1 ) 0 S −1, =S 0 µAu (A2 )

( [ ]) I Su (A) ⊆ R S r . 0

Fact 15.24.10. Let A ∈ Rn×n, and let S ∈ Rn×n be a nonsingular matrix such that

[

A A=S 1 A21

] 0 S −1, A2

1247

THE MATRIX EXPONENTIAL AND STABILITY THEORY

where A1 ∈ Rr×r satisfies spec(A1 ) ⊂ CRHP, A21 ∈ R(n−r)×r, and A2 ∈ R(n−r)×(n−r). Then, [ s ] µ (A ) 0 µAs (A) = S A 1 S −1, C12s µAs (A2 ) where C21s ∈ R(n−r)×r and µAs (A1 ) is nonsingular, and [ ] 0 0 u µA (A) = S S −1, C21u µAu (A2 ) where C21u ∈ R(n−r)×r. Consequently,

( [ ]) 0 Ss (A) ⊆ R S . In−r

If, in addition, A21 = 0, then [ s ] µ (A ) 0 µAs (A) = S A 1 S −1, 0 µAs (A2 )

[ µAu (A) = S

] 0 0 S −1, 0 µAu (A2 )

( [ ]) I R S r ⊆ Su (A). 0

Fact 15.24.11. Let A ∈ Rn×n, and let S ∈ Rn×n be a nonsingular matrix such that

[

A=S

] 0 S −1, A2

A1 A21

where A1 ∈ Rr×r satisfies spec(A1 ) ⊂ CRHP, A21 ∈ R(n−r)×r, and A2 ∈ R(n−r)×(n−r) is asymptotically stable. Then, [ s ] µA (A1 ) 0 −1 s µA (A) = S S , C21s 0 where C21s ∈ R(n−r)×r and µAs (A1 ) is nonsingular, and [ ] 0 0 µAu (A) = S S −1, C21u µAu (A2 ) where C21u ∈ R(n−r)×r and µAu (A2 ) is nonsingular. Consequently, ( [ ]) 0 Ss (A) = R S . In−r If, in addition, A21 = 0, then [ s ] µ (A ) 0 −1 µAs (A) = S A 1 S , 0 0

[ µAu (A) = S

] 0 0 S −1, 0 µAu (A2 )

( [ ]) I Su (A) = R S r . 0

15.25 Notes The Laplace transform (15.2.10) is given in [2469, p. 34]. Computational methods are discussed in [1391, 2066]. An arithmetic-mean–geometric-mean iteration for computing the matrix exponential and matrix logarithm is given in [2541]. The exponential function plays a central role in the theory of Lie groups, see [350, 648, 1299, 1471, 1495, 2379, 2781]. Applications to robotics and kinematics are given in [1997, 2104, 2200]. Additional applications are discussed in [647]. The real logarithm is discussed in [800, 1364, 2143, 2272]. The multiplicity and properties of logarithms are discussed in [1002]. An asymptotically stable polynomial is traditionally called Hurwitz. Semistability is defined in [624] and developed in [431, 444]. Stability theory is treated in [1292, 1782, 2257] and [1140,

1248

CHAPTER 15

Chapter XV]. Solutions of the Lyapunov equation under weak conditions are considered in [2475]. Structured solutions of the Lyapunov equation are discussed in [1586]. Linear and nonlinear difference equations are studied in [17, 622, 1710].

Chapter Sixteen Linear Systems and Control Theory This chapter considers linear state space systems with inputs and outputs. These systems are considered in both the time domain and frequency (Laplace) domain. Some basic results in control theory are also presented.

16.1 State Space Models Let A ∈ Rn×n and B ∈ Rn×m, and, for all t ≥ t0 , consider the state equation x˙(t) = Ax(t) + Bu(t),

(16.1.1)

x(t0 ) = x0 .

(16.1.2)

with the initial condition

In (16.1.1), x : [0, ∞) 7→ Rn is the state, and u : [0, ∞) 7→ Rm is the input. The function x is the solution of (16.1.1). The following result gives the solution of (16.1.1) and is known as the variation of constants formula. Proposition 16.1.1. For t ≥ t0 the state x(t) of the dynamical equation (16.1.1) with initial condition (16.1.2) is given by ∫t x(t) = e

x0 +

(t−t0 )A

e(t−τ)ABu(τ) dτ.

(16.1.3)

t0

Proof. Multiplying (16.1.1) by e

−tA

yields

e−tA[ x˙(t) − Ax(t)] = e−tABu(t), which is equivalent to

d −tA [e x(t)] = e−tABu(t). dt

Integrating over [t0 , t] yields e−tAx(t) = e−t0 Ax(t0 ) +

∫t

e−τABu(τ) dτ.

t0 tA

Now, multiplying by e yields (16.1.3). Alternatively, let x(t) be given by (16.1.3). Then, it follows from Leibniz’s rule Fact 12.16.7 that d d x˙(t) = e(t−t0 )Ax0 + dt dt

∫t e(t−τ)ABu(τ) dτ t0

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CHAPTER 16

∫t = Ae

x0 +

Ae(t−τ)ABu(τ) dτ + Bu(t)

(t−t0 )A

t0

= Ax(t) + Bu(t).



For convenience, we can reset the clock by replacing t0 by 0, and therefore assume without loss of generality that t0 = 0. In this case, x(t) for all t ≥ 0 is given by ∫t x(t) = e x0 + tA

e(t−τ)ABu(τ) dτ.

(16.1.4)

0

If u(t) = 0 for all t ≥ 0, then, for all t ≥ 0, x(t) is given by x(t) = etAx0 .

(16.1.5)

Now, let u(t) = δ(t)v, where δ(t) is the unit impulse occurring at t = 0 and v ∈ Rm. Loosely speaking, the unit impulse occurring at t = 0 is zero for all t , 0 and is infinite at t = 0. More precisely, let a < b. Then,  ∫b   0, a > 0 or b ≤ 0, (16.1.6) δ(τ) dτ =   1, a ≤ 0 < b. a

More generally, if g: [a, b] → R and g is continuous at t0 ∈ [a, b], then  ∫b   a > t0 or b ≤ t0 , 0, δ(τ − t0 )g(τ) dτ =   g(t0 ), a ≤ t0 < b. n

(16.1.7)

a

Consequently, for all t ≥ 0, x(t) is given by x(t) = etAx0 + etABv.

(16.1.8)

The unit impulse has the physical dimensions of 1/time. This convention makes the integral in (16.1.6) dimensionless. Alternatively, let the input u(t) be a step function; that is, u(t) = 0 for all t < 0 and u(t) = v for all t ≥ 0, where v ∈ Rm. Then, by replacing t − τ by τ in the integral in (16.1.4), it follows that, for all t ≥ 0, ∫t x(t) = etAx0 + eτA dτBv. (16.1.9) 0

Using Fact 15.14.10, (16.1.9) can be written for all t ≥ 0 as   ind ∑A t i   Ai−1  Bv. x(t) = etAx0 + AD (etA − I) + (I − AAD ) i!

(16.1.10)

i=1

If A is group invertible, then, for all t ≥ 0, (16.1.10) becomes x(t) = etA x0 + [A# (etA − I) + t(I − AA# )]Bv.

(16.1.11)

If, in addition, A is nonsingular, then, for all t ≥ 0, (16.1.11) becomes x(t) = etA x0 + A−1 (etA − I)Bv.

(16.1.12)

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LINEAR SYSTEMS AND CONTROL THEORY

Next, consider the output equation y(t) = Cx(t) + Du(t), where t ≥ 0, y(t) ∈ R is the output, C ∈ R , and D ∈ R of (16.1.1), (16.1.13) is l

l×n

(16.1.13) . Then, for all t ≥ 0, the total response

l×m

∫t y(t) = CetAx0 +

Ce(t−τ)ABu(τ) dτ + Du(t).

(16.1.14)

0

If u(t) = 0 for all t ≥ 0, then the free response is given by y(t) = CetA x0 ,

(16.1.15)

while, if x0 = 0, then the forced response is given by ∫t y(t) =

Ce(t−τ)ABu(τ) dτ + Du(t).

(16.1.16)

0

Setting u(t) = δ(t)v, where v ∈ R , yields, for all t > 0, the total response m

y(t) = CetAx0 + H(t)v,

(16.1.17)

where, for all t ≥ 0, the impulse response function H(t) is defined by △

H(t) = CetAB + δ(t)D.

(16.1.18)

The corresponding forced response is the impulse response y(t) = H(t)v = CetABv + δ(t)Dv.

(16.1.19)

Alternatively, if u(t) = v ∈ Rm for all t ≥ 0, then the total response is ∫t y(t) = Ce x0 + tA

CeτA dτBv + Dv,

(16.1.20)

0

and the forced response is the step response ∫t y(t) =

∫t H(τ) dτv =

0

CeτA dτBv + Dv.

(16.1.21)

0

For an arbitrary input u, the forced response can be written as the convolution integral ∫t y(t) =

H(t − τ)u(τ) dτ.

(16.1.22)

0

Setting u(t) = δ(t)v in (16.1.22) yields (16.1.19) by noting that ∫t δ(t − τ)δ(τ) dτ = δ(t).

(16.1.23)

0

Proposition 16.1.2. Let D = 0 and m = 1, and assume that x0 = Bv. Then, the free response and the impulse response are equal and given by

y(t) = CetAx0 = CetABv.

(16.1.24)

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CHAPTER 16

16.2 Laplace Transform Analysis and Transfer Functions Now, consider the linear system x˙(t) = Ax(t) + Bu(t),

(16.2.1)

y(t) = Cx(t) + Du(t),

(16.2.2)

with state x(t) ∈ Rn, input u(t) ∈ Rm, and output y(t) ∈ Rl, where t ≥ 0 and x(0) = x0 . Taking Laplace transforms yields s xˆ(s) − x0 = A xˆ(s) + Bˆu(s), yˆ (s) = C xˆ(s) + Dˆu(s),

(16.2.3) (16.2.4)

where ∫∞

△

xˆ(s) = L{x(t)} =

e−stx(t) dt,

(16.2.5)

0 △

uˆ (s) = L{u(t)}, Hence,

△

yˆ (s) = L{y(t)}.

(16.2.6)

xˆ(s) = (sI − A)−1 x0 + (sI − A)−1Bˆu(s),

(16.2.7)

yˆ (s) = C(sI − A)−1 x0 + [C(sI − A)−1B + D]ˆu(s).

(16.2.8)

and thus

We can also obtain (16.2.8) from the time-domain expression for y(t) given by (16.1.14). Using Proposition 15.2.2, it follows from (16.1.14) that   t ∫         (t−τ)A tA + Dˆu(s) Ce Bu(τ) dτ yˆ (s) = L{Ce x0 } + L        0

= CL{e }x0 + CL{etA }Bˆu(s) + Dˆu(s) tA

= C(sI − A)−1 x0 + [C(sI − A)−1B + D]ˆu(s),

(16.2.9)

which coincides with (16.2.8). We define △

G(s) = C(sI − A)−1B + D.

(16.2.10)

Note that G ∈ R (s), and thus, by Definition 6.7.2, G is a proper rational transfer function. Since L{δ(t)} = 1, it follows from the definition (16.1.18) of the impulse response function H that l×m

G(s) = L{H(t)}.

(16.2.11)

1 C(sI − A)AB + D. χA (s)

(16.2.12)

Using (6.7.2), G can be written as G(s) =

It follows from (6.7.3) that G is a proper rational transfer function. Furthermore, G is a strictly proper rational transfer function if and only if D = 0, whereas G is an exactly proper rational transfer function if and only if D , 0. Finally, if A is nonsingular, then G(0) = −CA−1B + D.

(16.2.13)

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LINEAR SYSTEMS AND CONTROL THEORY

Let A ∈ Rn×n. If |s| > ρmax (A), then Proposition 11.3.10 implies that ( )−1 ∑ ∞ 1 k 1 1 −1 I− A = (sI − A) = A, k+1 s s s k=0

(16.2.14)

where the Laurent series is absolutely convergent, and thus ∑ 1 1 1 Hk , G(s) = D + CB + 2 CAB + · · · = s s sk k=0 ∞

where, for all k ≥ 0, the Markov parameter Hk ∈ Rl×m is defined by    k = 0, △  D, Hk =   CAk−1B, k ≥ 1.

(16.2.15)

(16.2.16)

It follows from (16.2.14) that lim s→∞ (sI − A)−1 = 0 and from (16.2.15) that H0 = D = lim G(s),

(16.2.17)

H1 = CB = lim s[G(s) − H0 ],

(16.2.18)

s→∞

H2 = CAB =

s→∞ lim (s2 [G(s) s→∞

− H0 ] − sH1 ).

(16.2.19)

Furthermore, Definition 6.7.3 implies that reldeg G = min {k ≥ 0: Hk , 0}.

(16.2.20)

16.3 The Unobservable Subspace and Observability Let A ∈ Rn×n and C ∈ Rl×n, and, for all t ≥ 0, consider the linear system x˙(t) = Ax(t),

(16.3.1)

x(0) = x0 , y(t) = Cx(t).

(16.3.2) (16.3.3)

Definition 16.3.1. The unobservable subspace Utf(A, C) of (A, C) at time tf > 0 is the subspace △

Utf (A, C) = {x0 ∈ Rn : y(t) = 0 for all t ∈ [0, tf ]} .

(16.3.4)

Let tf > 0. Then, Definition 16.3.1 states that x0 ∈ Utf(A, C) if and only if y(t) = 0 for all t ∈ [0, tf ]. Since y(t) = 0 for all t ∈ [0, tf ] is the free response corresponding to x0 = 0, it follows that 0 ∈ Utf(A, C). Now, suppose there exists a nonzero vector x0 ∈ Utf (A, C). Then, with x(0) = x0 , the free response is given by y(t) = 0 for all t ∈ [0, tf ], and thus x0 cannot be determined from knowledge of y(t) for all t ∈ [0, tf ]. The following result provides explicit expressions for Utf (A, C). Lemma 16.3.2. Let tf > 0. Then, the following subspaces are equal: i) Utf (A, C). ∩ ii) t∈[0,tf ] N(CetA ). ∩ i iii) n−1 i=0 N(CA ).  C   CA  iv) N  ..  . .n−1 (∫ CA ) tf T v) N 0 etA CTCetA dt .

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CHAPTER 16

∫ tf T If, in addition, limtf →∞ 0 etA CTCetA dt exists, then the following subspace is equal to i)–v): (∫ ∞ T ) vi) N 0 etA CTCetA dt . Proof. See the proof of Lemma 16.6.2.  The expressions given by iii) and iv) of Lemma 16.3.2 show that Utf (A, C) is independent of tf . We thus write U(A, C) for Utf (A, C), and call U(A, C) the unobservable subspace of (A, C). (A, C) is observable if U(A, C) = {0}. For convenience, define the nl × n observability matrix    C    △  CA  (16.3.5) O(A, C) =  ..   .  CAn−1 so that Define

U(A, C) = N[O(A, C)].

(16.3.6)

p = n − dim U(A, C) = n − def O(A, C).

(16.3.7)

△

Corollary 16.3.3. For all tf > 0,

∫tf

⊥

p = dim U(A, C) = rank O(A, C) = rank If, in addition, limtf →∞

∫ tf 0

T

etA CTCetA dt.

(16.3.8)

0 tAT T

tA

e C Ce dt exists, then ∫∞ p = rank

T

etA CTCetA dt.

(16.3.9)

0

Corollary 16.3.4. U(A, C) is an invariant subspace of A.

The following result shows that the unobservable subspace U(A, C) is unchanged by output injection x˙(t) = Ax(t) + Fy(t). (16.3.10) Proposition 16.3.5. Let F ∈ Rn×l. Then,

U(A + FC, C) = U(A, C).

(16.3.11)

In particular, (A, C) is observable if and only if (A + FC, C) is observable. Proof. See the proof of Proposition 16.6.5.  ˜ ˜ Let U(A, C) ⊆ Rn be a subspace that is complementary to U(A, C). Then, U(A, C) is an observ˜ C) is nonzero and x0′′ ∈ U(A, C), able subspace in the sense that, if x0 = x0′ + x0′′ , where x0′ ∈ U(A, ′ then it is possible to determine x0 from knowledge of y(t) for all t ∈ [0, tf ]. Using Proposition 4.8.3, ˜ let P ∈ Rn×n be the unique idempotent matrix such that R(P) = U(A, C) and N(P) = U(A, C). Then, ′ x0 = Px0 . The following result constructs P and provides an expression for x0′ in terms of y(t) for ˜ U(A, C) = U(A, C)⊥. In this case, P is a projector. Lemma 16.3.6. Let tf > 0, and define P ∈ Rn×n by  tf + tf ∫  ∫ T T  △  P =  etA CTCetA dt etA CTCetA dt. (16.3.12)   0

0

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LINEAR SYSTEMS AND CONTROL THEORY

Then, P is the projector onto U(A, C)⊥, and P⊥ is the projector onto U(A, C). Hence, R(P) = N(P⊥ ) = U(A, C)⊥,

(16.3.13)

N(P) = R(P⊥ ) = U(A, C),

(16.3.14)

⊥

If x0 =

x0′

+

x0′′ ,

where

x0′

rank P = def P⊥ = dim U(A, C) = p,

(16.3.15)

def P = rank P⊥ = dim U(A, C) = n − p.

(16.3.16)

⊥

x0′′

∈ U(A, C) and ∈ U(A, C), then  + tf  tf  ∫ ∫ T T   etA CTy(t) dt. x0′ = Px0 =  etA CTCetA dt  

(16.3.17)

0

0

Finally, (A, C) is observable if and only if P = In . If these conditions hold, then, for all x0 ∈ Rn,  tf −1 tf ∫  ∫ T  tAT T tA  etA CTy(t) dt. (16.3.18) x0 =  e C Ce dt   0

0

Lemma 16.3.7. Let α ∈ R. Then,

U(A + αI, C) = U(A, C).

(16.3.19)

The following result uses a coordinate transformation to characterize the observable dynamics of a system. Theorem 16.3.8. There exists an orthogonal matrix S ∈ Rn×n such that [ ] A 0 A=S 1 S −1, C = [C1 0]S −1, (16.3.20) A21 A2 where A1 ∈ R p×p, C1 ∈ Rl×p, and (A1 , C1 ) is observable. Proof. See the proof of Theorem 16.6.8.  Proposition 16.3.9. Let S ∈ Rn×n, and assume that S is orthogonal. Then, the following statements are equivalent: i) A and C have the form (16.3.20), where A1 ∈ R p×p, C1 ∈ Rl×p, and (A1 , C1 ) is observable. ( [ 0 ]) ii) U(A, C) = R S In−p . ( [ I ]) ⊥ p iii) U(A, C) = R S 0 . [ ] Ip 0 T iv) P = S S . 0 0 Proposition 16.3.10. Let S ∈ Rn×n, and assume that S is nonsingular. Then, the following statements are equivalent: i) A and C have the form (16.3.20), where A1 ∈ R p×p, C1 ∈ Rl×p, and (A1 , C1 ) is observable. ( [ 0 ]) ( [ ]) ˜ ii) U(A, C) = R S In−p and U(A, C) = R S I0p . [ ] iii) P = S I0p 00 S −1. Definition 16.3.11. Let S ∈ Rn×n, assume that S is nonsingular, and let A and C have the form (16.3.20), where A1 ∈ R p×p, C1 ∈ Rl×p, and (A1 , C1 ) is observable. Then, λ ∈ C is an unobservable eigenvalue of (A, C) if λ ∈ spec(A2 ), the unobservable spectrum of (A, C) is spec(A2 ), and the unobservable multispectrum of (A, C) is mspec(A2 ). Finally, λ ∈ spec(A) is an observable eigenvalue of (A, C) if λ is not an unobservable eigenvalue of (A, C).

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Definition 16.3.12. The observability pencil OA,C (s) is the pencil

OA,C = P[ A ],[ I ] . −C

Equivalently,

(16.3.21)

0

[

] sI − A OA,C (s) = . C

(16.3.22)

Proposition 16.3.13. Let λ ∈ spec(A). Then, λ is an unobservable eigenvalue of (A, C) if and

[

] λI − A rank < n. C

only if

(16.3.23)

Proof. See the proof of Proposition 16.6.13.  Proposition 16.3.14. Let λ ∈ mspec(A) and F ∈ Rn×l. Then, λ is an unobservable eigenvalue

of (A, C) if and only if λ is an unobservable eigenvalue of (A + FC, C). Proof. See the proof of Proposition 16.6.14.  [ ] n Proposition 16.3.15. Assume that (A, C) is observable. Then, the Smith form of OA,C is 0Il×n . Proof. See the proof of Proposition 16.6.15.  Proposition 16.3.16. S ∈ Rn×n, assume that S is nonsingular, and let A and C have the form (16.3.20), where A1 ∈ R p×p, C1 ∈ Rl×p, and (A1 , C1 ) is observable. Furthermore, let p1 , . . . , pn−p be the similarity invariants of A2 , where, for all i ∈ {1, . . . , n − p − 1}, pi divides pi+1 . Then, there exist unimodular matrices S 1 ∈ R[s](n+l)×(n+l) and S 2 ∈ R[s]n×n such that, for all s ∈ C,     I p   p1 (s)   [ ] sI − A ..   . (16.3.24) = S 1 (s)   S 2 (s). C  pn−p (s)     0l×n Consequently, Szeros(OA,C ) = mSzeros(OA,C ) =

n−p ∪

roots(pi ) = roots(χA2 ) = spec(A2 ),

i=1 n−p ∪

mroots(pi ) = mroots(χA2 ) = mspec(A2 ).

(16.3.25) (16.3.26)

i=1

Proof. See the proof of Proposition 16.6.16. Proposition 16.3.17. Let s ∈ C. Then,

O(A, C) ⊆ Re R

 ([

sI − A C

]) .

Proof. See the proof of Proposition 16.6.17.

The next result characterizes observability in several equivalent ways. Theorem 16.3.18. The following statements are equivalent: i) (A, C) is observable. ∫t T ii) There exists t > 0 such that 0 eτA CTCeτA dτ is positive definite. ∫t T iii) For all t > 0, 0 eτA CTCeτA dτ is positive definite.

(16.3.27) 

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LINEAR SYSTEMS AND CONTROL THEORY

iv) rank O(A, C) = n. v) Every eigenvalue of (A, C) is observable. vi) If x ∈ Cn is an eigenvector of A, then BTx , 0. vii) There exists F ∈ Rn×l such that (A + FC, C) is observable. viii) For all F ∈ Rn×l ,∫(A + FC, C) is observable. t T If, in addition, limt→∞ 0 eτA CTCeτA dτ exists, then the following statement is equivalent to i)–viii): ∫∞ T ix) 0 etA CTCetA dt is positive definite. Proof. See the proof of Theorem 16.6.18.  The following result, which is a restatement of the equivalence of i) and v) of Theorem 16.3.18, is the PBH test for observability. Corollary 16.3.19. The following statements are equivalent: i) (A, C) is observable. ii) For all s ∈ C, [ ] sI − A rank = n. (16.3.28) C The following result implies that arbitrary eigenvalue placement is possible for (16.3.10) if (A, C) is observable. Proposition 16.3.20. The pair (A, C) is observable if and only if, for every polynomial p ∈ R[s] such that deg p = n, there exists F ∈ Rn×l such that mspec(A + FC) = mroots(p). Proof. See the proof of Proposition 16.6.20. 

16.4 Observable Asymptotic Stability △

Let A ∈ Rn×n and C ∈ Rl×n, and define p = n − dim U(A, C). Definition 16.4.1. (A, C) is observably asymptotically stable if Su (A) ⊆ U(A, C).

(16.4.1)

Proposition 16.4.2. Let F ∈ Rn×l. Then, (A, C) is observably asymptotically stable if and only if (A + FC, C) is observably asymptotically stable. Proposition 16.4.3. The following statements are equivalent: i) (A, C) is observably asymptotically stable. ii) There exists an orthogonal matrix S ∈ Rn×n such that (16.3.20) holds, where A1 ∈ R p×p is asymptotically stable and C1 ∈ Rl×p. iii) There exists a nonsingular matrix S ∈ Rn×n such that (16.3.20) holds, where A1 ∈ R p×p is asymptotically stable and C1 ∈ Rl×p. iv) limt→∞ CetA = 0. v) The positive-semidefinite matrix P ∈ Rn×n defined by

∫∞ T P = etA CTCetA dt △

(16.4.2)

0

exists. vi) There exists a positive-semidefinite matrix P ∈ Rn×n satisfying ATP + PA + CTC = 0.

(16.4.3)

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If these conditions hold, then the positive-semidefinite matrix P ∈ Rn×n defined by (16.4.2) satisfies (16.4.3). Proof. See the proof of Proposition 16.7.3.  The matrix P defined by (16.4.2) is the observability Gramian, and (16.4.3) is the observation Lyapunov equation. Proposition 16.4.4. Assume that (A, C) is observably asymptotically stable, let P ∈ Rn×n be the positive-semidefinite matrix defined by (16.4.2), and define P ∈ Rn×n by (16.3.12). Then, the following statements hold: i) PP+ = P. ii) R(P) = R(P) = U(A, C)⊥. iii) N(P) = N(P) = U(A, C). iv) rank P = rank P = p. v) P is the unique positive-semidefinite solution of (16.4.3) whose rank is p. Proof. See the proof of Proposition 16.7.4.  Proposition 16.4.5. Assume that (A, C) is observably asymptotically stable, let P ∈ Rn×n be the positive-semidefinite matrix defined by (16.4.2), and let Pˆ ∈ Rn×n. Then, the following statements are equivalent: i) Pˆ is positive semidefinite and satisfies (16.4.3). ii) There exists a positive-semidefinite matrix P0 ∈ Rn×n such that Pˆ = P+P0 and ATP0 +P0 A = 0. If these conditions hold, then rank Pˆ = p + rank P0 , ∑ gmultA (λ). rank P0 ≤

(16.4.4) (16.4.5)

λ∈spec(A) λ∈IA

Proof. See the proof of Proposition 16.7.5. Proposition 16.4.6. The following statements are equivalent:



i) (A, C) is observably asymptotically stable, every imaginary eigenvalue of A is semisimple, and A has no ORHP eigenvalues. ii) (16.4.3) has a positive-definite solution P ∈ Rn×n. Proof. See the proof of Proposition 16.7.6.  Proposition 16.4.7. The following statements are equivalent: i) (A, C) is observably asymptotically stable, and A has no imaginary eigenvalues. ii) (16.4.3) has exactly one positive-semidefinite solution P ∈ Rn×n. If these conditions hold, then P ∈ Rn×n is given by (16.4.2) and satisfies rank P = p. Proof. See the proof of Proposition 16.7.7.  Corollary 16.4.8. Assume that A is asymptotically stable. Then, the positive-semidefinite matrix P ∈ Rn×n defined by (16.4.2) is the unique solution of (16.4.3) and satisfies rank P = p. Proof. See the proof of Corollary 16.7.8.  Proposition 16.4.9. The following statements are equivalent: i) (A, C) is observable, and A is asymptotically stable. ii) (16.4.3) has exactly one positive-semidefinite solution P ∈ Rn×n, and P is positive definite. If these conditions hold, then P ∈ Rn×n is given by (16.4.2). Proof. See the proof of Proposition 16.7.9. 

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LINEAR SYSTEMS AND CONTROL THEORY

Corollary 16.4.10. Assume that A is asymptotically stable. Then, the positive-semidefinite matrix P ∈ Rn×n defined by (16.4.2) exists. Furthermore, P is positive definite if and only if (A, C) is observable.

16.5 Detectability △

Let A ∈ Rn×n and C ∈ Rl×n, and define p = n − dim U(A, C). Definition 16.5.1. The undetectable subspace Uu (A, C) of (A, C) is △

Uu (A, C) = U(A, C) ∩ Su (A).

(16.5.1)

Furthermore, (A, C) is detectable if Uu (A, C) = {0}. Proposition 16.5.2. (A, C) is detectable if and only if U(A, C) ⊆ Ss (A). Proposition 16.5.3. Let F ∈ R

(16.5.2)

. Then, (A, C) is detectable if and only if (A + FC, C) is

n×l

detectable. Proposition 16.5.4. The following statements are equivalent: i) (A, C) is detectable. ii) There exists an orthogonal matrix S ∈ Rn×n such that (16.3.20) holds, where A1 ∈ R p×p, C1 ∈ Rl×p, (A1 , C1 ) is observable, and A2 ∈ R(n−p)×(n−p) is asymptotically stable. iii) There exists a nonsingular matrix S ∈ Rn×n such that (16.3.20) holds, where A1 ∈ R p×p, C1 ∈ Rl×p, (A1 , C1 ) is observable, and A2 ∈ R(n−p)×(n−p) is asymptotically stable. iv) Every CRHP eigenvalue of (A, C) is observable. Proof. See the proof of Proposition 16.8.4.  The following result, which is a restatement of the equivalence of i) and iv) of Proposition 16.5.4, is the PBH test for detectability. Corollary 16.5.5. The following statements are equivalent: i) (A, C) is detectable. ii) For all s ∈ CRHP, [ ] sI − A rank = n. (16.5.3) C Proposition 16.5.6. The following statements are equivalent:

i) A is asymptotically stable. ii) (A, C) is observably asymptotically stable and detectable. Proof. See the proof of Proposition 16.8.6.  Corollary 16.5.7. The following statements are equivalent: i) There exists a positive-semidefinite matrix P ∈ Rn×n satisfying (16.4.3), and (A, C) is detectable. ii) A is asymptotically stable. Proof. See the proof of Proposition 16.8.7. 

16.6 The Controllable Subspace and Controllability Let A ∈ Rn×n and B ∈ Rn×m, and, for t ≥ 0, consider the linear system x˙(t) = Ax(t) + Bu(t),

(16.6.1)

x(0) = 0.

(16.6.2)

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Definition 16.6.1. The controllable subspace Ctf(A, B) of (A, B) at time tf > 0 is the subspace △

Ctf (A, B) = {xf ∈ Rn : there exists a continuous control u : [0, tf ] 7→ Rm such that the solution x(·) of (16.6.1), (16.6.2) satisfies x(tf ) = xf }.

(16.6.3)

Let tf > 0. Then, Definition 16.6.1 states that xf ∈ Ctf (A, B) if and only if there exists a continuous control u : [0, tf ] → 7 Rm such that ∫tf xf =

e(tf −t)ABu(t) dt.

(16.6.4)

0

The following result provides expressions for Ctf (A, B). Lemma 16.6.2. Let tf > 0. Then, the following subspaces are equal:

i) Ctf (A, B). ]⊥ [∩ T tAT ii) t∈[0,tf ] N(B e ) . ]⊥ [∩ n−1 T iT iii) i=0 N(B A ) . iv) R([B AB · · · An−1B]). (∫ tf ) T v) R 0 etABBTetA dt . ∫ tf T If, in addition, limtf →∞ 0 etABBTetA dt exists, then the following subspace is equal to i)–v): (∫ ∞ ) T vi) R 0 etABBTetA dt . ∩ T Proof. To prove i) ⊆ ii), let η ∈ t∈[0,tf ] N(BTetA ) so that ηTetAB = 0 for all t ∈ [0, tf ]. Now, let ∫ tf u : [0, tf ] 7→ Rm be continuous. Then, ηT 0 e(tf −t)ABu(t) dt = 0, which implies that η ∈ Ctf (A, B)⊥. ∩ T iT T i To prove ii) ⊆ iii), let η ∈ n−1 i=0 N(B A ) so that η A B = 0 for all i ∈ {0, 1, . . . , n − 1}. It follows from the Cayley-Hamilton theorem given by Theorem 6.4.7 that ηTAiB = 0 for all i ≥ 0. Now, let ∑ i −1 T i T tAT t ∈ [0, tf ]. Then, ηTetAB = ∞ i=0 t (i!) η A B = 0, and thus η ∈ N(B e ). To show that iii) ⊆ iv), let η ∈ R([B AB · · · An−1B])⊥. It thus follows that η ∈ N([B AB · · · An−1B]T ), which implies that ηTAiB = 0 for all i ∈ {0, 1, . . . , n − 1}. (∫ tf ) ∫ tf T T To prove iv) ⊆ v), let η ∈ N 0 etABBTetA dt . Therefore, ηT 0 etABBTetA dtη = 0, which implies that ηTetAB = 0 for all t ∈ [0, tf ]. Differentiating with respect to t and setting t = 0 implies that ηTAiB = 0 for all i ∈ {0, 1, . . . , n − 1}. Hence, η ∈ R([B AB · · · An−1B])⊥ . ∫ tf To prove v) ⊆ i), let η ∈ Ctf(A, B)⊥ . Then, ηT 0 e(tf −t)ABu(t) dt = 0 for all continuous u : [0, tf ] 7→ (∫ tf ) ∫ tf T T T Rm. Letting u(t) = BTe(tf −t)A η implies ηT 0 etABBTetA dtη = 0, and thus η ∈ N 0 etABBTetA dt .  The expressions given by iii) and iv) of Lemma 16.6.2 show that Ctf (A, B) is independent of tf . We thus write C(A, B) for Ctf (A, B), and call C(A, B) the controllable subspace of (A, B). (A, B) is controllable if C(A, B) = Rn. For convenience, define the n × nm controllability matrix △

K(A, B) = [B AB · · · An−1B]

(16.6.5)

C(A, B) = R[K(A, B)].

(16.6.6)

q = dim C(A, B) = rank K(A, B).

(16.6.7)

so that Define

△

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LINEAR SYSTEMS AND CONTROL THEORY

Corollary 16.6.3. For all tf > 0,

∫tf q = dim C(A, B) = rank K(A, B) = rank If, in addition, limtf →∞

∫ tf 0

T

etABBTetA dt.

(16.6.8)

0 T

etABBTetA dt exists, then ∫∞ q = rank

T

etABBTetA dt.

(16.6.9)

0

Corollary 16.6.4. C(A, B) is an invariant subspace of A.

The following result shows that the controllable subspace C(A, B) is unchanged by full-state feedback u(t) = Kx(t) + v(t). Proposition 16.6.5. Let K ∈ Rm×n. Then, C(A + BK, B) = C(A, B).

(16.6.10)

In particular, (A, B) is controllable if and only if (A + BK, B) is controllable. Proof. Note that C(A + BK, B) = R [K(A + BK, B)] = R([B AB + BKB A2B + ABKB + BKAB + BKBKB · · · ]) = R[K(A, B)] = C(A, B).  ˜ ˜ Let C(A, B) ⊆ Rn be a subspace that is complementary to C(A, B). Then, C(A, B) is an uncon′ ′′ n ′ ˜ trollable subspace in the sense that, if xf = xf + xf ∈ R , where xf ∈ C(A, B) and xf′′ ∈ C(A, B) is m ′ nonzero, then there exists a continuous control u: [0, tf ] → R such that x(tf ) = xf , but there exists no continuous control such that x(tf ) = xf . Using Proposition 4.8.3, let Q ∈ Rn×n be the unique ˜ idempotent matrix such that R(Q) = C(A, B) and N(Q) = C(A, B). Then, xf′ = Qxf . The following △ ˜ result constructs Q and a continuous control u(·) that yields x(tf ) = xf′ for C(A, B) = C(A, B)⊥. In this case, Q is a projector. Lemma 16.6.6. Let tf > 0, and define Q ∈ Rn×n by + tf  tf  ∫ ∫ T T  △  etABBTetA dt. (16.6.11) Q =  etABBTetA dt   0

0

Then, Q is the projector onto C(A, B), and Q⊥ is the projector onto C(A, B)⊥. Hence, R(Q) = N(Q⊥ ) = C(A, B), N(Q) = R(Q) = C(A, B)⊥,

(16.6.12) (16.6.13)

rank Q = def Q⊥ = dim C(A, B) = q, def Q = rank Q⊥ = dim C(A, B)⊥ = n − q.

(16.6.14) (16.6.15)

Now, define u: [0, tf ] 7→ Rm by △

T (tf −t)AT

u(t) = B e

 tf + ∫    τA T τAT e BB e dτ   xf . 0

(16.6.16)

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If xf = xf′ + xf′′ , where xf′ ∈ C(A, B) and xf′′ ∈ C(A, B)⊥, then xf′

∫tf = Qxf =

e(tf −t)ABu(t) dt.

(16.6.17)

0

Finally, (A, B) is controllable if and only if Q = In . If these conditions hold, then, for all xf ∈ Rn, ∫tf xf =

e(tf −t)ABu(t) dt,

(16.6.18)

0

where u : [0, tf ] 7→ Rm is given by u(t) = BTe(tf −t)A

T

−1  tf  ∫ T  eτABBTeτA dτ xf .  

(16.6.19)

0

Lemma 16.6.7. Let α ∈ R. Then,

C(A + αI, B) = C(A, B).

(16.6.20)

The following result uses a coordinate transformation to characterize the controllable dynamics of (16.6.1). Theorem 16.6.8. There exists an orthogonal matrix S ∈ Rn×n such that [ ] [ ] A A12 −1 B A=S 1 S , B=S 1 , (16.6.21) 0 A2 0 where A1 ∈ Rq×q, B1 ∈ Rq×m, and (A1 , B1 ) is controllable. △ Proof. Let α < 0 be such that Aα = A + αI is asymptotically stable, and let Q ∈ Rn×n be the positive-semidefinite solution of Aα Q + QATα + BBT = 0 (16.6.22) given by

∫∞ Q=

T

etAαBBTetAα dt. 0

It now follows from Lemma 16.6.2 and Lemma 16.6.7 that R(Q) = R[C(Aα , B)] = R[C(A, B)]. Hence,

rank Q = dim C(Aα , B) = dim C(A, B) = q. [ ] Next, let S ∈ Rn×n be an orthogonal matrix such that Q = S Q01 00 S T, where Q1 ∈ Rq×q is positive [ˆ ˆ ] [ ] definite. Writing Aα = S Aˆ 1 Aˆ12 S −1 and B = S BB1 , where Aˆ1 ∈ Rq×q and B1 ∈ Rq×m, it follows from (16.6.22) that

A21 A2

2

Aˆ1 Q1 + Q1 Aˆ1T + B1BT1 = 0, Aˆ 21 Q1 + B2 BT1 = 0, B2 BT2 = 0.

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LINEAR SYSTEMS AND CONTROL THEORY

Therefore, B2 = 0 and Aˆ 21 = 0, and thus [ Aˆ Aα = S 1 0 Furthermore,

[

Aˆ A=S 1 0

] Aˆ12 −1 S , Aˆ 2

[

] B1 B=S . 0

] ([ Aˆ12 −1 Aˆ1 S − αI = S Aˆ 2 0 [

Hence, A=S

A1 0

] ) Aˆ12 − αI S −1 . Aˆ 2

] A12 −1 S , A2

△ △ △ where A1 = Aˆ1 − αIq , A12 = Aˆ12 , and A2 = Aˆ 2 − αIn−q .  Proposition 16.6.9. Let S ∈ Rn×n, and assume that S is orthogonal. Then, the following statements are equivalent: i) A and B have the form (16.6.21), where A1 ∈ Rq×q, B1 ∈ Rq×m, and (A1 , B1 ) is controllable. ( [ ]) ii) C(A, B) = R S I0q . ( [ 0 ]) iii) C(A, B)⊥ = R S In−q . [ ] I 0 T iv) Q = S q S . 0 0 Proposition 16.6.10. Let S ∈ Rn×n, and assume that S is nonsingular. Then, the following statements are equivalent: i) A and B have the form (16.6.21), where A1 ∈ Rq×q, B1 ∈ Rq×m, and (A1 , B1 ) is controllable. ( [ ]) ( [ 0 ]) ˜ ii) C(A, B) = R S I0q and C(A, B) = R S In−q . [ I 0] −1 q iii) Q = S 0 0 S . Definition 16.6.11. Let S ∈ Rn×n, assume that S is nonsingular, and let A and B have the form (16.6.21), where A1 ∈ Rq×q, B1 ∈ Rq×m, and (A1 , B1 ) is controllable. Then, λ ∈ C is an uncontrollable eigenvalue of (A, B) if λ ∈ spec(A2 ), the uncontrollable spectrum of (A, B) is spec(A2 ), and the uncontrollable multispectrum of (A, B) is mspec(A2 ). Finally, λ ∈ spec(A) is a controllable eigenvalue of (A, B) if λ is not an uncontrollable eigenvalue of (A, B). Definition 16.6.12. The controllability pencil CA,B (s) is the pencil

CA,B = P [A −B],[I 0] .

(16.6.23)

CA,B (s) = [sI − A B].

(16.6.24)

Equivalently, Proposition 16.6.13. Let λ ∈ spec(A). Then, λ is an uncontrollable eigenvalue of (A, B) if and

only if

rank [λI − A B] < n.

(16.6.25)

Proof. Since (A1 , B1 ) is controllable, it follows from (16.6.21) that

[

rank [λI − A B] = rank

λI − A1 0

A12 λI − A2

B1 0

]

= rank [λI − A1 B1 ] + rank(λI − A2 ) = q + rank(λI − A2 ). Hence, rank [λI−A B] < n if and only if rank(λI−A2 ) < n−q; that is, if and only if λ ∈ spec(A2 ). 

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Proposition 16.6.14. Let λ ∈ mspec(A) and K ∈ Rm×n. Then, λ is an uncontrollable eigenvalue

of (A, B) if and only if λ is an uncontrollable eigenvalue of (A + BK, B). Proof. Theorem 16.6.8 implies that there exists an orthogonal matrix S ∈ Rn×n such that (16.6.21) holds, where A1 ∈ Rq×q , B1 ∈ Rq×m , and (A1 , B1 ) is controllable. Partitioning KS = [K1 K2 ], where K1 ∈ Rm×q and K2 ∈ Rm×(n−q) , it follows that [ ] A + B1 K1 A12 + B1 K2 −1 A + BK = S 1 S . 0 A2

Therefore, the uncontrollable eigenvalues of A + BK are the elements of spec(A2 ).  Proposition 16.6.15. Assume that (A, B) is controllable. Then, the Smith form of CA,B is [In 0n×m ]. Proof. First, note that, if λ ∈ C is not an eigenvalue of A, then n = rank(λI − A) = rank [λI − A B] = rank CA,B (λ). Therefore, rank CA,B = n, and thus CA,B has n Smith polynomials. Furthermore, since (A, B) is controllable, it follows that (A, B) has no uncontrollable eigenvalues. Therefore, it follows from Proposition 16.6.13 that, for all λ ∈ spec(A), rank [λI − A B] = n. Consequently, rank CA,B (λ) = n for all λ ∈ C. Thus, every Smith polynomial CA,B is 1.  Proposition 16.6.16. Let S ∈ Rn×n, assume that S is nonsingular, and let A and B have the form (16.6.21), where A1 ∈ Rq×q, B1 ∈ Rq×m, and (A1 , B1 ) is controllable. Furthermore, let p1 , . . . , pn−q be the similarity invariants of A2 , where, for all i ∈ {1, . . . , n − q − 1}, pi divides pi+1 . Then, there exist unimodular matrices S 1 ∈ R[s]n×n and S 2 ∈ R[s](n+m)×(n+m) such that, for all s ∈ C,     Iq  p1 (s) 0n×m   S (s). (16.6.26) [sI − A B] = S 1 (s)  ..  2  .   pn−q (s) Consequently, Szeros(CA,B ) = mSzeros(CA,B ) =

n−q ∪

roots(pi ) = roots(χA2 ) = spec(A2 ),

i=1 n−q ∪

mroots(pi ) = mroots(χA2 ) = mspec(A2 ).

(16.6.27) (16.6.28)

i=1

Proof. Define S ∈ Rn×n as in Theorem 16.6.8, let Sˆ1 ∈ R[s]q×q and Sˆ 2 ∈ R[s](q+m)×(q+m) be

unimodular matrices such that Sˆ1 (s)[sIq − A1 B1 ]Sˆ 2 (s) = [Iq 0q×m ], and let Sˆ 3 , Sˆ 4 ∈ R(n−q)×(n−q) be unimodular matrices such that ˆ Sˆ 3 (s)(sI − A2 )Sˆ 4 (s) = P(s), △ where Pˆ = diag(p1 , . . . , pn−q ). Then, [ −1 ][ Iq Sˆ (s) 0 [sI − A B] = S 1 −1 ˆ 0 S 3 (s) 0

  Iq  ×  0  0

0 0 Im

0 0q×m ˆ P(s) 0

 −Sˆ1 (s)A12  [ ˆ −1  S (s) Sˆ 4−1 (s)  2 0  0

]

 ]  Iq  0 In−q  0 0

0 0 In−q

 0q×m  [ −1  S Im   0 0

] 0 . Im



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Proposition 16.6.17. Let s ∈ C. Then,

C(A, B) ⊆ Re R([sI − A B]).

(16.6.29)

Proof. Using Proposition 16.6.9 and the notation in the proof of Proposition 16.6.16, it follows that, for all s ∈ C, ]) ( [ −1 ( [ ]) Sˆ (s) 0 = R([sI − A B]). C(A, B) = R S I0q ⊆ R S 1 ˆ 0 Sˆ −1 (s)P(s) 3

Finally, (16.6.29) follows from the fact that C(A, B) is a real subspace.  The next result characterizes controllability in several equivalent ways. Theorem 16.6.18. The following statements are equivalent: i) (A, B) is controllable. ∫t T ii) There exists t > 0 such that 0 eτABBTeτA dτ is positive definite. ∫t T iii) For all t > 0, 0 eτABBTeτA dτ is positive definite. iv) rank K(A, B) = n. v) Every eigenvalue of (A, B) is controllable. vi) If x ∈ Cn is an eigenvector of A, then Cx , 0. vii) There exists K ∈ Rm×n such that (A + BK, B) is controllable. viii) For all K ∈ Rm×n∫, (A + BK, B) is controllable. t T If, in addition, limt→∞ 0 eτABBTeτA dτ exists, then the following statement is equivalent to i)–viii): ∫∞ T ix) 0 etABBTetA dt is positive definite. Proof. The equivalence of i)–iv) follows from Lemma 16.6.2. To prove iv) =⇒ v), suppose that v) does not hold; that is, there exist λ ∈ spec(A) and a nonzero vector x ∈ Cn such that x∗A = λx∗ and x∗B = 0. It thus follows that x∗AB = λx∗B = 0. Similarly, x∗AiB = 0 for all i ∈ {0, 1, . . . , n −1}. Hence, (Re x)T K(A, B) = 0 and (Im x)T K(A, B) = 0. Since Re x and Im x are not both zero, it follows that dim C(A, B) < n. Conversely, to show that v) implies iv), suppose that rank K(A, B) < n. Then, there exists a nonzero vector x ∈ Rn such that xTAi B = 0 for all i ∈ {0, . . . , n − 1}. Now, let p ∈ R[s] be a nonzero polynomial of minimal degree such that xTp(A) = 0. Note that p is not a constant polynomial and that xTµA (A) = 0. Thus, 1 ≤ deg p ≤ deg µA . Now, let λ ∈ C satisfy p(λ) = 0, and let q ∈ C[s] △ satisfy p(s) = q(s)(s − λ). Since deg q < deg p, it follows that xTq(A) , 0. Therefore, η = q(A)∗ x is nonzero. Furthermore, η∗ (A − λI) = xTp(A) = 0. Since, for all i ∈ {0, . . . , n − 1}, xTAi B = 0, it follows that η∗B = xTq(A)B = 0. Consequently, v) does not hold.  The following result, which is a restatement of the equivalence of i) and v) of Theorem 16.6.18, is the PBH test for controllability. Corollary 16.6.19. The following statements are equivalent: i) (A, B) is controllable. ii) For all s ∈ C, rank [sI − A B] = n.

(16.6.30)

The following result implies that arbitrary eigenvalue placement is possible for (16.6.1) if (A, B) is controllable. Proposition 16.6.20. The pair (A, B) is controllable if and only if, for every polynomial p ∈ R[s] such that deg p = n, there exists a matrix K ∈ Rm×n such that mspec(A + BK) = mroots(p). △ △ Proof. For the case m = 1, define Ac = C(χA ) and Bc = en as in (16.9.6). Then, Proposition

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16.9.3 implies that K(Ac , Bc ) is nonsingular, while Corollary 16.9.21 implies that Ac = S −1AS and △ Bc = S −1B. Now, let mroots(p) = {λ1 , . . . , λn }ms ⊂ C. Letting K = eTn [C(p) − Ac ]S −1 it follows that A + BK = S (Ac + Bc KS )S −1 = S (Ac + En,n [C(p) − Ac ])S −1 = SC(p)S −1. The case m ≥ 2 requires the multivariable controllable canonical form. See [2349, p. 248].



16.7 Controllable Asymptotic Stability △

Let A ∈ Rn×n and B ∈ Rn×m, and define q = dim C(A, C). Definition 16.7.1. (A, B) is controllably asymptotically stable if C(A, B) ⊆ Ss (A).

(16.7.1)

Proposition 16.7.2. Let K ∈ Rm×n. Then, (A, B) is controllably asymptotically stable if and

only if (A + BK, B) is controllably asymptotically stable. Proposition 16.7.3. The following statements are equivalent: i) (A, B) is controllably asymptotically stable. ii) There exists an orthogonal matrix S ∈ Rn×n such that (16.6.21) holds, where A1 ∈ Rq×q is asymptotically stable and B1 ∈ Rq×m. iii) There exists a nonsingular matrix S ∈ Rn×n such that (16.6.21) holds, where A1 ∈ Rq×q is asymptotically stable and B1 ∈ Rq×m. iv) limt→∞ etAB = 0. v) The positive-semidefinite matrix Q ∈ Rn×n defined by △

∫∞

Q=

T

etABBTetA dt

(16.7.2)

0

exists. vi) There exists a positive-semidefinite matrix Q ∈ Rn×n satisfying AQ + QAT + BBT = 0.

(16.7.3)

If these conditions hold, then the positive-semidefinite matrix Q ∈ Rn×n defined by (16.7.2) satisfies (16.7.3). Proof. To prove i) =⇒ ii), note that, since (A, B) is controllably asymptotically stable, it follows that C(A, B) ⊆ Ss (A) = N[µAs (A)] = R[µAu (A)]. Then, Theorem 16.6.8 implies that there exists an orthogonal S ∈ Rn×n such that (16.6.21) holds, where A1 ∈ Rq×q and (A1 , B1 ) is controllable. ( [matrix ]) Iq Thus, R S 0 = C(A, B) ⊆ R[µAs (A)]. Next, note that  s   µA(A1 ) B12s  −1 s   S , µA(A) = S  0 µAs (A2 ) where B12s ∈ Rq×(n−q), and suppose that A1 is not asymptotically stable with CRHP eigenvalue λ. Then, λ < roots(µsA ), and thus µsA(A1 ) , 0. Let x1 ∈ Rn−q satisfy µsA(A1 )x1 , 0. Then, [ ] ( [ ]) [ ] [ s ] x1 Iq x1 µA(A1 )x1 s ∈R S = C(A, B), µA(A)S =S , 0 0 0 0 [ ] and thus x01 < N[µAs (A)] = Ss (A), which implies that C(A, B) is not contained in Ss (A). Therefore, A1 is asymptotically stable. [ tA ] To prove iii) =⇒ iv), note that etAB = S e 01 B1 → 0 as t → ∞.

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Next, to prove iv) =⇒ v), note that, since etAB → 0 as t → ∞, it follows that every entry of etAB involves exponentials of t, where the coefficients of t have negative real part. Hence, so does every ∫∞ T T entry of etABBTetA , which implies that 0 etABBTetA dt exists. ∫∞ T T To prove v) =⇒ vi), note that, since Q = 0 etABBTetA dt exists, it follows that etABBTetA → 0 as t → ∞. Thus, ∫∞ [ ∫∞ ] T tAT tA T tAT T tA T tAT tA T tAT T d tA − BBT = −BBT , AQ + QA = Ae BB e + e BB e A dt = dt e BB e dt = lim e BB e t→∞

0

0

which shows that Q satisfies (16.7.3). To prove vi) =⇒ i), note that, for all t ≥ 0, ∫t

∫t ∫t d τA τAT T T T eτABBTeτA dτ = − eτA (AQ + QAT )eτA dτ = − e Qe dτ = Q − etA QetA ≤ Q. dτ

0

0

0

Next, Theorem 16.6.8 implies that there exists an orthogonal matrix S ∈ Rn×n such that (16.6.21) holds, where A1 ∈ Rq×q, B1 ∈ Rq×m, and (A1 , B1 ) is controllable. Consequently, ∫t

τA1

T τA1T

e B1B1 e

∫t dτ = [I 0]S

T

0

[ ] [ ] I I ≤ [I 0]S TQS . 0 0

eτABBTeτA dτS T

0

∫∞ T Thus, Proposition 10.6.3 implies that Q1 = 0 etA1B1BT1etA1 dt exists. Since (A1 , B1 ) is controllable, it follows from vii) of Theorem 16.6.18 that Q1 is positive definite. Now, let λ be an eigenvalue of AT1 , and let x1 ∈ Cn be an associated eigenvector. Consequently, △ α = x1∗ Q1 x1 is positive, and △

α=

x1∗

∫∞

λt

T λt

e BB1e dtx1 = 0

∫∞

x1∗ B1BT1 x1

0

e2(Re λ)t dt.

∫∞ Hence, 0 e2(Re λ)t dt = α/(x1∗B1 BT1 x1 ) exists, and thus Re λ < 0. Consequently, A1 is asymptotically stable, and thus C(A, B) ⊆ Ss (A); that is, (A, B) is controllably asymptotically stable.  The matrix Q ∈ Rn×n defined by (16.7.2) is the controllability Gramian, and (16.7.3) is the control Lyapunov equation. Proposition 16.7.4. Assume that (A, B) is controllably asymptotically stable, let Q ∈ Rn×n be the positive-semidefinite matrix defined by (16.7.2), and define Q ∈ Rn×n by (16.6.11). Then, the following statements hold: i) QQ+ = Q. ii) R(Q) = R(Q) = C(A, B). iii) N(Q) = N(Q) = C(A, B)⊥. iv) rank Q = rank Q = q. v) Q is the unique positive-semidefinite solution of (16.7.3) whose rank is q. Proof. See [2475] for the proof of v).  n×n Proposition 16.7.5. Assume that (A, B) is controllably asymptotically stable, let Q ∈ R be the positive-semidefinite matrix defined by (16.7.2), and let Qˆ ∈ Rn×n. Then, the following statements are equivalent: i) Qˆ is positive semidefinite and satisfies (16.7.3).

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ii) There exists a positive-semidefinite matrix Q0 ∈ Rn×n such that Qˆ = Q + Q0 and AQ0 + Q0 AT = 0. If these conditions hold, then rank Qˆ = q + rank Q0 , ∑ rank Q0 ≤ gmultA (λ).

(16.7.4) (16.7.5)

λ∈spec(A) λ∈IA



Proof. See [2475]. Proposition 16.7.6. The following statements are equivalent:

i) (A, B) is controllably asymptotically stable, every imaginary eigenvalue of A is semisimple, and A has no ORHP eigenvalues. ii) (16.7.3) has a positive-definite solution Q ∈ Rn×n. Proof. See [2475].  Proposition 16.7.7. The following statements are equivalent: i) (A, B) is controllably asymptotically stable, and A has no imaginary eigenvalues. ii) (16.7.3) has exactly one positive-semidefinite solution Q ∈ Rn×n. If these conditions hold, then Q ∈ Rn×n is given by (16.7.2) and satisfies rank Q = q. Proof. See [2475].  Corollary 16.7.8. Assume that A is asymptotically stable. Then, the positive-semidefinite matrix Q ∈ Rn×n defined by (16.7.2) is the unique solution of (16.7.3) and satisfies rank Q = q. Proof. See [2475].  Proposition 16.7.9. The following statements are equivalent: i) (A, B) is controllable, and A is asymptotically stable. ii) (16.7.3) has exactly one positive-semidefinite solution Q ∈ Rn×n, and Q is positive definite. If these conditions hold, then Q ∈ Rn×n is given by (16.7.2). Proof. See [2475].  Corollary 16.7.10. Assume that A is asymptotically stable. Then, the positive-semidefinite matrix Q ∈ Rn×n defined by (16.7.2) exists. Furthermore, Q is positive definite if and only if (A, B) is controllable.

16.8 Stabilizability △

Let A ∈ Rn×n and B ∈ Rn×m, and define q = dim C(A, B). Definition 16.8.1. The stabilizable subspace Cs (A, B) of (A, B) is △

Cs (A, B) = C(A, B) + Ss (A).

(16.8.1)

Furthermore, (A, B) is stabilizable if Cs (A, B) = Rn . Proposition 16.8.2. (A, B) is stabilizable if and only if Su (A) ⊆ C(A, B). Proposition 16.8.3. Let K ∈ R

(16.8.2)

. Then, (A, B) is stabilizable if and only if (A + BK, B) is

m×n

stabilizable. Proposition 16.8.4. The following statements are equivalent:

i) (A, B) is stabilizable. ii) There exists an orthogonal matrix S ∈ Rn×n such that (16.6.21) holds, where A1 ∈ Rq×q, B1 ∈ Rq×m, (A1 , B1 ) is controllable, and A2 ∈ R(n−q)×(n−q) is asymptotically stable.

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iii) There exists a nonsingular matrix S ∈ Rn×n such that (16.6.21) holds, where A1 ∈ Rq×q, B1 ∈ Rq×m, (A1 , B1 ) is controllable, and A2 ∈ R(n−q)×(n−q) is asymptotically stable. iv) Every CRHP eigenvalue of (A, B) is controllable. Proof. To prove i) =⇒ ii), note that, since (A, B) is stabilizable, it follows that Su (A) = N[µAu (A)] = R[µAs (A)] ⊆ C(A, B). Using Theorem 16.6.8, it follows that there exists an orthogoq×q nal matrix S ∈ Rn×n such( that [ I ])(16.6.21) holds, where A1 ∈ R and (A1 , B1 ) is controllable. Thus, s q R[µA(A)] ⊆ C(A, B) = R S 0 . Next, note that [ s ] µ (A ) B12s µAs (A) = S A 1 S −1, 0 µAs (A2 ) where B12s ∈ Rq×(n−q), and suppose that A2 is not asymptotically stable with CRHP eigenvalue λ. Then, λ < roots(µsA ), and thus µsA(A2 ) , 0. Let x2 ∈ Rn−q satisfy µsA(A2 )x2 , 0. Then,   ( [ ]) [ ]  B12s x2  0 s  < R S Iq = C(A, B), µA(A)S = S  s 0 x2 µA(A2 )x2 which implies that Su (A) is not contained in C(A, B). Hence, A2 is asymptotically stable. The statement ii) =⇒ iii) is immediate. To prove iii) =⇒ iv), let λ ∈ spec(A) be a CRHP eigenvalue of A. Since A2 is asymptotically stable, it follows that λ < spec(A2 ). Consequently, Proposition 16.6.13 implies that λ is not an uncontrollable eigenvalue of (A, B), and thus λ is a controllable eigenvalue of (A, B). To prove iv) =⇒ i), let S ∈ Rn×n be nonsingular and such that A and B have the form (16.6.21), where A1 ∈ Rq×q, B1 ∈ Rq×m, and (A1 , B1 ) is controllable. Since every CRHP eigenvalue of (A, B) is controllable, it follows from 16.6.13 that A2 is asymptotically stable. From Fact 15.24.4 ( [Proposition ]) Iq it follows that Su (A) ⊆ R S 0 = C(A, B), which implies that (A, B) is stabilizable.  The following result, which is a restatement of the equivalence of i) and iv) of Proposition 16.8.4, is the PBH test for stabilizability. Corollary 16.8.5. The following statements are equivalent: i) (A, B) is stabilizable. ii) For all s ∈ CRHP, rank [sI − A B] = n.

(16.8.3)

Proposition 16.8.6. The following statements are equivalent: i) A is asymptotically stable. ii) (A, B) is controllably asymptotically stable and stabilizable. Proof. To prove i) =⇒ ii), note that, since A is asymptotically stable, it follows that Su (A) = {0}, and Ss (A) = Rn. Thus, Su (A) ⊆ C(A, B), and C(A, B) ⊆ Ss (A). To prove ii) =⇒ i), note that, since (A, B) is stabilizable and controllably asymptotically stable, it follows that Su (A) ⊆ C(A, B) ⊆ Ss (A), and thus Su (A) = {0}. As an alternative proof that ii) =⇒ i), note that, since (A, B) is stabilizable, it follows from Proposition 16.8.4 that there exists a nonsingular matrix S ∈ Rn×n such that (16.6.21) holds, where A1 ∈ Rq×q, B1 ∈ Rq×m, (A1 , B1 ) is controllable, and A2 ∈ R(n−q)×(n−q) is asymptotically stable. Then,  ∫ ∞ tA  T ∫∞  e 1B1BT1etA1 dt 0  −1 tA T tAT 0 e BB e dt = S   S . 0 0 0

Since the integral on the left-hand side exists by assumption, the integral on the right-hand side

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also (A1 , B1 ) is controllable, it follows from vii) of Theorem 16.6.18 that Q1 = ∫ ∞ exists. Since tA1 T tA1T e B B e dt is positive definite. 1 1 0 Now, let λ be an eigenvalue of AT1 , and let x1 ∈ Cq be an associated eigenvector. Consequently, α = x1∗ Q1 x1 is positive, and △

∫∞

α= ∫∞

x1∗

λt

T λt

e B1B1e dtx1 =

x1∗ B1BT1 x1

∫∞

e2(Re λ)t dt.

0

0 2(Re λ)t

Hence, 0 e dt exists, and thus Re λ < 0. Consequently, A1 is asymptotically stable, and thus A is asymptotically stable.  Corollary 16.8.7. The following statements are equivalent: i) There exists a positive-semidefinite matrix Q ∈ Rn×n satisfying (16.7.3), and (A, B) is stabilizable. ii) A is asymptotically stable. Proof. This result follows from Proposition 16.7.3 and Proposition 16.8.6. 

16.9 Realization Theory Given a proper rational transfer function G, we wish to determine (A, B, C, D) such that (16.2.10) holds. The following terminology is convenient. Definition 16.9.1. Let G ∈ Rl×m(s). If l = m = 1, then G is a single-input/single-output (SISO) rational transfer function; if l = 1 and m > 1, then G is a multiple-input/single-output (MISO) rational transfer function; if l > 1 and m = 1, then G is a single-input/multiple-output (SIMO) rational transfer function; and, if l > 1 and m > 1, then G is a multiple-input/multiple-output (MIMO) rational transfer function. Definition 16.9.2. Let G ∈ R(s)l×m that A ∈ Rn×n, B ∈ Rn×m, C ∈ Rl×n, and prop , and assume [ A B ] D ∈ Rl×m satisfy G(s) = C(sI − A)−1B + D. Then, C D is a realization of G, which is written as   A G ∼  C

 B   . D

(16.9.1)

The order of the realization (16.9.1) is the size of A. Finally, (A, B, C) is controllable and observable if (A, B) is controllable and (A, C) is observable. Let n = 0. Then, A, B, and C are empty matrices, and G ∈ R(s)l×m prop is given by G(s) = 0l×0 (sI0×0 − 00×0 )−1 00×m + D = 0l×m + D = D. [ 0 ] 00×m 0×0 Therefore, the order of the realization 0 is zero. D

(16.9.2)

l×0

Although the realization (16.9.1) is not unique, the matrix D is unique and is given by △

Furthermore, note that G ∼

[

A

B

C

D

D = G(∞) = lim G(s). s→∞ ] [ if and only if G − D ∼

(16.9.3) A

B

C

0

]

.

The following result shows that every proper, SISO rational transfer [ function ] G has a realization.

In fact, two realizations are the controllable canonical form G ∼ [ A B ] o o canonical form G ∼ C D . o

o

Ac

Bc

Cc

Dc

and the observable

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Proposition 16.9.3. Let G ∈ R(s)prop be the SISO proper rational transfer function

G(s) = Then, G ∼

[

Ac

Bc

Cc

Dc

]

αn sn + αn−1 sn−1 + · · · + α1 s + α0 . sn + βn−1 sn−1 + · · · + β1 s + β0

, where Ac , Bc , Cc , Dc are defined by

     △  Ac =    

0 0 .. .

1 0 .. .

0 1 .. .

0 −β0

0 −β1

0 −β2

··· ··· .. .

0 0 .. .

··· ···

1 −βn−1

      ,  

   0   0      △  Bc =  ...  ,    0    1

△

and G ∼

[

Ao

Bo

Co

Do

△

··· ··· ··· .. . ···

0 0 0

−β0 −β1 −β2 .. .

1 −βn−1

Co = [0 0 · · · 0 1],

      ,  

(16.9.5)

△

Cc = [α0 − β0 αn α1 − β1 αn · · · αn−1 − βn−1 αn ], ] , where Ao , Bo , Co , Do are defined by   0 0  1 0   △  Ao =  0 1  . .  .. ..  0 0

Furthermore,

(16.9.4)

Dc = αn ,

   α0 − β0 αn   α − β α  1 n   1   △  Bo =  α2 − β2 αn  ,   ..   .   αn−1 − βn−1 αn

△

D o = αn .

 0  0   0 0  . ..  . .  . K(Ac , Bc ) = O(Ao , Co ) =  0  0    0 1  1 −βn−1

where each “◦” denotes a possibly nonzero entry, and  β2  β1   β2 β3  ..  .. .  . −1 −1 K(Ac , Bc ) = O(Ao , Co ) =   βn−2 βn−1    βn−1 1  1 0

(16.9.6)

(16.9.7)

(16.9.8)

1

··· . .. . .. . ..

−βn−1

..

.

◦

◦

···

◦

0 0 . ..

β3

1

··· . .. . .. . ..

0

..

0

···

β4 . ..

.

0 1 . .. ◦

βn−1 1 . .. 0 0 0

   −βn−1   ..  .   , ◦    ◦   ◦ 1

 1    0   ..  .   . 0    0   0

(16.9.9)

(16.9.10)

Hence, (Ac , Bc ) is controllable, and (Ao , Co ) is observable. Proof. The realizations can be verified by forming the corresponding transfer functions. It follows from Fact 3.16.15 that det K(Ac , Bc ) = det O(Ao , Co ) = (−1)⌊n/2⌋ , which implies that (Ac , Bc ) is controllable and (Ao , Co ) is observable. 

1272

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The following result shows that every proper rational transfer function has a realization. n×n Theorem 16.9.4. Let G ∈ R(s)l×m , B ∈ Rn×m, C ∈ Rl×n, and prop . Then, there exist A ∈ R [ A B ] D ∈ Rl×m such that G ∼ C D . [ A B ] ij ij . Proof. By Proposition 16.9.3, every entry G(i, j) of G has a realization G(i, j) ∼ C D ij

Combining these realizations yields a realization of G. Proposition 16.9.5. Let G ∈ R(s)l×m prop have the nth-order realization

and assume that S is nonsingular. Then,

  SAS −1 G ∼  CS −1

 SB  . D

[

A

B

C

D

]

ij



, let S ∈ Rn×n,

(16.9.11)

If, in addition, (A, B, C) is (controllable, observable), then so is (SAS −1 , SB, CS −1 ). [ ] [ A B ] Bˆ Aˆ and be nth-order realizations of Definition 16.9.6. Let G ∈ R(s)l×m , and let prop C D D Cˆ [ A B ] [ ˆ ˆ ] B A G. Then, C D and ˆ are equivalent if there exists a nonsingular matrix S ∈ Rn×n such C

D

that Aˆ = SAS −1, Bˆ = SB, and Cˆ = CS −1. The following result shows that the Markov parameters of a proper rational transfer function are independent of the realization. [ A B ] Lemma 16.9.7. Let G ∈ R(s)l×m , and assume that G ∼ , where A ∈ Rn×n, and G ∼ prop C D [ ˆ ˆ ] A B ˆ ˆ and, for all k ≥ 0, CAkB = Cˆ AˆkB. , where Aˆ ∈ Rnˆ ׈n. Then, D = D, Cˆ

Dˆ

ˆ = G(∞), it follows that D = D. ˆ Next, it follows from Proof. Since D = G(∞) and D (16.2.11) and the definition (16.1.18) of the impulse response function H that, for all t ≥ 0, ˆ Setˆ t AˆB. ˆ t AˆBˆ + δ(t)D. Hence, for all t ≥ 0, CetAB = Ce H(t) = L−1 {G(s)} = CetAB + δ(t)D = Ce ˆ Furthermore, for all t ≥ 0 and k ≥ 1, ting t = 0 yields CB = Cˆ B. dk tA dk ˆ tAˆ ˆ ˆ ˆ Ce B = Ce B = Cˆ Aˆ k et AB. dtk dtk ˆ Setting t = 0 implies that, for all k ≥ 1, CAkB = Cˆ AˆkB. CAk etAB =

 The following result shows that, if two controllable and observable realizations of a proper rational transfer function have the same order, then they are equivalent. Proposition 16.9.8. Let G ∈ R(s)l×m , and assume that G has the nth-order controllable and [prop ] [ A B ] A2 B2 1 1 observable realizations C D and C D . Then, there exists a unique nonsingular matrix 1

2

S ∈ Rn×n such that A2 = SA1 S −1 , B2 = SB1 , and C2 = C1 S −1 . In fact, S and S −1 are given by S = (OT2 O2 )−1OT2 O1 = K2 KT2 (K1 KT2 )−1 , S △

−1

△

T −1

= K1 K2 (K2 K2 ) T

−1

(16.9.12)

= (O2 O1 ) O2 O2 , T

T

△

(16.9.13)

△

where K1 = K(A1 , B1 ), K2 = K(A2 , B2 ), O1 = O(A1 , C1 ), and O2 = O(A2 , C2 ). If, in addition, m = l = 1, then S and S −1 are given by −1 −1 S = K2 K−1 S −1 = K1 K−1 (16.9.14) 1 = O2 O1 , 2 = O1 O2 . [ A B ] [ A B ] 1 1 2 2 Proof. By Lemma 16.9.7, the realizations and C D generate the same Markov C D 1

2

parameters. Hence, O1 K1 = O2 K2 , and thus OT2 O1 K1 KT2 = OT2 O2 K2 KT2 . Since (A2 , B2 , C2 ) is con-

1273

LINEAR SYSTEMS AND CONTROL THEORY

trollable and observable, it follows that the n × n matrices K2 KT2 and OT2O2 are nonsingular. Therefore, (OT2 O2 )−1 OT2 O1 K1 KT2 (K2 KT2 )−1 = I. Defining S ∈ Rn×n as in (16.9.12), it follows that S −1 is given by (16.9.13). Since O2 A2 K2 = O1 A1 K1 , it follows that OT2 O2 A2 K2 KT2 = OT2 O1 A1 K1 KT2 , and thus A2 = SA1 S −1 . Likewise, it follows from O1 B1 = O2 B2 and C1K1 = C2 K2 that B2 = SB1 and C2 = C1 S −1. To prove uniqueness, suppose that there exists a nonsingular matrix Sˆ ∈ Rn×n such that A2 = ˆ ˆ 1 , and C2 = C1 Sˆ −1. Then, it follows that O1 = O2 Sˆ . Since O1 = O2 S , it follows SA1 Sˆ −1, B2 = SB ˆ ˆ  that O2 (S − S ) = 0. Consequently, S = S. n×n Proposition 16.9.9. Let A1 ∈ R and B1 ∈ Rn×m , assume that (A1 , B1 ) is controllable, let △ △ S ∈ Rn×n , assume that S is nonsingular, and define A2 = S A1 S −1 and B2 = SB1 . Then, (A2 , B2 ) △ T T −1 −1 T T −1 is controllable, S = K2 K2 (K1 K2 ) , and S = K1 K2 (K2 K2 ) , where K1 = K(A1 , B1 ) and △ −1 −1 −1 K2 = K(A2 , B2 ). If, in addition, m = 1, then S = K2 K1 and S = K1 K2 . Proof. Note that K2 = S K1 , and thus K2 KT2 = S K1 KT2 .  Proposition 16.9.10. Let A1 ∈ Rn×n and C1 ∈ R p×n , assume that (A1 , C1 ) is observable, let △ △ S ∈ Rn×n , assume that S is nonsingular, and define A2 = S A1 S −1 and C2 = CS −1 . Then, (A2 , C2 ) △ △ T −1 T −1 T −1 T is observable, S = (O2 O2 ) O2 O1 , and S = (O2 O1 ) O2 O2 , where O1 = O(A1 , C1 ) and O2 = −1 −1 −1 O(A2 , C2 ). If, in addition, m = 1, then S = O1 O2 and S = O2 O1 . Proof. Note that O2 = O1 S −1 , and thus OT2 O2 = OT2 O1 S −1 .  Proposition 16.9.11. Let G ∈ R(s)l×m , assume that G has the nth-order controllable and obprop [ A B ] 1 1 △ servable realization C D , let S ∈ Rn×n , assume that S is nonsingular, and define A2 = SA1 S −1 , 1 [ A B ] 2 2 △ △ B2 = SB1 , and C2 = C1 S −1 . Then, C D is an nth-order controllable and observable realization 2

△

△

of G, and S and S −1 are given by (16.9.12) and (16.9.13), where K1 = K(A1 , B1 ), K2 = K(A2 , B2 ), △ △ O1 = O(A1 , C1 ), and O2 = O(A2 , C2 ). If, in addition, m = l = 1, then S and S −1 are given by (16.9.14). The following result, known as the Kalman decomposition, is useful for constructing controllable and observable realizations. [ ] Proposition 16.9.12. Let G ∈ R(s)l×m prop , where G ∼

matrix S ∈ R

A

B

C

D

. Then, there exists a nonsingular

n×n

such that  0  A1  A A 21 2 A = S  0  0 0 0

   B1   B  B = S  2  , C = [C1 0 C3 0]S −1 ,  0  0 ([ A 0 ] [ B ]) ([ where, for all i ∈ {1, . . . , 4}, Ai ∈ Rni ×ni , A211 A2 , B12 is controllable, and A01 observable. Furthermore, the following statements hold: i) (A, B) is stabilizable if and only if A3 and A4 are asymptotically stable. ii) (A, B) is controllable if and only if A3 and A4 are empty. iii) (A, C) is detectable if and only if A2 and A4 are asymptotically stable. iv) (A, C) is observable if and only if A2 and A4 are empty. [ A B ] 1 1 v) G ∼ C D . A13 A23 A3 A43

 0   A24  −1 S , 0  A4

(16.9.15)

A13 A3

]

, [ C1

C3 ]

)

is

1

vi) (A1 , B1 , C1 ) is controllable and observable. Proof. Let α ≤ 0 be such that A + αI is asymptotically stable, and let Q ∈ Rn×n and P ∈ Rn×n denote the controllability and observability Gramians of the system (A + αI, B, C). Then, Theorem

1274

CHAPTER 16

10.3.5 implies that there exists a nonsingular matrix S ∈ Rn×n such that     P1 0 0  Q1 0 0 0   0 0 0  0 Q2 0 0  T − T  S , P = S  Q = S  0 0 0   0 0 P2  0 0 0 0 0 0 0 0

 0   0  −1 S , 0  0

where Q1 and P1 are the same size, and where Q1, Q2 , P1, and P2 are positive definite and diagonal. The form of SAS −1 , SB, and CS −1 given by (16.9.15) now follows from (16.7.3) and (16.4.3) with A −1 −1 replaced by A + αI, where, as in the proof of Theorem 16.6.8, [ SAS ]= S (A + αI)S − αI. Finally, statements i)–v) are immediate, while it can be verified that

A1

B1

C1

D1

is a realization of G.



Note that the uncontrollable multispectrum of (A, B) is given by mspec(A3 ) ∪ mspec(A4 ), while the unobservable multispectrum of (A, C) is given by mspec(A2 ) ∪ mspec(A4 ). Likewise, the uncontrollable-unobservable multispectrum of (A, B, C) is given by mspec(A4 ). [ A B ] Let G ∼ C D . Then, define the i-step observability matrix Oi (A, C) ∈ Ril×n by    C    △  CA  Oi (A, C) =  ..   .  CAi−1

(16.9.16)

and the j-step controllability matrix K j (A, B) ∈ Rn× jm by △

K j (A, B) = [B AB · · · A j−1B].

(16.9.17)

Note that O(A, C) = On (A, C) and K(A, B) = Kn (A, B). Furthermore, define the Markov blockHankel matrix Hi, j,k (G) ∈ Ril× jm of G by △

Hi, j,k (G) = Oi (A, C)Ak K j (A, B).

(16.9.18)

Note that Hi, j,k (G) is the block-Hankel matrix of Markov parameters given by   CAk+1B CAk+2B · · · CAk+ j−1B   CAkB    .. .. ..  CAk+1B CAk+2B . . .    . . . .   . . . . . . . . Hi, j,k (G) =  CAk+2B    .. . . . .  . . . .  . . . . .    . . .  . . . k+i−1 k+ j+i−2 . . . CA B CA B   Hk+1   H  k+2  =  Hk+3  .  ..   Hk+i

Hk+2 Hk+3 . .. . .. . ..

Hk+3 . .. ..

.

.

..

.

..

··· . .. ..

.

Hk+ j . .. ..

.

.

..

..

.

.

..

Hk+ j+i−1

       .    

(16.9.19)

Note that Hi, j,0 (G) = Oi (A, C)K j (A, B),

(16.9.20)

Hi, j,1 (G) = Oi (A, C)AK j (A, B).

(16.9.21)

1275

LINEAR SYSTEMS AND CONTROL THEORY

Furthermore, define △

H(G) = Hn,n,0 (G) = O(A, C)K(A, B).

(16.9.22)

The following result provides a MIMO extension of Fact 6.8.10. [ A B ] Proposition 16.9.13. Let G ∼ , where A ∈ Rn×n. Then, the following statements are C D equivalent: i) (A, B, C) is controllable and observable. ii) rank H(G) = n. iii) For all i, j ≥ n, rank Hi, j,0 (G) = n. iv) There exist i, j ≥ n such that rank Hi, j,0 (G) = n. Proof. The equivalence of ii), iii), and iv) follows from Fact 3.14.13. To prove i) =⇒ ii), note that, since the n × n matrices O(A, C)T O(A, C) and K(A, B)K(A, B)T are positive definite, it follows that n = rank O(A, C)T O(A, C)K(A, B)K(A, B)T ≤ rank H(G) ≤ n. Conversely, n = rank H(G) ≤ min {rank O(A, C), rank K(A, B)} ≤ n.  [ A B ] Proposition 16.9.14. Let G ∼ , where A ∈ Rn×n, assume that (A, B, C) is controllable C D [ ˆ ˆ ] B A , and observable, and let i, j ≥ 1 satisfy rank Oi (A, C) = rank K j (A, B) = n. Then, G ∼ ˆ C

D

where △ Aˆ = O+i (A, C)Hi, j,1 (G)K+j (A, B), [ ] Im △ Bˆ = K j (A, B) , 0( j−1)n×m

(16.9.23) (16.9.24)

△ Cˆ = [Il 0l×(i−1)l ]Oi (A, C).

(16.9.25) △

il×n Proposition 16.9.15. Let G ∈ R(s)l×m prop , let i, j ≥ 1, define n = rank Hi, j,0 (G), and let L ∈ R

and R ∈ Rn× jm satisfy Hi, j,0 (G) = LR. Then, the realization [ ]  Im  + +  L Hi, j,1 (G)R R 0 ( j−1)n×m G ∼   [Il 0l×(i−1)l ]L D

     

(16.9.26)

is controllable and observable. A rational transfer function G ∈ R(s)l×m prop can have realizations of different orders. For example, letting A = 1, ] 1 0 ˆ A= , 0 1 [

it follows that

B = 1, C = 1, D = 0, [ ] 1 ˆ B= , Cˆ = [1 0], Dˆ = 0, 0

1 . s −1 It is usually desirable to find realizations whose order is as small as possible. ˆ − A) ˆ −1 Bˆ + Dˆ = G(s) = C(sI − A)−1B + D = C(sI

1276

CHAPTER 16

Definition 16.9.16. Let G ∈ R(s)l×m prop , and assume that G ∼

[

A

B

C

D

]

. Then,

[

A

B

C

D

] is a

minimal realization of G if its order is either less than or equal to the order of every realization of G. In this case, we write    A B  min  (16.9.27) G ∼   . C D Note that the minimality of a realization is independent of D. The following result shows that every proper transfer function has a minimal realization. Proposition 16.9.17. Let G ∈ R(s)l×m prop . Then, G has a minimal realization. [ A B ] Proof. Theorem 16.9.4 implies that G has a realization G ∼ . Let n be the order of C D

this realization. For all i ∈ {1, . . . , n}, let Ri denote the set of realizations of G whose order is i. Let i0 denotes the smallest i ∈ {1, . . . , n} such that Ri is nonempty. Then, each realization in Ri0 is a minimal realization. Hence, G has a minimal realization.  The following result shows that a realization is controllable and observable if and only if it is minimal. ] [ ] [ Proposition 16.9.18. Let G ∈ Rl×m(s), and assume that G ∼

A

B

C

D

. Then,

A

B

C

D

is

minimal if and only if (A, B, C) is controllable and observable. Proof. To prove necessity, suppose that (A, B, C) is either not controllable or not observable. [ A B ] Then, Proposition 16.9.12 implies that G has a realization of order less than n. Hence, C D is not minimal. n×n To [ prove]sufficiency, let A ∈ R , where (A, B, C) is controllable and observable, but suppose A

B

C

D

is not a minimal realization of G. Hence, Proposition 16.9.12 implies that G has a [ ˆ ˆ ] B A of order nˆ < n. Since the Markov parameters of G are independent minimal realization ˆ

that

C

D

of the realization, it follows from Proposition 16.9.13 that rank H(G) = nˆ < n. However, since (A, B, C) is controllable and observable, it follows from Proposition 16.9.13 that rank H(G) = n, which is a contradiction.  ] [ ] [ Proposition 16.9.3 implies that the realizations

Ac

Bc

Cc

Dc

and

Ao

Bo

Co

Do

of G are such that

(Ac , Bc ) is controllable and (Ao , Co ) is observable. However, Proposition [ A B ] 16.9.3 [ AdoesB not ] imply c o c o that (Ac , Cc ) is observable and (Ao , Bo ) is controllable, and thus C D and C D are not c

c

o

o

necessarily minimal. It turns out that minimality of these realizations depends on whether or not the numerator and denominator of G given by (16.9.4) are coprime. Proposition 16.9.19. Let G ∈ R(s)prop , assume that G = p/q, where p, q ∈ R[s], assume that [ ]

G∼

A

B

C

D

, where A ∈ Rn×n , and consider the following statements:

i) n = deg q. ii) p and q are coprime. [ A B ] iii) C D is a minimal realization of G. Then, if two of the above statements are satisfied, then the remaining statement is also satisfied. [ A B ] of G is not minimal. Proof. To prove [i), ii)] =⇒ iii), suppose that the realization [ ˆ ˆ C] D B A Therefore, Proposition 16.9.17 implies that G has a realization ˆ , where Aˆ ∈ Rnˆ ׈n and nˆ < n. C

△

D

△ ˆ ˆ A Bˆ + Dq(s) Therefore, G = p/ ˆ q, ˆ where p(s) ˆ = C(sI − A) ˆ and qˆ = χAˆ . Since G = p/q = p/ ˆ qˆ and

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LINEAR SYSTEMS AND CONTROL THEORY

deg qˆ = nˆ < n = deg q, it follows that p and q are not coprime, which is a contradiction. △ To prove [ii), iii)] =⇒ i), note that G = p/q = p/ ˆ q, ˆ where p(s) ˆ = C(sI − A)A B + Dq(s) ˆ and △ qˆ = χA . Since p and q are coprime, it [follows that deg q ≤ deg q ˆ = n. Assume that deg q < n. ] [ ] Then, Proposition 16.9.3 implies that G ∼

Ac

Bc

Cc

D

, where Ac ∈ R(deg q)×(deg q) . Hence,

A

B

C

D

is not

minimal, which is a contradiction. To prove [iii), i)] =⇒ ii), suppose that p and q are not coprime. Hence, let p, ˆ[ qˆ ∈ R[s] ] satisfy Ac Bc △ G = p/ ˆ q, ˆ where nˆ = deg qˆ < n. Next, Proposition 16.9.3 implies that G ∼ C D , where [c A B ] [ A B ] nˆ ׈n Ac ∈ R . Since C D is an nth-order realization of G and nˆ < n, it follows that C D is not

minimal, which is a contradiction.  Proposition 16.9.20. Let G ∈ R(s)prop be the SISO proper rational transfer function defined in (16.9.4), define (Ac , Bc , Cc ) by (16.9.5) and (16.9.6), and define (Ao , Bo , Co ) by (16.9.7) and (16.9.8). Then, the following statements are equivalent: i) The numerator and denominator of G given in (16.9.4) are coprime. ii) (Ac , Cc ) is observable. iii) (Ac , Bc , Cc ) is controllable and observable. iv) (Ao , Bo ) is controllable. v) (Ao , Bo , Co ) is controllable and observable. Proof. Let p, q ∈ R[s] denote the numerator and denominator [ ] of G in (16.9.4). To prove i) =⇒ Ac

Bc

iii), note that, since p and q are coprime, deg q = n, and C D is a realization of G, Proposition c c [ A B ] c c 16.9.19 implies that C D is a minimal realization of G. Proposition 16.9.18 thus implies that c

c

(Ac , Bc , Cc ) is controllable and observable. Next, iii) =⇒ ii) is immediate. To prove ii) =⇒ i), note that Proposition 16.9.3 implies that (Ac , Bc ) is[ controllable. Since (Ac , Bc , Cc ) is controllable ] and observable, Proposition 16.9.18 implies that

Ac

Bc

Cc

Dc

is a minimal realization of G. Since

Ac ∈ R and deg q = n, Proposition 16.9.19 implies that p and q are coprime. Next, note that (ATo , CoT , BoT ) = (Ac , Bc , Cc ). Since i) ⇐⇒ ii) ⇐⇒ iii), it follows that i) ⇐⇒ [(ATo , BTo ) is observable] ⇐⇒ [(ATo , CoT , BoT ) is controllable and observable]. Equivalently, i) ⇐⇒ iv) ⇐⇒ v).  Note that n×n

K(Ac , Bc ) = K(ATo , CoT ) = O(Ao , Co )T ,

O(Ac , Cc ) = O(ATo , BTo ) = K(Ao , Bo )T .

(16.9.28)

The following results are consequences of Proposition 16.9.9, Proposition 16.9.10, and Proposition 16.9.11. Corollary Let G ∈ R(s)prop be given by (16.9.4), assume that G has the nth-order ] [ 16.9.21. realization

A

B

C

D

, where (A, B) is controllable, and define Ac , Bc , Cc , Dc by (16.9.5) and (16.9.6). △

Furthermore, define S c = K(Ac , Bc )K(A, B)−1 . Then,     Ac Bc   S c AS c−1   =  Cc Dc CS c−1

 S c B   . D

(16.9.29)

Corollary Let G ∈ R(s)prop be given by (16.9.4), assume that G has the nth-order [ 16.9.22. ]

realization

A

B

C

D

, where (A, C) is observable, and define Ao , Bo , Co , Do by (16.9.7) and (16.9.8).

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CHAPTER 16 △

Furthermore, define S o = O(Ao , Co )−1 O(A, C). Then,     Ao Bo   S o AS o−1  =   Co Do CS o−1

 S o B   . D

(16.9.30)

Corollary Let G ∈ R(s)prop be given by (16.9.4), assume that G has the nth-order [ 16.9.23. ]

realization

A

B

C

D

, where (A, B, C) is controllable and observable, and define Ac , Bc , Cc , Dc by △

(16.9.5) and (16.9.6) and Ao , Bo , Co , Do by (16.9.7) and (16.9.8). Furthermore, define S c = △ K(Ac , Bc )K(A, B)−1 and S o = O(Ao , Co )−1 O(A, C). Then, (16.9.29) and (16.9.30) hold. [ A B ] Theorem 16.9.24. Let G ∈ R(s)l×m and A ∈ Rn×n. Then, prop , where G ∼ C D poles(G) ⊆ spec(A),

(16.9.31)

mpoles(G) ⊆ mspec(A).

(16.9.32)

Furthermore, the following statements are equivalent: [ A B ] min i) G ∼ C D . ii) Mcdeg(G) = n. iii) mpoles(G) = mspec(A). Proof. See [2349, p. 319]. min

Definition 16.9.25. Let G ∈ R(s)l×m prop , where G ∼

[

A

B

C

D

]

 . Then, G is (asymptotically stable,

semistable, Lyapunov stable) if A is. Proposition 16.9.26. Let G = p/q ∈ R(s)prop , where p, q ∈ R[s], and assume that p and q are coprime. Then, G is (asymptotically stable, semistable, Lyapunov stable) if and only if q is. Proposition 16.9.27. Let G ∈ R(s)l×m prop . Then, G is (asymptotically stable, semistable, Lyapunov stable) if and only if every entry of G is. [ A B ] min and A is asymptotically stable. Definition 16.9.28. Let G ∈ R(s)l×m prop , where G ∼ C D [ A B ] Then, the realization C D is balanced if the controllability and observability Gramians (16.7.2) and (16.4.2) are diagonal and equal. min

Proposition 16.9.29. Let G ∈ R(s)l×m prop , where G ∼

Then, there exists a nonsingular matrix S ∈ Rn×n

[

A

B

C

D

]

and A is asymptotically stable. [ ] SAS −1 SB such that the realization G ∼ CS −1 D is

balanced. Proof. It follows from Corollary 10.3.4 that there exists a nonsingular matrix S ∈ Rn×n such

that SQS T and S −TPS −1 are diagonal, where Q and P are the positive-definite controllability and ] [ SAS −1 SB observability Gramians (16.7.2) and (16.4.2). Hence, the realization CS −1 D is balanced. 

16.10 Zeros In Section 4.7 the Smith-McMillan decomposition is used to define transmission zeros and blocking zeros of a transfer function G(s). We now define the invariant zeros of a realization of G(s) and relate these zeros to the transmission zeros. These zeros are related to the Smith zeros of a polynomial matrix as well as the spectrum of a pencil. [ A B ] Definition 16.10.1. Let G ∈ R(s)l×m . Then, the Rosenbrock system prop , where G ∼ C D

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LINEAR SYSTEMS AND CONTROL THEORY

matrix Z ∈ R[s](n+l)×(n+m) is the polynomial matrix [ ] sI − A B △ Z(s) = . C −D [ A Furthermore, z ∈ C is an invariant zero of the realization C

(16.10.1) B

] if

D

rank Z(z) < rank Z.

(16.10.2)

Note that Z is square if and only if G is square. [ ] A B Let G ∈ R(s)l×m , where G ∼ and A ∈ Rn×n, and note that Z is the pencil prop C D Z(s) =

P[

] ][ A −B In 0 (s) −C D , 0 0

Thus, Szeros(Z) = spec

([

A −C

([

 −B   . D

  0   A  −  −C 0

  In = s 0

] [ I 0 ]) , 0n 0 , ] [ I 0 ]) −B n D , 0 0 .

−B D

A −C

(16.10.4)

mSzeros(Z) = mspec [ A B ] Hence, we define the set of invariant zeros of C D by ([ A B ]) △ izeros C D = Szeros(Z) [ A B ] and the multiset of invariant zeros of C D by ([ A B ]) △ mizeros C D = mSzeros(Z). ] ][ −B In 0 −C D , 0 0

Note that P[ A

(16.10.3)

(16.10.5)

is regular if and only if rank Z = n + min {l, m}.

The following result shows that a strictly proper transfer function with either full-state observation or full-state actuation has no invariant zeros. ] [ Proposition 16.10.2. Let G ∈ R(s)l×m prop , where G ∼

A

B

C

0

statements hold: i) If m = n and B is nonsingular, then rank Z = n + rank C and ii) If l = n and C is nonsingular, then rank Z = n + rank B and

and A ∈ Rn×n. Then, the following

[ [

A

B

C

0

A

B

C

0

] ]

has no invariant zeros. has no invariant zeros.

iii) If m = n and B is nonsingular, then P[ In iv) If l = n and C is nonsingular, then

] 0 ,[ A −B ] is regular if and only if rank C = min {l, n}. 0 0 −C 0 P[ In 0 ] [ A −B ] is regular if and only if rank B = min {m, n}. , 0 0 −C 0

It is useful to note that, for all s < spec(A),   I Z(s) =  C(sI − A)−1

  0   sI − A B     I 0 −G(s)

   sI − A 0   I   =  C −G(s) 0

 (sI − A)−1B   . I

(16.10.6)

(16.10.7)

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Furthermore,

[

]

A

B

C

D

and A ∈ Rn×n . If s < spec(A), then

rank Z(s) = n + rank G(s).

(16.10.8)

rank Z = n + rank G.

(16.10.9)

Proposition 16.10.3. Let G ∈ R(s)l×m prop , where G ∼

Proof. Let s < spec(A). Then, it follows from Theorem 16.9.24 that s < poles(G). Hence, (16.10.6) implies (16.10.8). Furthermore, Proposition 6.3.7 and Proposition 6.7.8 imply that

rank Z = max rank Z(s) = s∈C

max

s∈C\spec(A)

rank Z(s) = n +

] ][ −B In 0 −C D , 0 0

Proposition 16.10.3 implies that P[ A

max

s∈C\spec(A)

rank G(s) = n + rank G.



is (regular, singular) for one realization of G

if and only if it is (regular, singular) for every realization of G. The following result shows that P[ A −B ] [ In 0 ] is regular if and only if G has full rank. −C D , 0 0 [ A B ] Corollary 16.10.4. Let G ∈ R(s)l×m , where G ∼ . Then, P[ A −B ] [ In 0 ] is regular if prop C D −C D , 0 0

and only if rank G = min {l, m}. Let G be SISO. Then, it follows from (16.10.6) and (16.10.7) that, for all s ∈ C\spec(A), det Z(s) = −[det(sI − A)]G(s).

(16.10.10)

det Z(s) = −C(sI − A)AB − [det(sI − A)]D.

(16.10.11)

Consequently, for all s ∈ C, The equality (16.10.11) also follows from Fact 3.17.2. If s ∈ spec(A), then (16.10.11) implies that det Z(s) = −C(sI − A)AB.

(16.10.12)

If, in addition, n ≥ 2 and rank(sI − A) ≤ n − 2, then Fact 3.19.3 implies that (sI − A) = 0, and thus A

det Z(s) = 0.

(16.10.13)

In the case n = 1, it follows that, for all s ∈ C, (sI − A)A = 1, and thus, for all s ∈ C, det Z(s) = −CB − (sI − A)D.

(16.10.14)

Finally, let s ∈ C\spec(A). Then, it follows from (16.10.10) and (16.10.11) that G(s) =

− det Z(s) C(sI − A)AB + [det(sI − A)]D = . det(sI − A) det(sI − A)

(16.10.15)

Consequently, G(s) , 0 if and only if det Z(s) , 0. These observations are summarized by the following result for scalar transfer functions. [ A B ] Corollary 16.10.5. Let G ∈ R(s)prop , where G ∼ . Then, the following statements are C D equivalent: i) P[ A −B ] [ In

0 −C D , 0 0

ii) iii) iv) v) vi)

]

is regular.

G , 0. rank G = 1. det Z , 0. rank Z = n + 1. C(sI − A)AB + [det(sI − A)]D is not the zero polynomial.

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LINEAR SYSTEMS AND CONTROL THEORY

If these conditions hold, then ([ mizeros [ A min If, in addition, G ∼ C

A C B D

([ A B ]) mizeros C D = mroots(det Z), ]) B = mtzeros(G) ∪ [mspec(A)\mpoles(G)]. D ] , then ([ A B ]) mizeros C D = mtzeros(G).

(16.10.16) (16.10.17)

(16.10.18)

Now, suppose that G is square; that is, l = m. Then, it follows from (16.10.6) and (16.10.7) that, for all s ∈ C\spec(A), det Z(s) = (−1)l [det(sI − A)] det G(s),

(16.10.19)

and thus det G(s) =

(−1)l det Z(s) . det(sI − A)

(16.10.20)

Furthermore, for all s ∈ C, [det(sI − A)]l−1 det Z(s) = (−1)l det[C(sI − A)AB + [det(sI − A)]D].

(16.10.21)

Hence, if l ≥ 2, then, for all s ∈ spec(A), det C(sI − A)AB = 0.

(16.10.22)

We thus have the following result for square transfer functions G that satisfy det G , 0. [ A B ] . Then, the following statements are Corollary 16.10.6. Let G ∈ R(s)l×l prop , where G ∼ C D equivalent: i) P[ A −B ] [ In

0 −C D , 0 0

]

is regular.

ii) det G , 0. iii) rank G = l. iv) det Z , 0. v) rank Z = n + l. vi) det[C(sI − A)AB + [det(sI − A)]D] is not the zero polynomial. If these conditions hold, then ([ A B ]) mizeros C D = mroots(det Z), ([ A B ]) [ ] mizeros C D = mtzeros(G) ∪ mspec(A)\mpoles(G) , ([ A B ]) [ ] izeros C D = tzeros(G) ∪ spec(A)\poles(G) . [ A B ] min If, in addition, G ∼ C D , then ([ A B ]) mizeros C D = mtzeros(G).

(16.10.23) (16.10.24) (16.10.25)

(16.10.26)

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Example 16.10.7. Consider G ∈ R2×2 (s) defined by

 0   . s+1 

 s−1  G(s) =  s+1 0 △

(16.10.27)

s−1

Then, the Smith-McMillan form of G is given by  1  2 △ G(s) = S 1 (s)  s −1 0

0 s2 − 1

   S 2 (s),

(16.10.28)

where S 1 , S 2 ∈ R[s]2×2 are the unimodular matrices   1    4 (s − 1)2 (s + 2) (s + 1)2  (s − 1)2 −1  △  △   .     , S 2 (s) =  S 1 (s) =   1 − 14 (s + 1)2 (s − 2) 41 (s + 2) (s − 2) 1 4

(16.10.29)

A minimal realization of G is given by   −1  0  min  G ∼   − 2  0

0 1

1 0

0 2

1 0

    . 0   1 0 1

(16.10.30)

Thus, mtzeros(G) = mpoles(G) = mspec(A) = {1, −1}. Finally, det Z(s) = (−1)2 [det(sI − A)] det G = s2 − 1, which confirms (16.10.26). ^ [ A B ] l×m Theorem 16.10.8. Let G ∈ R(s)prop , where G ∼ . Then, C D ([ A B ]) (16.10.31) izeros C D \spec(A) ⊆ tzeros(G), ([ A B ]) tzeros(G)\poles(G) ⊆ izeros C D . (16.10.32) [ A B ] min If, in addition, G ∼ C D , then ([ A B ]) (16.10.33) izeros C D \poles(G) = tzeros(G)\poles(G). Proof. To prove (16.10.31), let z ∈ izeros

([

A

B

C

D

]) \spec(A). Since z < spec(A), Theorem

16.9.24 implies that z < poles(G). It now follows from Proposition 16.10.3 that n + rank G(z) = rank Z(z) < rank Z = n + rank G, which implies that rank G(z) < rank G. Thus, z ∈ tzeros(G). To prove (16.10.32), let z ∈ tzeros(G)\poles(G). Then, Proposition([16.10.3 ]) implies that rank Z(z) = n + rank G(z) < n + rank G = rank Z, which implies that z ∈ izeros

A

B

C

D

. The last statement

follows from (16.10.31), (16.10.32), and Theorem 16.9.24.  The following result is a stronger form of Theorem 16.10.8. [ A B ] , let S ∈ Rn×n, assume that S is Theorem 16.10.9. Let G ∈ R(s)l×m , where G ∼ prop C D ([ ] [ ]) nonsingular, and let A, B, and C have the form (16.9.15), where AA211 A02 , BB12 is controllable and ([ A A ] ) 1 13 C C 0 A3 , [ 1 3 ] is observable. Then, ([ A B ]) 1 1 mtzeros(G) = mizeros C D , (16.10.34) 1 ([ A B ]) mizeros C D = mspec(A2 ) ∪ mspec(A3 ) ∪ mspec(A4 ) ∪ mtzeros(G). (16.10.35)

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LINEAR SYSTEMS AND CONTROL THEORY

Proof. Defining Z by (16.10.1) and using the notation of Proposition 16.9.12, Z has the same Smith form as   −A43 0 0 0   sI − A4    0 sI − A3 0 0 0    △  −A23 sI − A2 −A21 B2  . Z˜ =  −A24   −A13 0 sI − A1 B1   0   0 C3 0 C1 −D △

Hence, Proposition 16.10.3 implies that rank Z = rank Z˜ = n + r, where r = rank G. Let p˜ 1 , . . . , p˜ n+r ˜ Then, since p˜ n+r is the monic greatest common divisor of all (n + be the Smith polynomials of Z. ˜ it follows that p˜ n+r = χA χA χA pr , where pr is the rth Smith r) × (n + r) subdeterminants of Z, 1 2 3 [ sI−A B ] 1 1 polynomial of C1 −D . Therefore, ]) ([ 1 B1 mSzeros(Z) = mspec(A2 ) ∪ mspec(A3 ) ∪ mspec(A4 ) ∪ mSzeros sI−A C1 −D . Next, using the Smith-McMillan decomposition Theorem 6.7.5, there exist unimodular matrices S 1 ∈ R[s]l×l and S 2 ∈ R[s]m×m such that G = S 1 D−1 0 N0 S 2 , where     0 0    p1  q1      . . . . △  △  .  , N = . .  D0 =   0     p q    r r    0 0(l−r)×(m−r) 0 Il−r Now, define the polynomial matrix Zˆ ∈ R[s](n+l)×(n+m) by   0(n−l)×l 0(n−l)×m   In−l   △  D0 N0 S 2  . Zˆ =  0l×(n−l)   0l×(n−l) S1 0l×m Since S 1 is unimodular, it follows that the Smith form S of Zˆ is given by    In 0n×m  S =   . 0l×n N0 ˆ = mSzeros(S) = mtzeros(G). Consequently, mSzeros(Z) Next, note that

  0(n−l)×l    In−l   0(n−l)×l 0(n−l)×m  D0  = n,  = rank  0l×(n−l)   D0 N0 S 2 0l×(n−l) S1  −1 [ ] [ ]  In−l 0(n−l)×l  0(n−l)×m    G = 0l×(n−l) S 1 0l×m  .  N0 S 2 0l×(n−l) D0 [ A B ] [ ] 1 1 min 1 B1 Furthermore, G ∼ C D , Consequently, Zˆ and sI−A have no decoupling zeros [2339, pp. C D 1 1 [ ] 1 B1 64–70], and it thus follows from Theorem 3.1 of [2339, p. 106] that Zˆ and sI−A C1 D have the same Smith form. Thus, ]) ([ sI − A1 B1 ˆ = mtzeros(G). mSzeros = mSzeros(Z) C1 −D   In−l rank  0l×(n−l)

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CHAPTER 16

([

Consequently, mizeros

A1 C1

B1 D

])

= mSzeros

([ sI−A

1

C1

B1 −D

])

= mtzeros(G),

which proves (16.10.34). Finally, to prove (16.10.31) note that ([ A B ]) mizeros C D = mSzeros(Z) = mspec(A2 ) ∪ mspec(A3 ) ∪ mspec(A4 ) ∪ mSzeros

([ sI−A

1 B1 −C1 −D

])

= mspec(A2 ) ∪ mspec(A3 ) ∪ mspec(A4 ) ∪ mtzeros(G).



Proposition 16.10.10. Equivalent realizations have the same invariant zeros. Furthermore, invariant zeros are not changed by output feedback. [ A B ] and u = Ky + v, where det(I − KD) , 0, the closed-loop transfer Proof. Letting G ∼ C D

function from v to y has the realization   A + BK0C ˜ G ∼  C + DK0C

 B(I − KD)−1   , D(I − KD)−1

△

where K0 = (I − KD)−1K. Since [ ] [ ][ ] zI − (A + BK0C) B(I − KD)−1 zI − A B I 0 = , C + DK0C −D(I − KD)−1 C −D −K0C (I − KD)−1 ] [ A B ] [ B(I − KD)−1 A + BK0 C it follows that C D and C + DK C D(I − KD)−1 have the same invariant zeros. 0



The special case C = I is full-state feedback. Fact 16.24.15 also concerns invariant zeros under output feedback. The following result provides an interpretation of i) of Theorem 16.17.9. [ A B ] △ Proposition 16.10.11. Let G ∈ R(s)l×m , where G ∼ , and assume that R = DTD is prop C D positive definite. Then, the following statements hold: i) rank Z = n + m. [ A B ] ii) z ∈ C is an invariant zero of C D if and only if z is an unobservable eigenvalue of (A − BR−1DTC, [I − DR−1DT ]C). Proof. To prove i), suppose that rank Z < n + m. Then, for all s ∈ C, there exists a nonzero [ ] vector yx ∈ N[Z(s)]; that is, [ ][ ] sI − A B x = 0. C −D y Consequently, Cx − Dy = 0, which implies that DTCx − Ry = 0, and thus y = R−1DTCx. Furthermore, since (sI − A + BR−1DTC)x = 0, choosing s < spec(A − BR−1DTC) implies x = 0, and thus y = 0, which is a contradiction. [ ] A

B

To prove ii), note that it follows from i) that z is an invariant zero of C D if and only if [ ] rank Z(z) < n + m, which holds if and only if there exists a nonzero vector yx ∈ N[Z(z)]. Therefore,    zI − A + BR−1DTC    x = 0, (I − DR−1DT )C where x , 0, which holds if and only if z is an unobservable eigenvalue of (A − BR−1DTC, [I − DR−1DT ]C). 

1285

LINEAR SYSTEMS AND CONTROL THEORY △

Corollary 16.10.12. Assume that[ R = DT]D is positive definite, and assume that (A − BR−1DTC, B D

A C

[I − DR−1DT ]C) is observable. Then,

has no invariant zeros.

16.11 H2 System Norm Consider the system x˙(t) = Ax(t) + Bu(t), y(t) = Cx(t),

(16.11.1) (16.11.2)

where A ∈ Rn×n is asymptotically stable, B ∈ Rn×m, and C ∈ Rl×n. Then, for all t ≥ 0, the impulse response function defined by (16.1.18) is given by H(t) = CetAB. The L2 norm of H is defined by ∥H∥L2

 ∞ 1/2 ∫    2 =  ∥H(t)∥F dt .   △

(16.11.3)

(16.11.4)

0

The following result provides expressions for ∥H∥L2 in terms of the controllability and observability Gramians. Theorem 16.11.1. Assume that A is asymptotically stable. Then, the L2 norm of H is given by where Q, P ∈ R

n×n

∥H∥2L2 = tr CQCT = tr BTPB,

(16.11.5)

AQ + QAT + BBT = 0,

(16.11.6)

A P + PA + C C = 0.

(16.11.7)

satisfy T

Proof. Note that

T

∫∞ ∥H∥2L2

=

T

tr CetABBTetA CT dt = tr CQCT, 0

where Q satisfies (16.11.6). The dual expression (16.11.7) follows either in a similar manner or by noting that tr CQCT = tr CTCQ = − tr (ATP + PA)Q = − tr (AQ + QAT )P = tr BBTP = tr BTPB.



For the following definition, note that, for all s ∈ C\poles(G), ∥G(s)∥F = [tr G(s)G∗(s)]1/2. Definition 16.11.2. Let G ∈ R norm of G ∈ Rl×m(s) is given by

l×m

(16.11.8)

(s), and assume that G is asymptotically stable. Then, the H2

∥G∥H2

 1/2  ∫∞  1  △  =  ∥G(ω ȷ)∥2F dω .  2π 

(16.11.9)

−∞

The following result is Parseval’s theorem, which relates the L2 norm of the impulse response function to the H2 norm of its Fourier transform.

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CHAPTER 16

Theorem 16.11.3. Let G ∈ R(s)l×m prop , where G ∼

that A ∈ R

n×n

[

A

B

C

0

] , define H by (16.11.3), and assume

is asymptotically stable. Then, ∫∞

1 H(t)H (t) dt = 2π T

∫∞ G(ω ȷ)G∗(ω ȷ) dω.

(16.11.10)

−∞

0

Therefore, ∥H∥L2 = ∥G∥H2 . Proof. First note that

∫∞ G(s) = L{H(t)} =

(16.11.11)

H(t)e−st dt

0

and that 1 H(t) = 2π

∫∞ G(ω ȷ)eω ȷt dω.

−∞

Hence, ∫∞ 0

  ∫∞  ∫∞   1   T −st ω ȷt  H(t)HT (t)e−st dt = G(ω ȷ)e dω  H (t)e dt  2π −∞ 0   ∞ ∫∞  ∫ 1  T −(s−ω ȷ)t  dt dω = G(ω ȷ)  H (t)e   2π −∞ ∫∞

=

1 2π

0

G(ω ȷ)GT(s − ω ȷ) dω.

−∞

Setting s = 0 yields (16.11.7), while taking the trace of (16.11.10) yields (16.11.11).  [ A B ] l×m n×n Corollary 16.11.4. Let G ∈ R(s)prop , where G ∼ , and assume that A ∈ R is asympC 0 totically stable. Then, ∥G∥2H2 = ∥H∥2L2 = tr CQCT = tr BTPB,

(16.11.12)

where Q, P ∈ Rn×n satisfy (16.11.6) and (16.11.7), respectively. The following corollary of Theorem 16.11.3 provides a frequency-domain expression for the solution of the Lyapunov equation. Corollary 16.11.5. Let A ∈ Rn×n, assume that A is asymptotically stable, let B ∈ Rn×m, and define Q ∈ Rn×n by ∫∞ 1 (ω ȷI − A)−1BBT(ω ȷI − A)−∗ dω. Q= 2π

(16.11.13)

AQ + QAT + BBT = 0.

(16.11.14)

−∞

Then, Q satisfies

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LINEAR SYSTEMS AND CONTROL THEORY

Proof. The result follows from (16.11.10) with H(t) = etAB and G(s) = (sI−A)−1B. Alternatively,

to show that the solution Q of (16.11.14) is given by (16.11.13), note that ∫∞ ∫∞ ∫∞ −1 −∗ (ω ȷI − A) dωQ + Q (ω ȷI − A) dω = (ω ȷI − A)−1BBT(ω ȷI − A)−∗ dω. −∞

−∞

−∞

Assuming that A is diagonalizable with eigenvalues λi = −σi + ωi ȷ, it follows that ∫∞ −∞

dω = ω ȷ − λi

∫∞ −∞

which implies that

σi π σi − ω ȷ dω = − ȷ lim 2 2 r→∞ |σi | σi + ω

∫r −r

σ2i

ω dω = π, + ω2

∫∞ (ω ȷI − A)−1 dω = πIn , −∞

which yields (16.11.13). See [685] for the case where A is not diagonalizable.  Proposition 16.11.6. Let G1, G2 ∈ Rl×m (s) be asymptotically stable rational transfer functions. prop Then,

min

Proof. Let G1 ∼

[

A1 C1

B1 0

∥G1 + G2 ∥H2 ≤ ∥G1 ∥H2 + ∥G2 ∥H2 . (16.11.15) ] [ A B ] 2 2 min and G2 ∼ C , where A1 ∈ Rn1 ×n1 and A2 ∈ Rn2 ×n2 . It 0 2

follows from Proposition 16.13.2 that

   A1 0 B1    G1 + G2 ∼  0 A2 B2  .   C1 C2 0 √ √ Now, Corollary 16.11.4 implies that ∥G1∥H2 = tr C1Q1C1T and ∥G2 ∥H2 = tr C2 Q2C2T, where Q1 ∈

Rn1 ×n1 and Q2 ∈ Rn2 ×n2 are the unique positive-definite matrices satisfying A1Q1 + Q1 A1T + B1BT1 = 0 and A2 Q2 + Q2 AT2 + B2 BT2 = 0. Furthermore,  T  C1  2 ∥G1 + G2 ∥H2 = tr [C1 C2 ]Q  T  , C2 where Q ∈ R(n1 +n2 )×(n1 +n2 ) is the unique, positive-semidefinite matrix satisfying [ ] [ ]T [ ] [ ]T A1 0 A 0 B B1 Q+Q 1 + 1 = 0. 0 A2 0 A2 B2 B2 [Q Q ] It can be seen that Q = QT121 Q122 , where Q1 and Q2 are as given above and where Q12 satisfies A1 Q12 + Q12 AT2 + B1BT2 = 0. Now, using the Cauchy-Schwarz inequality (11.3.17) and iii) of Proposition 10.2.5, it follows that ∥G1 + G2 ∥2H2 = tr(C1 Q1C1T + C2 Q2C2T + C2 QT12C1T + C1 Q12C2T ) T = ∥G1∥2H2 + ∥G2 ∥2H2 + 2 tr C1Q12 Q−1/2 Q1/2 2 2 C2 T T T 1/2 ≤ ∥G1∥2H2 + ∥G2 ∥2H2 + 2[tr(C1Q12 Q−1 2 Q12 C 1 ) tr(C 2 Q2 C 2 )]

≤ ∥G1∥2H2 + ∥G2 ∥2H2 + 2[tr(C1Q1C1T ) tr(C2 Q2C2T )]1/2

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CHAPTER 16

= (∥G1∥H2 + ∥G2 ∥H2 )2.



16.12 Harmonic Steady-State Response The following result concerns the response of a linear system to a harmonic input. Theorem 16.12.1. For t ≥ 0, consider the linear system x˙(t) = Ax(t) + Bu(t),

(16.12.1)

u(t) = Re u0 eω0 ȷt ,

(16.12.2)

with harmonic input where u0 ∈ Cm and ω0 ∈ R is such that ω0 ȷ < spec(A). Then, x(t) is given by x(t) = etA (x(0) − Re[(ω0 ȷI − A)−1Bu0 ]) + Re[(ω0 ȷI − A)−1Bu0 eω0 ȷt ].

(16.12.3)

Proof. We have

∫t x(t) = e x(0) + tA

(t−τ)A

e

BRe(u0 e

ω0 ȷτ

∫t )dτ = e x(0) + e Re tA

tA

0

eτ(ω0 ȷI−A) dτBu0

0

= etA x(0) + etA Re[(ω0 ȷI − A)−1 (et(ω0 ȷI−A) − I)Bu0 ] = etA x(0) + Re[(ω0 ȷI − A)−1 (eω0 ȷtI − etA )Bu0 ] = etA x(0) + Re[(ω0 ȷI − A)−1 (−etA )Bu0 ] + Re[(ω0 ȷI − A)−1eω0 ȷtBu0 ] = etA (x(0) − Re[(ω0 ȷI − A)−1Bu0 ]) + Re[(ω0 ȷI − A)−1Bu0 eω0 ȷt ].

[ A Theorem 16.12.1 shows that the total response y(t) of the linear system G ∼ C harmonic input can be written as y(t) = ytrans (t) + yhss (t), where the transient component △

ytrans (t) = CetA (x(0) − Re[(ω0 ȷI − A)−1Bu0 ])

B 0

]

 to a

(16.12.4)

depends on the initial condition and the input, and the harmonic steady-state component yhss (t) = Re[G(ω0 ȷ)u0 eω0 ȷt ]

(16.12.5)

depends only on the input. If A is asymptotically stable, then limt→∞ ytrans (t) = 0, and thus y(t) approaches its harmonic steady-state component yhss (t) for large t. Since the harmonic steady-state component is sinusoidal, it follows that y(t) does not converge in the usual sense. Finally, if A is semistable, then it follows from Proposition 15.9.2 that lim ytrans (t) = C(I − AA# )(x(0) − Re[(ω0 ȷI − A)−1Bu0 ]),

t→∞

(16.12.6)

which represents a constant offset to the harmonic steady-state component. △ In the SISO case, let u0 = a0 (sin ϕ0 − cos ϕ0 ȷ), where a0 , ϕ0 ∈ R, and consider the input u(t) = a0 sin(ω0 t + ϕ0 ) = Re u0 eω0 ȷt.

(16.12.7)

θȷ

Then, writing G(ω0 ȷ) = Me , where M ∈ R, it follows that yhss (t) = a0 Msin(ω0 t + ϕ0 + θ).

(16.12.8)

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LINEAR SYSTEMS AND CONTROL THEORY

16.13 System Interconnections ∼ Let G ∈ R(s)l×m prop . We define the parahermitian conjugate G of G by △

G∼ (s) = GT(−s).

(16.13.1)

The following result provides realizations for GT, G∼, and G−1. Proposition 16.13.1. Let G ∈ R(s)l×m prop , and assume that G ∼

 T  A G ∼  T B T

  −AT G ∼  T B

 CT   , DT

∼

Furthermore, if G is square and D is nonsingular, then   A − BD−1C −1 G ∼  −D−1C

[

]

A

B

C

D

 −CT   . DT

. Then, (16.13.2)

 BD−1   . D−1

(16.13.3)

Proof. Since y = Gu, it follows that G−1 satisfies u = G−1 y. Since x˙ = Ax + Bu and y = Cx + Du,

it follows that u = −D−1Cx+D−1y, and thus x˙ = Ax+B(−D−1Cx+D−1 y) = (A−BD−1C)x+BD−1 y.  [ ] [ AT CT ] A B Note that, if G ∈ R(s)prop and G ∼ C D , then G ∼ T . B

1 ×m1 R(s)lprop

D

2 ×m2 R(s)lprop .

Let G1 ∈ and G2 ∈ If m2 = l2 , then the cascade interconnection of G1 and G2 shown in Figure 16.13.1 is the product G2G1, while, if m1 = m2 and l1 = l2 , then the parallel interconnection shown in Figure 16.13.2 is the sum G1 + G2. u1 -

y1 = u2

G1

-

y2

G2

-

Figure 16.13.1

Cascade Interconnection of Linear Systems

-

G1 + ? f y2 + 6

u1 -

-

G2

Figure 16.13.2

Parallel Interconnection of Linear Systems i ×mi Proposition 16.13.2. For i ∈ {1, 2}, let Gi ∈ R(s)lprop , where Gi ∼

  A1  G2G1 ∼  B2C1  D2C1

0 A2

B1 B2 D1

C2

D2 D1

    . 

[

Ai Ci

Bi Di

]

. If l1 = m2 , then

(16.13.4)

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CHAPTER 16

If l1 = l2 and m1 = m2 , then

  A1  G1 + G2 ∼  0  C1

0 A2

B1 B2

C2

D1 + D2

    . 

(16.13.5)

Proof. Consider the state space equations

x˙1 = A1 x1 + B1 u1, y1 = C1 x1 + D1 u1 ,

x˙2 = A2 x2 + B2 u2 , y2 = C2 x2 + D2 u2 .

Since u2 = y1 , it follows that x˙2 = A2 x2 + B2C1 x1 + B2 D1 u1 , y2 = C2 x2 + D2C1 x1 + D2 D1 u1 , and thus

[

] [ x˙1 A1 = x˙2 B2C1

] [ ] x1 B1 + u, x2 B2 D1 1 [ ] x y2 = [D2C1 C2 ] 1 + D2 D1 u1 , x2 0 A2

][

which yields the realization of G2G1. The realization of G1 + G2 can be obtained similarly.  It is often useful to combine transfer functions by concatenating them into either row, column, or block-diagonal transfer functions. [ ] [ A B ] A1 B1 2 2 Proposition 16.13.3. Let G1 ∼ C D and G2 ∼ . Then, C D 1

  A1  [G1 G2 ] ∼  0  C1

1

0 A2

B1 0

0 B2

C2

D1

D2

[

G1 0

0 G2

]

2

    , 

  A1  0  ∼   C1  0

2

  A [ ]  1  0 G1 ∼  G2  C1  0 0 A2

B1 0

0 C2

D1 0

    , D1   D2 B1 B2

0 A2

0 C2  0   B2   . 0   D2

(16.13.6)

(16.13.7)

Next, we interconnect a pair of systems G1, G2 by means of feedback as shown in Figure 16.13.3. It can be seen that u and y are related by yˆ = (I + G1G2 )−1G1 uˆ ,

(16.13.8)

−1

yˆ = G1(I + G2G1 ) uˆ .

(16.13.9)

The equivalence of (16.13.8) and (16.13.9) follows from the push-through identity given by Fact 3.20.6, (I + G1G2 )−1G1 = G1(I + G2G1 )−1.

(16.13.10)

A realization of this rational transfer function is given by the following result. [ A B ] [ A B ] 1 1 2 2 Proposition 16.13.4. Let G1 ∼ and G ∼ , and assume that det(I + 2 C D C D 1

1

2

2

1291

LINEAR SYSTEMS AND CONTROL THEORY

u -f − 6

-

y

G1 

G2

Figure 16.13.3

Feedback Interconnection of Linear Systems

D1D2 ) , 0. Then,

  A1 − B1 (I + D2 D1 )−1D2C1  B2 (I + D1D2 )−1C1 (I + G1G2 )−1G1 ∼   (I + D1D2 )−1C1

−B1 (I + D2 D1 )−1C2 A2 − B2 (I + D1D2 )−1D1C2 −(I + D1D2 )−1D1C2

B1(I + D2 D1 )−1 B2 (I + D1D2 )−1D1

    .  

(I + D1D2 )−1D1 (16.13.11)

16.14 Standard Control Problem The standard control problem shown in Figure 16.14.1 involves four distinct signals, namely, an exogenous input w, a control input u, a performance variable z, and a feedback signal y. This system can be written as [ ] [ ] zˆ(s) w(s) ˆ = G(s) , (16.14.1) yˆ (s) uˆ (s) where G(s) is partitioned as △

G= with the realization

[

G11 G21

  A  G ∼  E1  C

D1 E0 D2

G12 G22

] (16.14.2)

 B   E2  ,  D

(16.14.3)

which represents the equations

Consequently,

x˙ = Ax + D1 w + Bu,

(16.14.4)

z = E1 x + E0 w + E2 u, y = Cx + D2 w + Du.

(16.14.5) (16.14.6)

  E1 (sI − A)−1D1 + E0 G(s) =  C(sI − A)−1D1 + D2

 E1 (sI − A)−1B + E2   , C(sI − A)−1B + D

which shows that G11, G12 , G21, and G22 have the realizations       A B   A D1  A D1      G11 ∼   , G12 ∼   , G21 ∼  E1 E0 E1 E2 C D2

   ,

G22

  A ∼  C

(16.14.7)  B   . D

(16.14.8)

1292

CHAPTER 16

w u

-

G11

G12

G21

G22

Gc

z

-

y



Figure 16.14.1

Standard Control Problem

Letting Gc denote a feedback controller with realization    Ac Bc  Gc ∼   , Cc Dc

(16.14.9)

we interconnect G and Gc according to uˆ (s) = Gc (s)ˆy(s).

(16.14.10)

˜ w(s) The resulting rational transfer function G˜ satisfying zˆ(s) = G(s) ˆ is thus given by G˜ = G11 + G12Gc (I − G22Gc )−1G21 ,

(16.14.11)

= G11 + G12 (I − GcG22 )−1GcG21.

(16.14.12)

A realization of G˜ is given by the following result. △ Proposition 16.14.1. Let G and Gc have the realizations (16.14.3) and (16.14.9), define Ω = I − DDc , and assume that Ω is nonsingular. Then,   B(I + Dc Ω−1D)Cc D1 + BDc Ω−1D2   A + BDc Ω−1C    . Bc Ω−1C Ac + Bc Ω−1DCc Bc Ω−1D2 (16.14.13) G˜ ∼     E1 + E2Dc Ω−1C E2 (I + Dc Ω−1D)Cc E0 + E2Dc Ω−1D2 The realization (16.14.13) can be simplified if DDc = 0. For example, if D = 0, then   BCc D1 + BDc D2   A + BDcC   BcC Ac Bc D2  , G˜ ∼  (16.14.14)   E1 + E2 DcC E2Cc E0 + E2 Dc D2 whereas, if Dc = 0, then

  A ˜G ∼  BcC  E1

BCc Ac + Bc DCc

D1 Bc D2

E2Cc

E0

    . 

(16.14.15)

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LINEAR SYSTEMS AND CONTROL THEORY

Finally, if both D = 0 and Dc = 0, then

  A  G˜ ∼  BcC  E1

BCc Ac

D1 Bc D2

E2Cc

E0

    . 

(16.14.16)

The feedback interconnection shown in Figure 16.14.1 forms the basis for the standard control problem in feedback control. For this problem the signal w is an exogenous signal representing either a command or a disturbance, while the signal z is the performance variable, which is the variable whose behavior reflects the performance of the closed-loop system. The performance variable may or may not be physically measured. The controlled input (also called the control) u is the output of the feedback controller Gc , while the measurement signal y serves as the input to the feedback controller Gc . The standard control problem is the following: Given knowledge of w, determine Gc that minimizes a performance criterion J(Gc ).

16.15 Linear-Quadratic Control Let A ∈ Rn×n and B ∈ Rn×m, and consider the system x˙(t) = Ax(t) + Bu(t), x(0) = x0 ,

(16.15.1) (16.15.2)

where t ≥ 0. Furthermore, let K ∈ Rm×n, and consider the full-state-feedback control law u(t) = Kx(t).

(16.15.3)

The objective of the linear-quadratic control problem is to minimize the quadratic performance measure ∫∞ J(K, x0 ) = [xT(t)R1 x(t) + 2xT(t)R12 u(t) + uT(t)R2 u(t)] dt, (16.15.4) 0

with respect[to the]feedback gain K ∈ Rm×n , where R1 ∈ Rn×n, R12 ∈ Rn×m, and R2 ∈ Rm×m. We R R assume that RT1 R122 is positive semidefinite and R2 is positive definite. 12 The performance measure (16.15.4) indicates the desire to maintain the state vector x(t) close to the zero equilibrium without an excessive expenditure of control effort. Specifically, the term xT(t)R1x(t) is a measure of the deviation of the state x(t) from the zero state, where the n × n positivesemidefinite matrix R1 determines how much weighting is associated with each component of the state. Likewise, the m × m positive-definite matrix R2 weights the magnitude of the control input. Finally, the cross-weighting term R12 arises due to the use of filters to shape the system response. Using (16.15.1) and (16.15.3), the closed-loop dynamic system can be written as x˙(t) = (A + BK)x(t)

(16.15.5)

so that ˜

x(t) = et A x0 ,

(16.15.6)

△

where A˜ = A + BK. Thus, the quadratic performance measure (16.15.4) becomes ∫∞ J(K, x0 ) =

∫∞ ˜ xT(t)Rx(t) dt =

0

0

∫∞ ∫∞ ˜T t A˜ ˜ x0 dt = tr xT0 et A˜TRe ˜ t A˜ dtx0 = tr et A˜TRe ˜ t A˜ dtx0 x0T, xT0 et A Re 0

0

(16.15.7)

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where

△ R˜ = R1 + R12 K + K TRT12 + K TR2 K.

Now, consider the standard control problem with plant   A D1 B  G ∼  E1 0 E2  0 0 In

    

and full-state feedback u = Kx. Then, the closed-loop transfer function is given by    A + BK D1  ˜G ∼   .  E1 + E2 K 0

(16.15.8)

(16.15.9)

(16.15.10)

The following result shows that the quadratic performance measure (16.15.4) is equal to the H2 norm of a transfer function. Proposition 16.15.1. Assume that D1 = x0 and      R1 R12   E1T   =   [E1 E2 ],  (16.15.11) E2T RT12 R2 and let G˜ be given by (16.15.10). Then, ˜ 2. J(K, x0 ) = ∥G∥ H2

(16.15.12)

 For the following development, we assume that (16.15.11) holds so that R1, R12 , and R2 are given by Proof. This result follows from Proposition 16.1.2.

R1 = E1TE1,

R12 = E1TE2 ,

R2 = E2TE2 .

(16.15.13)

To develop necessary conditions for the linear-quadratic control problem, we restrict K to the set of stabilizing gains △

S = {K ∈ Rm×n : A + BK is asymptotically stable}.

(16.15.14)

Obviously, S is nonempty if and only if (A, B) is stabilizable. The following result gives necessary conditions that characterize a stabilizing solution K of the linear-quadratic control problem. Theorem 16.15.2. Assume that (A, B) is stabilizable, assume that K ∈ S solves the linearquadratic control problem, and assume that (A + BK, D1 ) is controllable. Then, there exists an n × n positive-semidefinite matrix P such that K is given by T T K = −R−1 2 (B P + R12 )

(16.15.15)

T AˆTRP + PAˆ R + Rˆ 1 − PBR−1 2 B P = 0,

(16.15.16)

and such that P satisfies

where △ T Aˆ R = A − BR−1 2 R12 ,

△ T Rˆ 1 = R1 − R12 R−1 2 R12.

(16.15.17)

Furthermore, the minimal cost is given by J(K) = tr PV, △

where V = D1DT1 .

(16.15.18)

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Proof. Since K ∈ S, it∫follows that A˜ is asymptotically stable. It then follows that J(K) is given ∞ ˜T t A˜ △ ˜ dt is positive semidefinite and satisfies the Lyapunov equation by (16.15.18), where P = 0 et A Re

A˜TP + PA˜ + R˜ = 0.

(16.15.19)

Note that (16.15.19) can be written as (A + BK)TP + P(A + BK) + R1 + R12 K + K TRT12 + K TR2 K = 0.

(16.15.20)

To optimize (16.15.18) subject to the constraint (16.15.19) over the open set S, form the Lagrangian △ ˜ L(K, P, Q, λ0 ) = tr [λ0 PV + Q(A˜TP + PA˜ + R)],

(16.15.21)

where the Lagrange multipliers λ0 ≥ 0 and Q ∈ R are not both zero. Note that the n × n Lagrange multiplier Q accounts for the n × n constraint equation (16.15.19). The necessary condition ∂L/∂P = 0 implies n×n

˜ + QA˜T + λ0V = 0. AQ

(16.15.22)

Since A˜ is asymptotically stable, it follows from Proposition 15.10.3 that, for all λ0 ≥ 0, (16.15.22) has a unique solution Q and, furthermore, Q is positive semidefinite. In particular, if λ0 = 0, then Q = 0. Since λ0 and Q are not both zero, we can set λ0 = 1 so that (16.15.22) becomes ˜ + QA˜T + V = 0. AQ

(16.15.23)

˜ D1 ) is controllable, it follows from Corollary 16.7.10 that Q is positive definite. Since (A, Next, evaluating ∂L/∂K = 0 yields R2 KQ + (BTP + RT12 )Q = 0.

(16.15.24)

Since Q is positive definite, it follows from (16.15.24) that (16.15.15) holds. Furthermore, using (16.15.15), it follows that (16.15.19) is equivalent to (16.15.16).  With K given by (16.15.15), the closed-loop dynamics matrix A˜ = A + BK is given by T T A˜ = A − BR−1 2 (B P + R12 ),

(16.15.25)

where P is the solution of the Riccati equation (16.15.16).

16.16 Solutions of the Riccati Equation For convenience in the following development, we assume that R12 = 0. With this assumption, the gain K given by (16.15.15) becomes T K = −R−1 2 B P.

Defining (16.15.25) becomes

△

(16.16.1)

T Σ = BR−1 2 B ,

(16.16.2)

A˜ = A − ΣP,

(16.16.3)

while the Riccati equation (16.15.16) can be written as ATP + PA + R1 − PΣP = 0.

(16.16.4)

Note that (16.16.4) has the alternative representation (A − ΣP)TP + P(A − ΣP) + R1 + PΣP = 0,

(16.16.5)

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which is equivalent to the Lyapunov equation A˜TP + PA˜ + R˜ = 0,

(16.16.6)

△ R˜ = R1 + PΣP.

(16.16.7)

where

By comparing (16.15.16) and (16.16.4), it can be seen that the linear-quadratic control problems with (A, B, R1, R12 , R2 ) and (Aˆ R , B, Rˆ 1, 0, R2 ) are equivalent. Hence, there is no loss of generality in assuming that R12 = 0 in the following development, where A and R1 take the place of Aˆ R and Rˆ 1, respectively. To motivate the subsequent development, the following examples demonstrate the existence of solutions under various assumptions on (A, B, E1 ). In the following four examples, (A, B) is not stabilizable. Example 16.16.1. Let n = 1, A = 1, B = 0, E1 = 0, and R2 > 0. Hence, (A, B, E1 ) has an ORHP eigenvalue that is uncontrollable and unobservable. In this case, (16.16.4) has the unique solution P = 0. Furthermore, since B = 0, it follows that A˜ = A. ^ Example 16.16.2. Let n = 1, A = 1, B = 0, E1 = 1, and R2 > 0. Hence, (A, B, E1 ) has an ORHP eigenvalue that is uncontrollable and observable. In this case, (16.16.4) has the unique solution P = −1/2 < 0. Furthermore, since B = 0, it follows that A˜ = A. ^ Example 16.16.3. Let n = 1, A = 0, B = 0, E1 = 0, and R2 > 0. Hence, (A, B, E1 ) has an imaginary eigenvalue that is uncontrollable and unobservable. In this case, (16.16.4) has infinitely many solutions P ∈ R. Hence, (16.16.4) has no maximal solution. Furthermore, since B = 0, it follows that, for every solution P, A˜ = A. ^ Example 16.16.4. Let n = 1, A = 0, B = 0, E1 = 1, and R2 > 0. Hence, (A, B, E1 ) has an imaginary eigenvalue that is uncontrollable and observable. In this case, (16.16.4) becomes R1 = 0. Thus, (16.16.4) has no solution. ^ In the remaining examples, (A, B) is controllable. Example 16.16.5. Let n = 1, A = 1, B = 1, E1 = 0, and R2 > 0. Hence, (A, B, E1 ) has an ORHP eigenvalue that is controllable and unobservable. In this case, (16.16.4) has the solutions P = 0 and P = 2R2 > 0. The corresponding closed-loop dynamics matrices are A˜ = 1 > 0 and A˜ = −1 < 0. Hence, the solution P = 2R2 is stabilizing, and the closed-loop eigenvalue −1, which does not depend on R2 , is the reflection of the open-loop eigenvalue 1 across the imaginary axis. ^ Example 16.16.6. Let n = 1, A = 1, B = 1, E1 = 1, and R2 > 0. Hence, (A, B, E1 ) has an ORHP eigenvalue that is controllable √ and observable. In this case, (16.16.4) has the solutions √ P = R2 −

R22 + R2 < 0 and P = R2 + R22 + R2 > 0. The corresponding closed-loop dynamics √ √ ˜ ˜ matrices are √ A = 1 + 1/R2 > 0 and A = − 1 + 1/R2 < 0. Hence, the positive-definite solution P = R2 + R22 + R2 is stabilizing. ^

Example 16.16.7. Let n = 1, A = 0, B = 1, E1 = 0, and R2 > 0. Hence, (A, B, E1 ) has an imaginary eigenvalue that is controllable and unobservable. In this case, (16.16.4) has the unique solution P = 0, which is not stabilizing. ^ Example 16.16.8. Let n = 1, A = 0, B = 1, E1 = 1, and R2 > 0. Hence, (A, B, E1 ) has an imaginary √ eigenvalue that is√controllable and observable. In this case, (16.16.4) has the solutions ˜ R2 < 0 and P = P = − √ √ R2 > 0. The corresponding closed-loop dynamics √ matrices are A = ˜ 1/ R2 > 0 and A = −1/ R2 < 0. Hence, the positive-definite solution P = R2 is stabilizing. ^ [ ] 0 1 , B = I , E = 0, and R = 1. Hence, as in Example Example 16.16.9. Let n = 2, A = −1 2 1 2 0 16.16.7, both eigenvalues of (A, B, E1 ) are imaginary, controllable, and unobservable. Taking the

LINEAR SYSTEMS AND CONTROL THEORY

1297

trace of (16.16.4) yields tr P2 = 0. Thus, the unique symmetric matrix P that satisfies (16.16.4) is P = 0, which implies that A˜ = A. Consequently, the open-loop eigenvalues ± ȷ are unaffected by the feedback gain (16.15.15) even though (A, B) is controllable. ^ Example 16.16.10. Let n = 2, A = 0, B = I2 , E1 = I2 , and R2 = I. Hence, as in Example 16.16.8, both eigenvalues of (A, B, E1 ) are imaginary, controllable, and observable. Furthermore, (16.16.4) becomes P2 = I. Requiring that P be symmetric, it follows that P is a reflector. Hence, P = I is the unique positive-semidefinite solution. In fact, P is positive definite and stabilizing since A˜ = −I. ^ [ ] [ ] Example 16.16.11. Let A = 10 02 , B = 11 , E1 = 0, and R2 = 1 so that (A, B) is controllable, although neither of the states is weighted. In this case, (16.16.4) has four positive-semidefinite solutions, which are given by [ ] [ ] [ ] [ ] 18 −24 2 0 0 0 0 0 P1 = , P2 = , P3 = , P4 = . −24 36 0 0 0 4 0 0 The corresponding feedback matrices are K1 = [6 −12], K2 = [−2 0], K3 = [0 −4], and K4 = [0 0]. Letting A˜ i = A − ΣPi , it follows that spec(A˜1 ) = {−1, −2}, spec(A˜ 2 ) = {−1, 2}, spec(A˜ 3 ) = {1, −2}, and spec(A˜ 4 ) = {1, 2}. Thus, P1 is the unique solution that stabilizes the closed-loop system, while the solutions P2 and P3 partially stabilize the closed-loop system. Note also that the closed-loop poles that differ from those of the open-loop system are mirror images of the open-loop poles as reflected across the imaginary axis. Finally, note that these solutions satisfy the partial orderings P1 ≥ P2 ≥ P4 and P1 ≥ P3 ≥ P4 , and that “larger” solutions are more stabilizing than “smaller” solutions. Moreover, letting J(Ki ) = tr PiV, it can be seen that larger solutions incur a greater closedloop cost, with the greatest cost incurred by the stabilizing solution P1 . However, the cost expression J(K) = tr PV does not follow from (16.15.4) if A + BK is not asymptotically stable. ^ The following definition concerns solutions of the Riccati equation. Definition 16.16.12. A matrix P ∈ Rn×n is a solution of the Riccati equation (16.16.4) if P is symmetric and satisfies (16.16.4). Furthermore, P is the stabilizing solution of (16.16.4) if A˜ = A − ΣP is asymptotically stable. Finally, a solution Pmax of (16.16.4) is the maximal solution to (16.16.4) if P ≤ Pmax for every solution P to (16.16.4). Since the ordering “≤” is antisymmetric, it follows that (16.16.4) has at most one maximal solution. The uniqueness of the stabilizing solution is shown in the following section. Next, define the 2n × 2n Hamiltonian [ ] A Σ △ H= . (16.16.8) R1 −AT Proposition 16.16.13. The following statements hold: i) The Hamiltonian H is a Hamiltonian matrix. ii) χH has a spectral factorization; that is, there exists a monic polynomial p ∈ R[s] such that, for all s ∈ C, χH (s) = p(s)p(−s). iii) χH (ω ȷ) ≥ 0 for all ω ∈ R. iv) If either R1 = 0 or Σ = 0, then mspec(H) = mspec(A) ∪ mspec(−A). v) χH is even. vi) λ ∈ spec(H) if and only if −λ ∈ spec(H). vii) If λ ∈ spec(H), then amultH (λ) = amultH (−λ). viii) Every imaginary root of χH has even multiplicity. ix) Every imaginary eigenvalue of H has even algebraic multiplicity. Proof. This result follows from Proposition 6.1.1 and Fact 6.9.33. 

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It is helpful to keep in mind that spectral factorizations are not unique. For example, if χH (s) = (s + 1)(s + 2)(−s + 1)(−s + 2), then χH (s) = p(s)p(−s) = p(s) ˆ p(−s), ˆ where p(s) = (s + 1)(s + 2) and p(s) ˆ = (s + 1)(s − 2). Thus, the spectral factors p(s) and p(−s) can “trade” roots. These roots are the eigenvalues of H. The following result shows that the Hamiltonian H is closely linked to the Riccati equation (16.16.4). Proposition 16.16.14. Let P ∈ Rn×n be symmetric. Then, the following statements are equivalent: i) P is a solution of (16.16.4). ii) P satisfies [ ] I [P I]H = 0. (16.16.9) −P [

iii) P satisfies H iv) P satisfies

[

I 0 H= −P I

] [ ] I I = (A − ΣP). −P −P

][

A − ΣP Σ 0 −(A − ΣP)T

(16.16.10) ][

] I 0 . P I

(16.16.11)

If these conditions hold, then the following statements hold: v) mspec(H) = mspec(A − ΣP) ∪ mspec[−(A − ΣP)]. vi) χH (s) = (−1)n χA−ΣP (s)χA−ΣP (−s). ([ I ]) vii) R −P is an invariant subspace of H. Corollary 16.16.15. Assume that (16.16.4) has a stabilizing solution. Then, H has no imaginary eigenvalues. For the next two results, P is not necessarily a solution of (16.16.4). Lemma 16.16.16. Assume that λ ∈ spec(A) is an observable eigenvalue of (A, R1 ), and let ˜ R). ˜ P ∈ Rn×n be symmetric. Then, λ ∈ spec(A) is an observable eigenvalue of (A, [ ˜] λI− A Proof. Suppose that rank R˜ < n. Then, there exists a nonzero vector v ∈ Cn such that ˜ = λv and Rv ˜ = 0. Hence, v∗R1v = −v∗PΣPv ≤ 0, which implies that R1v = 0 and PΣPv = 0. Av [ ] Hence, ΣPv = 0, and thus Av = λv. Therefore, rank λI−A < n.  R1 Lemma 16.16.17. Assume that (A, R1 ) is (observable, detectable), and let P ∈ Rn×n be sym˜ R) ˜ is (observable, detectable). metric. Then, (A, Lemma 16.16.18. Assume that (A, E1 ) is observable, and assume that (16.16.4) has a solution P. Then, the following statements hold: ˜ = ν+ (P). i) ν− (A) ˜ ii) ν0 (A) = ν0 (P) = 0. ˜ = ν− (P). iii) ν+ (A) ˜ R) ˜ is observable. By Proof. Since (A, R1 ) is observable, it follows from Lemma 16.16.17 that (A, writing (16.16.4) as the Lyapunov equation (16.16.6), the result now follows from Fact 16.22.1. 

16.17 The Stabilizing Solution of the Riccati Equation Proposition 16.17.1. The following statements hold:

i) (16.16.4) has at most one stabilizing solution. ii) If P is the stabilizing solution of (16.16.4), then P is positive semidefinite.

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iii) If P is the stabilizing solution of (16.16.4), then ˜ R). ˜ rank P = rank O(A,

(16.17.1)

Proof. To prove i), suppose that (16.16.4) has stabilizing solutions P1 and P2 . Then,

ATP1 + P1 A + R1 − P1ΣP1 = 0, ATP2 + P2 A + R1 − P2 ΣP2 = 0. Subtracting these equations and rearranging yields (A − ΣP1 )T (P1 − P2 ) + (P1 − P2 )(A − ΣP2 ) = 0. Since A − ΣP1 and A − ΣP2 are asymptotically stable, it follows from Proposition 15.10.3 and Fact 15.19.33 that P1 − P2 = 0. Hence, (16.16.4) has at most one stabilizing solution. Next, to prove ii), let P be a stabilizing solution of (16.16.4). Then, it follows from (16.16.4) that ∫∞ T P = et(A−ΣP) (R1 + PΣP)et(A−ΣP) dt, 0

which shows that P is positive semidefinite. Finally, iii) follows from Corollary 16.3.3.  Theorem 16.17.2. Assume that (16.16.4) has a positive-semidefinite solution P, and assume that (A, E1 ) is detectable. Then, P is the stabilizing solution of (16.16.4) and the unique positivesemidefinite solution of (16.16.4). If, in addition, (A, E1 ) is observable, then P is positive definite. ˜ R) ˜ is detectable. Next, Proof. Since (A, R1 ) is detectable, Lemma 16.16.17 implies that (A, since (16.16.4) has a positive-semidefinite solution P, it follows from Corollary 16.5.7 that A˜ is asymptotically stable. Hence, by Proposition 16.17.1, P is the stabilizing solution of (16.16.4) and thus the unique positive-semidefinite solution of (16.16.4). The last statement follows from Lemma 16.16.18.  Corollary 16.17.3. Assume that (A, E1 ) is detectable. Then, (16.16.4) has at most one positivesemidefinite solution. Lemma 16.17.4. Let λ ∈ C, and assume that λ is either an uncontrollable eigenvalue of (A, B) or an unobservable eigenvalue of (A, E1 ). Then, λ ∈ spec(H). Proof. Note that    λI − A −Σ  λI − H =   . −R1 λI + AT In the case where λ is an uncontrollable eigenvalue of (A, B), the first n rows of λI − H are linearly dependent, and thus λ ∈ spec(H). On the other hand, in the case where λ is an unobservable eigenvalue of (A, E1 ), the first n columns of λI−H are linearly dependent, and thus λ ∈ spec(H).  The following result is a consequence of Lemma 16.17.4. Proposition 16.17.5. Let S ∈ Rn×n be a nonsingular matrix such that     0 A13 0   A1  B1    A21 A2 A23 A24  −1  B   S , B = S  2  , E1 = [E11 0 E13 0]S −1 , A = S  (16.17.2) 0 A3 0   0  0  0 0 A43 A4 0 ([ A 0 ] [ B ]) ([ A A ] ) where A211 A2 , B12 is controllable and 01 A133 , [ E11 E13 ] is observable. Then, mspec(A2 ) ∪ mspec(−A2 ) ⊆ mspec(H),

(16.17.3)

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mspec(A3 ) ∪ mspec(−A3 ) ⊆ mspec(H), mspec(A4 ) ∪ mspec(−A4 ) ⊆ mspec(H).

(16.17.4) (16.17.5)

Next, we present a partial converse of Lemma 16.17.4. Lemma 16.17.6. Let λ ∈ spec(H), and assume that Re λ = 0. Then, λ is either an uncontrollable eigenvalue of (A, B) or an unobservable eigenvalue of (A, E1 ). Proof. Let λ = ω ȷ be an eigenvalue of H, where ω ∈ R. Then, there exist x, y ∈ Cn such that [ x] [ x] [ x] y , 0 and H y = ω ȷ y . Consequently, Ax + Σy = ω ȷx,

R1 x − ATy = ω ȷy.

Rewriting these equalities as (A − ω ȷI)x = −Σy, yields

y∗ (A − ω ȷI)x = −y∗Σy,

(A − ω ȷI)∗ y = R1 x x∗(A − ω ȷI)∗ y = x∗R1 x.

Since x∗(A − ω ȷI)∗ y is real, it follows that x∗R1 x = −y∗Σy. Since 0 ≤ x∗R1 x = −y∗Σy ≤ 0, it follows that x∗R1 x = y∗Σy = 0, and thus BTy = 0 and E1 x = 0. Therefore, (A − ω ȷI)∗ y = 0,

(A − ω ȷI)x = 0, and hence

[

] A − ω ȷI x = 0, E1

y∗ [A − ω ȷI B] = 0.

[ ] [ ] ȷI Since yx , 0, it follows that either x , 0 or y , 0. Hence, either rank A−ω < n or rank [A − E1 ω ȷI B] < n.  The following result is a restatement of Lemma 16.17.6. ∈ Rn×n] be a nonsingular matrix such that (16.17.2) holds, where ([ A Proposition ] [ B ]) 16.17.7. Let S ([ ) A1 A13 0 1 1 E E is controllable and 0 A3 , [ 11 13 ] is observable. Then, A21 A2 , B2 mspec(H) ∩ IA ⊆ mspec(A2 ) ∪ mspec(−A2 ) ∪ mspec(A3 ) ∪ mspec(−A3 ) ∪ mspec(A4 ) ∪ mspec(−A4 ). (16.17.6) Combining Lemma 16.17.4 and Lemma 16.17.6 yields the following result. Proposition 16.17.8. Let λ ∈ C, assume that Re λ = 0, and let S ∈ Rn×n be a nonsingular matrix such that (16.17.2) holds, where (A1 , B1, E11 ) is controllable and observable, (A2 , B2 ) is controllable, and (A3 , E13 ) is observable. Then, the following statements are equivalent: i) λ is either an uncontrollable eigenvalue of (A, B) or an unobservable eigenvalue of (A, E1 ). ii) λ ∈ mspec(A2 ) ∪ mspec(A3 ) ∪ mspec(A4 ). iii) λ is an eigenvalue of H. The next result gives necessary and sufficient conditions under which (16.16.4) has a stabilizing solution. This result also provides a constructive characterization of the stabilizing solution. ii) of Proposition 16.10.11 shows that the statement [ in i) that ] every imaginary eigenvalue of (A, E1 ) is observable is equivalent to the statement that

A

B

E1

E2

has no imaginary invariant zeros.

Theorem 16.17.9. The following statements are equivalent: i) (A, B) is stabilizable, and every imaginary eigenvalue of (A, E1 ) is observable.

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LINEAR SYSTEMS AND CONTROL THEORY

n×n ii) ([ There ]exists such that (16.17.2) holds, where ([ A A ] S ∈ R) [ B ]) a nonsingular matrix A1 0 1 13 1 E11 E13 ] is observable, ν0 (A2 ) = 0, and A3 and , [ is controllable, , B2 0 A3 A21 A2 A4 are asymptotically stable. iii) (16.16.4) has a stabilizing solution. Now, assume that these conditions hold, and let [ ] M1 M12 M= ∈ R2n×2n (16.17.7) M21 M2

be a nonsingular matrix such that H = MZ M −1, where [ ] Z Z12 Z= 1 ∈ R2n×2n 0 Z2

(16.17.8)

and Z1 ∈ Rn×n is asymptotically stable. Then, M1 is nonsingular, and △

P = −M21 M1−1

(16.17.9)

is the stabilizing solution of (16.16.4). Proof. The equivalence of i) and ii) is immediate. To prove i) =⇒ iii), first note that Lemma 16.17.6 implies that H has no imaginary eigenvalues. Hence, since H is Hamiltonian, it follows that there exists M ∈ R2n×2n of the form (16.17.7) such that M is nonsingular and H = MZ M −1, where Z ∈ R2n×2n is of the form (16.17.8) and Z1 ∈ Rn×n is asymptotically stable. Next, note that HM = MZ implies that [ ] [ ] [ ] M1 Z1 M1 H =M = Z . M21 0 M21 1 Therefore,

[

]T [ ] [ ]T [ ] [ M1 M1 M1 M1 JH = J Z = M1T M21 n M21 M21 n M21 1

] [ M21 ] M21 Z = LZ1 , −M1 1 T

△

T where L = M1TM21 − M21 M1 . Since Jn H = (Jn H)T, it follows that LZ1 is symmetric; that is, LZ1 = Z1TLT. Since, in addition, L is skew symmetric, it follows that 0 = Z1TL + LZ1. Now, since Z1 is T T asymptotically stable, it follows that L = 0. Hence, M1TM21 = M21 M1 , which shows that M21 M1 is symmetric. To show that M1 is nonsingular, note that [ ] [ ] M1 M1 [I 0]H = [I 0] Z M21 M21 1

implies that

AM1 + ΣM21 = M1 Z1 .

Now, let x ∈ Rn satisfy M1 x = 0. Then, T T T xTM21 Σ M21 x = xT M21 (AM1 + Σ M21 )x = xTM21 M1 Z1 x = xTM1TM21 Z1 x = 0,

which implies that BTM21 x = 0. Hence, M1 Z1 x = (AM1 + Σ M21 )x = 0. Thus, Z1 N(M1 ) ⊆ N(M1 ). Now, suppose that M1 is singular. Since Z1 N(M1 ) ⊆ N(M1 ), it follows that there exist λ ∈ spec(Z1 ) and x ∈ Cn such that Z1 x = λx and M1 x = 0. Noting [ ] [ ] M1 M1 [0 I]H x = [0 I] Z x M21 M21 1

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yields −ATM21 x = M21λx, and thus (λI + AT )M21 x = 0. Since, in addition, as shown above, BTM21 x = T 0, it follows that x∗M21 [−λI −A B] = 0. Since λ ∈ spec(Z1 ), it follows that Re(−λ) > 0. Furthermore, since, by assumption, (A, B) is stabilizable, it[follows that rank [λI −A B] = n. Therefore, M21 x = 0. ] Combining this fact with M1 x = 0 yields MM211 x = 0. Since x is nonzero, it follows that M is △

singular, which is a contradiction. Consequently, M1 is nonsingular. Next, define P = −M21M1−1 and note that, since M1TM21 is symmetric, it follows that P = −M1−T (M1TM21 )M1−1 is also symmetric. [ ] [ ] Since H MM211 = MM211 Z1 , it follows that [ ] [ ] I I H = M1 Z1 M1−1, M21 M1−1 M21 M1−1 [

and thus H

] [ ] I I = M1 Z1 M1−1. −P −P

Multiplying on the left by [P I] yields 0 = [P I]H

[

] I = ATP + PA + R1 − PΣP, −P

which shows that P is a solution of (16.16.4). Similarly, multiplying on the left by [I 0] yields A − ΣP = M1 Z1 M1−1. Since Z1 is asymptotically stable, it follows that A − ΣP is also asymptotically stable. To prove iii) =⇒ i), note that the existence of a stabilizing solution P implies that (A, B) is stabilizable, and that (16.16.11) implies that H has no imaginary eigenvalues.  Corollary 16.17.10. Assume that (A, B) is stabilizable and (A, E1 ) is detectable. Then, (16.16.4) has a stabilizing solution.

16.18 The Maximal Solution of the Riccati Equation In this section we consider the existence of the maximal solution of (16.16.4). Example 16.16.3 shows that (16.16.4) may not have a maximal solution. Theorem 16.18.1. The following statements are equivalent: i) (A, B) is stabilizable. ii) (16.16.4) has a solution Pmax that is positive semidefinite, maximal, and satisfies spec(A − ΣPmax ) ⊂ CLHP.

(16.18.1)

Proof. i) =⇒ ii) is given by Theorem 2.1 and Theorem 2.2 of [1188]. See also (i) of Theorem 13.11 of [2999]. The converse follows from Corollary 3 of [2386].  Proposition 16.18.2. Assume that (16.16.4) has a maximal solution Pmax , let P be a solution of (16.16.4), and assume that spec(A − ΣPmax ) ⊂ CLHP and spec(A − ΣP) ⊂ CLHP. Then, P = Pmax . Proof. It follows from i) of Proposition 16.16.14 that spec(A − ΣP) = spec(A − ΣPmax ). Since Pmax is the maximal solution of (16.16.4), it follows that P ≤ Pmax . Consequently, it follows from the contrapositive form of the second statement of Theorem 10.4.9 that P = Pmax .  Proposition 16.18.3. Assume that (16.16.4) has a solution P such that spec(A − ΣP) ⊂ CLHP. Then, P is stabilizing if and only if H has no imaginary eigenvalues. It follows from Proposition 16.18.2 that (16.16.4) has at most one positive-semidefinite solution P such that spec(A − ΣP) ⊂ CLHP. Consequently, (16.16.4) has at most one positive-semidefinite stabilizing solution. Theorem 16.18.4. The following statements hold:

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LINEAR SYSTEMS AND CONTROL THEORY

i) (16.16.4) has at most one stabilizing solution. ii) If P is the stabilizing solution of (16.16.4), then P is positive semidefinite. iii) If P is the stabilizing solution of (16.16.4), then P is maximal. Proof. To prove i), suppose that (16.16.4) has stabilizing solutions P1 and P2 . Then, (A, B) is stabilizable, and Theorem 16.18.1 implies that (16.16.4) has a maximal solution Pmax such that spec(A − ΣPmax ) ⊂ CLHP. Now, Proposition 16.18.2 implies that P1 = Pmax and P2 = Pmax . Hence, P1 = P2 . Alternatively, suppose that (16.16.4) has the stabilizing solutions P1 and P2 . Then, ATP1 + P1 A + R1 − P1 ΣP1 = 0, ATP2 + P2 A + R1 − P2 ΣP2 = 0. Subtracting these equations and rearranging yields (A − ΣP1 )T (P1 − P2 ) + (P1 − P2 )(A − ΣP2 ) = 0. Since A − ΣP1 and A − ΣP2 are asymptotically stable, it follows from Proposition 15.10.3 and Fact 15.19.33 that P1 − P2 = 0. Hence, (16.16.4) has at most one stabilizing solution. Next, to prove ii), suppose that P is a stabilizing solution of (16.16.4). Then, it follows from (16.16.4) that ∫∞ T P = et(A−ΣP) (R1 + PΣP)et(A−ΣP) dt, 0

which shows that P is positive semidefinite. To prove iii), let P′ be a solution of (16.16.4). Then, (A − ΣP)T(P − P′ ) + (P − P′ )(A − ΣP) + (P − P′ )Σ(P − P′ ) = 0, which implies that P′ ≤ P. Thus, P is also the maximal solution of (16.16.4).  The following result concerns the monotonicity of solutions of the Riccati equation (16.16.4). Proposition 16.18.5. Assume that (A, B) is stabilizable, and let Pmax denote the maximal solution of (16.16.4). Furthermore, let Rˆ 1 ∈ Rn×n be positive semidefinite, let Rˆ 2 ∈ Rm×m be positive △ T definite, let Aˆ ∈ Rn×n, let Bˆ ∈ Rn×m, define Σˆ = Bˆ Rˆ −1 2 B , assume that ] [ ] [ Rˆ 1 AˆT R1 AT ≤ , A −Σ Aˆ −Σˆ and let Pˆ be a solution of

Then,

AˆTPˆ + Pˆ Aˆ + Rˆ 1 − Pˆ Σˆ Pˆ = 0.

(16.18.2)

Pˆ ≤ Pmax .

(16.18.3)

Proof. See Theorem 1 of [2895].  Corollary 16.18.6. Assume that (A, B) is stabilizable, let Rˆ 1 ∈ Rn×n be positive semidefinite,

assume that Rˆ 1 ≤ R1, and let Pmax and Pˆ max denote, respectively, the maximal solutions of (16.16.4) and

Then,

ATP + PA + Rˆ 1 − PΣP = 0.

(16.18.4)

Pˆ max ≤ Pmax .

(16.18.5)

Proof. This result follows from either Proposition 16.18.5 or Theorem 2.3 of [1188].



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The following result shows that, if R1 = 0, then the closed-loop eigenvalues of the closedloop dynamics obtained from the maximal solution consist of the CLHP open-loop eigenvalues and reflections of the ORHP open-loop eigenvalues. Proposition 16.18.7. Assume that (A, B) is stabilizable, assume that R1 = 0, and let P ∈ Rn×n be a positive-semidefinite solution of (16.16.4). Then, P is the maximal solution of (16.16.4) if and only if [ ] [ ] mspec(A − ΣP) = mspec(A) ∩ CLHP ∪ mspec(−A) ∩ OLHP . (16.18.6) Proof. Sufficiency follows from Proposition 16.18.2. To prove necessity, note that it follows from the definition (16.16.8) of H with R1 = 0 and from iv) of Proposition 16.16.14 that

mspec(A) ∪ mspec(−A) = mspec(A − ΣP) ∪ mspec[−(A − ΣP)]. Now, Theorem 16.18.1 implies that mspec(A−ΣP) ⊆ CLHP, which implies that (16.18.6) holds.  Corollary 16.18.8. Let R1 = 0, and assume that spec(A) ⊂ CLHP. Then, P = 0 is the unique positive-semidefinite solution of (16.16.4).

16.19 Positive-Semidefinite and Positive-Definite Solutions of the Riccati Equation The following result gives sufficient conditions under which (16.16.4) has a positive-semidefinite solution. Proposition 16.19.1. matrix S ∈ Rn×n such that ([ A Assume ] [ B ])that there exists a([nonsingular ] ) A1 A13 1 1 0 (16.17.2) holds, where A21 A2 , B2 is controllable, 0 A3 , [ E11 E13 ] is observable, and A3 is asymptotically stable. Then, (16.16.4) has a positive-semidefinite solution. Proof. First, rewrite (16.17.2) as     0   B1   A1 A13 0   0   0 A3 0 0  −1  S , B = S   , A = S   B2   A21 A23 A2 A24  0 0 A43 0 A4

([ A

[ E1 = E11

] [ ])

E13 ([ A

B1 1 1 0 A21([A2 , B]2 [ is])controllable, 0 A1 A13 B1 Since 0 A3 , 0 is stabilizable, it

where

] 0 0 S −1 , ] ) A13 E E A3 , [ 11 13 ] is observable, and A3 is asymptotically sta-

ble. follows from Theorem 16.18.1 that there exists a positiveˆ semidefinite matrix P1 that satisfies    [ ]T [ ]  T T  E11E11 E11  B1R−1 E13  BT1 0  A1 A13 ˆ A A 2 1 13  − Pˆ 1   Pˆ 1 = 0. P1 + Pˆ 1 +  T 0 A3 0 A3 E E11 E T E13 0 0 13

△

13

Consequently, P = S diag(Pˆ 1, 0, 0)S is a positive-semidefinite solution of (16.16.4).  Corollary 16.19.2. Assume that (A, B) is stabilizable. Then, (16.16.4) has a positive-semidefinite solution P. If, in addition, (A, E1 ) is detectable, then P is the stabilizing solution of (16.16.4), and thus P is the unique positive-semidefinite solution of (16.16.4). Finally, if (A, E1 ) is observable, then P is positive definite. Proof. The first statement is given by Theorem 16.18.1. Next, since (A, E1 ) is detectable, Theorem 16.17.2 implies that P is a stabilizing solution of (16.16.4), which is the unique positivesemidefinite solution of (16.16.4). Finally, using Theorem 16.17.2, (A, E1 ) observable implies that P is positive definite.  T

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The next result gives necessary and sufficient conditions under which (16.16.4) has a positivedefinite solution. Proposition 16.19.3. The following statements are equivalent: i) (16.16.4) has a positive-definite solution. n×n ii) ([ There ]exists such that (16.17.2) holds, where ] S ∈ )R [ B ]) a nonsingular ([matrix A1 A13 A1 0 1 E11 E13 ] is observable, A3 is asymptotically stable, [ , is controllable, , B2 0 A3 A21 A2 −A2 is asymptotically stable, spec(A4 ) ⊂ IA, and A4 is semisimple. If these conditions hold, then (16.16.4) has exactly one positive-definite solution if and only if A4 is empty, and infinitely many positive-definite solutions if and only if A4 is nonempty. Proof. See [2306].  Proposition 16.19.4. Assume that (16.16.4) has a stabilizing solution P, and let S ∈ Rn×n be a nonsingular matrix such that (16.17.2) holds, where (A1 , B1, E11 ) is controllable and observable, (A2 , B2 ) is controllable, (A3 , E13 ) is observable, ν0 (A2 ) = 0, and A3 and A4 are asymptotically stable. Then, def P = ν− (A2 ). (16.19.1) Hence, P is positive definite if and only if spec(A2 ) ⊂ ORHP.

16.20 Facts on Linear Differential Equations △

Fact 16.20.1. Let A ∈ Fn×n and b ∈ Fn , assume that A is nonsingular, define xs = −A−1b, and

let x : [0, ∞) 7→ Fn satisfy x˙(t) = Ax(t) + b. Then, for all t ≥ 0,

x(t) = etA x(0) + (I − etA )xs = xs + etA [x(0) − xs ]. If, in addition, A is asymptotically stable, then

∫

∞

lim x(t) = xs =

t→∞

eτA dτb.

0

Related: Fact 16.20.2. Fact 16.20.2. Let A1 ∈ Fn×n and A2 ∈ Fm×m , assume that A1 ⊕ A2 is nonsingular, let R ∈ Fn×m , △

define Ps = − vec−1 [(AT2 ⊗ A1 )−1 vec R] ∈ Fn×m , and let P : [0, ∞) 7→ Fn×m satisfy ˙ = A1 P(t) + P(t)A2 + R. P(t) Then, for all t ≥ 0,

P(t) = etA1 P(0)etA2 + Ps − etA1 Ps etA2 = Ps + etA1 [P(0) − Ps ]etA2 . If, in addition, AT2 ⊕ A1 is asymptotically stable, then ∫ ∞ lim P(t) = Ps = eτA1 ReτA2 dτ. t→∞

0

Related: Fact 16.20.1. Fact 16.20.3. Let A ∈ Fn×n and X0 ∈ Fn×n. Then, the matrix differential equation

˙ = AX(t), X(t)

X(0) = X0 ,

where t ≥ 0, has the unique solution X(t) = etAX0 . Fact 16.20.4. Let A ∈ Cn×n, let λ ∈ spec(A), and let v ∈ Cn be an eigenvector of A associated △ with λ. Then, for all t ≥ 0, x(t) = Re(eλt v) satisfies x˙(t) = Ax(t). Remark: x(t) is an eigensolution.  Fact 16.20.5. Let A ∈ Rn×n, let λ ∈ spec(A), let (v1, . . . , vk ) ∈ ki=1 Cn be a Jordan chain of A

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associated with λ, let i ∈ {1, . . . , k}, and define ( [ ]) △ 1 x(t) = Re eλt (i−1)! ti−1 v1 + · · · + tvi−1 + vi . Then, for all t ≥ 0, x(t) satisfies x˙(t) = Ax(t). Related: Fact 7.15.12 for the definition of a Jordan chain. Remark: x(t) is a generalized eigensolution. [ ] [ ] For i[ =] 1, x(t) is an eigensolution. See Fact [ ] 16.20.4. Example: Let A = 00 10 , λ = 0, v1 = β0 , v2 = β0 . If i = 2, then x(t) = tv1 + v2 = βtβ [ ] is a generalized eigensolution. Alternatively, i = 1 yields the eigensolution x(t) = v1 = β0 . Note that β represents velocity for the generalized eigensolution, and position for the eigensolution. See [2187]. Fact 16.20.6. Let A: [0, T ] 7→ Rn×n, assume that A is continuous, and let X0 ∈ Rn×n. Then, the matrix differential equation ˙ = A(t)X(t), X(t)

X(0) = X0

has a unique solution X: [0, T ] 7→ Rn×n. For all t ∈ [0, T ], ∫t

det X(t) = e 0 tr A(τ) dτ det X0 . If X0 is nonsingular, then, for all t ∈ [0, T ], X(t) is nonsingular. If, for all t1, t2 ∈ [0, T ], ∫ t2 ∫ t2 A(t2 ) A(τ) dτ = A(τ) dτA(t2 ), t1

then, for all t ∈ [0, T ],

t1 ∫t

X(t) = e 0 A(τ) dτX0 .

Source: Fact 12.16.22 implies that (d/dt) det X = tr XAX˙ = tr XAAX = tr XXAA = (det X) tr A. See See [1190], [1450, pp. 507, 508], and [2349, pp. 64–66]. Remark: The first result is Jacobi’s identity. Remark: If the commutativity assumption does not hold, then the solution is given by the Peano-Baker series. See [2349, Chapter 3]. Alternative expressions for X(t) are given by the Magnus, Fer, Baker-Campbell-Hausdorff-Dynkin, Wei-Norman, Goldberg, and Zassenhaus expansions. See [491, 963], [1293, pp. 118–120], and [1505, 1506, 1654, 1925, 2178, 2560, 2616, 2851, 2852, 2857]. Related: Fact 16.20.6. Fact 16.20.7. Let A: [t0 , t1 ] 7→ Rn×n, assume that A is continuous, let B: [t0 , t1 ] 7→ Rn×m, assume that B is continuous, let X: [t0 , t1 ] 7→ Rn×n be the unique solution of the matrix differential equation

˙ = A(t)X(t), X(t)

X(t0 ) = I,

△

define Φ(t, τ) = X(t)X −1(τ), let u : [t0 , t1 ] 7→ Rm, and assume that u is continuous. Then, the unique solution of the vector differential equation x˙(t) = A(t)x(t) + B(t)u(t), ∫

is

t

x(t) = Φ(t, t0 )x0 +

x(t0 ) = x0

Φ(t, τ)B(τ)u(τ) dτ.

t0

If, in addition, x1 = x(t1 ), then, for all t ∈ [t0 , t1 ], ∫ t x(t) = Φ(t, t1 )x1 + Φ(t, τ)B(τ)u(τ) dτ. t1

Remark: Fact 16.20.6 implies that, for all t ∈ [t0 , t1 ], X(t) is nonsingular. Remark: Φ(t, τ) is the

state transition matrix.

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LINEAR SYSTEMS AND CONTROL THEORY

Fact 16.20.8. Let A: [0, ∞) 7→ Cn×n, assume that A is continuous, assume that, for all t ≥ 0,

A(t + T ) = A(t), let X0 ∈ Cn×n, and let X : [0, ∞) 7→ Cn×n be the unique solution of the matrix differential equation ˙ = A(t)X(t), X(t) Then, there exist P : [0, ∞) 7→ C P(t) and

n×n

and B ∈ C

n×n

X(0) = X0 .

such that P(0) = X0 and, for all t ≥ 0, P(t + T ) =

X(t) = P(t)eBt .

Now, assume that X0 is nonsingular. Then, for all t ≥ 0, X(t + T )X −1 (t) = eBT . Source: [1297, pp. 118, 119]. Remark: eBT is a monodromy matrix, the eigenvalues of eBT are characteristic multipliers, and the eigenvalues of B are characteristic exponents. See [1297, pp. 118, 119] and [2936, pp. 70–109]. Fact 16.20.9. Let A: [0, ∞) 7→ Rn×n, assume that A is continuous, assume that, for all t ≥ 0, A(t + T ) = A(t), let X0 ∈ Rn×n, and let X : [0, ∞) 7→ Rn×n be the unique solution of the matrix differential equation ˙ = A(t)X(t), X(t)

X(0) = X0 .

Then, there exist P : [0, ∞) 7→ Rn×n and B ∈ Rn×n such that P(0) = X0 and such that, for all t ≥ 0, P(t + 2T ) = P(t) and X(t) = P(t)eBt . Now, assume that X0 is nonsingular. Then, for all t ≥ 0, X(t + T )X −1 (t) = eBT . Source: [1297, pp. 118, 119].

16.21 Facts on Stability, Observability, and Controllability Fact 16.21.1. Let A ∈ Rn×n , let x : [0, ∞) 7→ Rn satisfy x˙ = Ax, let C ∈ R p×n , and define △ y : [0, ∞) 7→ R p by y = Cx. Then, for all t ≥ 0, χA ( dtd )y(t) = 0. Remark: (d/dt)k = dk /dtk . Fact 16.21.2. Let A ∈ Rn×n, B ∈ Rn×m, and C ∈ R p×n, and assume that (A, B) is controllable and

(A, C) is observable. Then, for all v ∈ Rm, the step response ∫t y(t) =

CetA dτBv + Dv 0

is bounded on [0, ∞) if and only if A is Lyapunov stable and nonsingular. Fact 16.21.3. Let A ∈ Rn×n and C ∈ R p×n, assume that (A, C) is detectable, and let x(t) and y(t) satisfy x˙(t) = Ax(t) and y(t) = Cx(t) for all t ∈ [0, ∞). Then, the following statements hold: i) y is bounded if and only if x is bounded. ii) limt→∞ y(t) exists if and only if limt→∞ x(t) exists. iii) y(t) → 0 as t → ∞ if and only if x(t) → 0 as t → ∞. Fact 16.21.4. Let x(0) = x0 , and let xf − etf A x0 ∈ C(A, B). Then, for all t ∈ [0, tf ], the control u : [0, tf ] 7→ Rm defined by  tf + ∫  T △ T (tf −t)AT  τA T τA  e BB e dτ (xf − etf A x0 ) u(t) = B e   0

yields x(tf ) = xf . Fact 16.21.5. Let x(0) = x0 , let xf ∈ Rn, and assume that (A, B) is controllable. Then, for all

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t ∈ [0, tf ], the control u : [0, tf ] 7→ Rm defined by −1  tf  ∫ T T   △ u(t) = BTe(tf −t)A  eτABBTeτA dτ (xf − etf A x0 )   0

yields x(tf ) = xf . Fact 16.21.6. Let A ∈ Rn×n, let B ∈ Rn×m, assume that A is skew symmetric, assume that (A, B)

is controllable, and let α > 0. Then, A − αBBT is asymptotically stable. Fact 16.21.7. Let A ∈ Rn×n and B ∈ Rn×m. Then, (A, B) is (controllable, stabilizable) if and only if (A, BBT ) is (controllable, stabilizable). Now, assume that B is positive semidefinite. Then, (A, B) is (controllable, stabilizable) if and only if (A, B1/2 ) is (controllable, stabilizable). Fact 16.21.8. Let A ∈ Rn×n, B ∈ Rn×m, and Bˆ ∈ Rn×mˆ , and assume that (A, B) is (controllable, ˆ Then, (A, B) ˆ is also (controllable, stabilizable). stabilizable) and R(B) ⊆ R( B). n×n n×m Fact 16.21.9. Let A ∈ R , B ∈ R , and Bˆ ∈ Rn×mˆ , and assume that (A, B) is (controllable, ˆ is also (controllable, stabilizable). Source: Lemma stabilizable) and BBT ≤ Bˆ Bˆ T. Then, (A, B) 10.6.1 and Fact 16.21.8. ˆ Fact 16.21.10. Let A ∈ Rn×n, B ∈ Rn×m, Bˆ ∈ Rn×mˆ , and Cˆ ∈ Rm×n , and assume that (A, B) is (controllable, stabilizable). Then, ( ˆ [BBT + Bˆ Bˆ T ]1/2 ) A + Bˆ C, is also (controllable, stabilizable). Source: [2912, p. 79]. Fact 16.21.11. Let A ∈ Rn×n and B ∈ Rn×m. Then, the following statements are equivalent: i) (A, B) is controllable. ii) There exists α ∈ R such that (A + αI, B) is controllable. iii) (A + αI, B) is controllable for all α ∈ R. Fact 16.21.12. Let A ∈ Rn×n and B ∈ Rn×m. Then, the following statements are equivalent: i) (A, B) is stabilizable. ii) There exists α ≤ max {0, −αmax (A)} such that (A + αI, B) is stabilizable. iii) (A + αI, B) is stabilizable for all α ≤ max {0, −αmax (A)}. Fact 16.21.13. Let A ∈ Rn×n, assume that A is diagonal, and let B ∈ Rn×1. Then, (A, B) is controllable if and only if the diagonal entries of A are distinct and every entry of B is nonzero. Source: Note that     0   1 a1 · · · an−1  b1  ∏ n 1     ∏    . . . .  .    ..  =  ..   .. .. · .· · bi  (ai − a j ). det K(A, B) = det      i< j i=1 n−1  0 bn 1 an · · · an Fact 16.21.14. Let A ∈ Rn×n and B ∈ Rn×1, and assume that (A, B) is controllable. Then, A is cyclic. Source: Fact 7.15.10. Related: Fact 16.21.15. Fact 16.21.15. Let G ∈ R(s)prop , where G ∼

[

A

B

C

D

]

and A ∈ Rn×n. Then, the following

statements are equivalent: [ A B ] min i) G ∼ C D . ii) There exist p, q ∈ R[s] such that p and q are coprime and deg q = n. iii) mpoles(G) = mspec(A). Now, assume that i)–iii) hold. Then, the following statements hold:

1309

LINEAR SYSTEMS AND CONTROL THEORY

iv) A has exactly one Jordan block associated with each pole of G. v) Let λ ∈ spec(A). Then, λ is semisimple if and only if λ is a nonrepeated pole of G. vi) Let λ ∈ spec(A). Then, λ is defective if and only if λ is a repeated pole of G. Related: Theorem 16.9.24 and Fact 16.21.14. Fact 16.21.16. Let A ∈ Rn×n and B ∈ Rn×m, and assume that (A, B) is controllable. Then, max gmultA (λ) ≤ m.

λ∈spec(A)

Fact 16.21.17. Let A ∈ Rn×n and B ∈ Rn×m. Then, the following statements are equivalent:

i) (A, B) is (controllable, stabilizable) and A is nonsingular. ii) (A, AB) is (controllable, stabilizable). Fact 16.21.18. Let A ∈ Rn×n and B ∈ Rn×m, and assume that (A, B) is controllable. Then, (A, BTS −T ) is observable, where S ∈ Rn×n is a nonsingular matrix that satisfies AT = S −1AS. Fact 16.21.19. Let (A, B) be controllable, let t1 > 0, and define −1  t1  ∫  −tA T −tAT  P =  e BB e dt .   0

Then, A − BB P is asymptotically stable. Source: P satisfies T

T

(A − BBTP)TP + P(A − BBTP) + P(BBT + et1 ABBTet1A )P = 0. T

Since (A − BBTP, BBT + et1 ABBTet1 A ) is observable and P is positive definite, it follows from Proposition 15.10.5 that A − BBTP is asymptotically stable. Credit: D. L. Lukes and D. Kleinman. See [2351, pp. 113, 114]. Fact 16.21.20. Let A ∈ Rn×n and B ∈ Rn×m, and assume that A is asymptotically stable. Furthermore, let u : [0, ∞) 7→ Rm , assume that u is continuous and bounded, let x : [0, ∞) 7→ Rn , and assume that x satisfies x˙ = Ax + Bu. Then, x is bounded. If, in addition, u(t) → 0 as t → ∞, then x(t) → 0 as t → ∞. Source: [2496, p. 330]. Remark: These are consequences of input-to-state stability. Fact 16.21.21. Let A ∈ Rn×n and C ∈ R p×n, assume that (A, C) is observable, and let k ≥ n. Then, +   0 p×n  A =   Ok+1 (A, C). Ok (A, C) Credit: H. Palanthandalam-Madapusi. min

Fact 16.21.22. Let G1 , G2 ∈ R(s)prop , where G2 ∼ △

[

]

[

A1

B1

C1

0

] [ min , G2 ∼

A2

B2

C2

0

]

, let λ ∈ spec(A1 ),

˜ Remark: This result implies assume that G2 (λ) = 0, and define A˜ = BA2 C1 1 BA1 C2 2 . Then, λ ∈ spec(A). that CRHP pole-zero cancellation cannot occur in an asymptotically stable feedback loop. △ Fact 16.21.23. Let β0 , . . . , βn−1 ∈ F, define A = Cb (p) as in Fact 7.18.3, and let b, c ∈ Fn . Then, K(A, b)c = K(A, c)b. Source: [600].

16.22 Facts on the Lyapunov Equation and Inertia Fact 16.22.1. Let A, P ∈ Fn×n, assume that P is Hermitian, let C ∈ Fl×n, and assume that A∗P + PA + C ∗C = 0. Then, the following statements hold: i) |ν− (A) − ν+ (P)| ≤ n − rank O(A, C). ii) |ν+ (A) − ν− (P)| ≤ n − rank O(A, C). iii) If ν0 (A) = 0, then |ν− (A) − ν+ (P)| + |ν+ (A) − ν− (P)| ≤ n − rank O(A, C).

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CHAPTER 16

If, in addition, (A, C) is observable, then the following statements hold: iv) ν− (A) = ν+ (P). v) ν0 (A) = ν0 (P) = 0. vi) ν+ (A) = ν− (P). vii) If P is positive definite, then A is asymptotically stable. Source: [147, 695], [1738, p. 448], and [1883, 2891]. Remark: v) does not follow from i)–iii). Remark: For related results, see [638, 833, 1883, 2176]. Fact 16.22.2. Let A, P ∈ Fn×n, assume that P is nonsingular and Hermitian, and assume that A∗P + PA is negative semidefinite. Then, the following statements hold: i) ν− (A) ≤ ν+ (P). ii) ν+ (A) ≤ ν− (P). iii) If P is positive definite, then spec(A) ⊂ CLHP. Source: [1738, p. 447]. Remark: If P is positive definite, then A is Lyapunov stable, although this result does not follow from i) and ii). Fact 16.22.3. Let A, P ∈ Fn×n, and assume that ν0 (A) = 0, P is Hermitian, and A∗P + PA is negative semidefinite. Then, the following statements hold: i) ν− (P) ≤ ν+ (A). ii) ν+ (P) ≤ ν− (A). iii) If P is nonsingular, then ν− (P) = ν+ (A) and ν+ (P) = ν− (A). iv) If P is positive definite, then A is asymptotically stable. Source: [1738, p. 447]. Fact 16.22.4. Let A, P ∈ Fn×n, and assume that ν0 (A) = 0, P is nonsingular and Hermitian, and A∗P + PA is negative semidefinite. Then, the following statements hold: i) ν− (A) = ν+ (P). ii) ν+ (A) = ν− (P). Source: Fact 16.22.2 and Fact 16.22.3. See [1738, p. 448]. Credit: D. Carlson and H. Schneider. Fact 16.22.5. Let A, P ∈ Fn×n, assume that P is Hermitian, and assume that A∗P+ PA is negative definite. Then, the following statements hold: i) ν− (A) = ν+ (P). ii) ν0 (A) = 0. iii) ν+ (A) = ν− (P). iv) P is nonsingular. v) If P is positive definite, then A is asymptotically stable. Source: [970, pp. 441, 442], [1738, p. 445], and [2176]. This result follows from Fact 16.22.1 with positive-definite C = −(A∗P + PA)1/2. Remark: These statements are the classical constraints. An analogous result holds for the discrete-time Lyapunov equation, where the analogous definition of inertia counts the numbers of eigenvalues inside the open unit disk, outside the open unit disk, and on the unit circle. See [613, 862]. Credit: M. G. Krein, A. Ostrowski, and H. Schneider. Fact 16.22.6. Let A ∈ Fn×n. Then, the following statements are equivalent: i) ν0 (A) = 0. ii) There exists a nonsingular Hermitian matrix P ∈ Fn×n such that A∗P+PA is negative definite. iii) There exists a Hermitian matrix P ∈ Fn×n such that A∗P + PA is negative definite. If these conditions hold, then the following statements hold for P given by ii) and iii):

LINEAR SYSTEMS AND CONTROL THEORY

1311

ν− (A) = ν+ (P). ν0 (A) = ν0 (P) = 0. ν+ (A) = ν− (P). P is nonsingular. If P is positive definite, then A is asymptotically stable. Source: For i) =⇒ ii), see [1738, p. 445]. iii) =⇒ i) follows from Fact 16.22.5. See [97, 613, 640]. Fact 16.22.7. Let A ∈ Fn×n. Then, the following statements are equivalent: i) A is Lyapunov stable. ii) There exists a positive-definite matrix P ∈ Fn×n such that A∗P + PA is negative semidefinite. Furthermore, the following statements are equivalent: iii) A is asymptotically stable. iv) There exists a positive-definite matrix P ∈ Fn×n such that A∗P + PA is negative definite. v) For every positive-definite matrix R ∈ Fn×n, there exists a positive-definite matrix P ∈ Fn×n such that A∗P + PA is negative definite. Remark: See Proposition 15.10.5 and Proposition 15.10.6. Fact 16.22.8. Let A, P ∈ Fn×n, and assume that P is Hermitian. Then, the following statements hold: i) ν+ (A∗P + PA) ≤ rank P. ii) ν− (A∗P + PA) ≤ rank P. If, in addition, A is asymptotically stable, then the following statement holds: iii) 1 ≤ ν− (A∗P + PA) ≤ rank P. Source: [266, 862]. Fact 16.22.9. Let A, P ∈ Rn×n, assume that ν0 (A) = n, and assume that P is positive semidefinite. Then, exactly one of the following statements holds: i) ATP + PA = 0. ii) ν− (ATP + PA) ≥ 1 and ν+ (ATP + PA) ≥ 1. Source: [2755]. Fact 16.22.10. Let R ∈ Fn×n, and assume that R is Hermitian and ν+ (R) ≥ 1. Then, there exist an asymptotically stable matrix A ∈ Fn×n and a positive-definite matrix P ∈ Fn×n such that A∗P + PA + R = 0. Source: [266]. Fact 16.22.11. Let A ∈ Fn×n, assume that A is cyclic, and let a, b, c, d, e be nonnegative integers such that a + b = c + d + e = n, c ≥ 1, and e ≥ 1. Then, there exists a nonsingular, Hermitian matrix P ∈ Fn×n such that      a   c      ∗ In P =  0  , In(A P + PA) =  d  .     b e iv) v) vi) vii) viii)

Source: [2452, 2453]. Fact 16.22.12. Let P, R ∈ Fn×n, and assume that P is positive and R is Hermitian. Then, the

following statements are equivalent: i) tr RP−1 > 0. ii) There exists an asymptotically stable matrix A ∈ Fn×n such that A∗P + PA + R = 0. Source: [266]. Fact 16.22.13. Let A1 ∈ Rn1 ×n1 , A2 ∈ Rn2 ×n2 , B ∈ Rn1 ×m, and C ∈ Rm×n2 , assume that A1 ⊕ A2 is nonsingular, and assume that rank B = rank C = m. Furthermore, let X ∈ Rn1 ×n2 be the unique

1312

CHAPTER 16

solution of Then,

A1X + XA2 + BC = 0. rank X ≤ min {rank K(A1 , B), rank O(A2 , C)}.

Furthermore, if m = 1, then equality holds. Source: [858]. Remark: Related results are given in [2891, 2897]. Fact 16.22.14. Let A1 , A2 ∈ Rn×n, B ∈ Rn, C ∈ R1×n, assume that A1 ⊕ A2 is nonsingular, let X ∈ Rn×n satisfy A1X + XA2 + BC = 0, and assume that (A1 , B) is controllable and (A2 , C) is observable. Then, X is nonsingular. Source: Fact 16.22.13 and [2897]. Fact 16.22.15. Let A, P, R ∈ Rn×n, and assume that P and R are positive semidefinite, ATP + PA + R = 0, and N[O(A, R)] = N(A). Then, A is semistable. Source: [444]. Fact 16.22.16. Let A, V ∈ Rn×n, assume that A is asymptotically stable, assume that V is positive semidefinite, and let Q ∈ Rn×n be the unique, positive-definite solution to AQ+ QAT +V = 0. Furthermore, let C ∈ Rl×n, and assume that CVCT is positive definite. Then, CQCT is positive definite. Fact 16.22.17. Let A, R ∈ Rn×n, assume that A is asymptotically stable, assume that R ∈ Rn×n is positive semidefinite, and let P ∈ Rn×n satisfy ATP + PA + R = 0. Then, for all i, j ∈ {1, . . . , n}, there exist αi j ∈ R such that n ∑ P= αi j A(i−1)TRA j−1. i, j=1

ˆ Pˆ A+ ˆ Rˆ = 0, where Aˆ = In particular, for all i, j ∈ {1, . . . , n}, αi j = Pˆ (i, j) , where Pˆ ∈ Rn×n satisfies Aˆ TP+ ˆ C(χA ) and R = E1,1 . Source: [2472]. Remark: This is Smith’s method. See [859, 911, 1330, 1911] for finite-sum solutions of linear matrix equations. △ Fact 16.22.18. Let λ1 , . . . , λn ∈ C, assume that, for all i ∈ {1, . . . , n}, Re λi < 0, define Λ = n×n diag(λ1 , . . . , λn ), let k be a nonnegative integer, and, for all i, j ∈ {1, . . . , n}, define P ∈ C by ∫ ∞ △ 1 P= tk eΛt eΛt dt. k! 0 Then, P is positive definite, P satisfies the Lyapunov equation ΛP + PΛ + I = 0, and, for all i, j ∈ {1, . . . , n},    −1 k+1   . P(i, j) =  λi + λ j Source: For all nonzero x ∈ Cn,

x∗Px =

∫ 0

∞

tk ∥eΛt x∥22 dt

is positive. Hence, P is positive definite. Furthermore, note that ∫ ∞ (−1)k+1 k! P(i, j) = tk eλi t eλ j t dt = . (λi + λ j )k+1 0 Remark: See [575] and [1450, p. 348]. Related: Fact 10.9.20 and Fact 16.22.19. △ Fact 16.22.19. Let λ1 , . . . , λn ∈ C, assume that, for all i ∈ {1, . . . , n}, Re λi < 0, define Λ =

diag(λ1 , . . . , λn ), let k be a nonnegative integer, let R ∈ Cn×n, assume that R is positive semidefinite,

1313

LINEAR SYSTEMS AND CONTROL THEORY

and, for all i, j ∈ {1, . . . , n}, define P ∈ Cn×n by ∫ ∞ △ 1 tk eΛt ReΛt dt. P= k! 0 Then, P is positive semidefinite, P satisfies the Lyapunov equation ΛP + PΛ + R = 0, and, for all i, j ∈ {1, . . . , n},    −1 k+1 P(i, j) = R(i, j)   . λi + λ j If, in addition, I ⊙ R is positive definite, then P is positive definite. Source: Fact 10.25.16 and Fact 16.22.18. Related: Fact 10.9.20 and Fact 16.22.18. Note that P = Pˆ ⊙ R, where Pˆ is the solution to the Lyapunov equation with R = I. Fact 16.22.20. Let A, R ∈ Rn×n, assume that R ∈ Rn×n is positive semidefinite, let q, r ∈ R, where r > 0, and assume that there exists a positive-definite matrix P ∈ Rn×n that satisfies [A − (q + r)I]TP + P[A − (q + r)I] + 1r ATPA + R = 0. Then, the spectrum of A is contained in a disk centered at q + ȷ0 with radius r. Remark: The disk is an eigenvalue inclusion region. See [297, 1285, 2833] for related results concerning elliptical, parabolic, hyperbolic, sector, and vertical strip regions.

16.23 Facts on the Discrete-Time Lyapunov Equation Fact 16.23.1. Let A, R ∈ Fn×n. Then, the following statements are equivalent:

i) There exists a unique matrix Q ∈ Fn×n that satisfies Q = AQA∗ + R. ii) For all λ, µ ∈ spec(A), λµ , 1. iii) I − A ⊗ A is nonsingular. Now, assume that these conditions hold.Then, the following statements hold: iv) Q = vec−1 [(I − A ⊗ A)−1 vec R]. v) If R is Hermitian, then Q is Hermitian. vi) If R is positive semidefinite and A is discrete-time asymptotically stable, then A is positive semidefinite. In addition, (A, R) is controllable if and only if Q is positive definite. vii) If A is in companion form and R = En,n , then Q is positive definite and Toeplitz.

16.24 Facts on Realizations and the H2 System Norm Fact 16.24.1. Let x: [0, ∞) 7→ Rn and y: [0, ∞) 7→ Rn, assume that the integrals

∫∞

xT(t)x(t) dt and 0 yT(t)y(t) dt exist, and let xˆ : IA → 7 Cn and yˆ : IA → 7 Cn denote the Fourier transforms of x and y, respectively. Then, ∫ ∞ ∫ ∞ ∫ ∞ ∫ ∞ T ∗ T x (t)x(t) dt = xˆ (ω ȷ) xˆ(ω ȷ) dω, x (t)y(t) dt = Re xˆ∗ (ω ȷ)ˆy(ω ȷ) dω. ∫∞

0

−∞

0

−∞

0

Remark: These equalities are equivalent versions of Parseval’s theorem. The second equality fol-

lows from the first equality by replacing x with x + y. [ A min Fact 16.24.2. Let G ∈ R(s)l×m prop , where G ∼ C

B D

]

, and assume that, for all i ∈ {1, . . . , l}

and j ∈ {1, . . . , m}, G(i, j) = pi, j/qi, j , where pi, j , qi, j ∈ R[s] are coprime. Then, spec(A) =

l,m ∪ i, j=1

roots(pi, j ).

1314

CHAPTER 16

Fact 16.24.3. Let G ∼

Fact 16.24.4. Let G ∼

[

[

A

B

C

D

A

B

C

D

]

△

, let a, b ∈ R, where a , 0, and define H(s) = G(as + b). Then,

 −1   a (A − bI) B    . H ∼  a−1C D ] △ , where A is nonsingular, and define H(s) = G(1/s). Then,  −1  A H ∼  CA−1

−A−1B D − CA−1B

   .

Fact 16.24.5. Let G(s) = C(sI − A)−1B. Then,

G(ω ȷ) = −CA(ω2I + A2 )−1B − ω ȷC(ω2I + A2 )−1B. Fact 16.24.6. Let G(s) = C(sI − A)−1B, and define H(s) = sG(s). Then,

  A B H ∼  CA CB

Consequently,

   .

sC(sI − A)−1B = CA(sI − A)−1B + CB.

Fact 16.24.7. Let G(s) = C(sI − A)−1B, and assume that A is nonsingular. Then,

Fact 16.24.8. Let G =

[

[G

   ]   G12 ∼  G22   

G11 G21

Fact 16.24.9. Let G ∼

11

G21

G(s) = sCA−1 (sI − A)−1 B + G(0). [ A B ] ] ij ij G12 for all i, j = 1, 2. Then, , where G ∼ i j G22 C D

[

A

B

C

0

]

ij

ij

A11 0 0 0

0 A12 0 0

0 0 A21 0

0 0 0 A22

B11 0 B21 0

0 B12 0 B22

C11 0

C12 0

0 C21

0 C22

D11 D21

D12 D22

      .    

, where G ∈ Rl×m(s), and let M ∈ Rm×l . Then,

     A − BMC B   A − BMC BM  −1  , (I + GM) G ∼   . (I + GM) ∼  −C I C 0 [ A B ] Fact 16.24.10. Let G ∼ , where G ∈ Rl×m(s). If D has a left inverse DL ∈ Rm×l, then C D −1

  A − BDLC G ∼  − DL C L

 BDL   DL

satisfies GLG = I. If D has a right inverse DR ∈ Rm×l, then   A − BDRC BDR R G ∼  − DR C DR satisfies GGR = I.

  

1315

LINEAR SYSTEMS AND CONTROL THEORY

[

A C

B 0

]

be a SISO rational transfer function, and let λ ∈ C. Then, there exists a rational function H such that 1 G(s) = H(s) (s + λ)r Fact 16.24.11. Let G ∼

and such that λ is neither a pole nor a zero of H if and only if the Jordan form of A has exactly one block associated with λ, which ] r. [ is of size Fact 16.24.12. Let G ∼

A

C

B

. Then, G(s) is given by [ ] A − sI B . G(s) = (A − sI) C D D

Remark: See [307]. Remark: The vertical bar denotes the Schur complement. min

Fact 16.24.13. Let G ∈ F(s)n×m , where G ∼

[

A

B

C

D

] , and, for all i ∈ {1, . . . , n} and j ∈

{1, . . . , m}, let G(i, j) = pi j /qi j, where pi j, qi j ∈ F[s] are coprime. Then, n,m ∪

roots(qi j ) = spec(A).

i, j=1

Fact 16.24.14. Let A ∈ Rn×n, B ∈ Rn×m, and C ∈ Rm×n. If s ∈ C\spec(A), then

det[sI − (A + BC)] = det[I − C(sI − A)−1B] det(sI − A). If, in addition, n = m = 1 and s ∈ C, then det[sI − (A + BC)] = det(sI − A) − C(sI − A)AB. Source: Note that

] [ ][ ] sI − A B sI − A B I 0 det[I − C(sI − A) B] det(sI − A) = det = det C I C I −C I [ ] sI − A − BC B = det = det(sI − A − BC). 0 I −1

[

Remark: The last expression is used in [2054] to compute the frequency response of a transfer

function. Fact 16.24.15. Let A ∈ Rn×n, B ∈ Rn×m, C ∈ R p×n, and K ∈ Rm×p , assume that either m = 1 or

p = 1, and let s ∈ C. Then,

C(sI − A)A B = C[sI − (A + BKC)]A B. Source: Fact 3.19.11. Remark: This result shows that the invariant zeros of SISO, SIMO, and MISO transfer function are invariant under feedback. Related: Proposition 16.10.10. Problem:

Consider the case of MIMO transfer functions. Fact 16.24.16. Let A ∈ Rn×n, B ∈ Rn×m, C ∈ Rm×n, and K ∈ Rm×n, and assume that A + BK is nonsingular. Then, [ ] A B det = (−1)m det(A + BK)det[C(A + BK)−1B]. C 0 [ ] Hence, CA B0 is nonsingular if and only if C(A + BK)−1B is nonsingular. Source: Note that [ ] [ ][ ] [ ] A B A B I 0 A + BK B det = det = det = det(A + BK)det[−C(A + BK)−1B]. C 0 C 0 K I C 0

1316

CHAPTER 16

Fact 16.24.17. Let A1 ∈ Rn×n, C1 ∈ R1×n, A2 ∈ Rm×m, and B2 ∈ Rm×1, let λ ∈ C, assume that

λ is an observable eigenvalue of (A1 , C1 ) and a controllable eigenvalue of (A2 , B2 ), and define the dynamics matrix A of the cascaded system by [ ] A1 0 △ A= . B2C1 A2 Then, amultA(λ) = amultA1 (λ) + amultA2 (λ) and gmultA(λ) = 1. Remark: The eigenvalue λ is a cyclic eigenvalue of both subsystems as well as the cascaded system. In other words, λ, which occurs in a single Jordan block of each subsystem, occurs in a single Jordan block in the cascaded system. The Jordan blocks of the subsystems corresponding to λ are merged. Fact 16.24.18. Let G1 ∈ Rl1 ×m (s) and G2 ∈ Rl2 ×m (s) be strictly proper. Then,

[ ]

2

G1

2 2

G

= ∥G1∥H2 + ∥G2 ∥H2 . 2 H 2

Fact 16.24.19. Let G1 ∈ R

l×m1

(s) and G2 ∈ Rl×m2 (s) be strictly proper. Then,

∥[G1 G2 ]∥H2 = ∥G1∥2H2 + ∥G2 ∥2H2 . Fact 16.24.20. Let G1, G2 ∈ Rm×m (s) be strictly proper. Then,

[ ]

G1

G

= ∥[G1 G2 ]∥H2 . 2 H 2

△

Fact 16.24.21. Let G(s) =

α s+β ,

where β > 0. Then, |α| ∥G∥H2 = √ . 2β

△

Fact 16.24.22. Let G(s) =

Fact 16.24.23. Let G1 (s) =

α1 s+α0 , s2 +β1 s+β0

where β0 , β1 > 0. Then, √ α20 α2 ∥G∥H2 = + 1. 2β0 β1 2β1

α1 s+β1

and G2 (s) =

α2 s+β2 ,

where β1 , β2 > 0. Then,

∥G1G2 ∥H2 ≤ ∥G1∥H2 ∥G2 ∥H2 if and only if β1 + β2 ≥ 2. Remark: The H2 norm is not submultiplicative.

16.25 Facts on the Riccati Equation Fact 16.25.1. Assume that (A, B) is stabilizable, and assume that H defined by (16.16.8) has an imaginary eigenvalue λ. Then, every Jordan block of H associated with λ has even size. Source: Let P be a solution of (16.16.4), and let J denote the Jordan form of A[ − ΣP.] Then, there exists a △ Σˆ ˆ = ˆ nonsingular 2n × 2n block-diagonal matrix S such that H S−1 HS = J0 −J T , where Σ is positive △

semidefinite. Next, let Jλ = λIr + Nr be a Jordan block of J associated with λ, and consider the ˆ consisting of the rows and columns of λI − Jλ and λI + JT. Since (A, B) submatrix of λI − H λ is stabilizable, it follows that the rank of this submatrix is 2r − 1. Hence, every Jordan block of H associated with λ has even size. Remark: Canonical forms for symplectic and Hamiltonian matrices are discussed in [1763]. Fact 16.25.2. Let A, B ∈ Cn×n, assume that A and B are positive definite, let S ∈ Cn×n satisfy A = S ∗S, and define △ X = S −1 (SBS ∗ )1/2 S −∗.

1317

LINEAR SYSTEMS AND CONTROL THEORY

Then, X satisfies XAX = B. Source: [1391, p. 52]. Fact 16.25.3. Let A, B ∈ Cn×n, assume that either A or B is nonsingular. Then, there exist X, Y ∈ Cn×n such that XAY − YBX = I. Source: [107, pp. 37, 204–206]. Fact 16.25.4. Let A, B ∈ Cn×n, and assume that the 2n × 2n matrix ] [ A −2I A 2B − 21 A2 is simple. Then, there exists X ∈ Cn×n that satisfies X 2 + AX + B = 0. Source: [2727]. Fact 16.25.5. Let P, K, X ∈ Rn×n, and assume that P is positive definite and K is skew symmetric, and assume that XP − PX T = 2K. Then, there exists a symmetric matrix S ∈ Rn×n such that X = (K + S )P−1 . Furthermore, X is orthogonal if and only if there exists a symmetric matrix S ∈ Rn×n such that X = (K + S )P−1 and S 2 + SK + K TS = P2 + K 2 . Source: [631]. Remark: XP − PX T = 2K is the Moser-Veselov equation. Fact 16.25.6. Let A, B ∈ Fn×n, and assume that A and B are positive semidefinite. Then, the following statements hold: i) If A is positive definite, then X = A#B is the unique positive-definite solution of XA−1X − B = 0. ii) If A is positive definite, then X = 21 [−A+A#(A+4B)] is the unique positive-definite solution of XA−1X + X − B = 0. iii) If A is positive definite, then X = 21 [A + A#(A + 4B)] is the unique positive-definite solution of XA−1X − X − B = 0. iv) If B is positive definite, then X = A#B is the unique positive-definite solution of XB−1X = A. v) If A is positive definite, then X = 21 [A + A#(A + 4BA−1B)] is the unique positive-definite solution of BX −1B − X + A = 0. vi) If A is positive definite, then X = 21 [−A + A#(A + 4BA−1B)] is the unique positive-definite solution of BX −1B − X − A = 0. vii) If 0 < A ≤ B, then X = 12 [A + A#(4B − 3A)] is the unique positive-definite solution of XA−1X − X − (B − A) = 0. viii) If 0 < A ≤ B, then X = 21 [−A + A#(4B − 3A)] is the unique positive-definite solution of XA−1X + X − (B − A) = 0. ix) If 0 < A < B, X(0) is positive definite, and X(t) satisfies X˙ = −XA−1X + X + B − A, then

lim X(t) = 21 [A + A#(4B − 3A)].

t→∞

x) If 0 < A < B, X(0) is positive definite, and X(t) satisfies X˙ = −XA−1X − X + B − A,

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CHAPTER 16

then

lim X(t) = 12 [A + A#(4B − 3A)].

t→∞

xi) If 0 < A < B, X(0) and Y(0) are positive definite, and X(t) and Y(t) satisfy X˙ = −XA−1X + X + B − A, Y˙ = −YA−1Y − Y + B − A, then

lim X(t)#Y(t) = A#(B − A).

t→∞

Source: [1824]. Remark: A#B is the geometric mean of A and B. See Fact 10.11.68. Remark:

The solution X given by vii) is the golden mean of A and B.√In the scalar case with A = 1 and B = 2, the solution X of X 2 − X − 1 = 0 is the golden ratio 12 (1 + 5). See Fact 1.17.1. Fact 16.25.7. Let P0 ∈ Rn×n, assume that P0 is positive definite, let V ∈ Rn×n be positive semidefinite, and, for all t ≥ 0, let P(t) ∈ Rn×n satisfy ˙ = ATP(t) + P(t)A + P(t)VP(t), P(t) Then, for all t ≥ 0,

[ P(t) = e

tAT

P−1 0

∫

t

−

τA

e Ve

τAT

P(0) = P0 . ]−1 dτ etA.

0

Remark: P(t) satisfies a Riccati differential equation. Fact 16.25.8. Let Gc ∼

[

Ac

Bc

]

denote an nth-order dynamic controller for the standard 0 ˜ control problem. If Gc minimizes ∥G∥2 , then Gc is given by Cc

△

Ac = A + BCc − BcC − Bc DCc , △

Bc = (QC T + V12 )V2−1, △

T T Cc = −R−1 2 (B P + R12 ),

where P and Q are positive-semidefinite solutions to the algebraic Riccati equations T ˆ AˆTRP + PAˆ R − PBR−1 2 B P + R1 = 0, Aˆ E Q + QAˆTE − QC TV2−1CQ + Vˆ 1 = 0,

where Aˆ R and Rˆ 1 are defined by △ T Aˆ R = A − BR−1 2 R12 ,

△ T Rˆ 1 = R1 − R12 R−1 2 R12 ,

and Aˆ E and Vˆ 1 are defined by △ Aˆ E = A − V12V2−1C,

△ T Vˆ 1 = V1 − V12V2−1V12 .

Furthermore, the eigenvalues of the closed-loop system are given by ([ ]) A BCc mspec = mspec(A + BCc ) ∪ mspec(A − BcC). BcC Ac + Bc DCc [ A B ] c c Fact 16.25.9. Let Gc ∼ denote an nth-order dynamic controller for the discrete-time Cc 0 ˜ 2 , then Gc is given by standard control problem. If Gc minimizes ∥G∥ △

Ac = A + BCc − BcC − Bc DCc , △

Bc = (AQC T + V12 )(V2 + CQC T )−1,

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LINEAR SYSTEMS AND CONTROL THEORY △

Cc = −(R2 + BTPB)−1 (RT12 + BTPA), and the eigenvalues of the closed-loop system are given by ([ ]) A BCc mspec = mspec(A + BCc ) ∪ mspec(A − BcC). BcC Ac + Bc DCc [ A B ] c c Now, assume that D = 0 and Gc ∼ C D . Then, c

c

△

Ac = A + BCc − BcC − BDcC, △

Bc = (AQC T + V12 )(V2 + CQC T )−1 + BDc , △

Cc = −(R2 + BTPB)−1 (RT12 + BTPA) − DcC, △

Dc = (R2 + BTPB)−1 [BTPAQC T + RT12 QC T + BTPV12 ](V2 + CQC T )−1, and the eigenvalues of the closed-loop system are given by ([ ]) A + BDcC BCc mspec = mspec(A + BCc ) ∪ mspec(A − BcC). BcC Ac In both cases, P and Q are positive-semidefinite solutions to the discrete-time algebraic Riccati equations P = AˆTRPAˆ R − AˆTRPB(R2 + BTPB)−1BTPAˆ R + Rˆ 1 , Q = Aˆ E QAˆTE − Aˆ E QC T (V2 + CQC T )−1CQAˆTE + Vˆ 1, where Aˆ R and Rˆ 1 are defined by △ T Aˆ R = A − BR−1 2 R12 ,

△ T Rˆ 1 = R1 − R12 R−1 2 R12 ,

and Aˆ E and Vˆ 1 are defined by △ Aˆ E = A − V12V2−1C,

△ T Vˆ 1 = V1 − V12V2−1V12 .

Source: [1289].

16.26 Notes Linear system theory is treated in [574, 2349, 2726, 2907]. Time-varying linear systems are considered in [827, 2349], while discrete-time systems are emphasized in [1360]. The PBH test is given in [1349]. Spectral factorization results are given in [760]. Stabilization aspects are discussed in [934]. Observable asymptotic stability and controllable asymptotic stability were introduced and used to analyze Lyapunov equations in [2475]. Zeros are treated in [39, 1037, 1573, 1583, 1915, 2205, 2359, 2404]. Matrix-based methods for linear system identification are developed in [2778], while stochastic theory is considered in [1310]. Solutions of the LQR problem under weak conditions are given in [1156]. Solutions of the Riccati equation are considered in [901, 1189, 1701, 1707, 1735, 1736, 1973, 2306, 2883, 2895, 2900]. Proposition 16.16.16 is based on Theorem 3.6 of [2912, p. 79]. A variation of Theorem 16.18.1 is given without proof by Theorem 7.2.1 of [1511, p. 125]. There are numerous extensions to the results given in this chapter relating to various generalizations of (16.16.4). These generalizations include the case where R1 is indefinite [1188, 2892, 2894] as well as the case where Σ is indefinite [2386]. The latter case is relevant to H∞ optimal control theory [432]. Additional extensions include the Riccati inequality ATP + PA + R1 − PΣP ≥ 0 [2297, 2385, 2386, 2387], the discrete-time Riccati equation [18, 1360, 1503, 1735, 2297, 2899], and fixed-order control [1493].

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100

80

60

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20

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1970

1980

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2000

2010

Author Index Abdessemed, A. 885 Abel, U. 68, 76, 86, 249 Ablamowicz, R. 1206 Abou-Kandil, H. xxi Abramovich, S. 47, 747 Abu-Omar, A. 866 Aceto, L. 613, 619, 727, 1205 Acton, F. S. 242 Adamchik, V. 1170 Adegoke, K. 103, 1079 Aebi, C. 31, 49, 55, 104, 613 Afriat, S. 415, 595 Agaev, R. 1233 Agarwal, R. P. 1248 Ahlbrandt, C. D. 1319 Ahn, E. 1213 Aigner, M. 35, 52 Aitken, A. C. xxii Aizenberg, L. 948 Ajibade, A. O. 341 Akdeniz, F. 817 Al-Ahmar, M. 389 Al-Rashed, A. M. 852 Al Zhour, Z. 700, 821, 827, 873 Alaca, A. 119, 1092 Alaca, S. 119, 1092 Albadawi, H. 765, 766, 772, 865, 866, 870, 871, 876–879, 903 Albert, A. A. 571 Albert, A. E. 633, 637, 638, 665, 831, 911 Aldaz, J. M. 210, 855 Aldrovandi, R. xxi, 108, 112, 539, 615, 976, 1009 Aleksiejczuk, M. 667 Alexanderson, G. L. 530 Alexandrescu, P. 153 Alfakih, A. Y. 861 Alic, M. 744, 745 Alikhani, A. 855 Aling, H. 1319 Aljanaideh, K. 346, 948 Almkvist, G. 44, 45, 1070, 1080 Alpargu, G. 212, 220, 792 Alperin, R. C. 396 Alsina, C. 126, 133, 135, 141, 149, 157, 161, 168, 177, 212, 215, 222, 225, 250, 252, 446, 452, 456, 466, 472, 480, 484–486, 489, 491, 495, 1095 Alt, A. 177, 186, 480, 484 Altmann, S. L. xxi, 394, 438, 439 Altshiller-Court, N. 472, 480, 491 Alzer, H. 203, 250, 254, 1025, 1032, 1050 Amdeberhan, T. 86, 334, 1032, 1070, 1083, 1104 Amghibech, S. 776 Aminghafari, M. 608 Andalafte, E. Z. 856 Andelic, M. 424

Anderson, B. D. O. xxi, 605, 745, 793, 959, 1230, 1319 Anderson, E. C. 245 Anderson, G. 123, 124, 227, 249 Anderson, G. D. 1000 Anderson, M. 28, 370 Anderson, T. W. 740 Anderson, W. N. 590, 657, 658, 744, 760, 819, 831 Ando, T. 151, 152, 154, 177, 180, 592, 719, 721, 738, 740–742, 744–746, 757, 766, 777, 779, 780, 806, 807, 819, 824, 825, 827, 830, 831, 870, 871, 883, 893, 904, 907, 1213–1215, 1218, 1221–1223, 1311 Andreescu, T. 25, 28, 29, 31–33, 35, 36, 39, 42, 45–52, 55, 68, 75, 86, 92, 100, 103, 105, 119, 122, 124, 132, 133, 135–138, 144, 148–153, 155, 156, 160, 161, 164, 166–169, 173, 177, 178, 180, 183, 185, 186, 190, 191, 194, 197, 198, 201, 202, 207–211, 213, 215–217, 225–229, 232, 238, 239, 242–244, 249, 250, 252, 258, 261, 266, 269, 271–273, 327, 329, 355, 380, 386, 428, 429, 442, 443, 452, 456, 466, 472, 480, 484, 486, 489, 491–496, 736, 849, 855, 934, 944, 949–951, 954, 955, 958–963, 967, 968, 971, 986, 1003, 1008, 1023, 1025, 1031, 1047, 1051, 1057, 1073, 1079, 1081, 1087, 1089, 1090, 1115, 1127–1129, 1131, 1141, 1145, 1154, 1161–1163, 1226, 1239, 1317 Andreoli, M. 1027, 1100 Andrews, G. E. 33, 35, 40, 86, 92, 114, 122, 203, 208, 210, 238, 239, 254, 616, 930, 943, 960, 968, 971, 976, 989, 992, 996, 1000, 1003, 1021, 1039, 1044, 1049, 1053, 1056, 1058, 1060, 1091, 1121, 1157, 1161, 1162, 1176–1178 Andrica, D. 266, 271–273, 442, 443, 452, 472, 491, 493, 496 Andriopoulos, S. P. 466 Andruchow, E. 865, 882 Angel, E. xxi, 396 Anghel, N. 329 Anglesio, J. 249 Anh, T. Q. 134, 139, 145, 151, 153–155, 157, 160, 161, 164, 167, 169, 173, 177, 180, 183–185, 187, 193, 196–198, 201, 202, 206, 217, 220, 489 Anonymous 36, 39, 40, 49, 75, 105, 154, 173, 188, 269, 276, 334, 341, 466, 523, 619, 962, 966, 974, 1000, 1001, 1080 Antoulas, A. C. 1310

Apagodu, M. 32, 68 Apelblat, A. 1105, 1113 Aplevich, J. D. xxi, 563 Apostol, T. M. xl, 27, 75, 116, 215, 226, 727, 935, 937, 939, 949, 979, 980, 983, 1058, 1091, 1206 Apostolopoulos, G. 160, 456, 466, 472, 480, 1110 Araki, H. 767, 871, 875 Araujo, J. 609 Ardila, F. 334, 974 Arimoto, A. 615 Arnold, B. 202, 216, 276, 360–362, 384, 466, 489, 582, 583, 593, 594, 694, 721, 722, 752, 805, 831, 904, 911, 912 Arponen, T. 1205 Arsigny, V. 1215 Artin, M. 436, 440 Artzrouni, M. 1242 Arvanitoyeorgos, A. 440 Ash, A. 436 Ash, P. 149 Ashkartizabi, M. 608 Asiba, I. 137, 164, 191 Asiru, M. A. 39 Askey, R. 35, 86, 92, 114, 203, 208, 210, 238, 239, 254, 930, 943, 960, 968, 971, 989, 992, 996, 1000, 1003, 1039, 1044, 1049, 1053, 1056, 1058, 1060, 1121, 1157, 1161, 1162, 1176–1178 Askey, R. A. 1178 Aslaksen, H. 357, 382, 438, 440, 528, 529 Asner, B. A. 1225, 1229 Au-Yeung, Y.-H. 619, 738, 797, 799 Audenaert, K. M. R. 699, 767, 770, 882, 890 Aujla, J. S. 128, 271, 274, 739, 806, 813, 822, 827, 828, 830, 855, 870, 876, 877, 880 Aupetit, B. 1220 Avriel, M. 798 Axelsson, O. xxi Axler, C. 35, 966 Ayache, N. 1215 Ayres, F. 236 Azar, L. E. 220 Aziz, A. 1237, 1238 Baez, J. C. 434, 438 Bagby, R. 1000, 1170 Bagdasar, O. 117, 142, 143, 165, 199, 208 Bai, Z. 762 Bai, Z.-J. 340, 661 Bai, Z.-Z. 340, 661 Bailey, D. H. 50, 51, 54, 68, 94, 117, 210, 242, 495, 540, 963, 1000, 1015,

1434 1017, 1049, 1060, 1070, 1087, 1097, 1128, 1141, 1154 Bailey, D. W. 535 Bailey, H. 446, 452 Baillie, R. 1060 Bak, J. 94, 276, 1070, 1081, 1083 Baker, A. 382, 428, 438, 440, 607, 1193, 1212 Bakke, M. 200, 201 Baksalary, J. K. 320, 400, 402, 406, 415, 431, 568, 635, 662, 667, 673, 674, 738, 740, 814, 816, 817, 825 Baksalary, O. M. 313, 314, 317–320, 331, 376, 377, 379, 380, 397, 400, 402, 404–413, 415, 418, 431, 573, 589–591, 595, 630, 631, 633, 635, 638, 641–644, 647, 648, 650–654, 656, 657, 662, 667, 672–674, 677, 733, 736, 772, 796, 815 Bakula, M. K. 205 Ball, K. 859, 879 Ballantine, C. S. 377, 571, 608, 609, 831, 1228, 1311 Banerjee, A. 528 Banerjee, S. 497 Bang-Jensen, J. xxi, 26 Bani-Domi, W. 887–889 Bapat, R. 723 Bapat, R. B. 225, 320, 329, 423, 426, 539, 544, 628, 667, 821, 910 Bar-Itzhack, I. Y. 396, 1206 Barbara, R. 164, 167, 169, 173, 472, 489 Barbeau, E. J. 29, 31, 32, 36, 41, 43, 50, 56, 119, 124, 131, 133, 134, 146, 148, 149, 173, 177–180, 183, 184, 466, 480, 492, 1234, 1235 Barboianu, C. 1224 Barbu, C. 452 Baric, J. 47, 747 Barnes, E. R. 536 Barnett, J. K. R. 149, 456, 491 Barnett, S. xxi, 337, 350, 420, 520, 544, 610–612, 614, 619, 632, 669, 730, 734, 911, 1229, 1313 Barrett, W. 324, 424, 723, 729, 782, 822 Barria, J. 577 Bart, H. 1315 Baruh, H. 393, 438, 1208 Barvinok, A. 178–180, 307, 310, 788, 789, 803, 937–939 Barza, S. 213 Basarab, M. 480 Basor, E. L. 1060 Bataille, M. 48, 92, 398, 452, 480, 489 Bates, R. G. 427 Batinetu-Giurgiu, D. M. 177, 183, 218, 466, 480, 484, 1131 Batir, N. 51, 128, 216, 1003, 1014, 1032, 1048, 1050, 1070, 1071 Batra, P. 1224 Bau, D. xxi Bauer, F. L. 912 Bayard, D. S. 593 Bayat, M. 523 Bayless, J. 966 Bazaraa, M. S. 543, 922, 924 Beams, R. 608 Beardon, A. F. 38, 56 Beauregard, R. A. 41, 1008 Beavers, A. N. 608 Bebiano, N. xxi, 594, 777, 1215–1217 Beckenbach, E. F. 276, 834 Becker, R. I. 619, 797

AUTHOR INDEX

Beckner, W. 879 Beckwith, D. 1008, 1102 Beer, G. 940 Bekjan, T. N. 790 Bekker, P. A. 751, 772, 831 Belbachir, H. 112, 979, 980, 1009 Belinfante, J. G. 1247 Belitskii, G. R. 912 Bell, W. W. 986, 989, 990, 992, 1000, 1113, 1162, 1163 Bellman, R. 276, 350, 734, 741, 834, 1212, 1239 Belmega, E.-V. 181, 773 Ben-Ari, I. 124, 195, 245, 1084–1086 Ben-Israel, A. 628, 631, 642, 667, 670, 674, 679, 799 Ben Taher, B. 1206 Ben-Tal, A. 308, 362 Bencze, M. 40, 41, 45, 46, 48, 49, 58, 68, 100, 103, 120, 121, 125, 128, 132–134, 137, 138, 143, 144, 149, 157, 165, 173, 177, 180, 183, 189, 194, 200, 211, 214, 223, 225, 228, 229, 237, 250, 252, 261, 271, 446, 452, 456, 466, 472, 480, 529, 1073, 1131, 1238 Benitez, J. 413, 414, 440, 575, 632, 649, 655, 669, 670, 677, 678 Benjamin, A. T. 23, 32, 37, 48, 50, 58, 67, 68, 75–77, 86, 92, 94, 100, 103, 106–108, 111, 114, 115, 425, 1009, 1070, 1079 Bennett, G. 47, 51, 141, 186, 1223 Benson, C. T. 436 Bercu, G. 249 Berele, A. 53 Berg, L. 330 Berge, C. xxi Berinde, V. 135, 138 Berkovitz, L. D. 362, 940 Berman, A. 375, 440, 539, 542, 544, 1232, 1233 Berndt, B. 1080 Berndt, B. C. 242 Bernhardsson, B. 1228 Bernstein, D. S. 327, 346, 382, 387, 408, 410, 411, 424, 439, 440, 573, 597, 604, 647, 650–652, 779, 789, 858, 912, 948, 1206, 1212, 1218, 1220, 1228, 1232, 1233, 1247, 1306, 1309, 1311–1313, 1319 Bernstein, H. J. 892 Besenyei, A. 220, 751 Beumer, M. G. 1152, 1154 Beyer, W. A. 245 Bhagwat, K. V. 716, 741, 742, 1213 Bhat, S. P. 439, 597, 697, 1207, 1232, 1247, 1312 Bhatia, A. 697 Bhatia, R. 126, 328, 361, 362, 384, 427, 537, 578, 579, 583, 584, 716, 718, 719, 721, 722, 725–729, 737, 739, 742, 744, 760, 766, 767, 771, 777, 802, 803, 805–807, 809, 812, 820, 831, 847, 853, 862, 866, 867, 870–876, 879–883, 885–887, 893, 894, 898, 903, 905, 941, 942, 952, 1215, 1217, 1219, 1221, 1223 Bhattacharya, R. 576 Bhattacharyya, S. P. 1224, 1312, 1319 Bhattarcharyya, S. P. 1312 Bhaya, A. 1248 Bhimasankaram, P. 431, 640, 648 Bibak, Kh. 100, 103

Bicknell, M. R. 346 Bidkham, M. 1235 Biggs, N. xxi, 596 Binding, P. 619 Binmore, K. xxi Bjorck, A. 679, 911 Blanes, S. 1189, 1213, 1306 Blizard, W. D. 118 Bloch, E. D. 22, 118 Blondel, V. 1242 Blumenthal, L. M. 495 Blundon, W. J. 452 Boas, H. P. 118 Boche, H. xxi, 360 Boehm, W. 1241 Boese, F. G. 1235 Bogosel, B. 214 Bojanczyk, A. W. 532 Bollobas, B. xxi Bona, M. 68, 86, 94, 976 Bonar, D. D. 45, 68, 197, 217, 963, 965, 966, 1022–1025, 1027, 1044, 1046, 1047, 1056, 1062, 1070, 1074, 1097 Bondar, J. V. 216, 489 Bonfuglioli, A. 1189, 1213 Boros, G. 67, 68, 75, 86, 92, 239, 262, 968, 979, 980, 1008, 1024, 1032, 1036, 1047, 1061–1063, 1070, 1099, 1104, 1109, 1110, 1115, 1130, 1133, 1134, 1141, 1148, 1149, 1152, 1159, 1160, 1170, 1174 Borre, K. xxi Borwein, D. 1119, 1141 Borwein, J. M. 50, 51, 54, 68, 94, 117, 210, 219, 228, 229, 231, 242, 362, 495, 540, 543, 592, 593, 721, 741, 787, 935, 938, 939, 963, 968, 969, 1000, 1015, 1017, 1024, 1049, 1060, 1070, 1087, 1094, 1097, 1119, 1128, 1141, 1154 Borwein, P. 36, 239, 996, 997 Borwein, P. B. 968, 969, 1049, 1070 Bosch, A. J. 608, 619 Bostan, A. 529, 953 Bottcher, A. 595, 737, 884 Bottema, O. 446, 452, 466, 484 Boullion, T. L. 638, 666, 679 Bourin, J.-C. 216, 593, 739, 746, 759, 768, 769, 771–773, 779, 805, 813, 827, 865, 869, 870, 876, 886, 888, 900, 1216 Bourque, K. 727 Bowen, R. M. 691 Boyadzhiev, K. N. 57, 68, 75, 111, 975, 980, 1008, 1009 Boyd, S. xxi, 362, 756, 866 Bozkurt, D. 907 Bracken, P. 1036 Bradie, B. 1098, 1104, 1144 Bradley, C. J. 492 Bradley, D. 1070 Braslavsky, J. H. 1156 Brassesco, S. 971 Bremner, A. 34 Brennan, M. 436 Brenner, J. L. 359, 440, 535, 606 Bresler, Y. 824, 828 Bressoud, D. M. 67, 121, 617, 1070 Bressoud, M. 112, 114, 967, 976, 1009 Brewer, J. W. 701 Brickman, L. 789 Brockett, R. 1319 Brockett, R. W. 803, 1211, 1312 Brookfield, G. 269

AUTHOR INDEX

Brooks, B. P. 952 Brothers, H. 1023 Brothers, H. J. 127, 955 Broughan, K. A. 35 Brown, G. 224 Brown, J. W. 271, 276, 927–929 Browne, E. T. 831 Bru, R. 418, 604 Brualdi, R. A. xxi, 326, 330, 332, 362, 527, 535, 540, 792, 1230 Bruckman, P. 1044 Bruckman, P. S. 36, 68, 75, 86, 94, 993 Brychkov, Yu. A 1178 Brzezinski, A. 1309 Buckholtz, D. 415, 595 Bukor, J. 963 Bullen, P. S. 123, 127, 144, 173, 203–205, 222, 227, 228, 249, 252, 264, 276, 489, 745, 835, 850, 860 Bullo, F. xxi, 541 Bultheel, A. 544 Burch, J. M. xxi Burns, F. 667 Bushell, P. J. 810 Butzer, P. L. 1031

Cater, F. S. 607 Catoiu, S. 53 Catral, M. 675, 678 Ceausu, T. 124, 138, 195, 209, 225 Cerny, A. 396, 651 Cerone, P. 222 Chabrillac, Y. 798 Chakerian, G. D. 496 Chamberland, M. 86, 100, 103, 105, 122, 130, 132, 148, 179, 244, 245, 491, 986, 987, 1000, 1046, 1086, 1087, 1089 Chan, N. N. 738 Chandrasekar, J. 424 Chandrupatla, T. R. 114, 115, 1008, 1080 Chang, G. 86 Chang, X.-Q. 268, 801, 1058 Changjian, Z. 125, 138, 225, 237, 252, 452, 852 Chapellat, H. 1224 Chapman, S. T. 1087, 1094 Chartrand, G. xxi Chatelin, F. xxi Chattot, J.-J. xxi Chaturvedi, N. A. 387 Chebotarev, P. 1233 Caccia, D. 249 Chehab, J.-P. 762 Cahill, N. D. 425 Cai, K.-Y. 773, 774, 781, 782, 811, 826 Chellaboina, V. xxi, 440, 697, 1232, 1233 Cai, L.-x. 881 Chellaboina, V.-S. 868, 912, 1287 Cain, B. E. 567, 1310, 1311 Chen, A. 100, 103 Cairns, G. 31, 49, 55, 104, 613 Chen, B. 76, 1070 Calkin, N. J. 86 Chen, B. M. xxi, 311, 312, 315, 345, Callan, D. 55, 86, 321 582, 592, 619 Callebaut, D. K. 221 Chen, C.-J. 852 Callier, F. M. 632 Chen, C.-P. 52, 249, 964 Cameron, N. T. 425 Chen, C. T. 1310 Cameron, P. J. 49, 104, 1090, 1091 Chen, D. 699 Camouzis, E. 1234, 1248 Chen, G. 699 Campbell, S. L. 632, 639, 667, 671, Chen, H. 86, 100, 103, 123, 203, 217, 675, 679, 1227, 1247 243, 959, 980, 1036, 1053, 1070, Can, M. 466 1132, 1143, 1152 Can, V. Q. B. 134, 139, 145, 151, Chen, J. 891 153–155, 157, 160, 161, 164, 167, Chen, J.-Q. 436 169, 173, 177, 180, 183–185, 187, Chen, K.-W. 179 193, 196–198, 201, 202, 206, 217, Chen, L. 765, 905 220, 489 Chen, S. 824 Cancan, M. 128, 216 Chen, W. Y. C. 86 Cao, J. 144, 228 Chen, Z. 1052, 1056 Cao, L. 704, 737 Cheng, C.-M. 776, 777, 807, 907 Capelas de Oliveira, E. 994, 1052, Cheng, D. 701 1130, 1144, 1166 Cheng, H.-W. 1206 Cardoso, J. R. 1317 Cheng, S. 321, 377, 379, 636, 638, 641, Carlen, E. 859, 879 650, 651, 653, 654, 660, 668, 672, Carlen, E. A. 721, 790, 791 820 Carlson, B. C. 143, 968, 969, 1083, Cheung, W.-S. 249, 852 1102 Chiacchio, A. O. 1166 Carlson, D. 661, 667, 721, 831, 1229, Chien, M.-T. 424 1310, 1311 Chlebus, E. 963, 964 Carothers, N. L. 27 Choi, D. 856 Carpenter, J. A. 585 Carr, G. S. 144, 161, 212, 238, 456, 971 Choi, J. 75, 76, 979, 980, 996, 1001, 1003, 1024, 1030, 1032–1036, 1071, Carrier, G. F. 276, 983, 1015, 1017, 1074, 1130, 1132, 1146, 1151, 1152, 1032, 1040, 1047, 1062, 1104, 1105, 1154, 1166, 1170, 1177 1109, 1161 Choi, M.-D. 413, 420, 619 Carter, N. 440 Choi, S. 36, 996, 997 Cartier, P. 1247 Chollet, J. 758 Cartwright, D. I. 203, 1247 Choudhry, A. 34, 148, 608 Casas, F. 1189, 1213, 1306 Christodoulou, M. A. 529 Castellanos, D. 34, 51, 52, 54, 257, Chrystal, G. 120, 130, 131, 148, 149, 259, 986, 987, 1014, 1024, 1036, 178, 179, 974 1049, 1077, 1144 Castro-Gonzalez, N. 667, 675, 676, 678 Chu, M. T. 635, 662, 663, 737 Caswell, H. xxi, 610 Chu, W. 68, 75, 76, 86, 92, 1073

1435 Chu, X.-G. 58, 67, 68, 94, 161, 452, 472 Chu, Y. 143 Chu, Y.-M. 143, 266, 1080 Chuai, J. 688, 692 Chuang, I. L. xxi Chui, N. L. C. 1240 Chung, F. R. K. xxi Churchill, R. V. 271, 276, 927–929 Ciesielski, K. C. 943 Cilleruelo, J. 976 Cirtoaje, V. 134, 139, 145, 148, 150–157, 160, 161, 164, 167–169, 173, 177, 180, 183–187, 193, 194, 196–198, 201, 202, 206, 217, 218, 220, 452, 489 Cizmesija, A. 212, 213 Clark, J. 26, 576, 577, 619 Clarke, G. 100 Clarke, R. J. 75, 446, 1119 Clements, D. J. 1319 Climent, J. J. 604 Cline, R. E. 319, 321, 430, 638, 639, 666 Cloot, A. H. J. J. 189 Cloud, M. J. 276 Coakley, E. S. 434 Cofman, J. 105 Cohen, J. E. 1218, 1220 Cohn, P. M. 328 Cohoon, D. K. 327 Collins, E. G. 700, 1319 Combot, T. 529 Comtet, L. 37, 42, 43, 52, 55, 67, 68, 75, 86, 94, 103, 106, 108, 114, 130, 131, 274, 424, 441, 493, 495, 613, 950, 968, 971, 981, 983, 984, 989, 992, 1014, 1020, 1021, 1090 Conrad, K. 124, 195 Constales, D. 630 Contreras, M. xxi, 1233 Conway, J. C. 178, 187, 436, 438 Cook, W. J. 387 Coope, I. D. 593, 594, 911 Cooper, C. 130 Cooperstein, B. 701 Corach, G. 399, 595, 650, 651, 656, 865, 881, 882 Corduneanu, C. 957 Corless, M. J. xxi Coronel, A. 145, 161, 209 Cortes, J. xxi, 541 Cossali, G. E. 86 Costa, P. J. 953 Cottle, R. W. 788, 798, 1224 Cover, T. M. xxi, 231, 721, 776, 782, 1172 Cox, D. A. 92 Crabtree, D. E. 535, 601 Craciun, L. 200 Crassidis, J. L. 386 Crasta, N. 1207 Crawford, C. R. 798 Crilly, T. 529 Cross, G. W. 608 Crossley, M. D. 392 Crouzeix, J.-P. 798 Cui, J. 608, 609 Cullen, C. G. 604, 932, 952 Culver, W. J. 1247 Curgus, B. 614 Curtain, R. F. 1219 Curtis, M. L. 394, 438, 440 Cusamano, A. 1079

1436 Cvetkovic, D. xxi, 362, 527 Cvetkovski, Z. 47, 48, 125, 134, 137, 144, 145, 148, 151, 153–157, 160–162, 164, 165, 167, 169, 173, 177, 183, 185, 187, 190, 193, 195, 199, 201, 224, 225, 227, 237, 249, 452, 456, 466, 472, 480, 484, 485, 489, 492 Cvijovic, D. 1033, 1070

196, 198, 199, 216, 218, 224, 226, 252, 361, 446, 452, 466, 472, 480, 485, 489, 492 Delorme, J.-J. 34 DeMaio, J. 44, 56 DeMarco, C. L. 536 Dembo, A. 782 Demirturk, B. 100, 103 Demmel, J. W. xxi Dence, J. B. 1044 Dence, T. P. 1044 da Fonseca, C. M. 243, 424 da Providencia, J. xxi, 777, 1215–1217 Deng, C. 675, 740 Deng, C. Y. 402, 407, 674–678 Da Silva, J. A. D. 699 Denkowski, Z. 922, 923 Da Silva, R. 114, 1091 Denman, E. D. 608 Daboul, P. J. 438 DePrima, C. R. 605 Daboul, S. 111 D’Errico, J. R. 425 Dadipour, F. 748, 850–852, 856 Desbrow, D. 238 Dahlquist, G. 1232 Deshpande, M. N. 100, 103 Dai, H.-H. 43, 67, 75, 76, 236, 237, Desoer, C. A. 1219 262, 981, 984, 1019, 1046, 1109, 1114, 1120, 1123, 1154, 1161, 1163, DeTemple, D. W. 963 Deutsch, E. 887, 1219, 1235, 1236 1166, 1168 Devaney, R. L. xxi Dale, P. xxi, 362 Dey, A. 723 Dalyay, P. P. 183, 480 Dhillon, I. S. 791 Dana-Picard, T. 1097, 1101, 1104, 1112–1115, 1118, 1130, 1133, 1157 Dhrymes, P. J. xxi Diaz-Barrero, J. L. 40, 46, 76, 92, 94, Dancis, J. 569 100, 103, 127, 153, 180, 189, 197, D’Andrea, R. 902, 1223 202, 252, 452, 1235 Daners, D. 1117 Dieci, L. 1193, 1214 D’Angelo, H. 1319 Diestel, R. xxi D’Angelo, J. P. 271, 855 Dil, A. 113, 976, 985, 992, 1010 Daniel, J. W. xxii, 362 Dilcher, K. 75, 1008 Dankovic, B. M. 964 Dilip, A. S. A. 1319 Dannan, F. M. 214, 768, 778, 779 Diminnie, C. R. 856 Darus, M. 275 Dinca, M. 452, 480 Dasgupta, S. 1210 Dines, L. L. 789 Datko, R. 1231 Ding, F. 1205 Datta, B. N. xxi, 1310 Ding, J. 527, 909 Dattorro, J. xxi, 318, 326, 581, 1092 Dittmer, A. 386, 387, 691 Daubechies, I. 1242 Dixon, G. M. 438 Davidson, K. R. 922–924 Dixon, J. D. 775 Davies, E. B. 885 Djaferis, T. E. 1312 Davis, C. 875, 882 Djokovic, D. Z. 355, 575, 607, 609, 776 Davis, P. J. xxii, 446, 615, 616 Djordjevic, D. S. 377 Davison, E. J. 1228 Djordjevic, R. Z. 452, 466, 484 Dawlings, R. J. H. 609 Dobrushkin, V. A. 41, 1008 Day, J. 805, 1189, 1213 Dokovic, D. Z. 570, 572, 577, 605 Day, P. W. 216 Dolotin, V. 701 de Andrade Bezerra, J. 377 Dombre, E. xxi De Angelis, V. 971 Donoghue, W. F. 726, 727, 822, 831 de Boor, C. 362 Donsig, A. P. 922–924 de Groen, P. P. N. 868 Dopazo, E. 667 de Hoog, F. R. 636 Dopico, F. M. 428, 434, 606 De Hoyos, I. 945 Doran, C. 387, 438, 691, 858 de Launey, W. 700 Doran, C. J. L. 387, 438, 691, 858 De Moor, B. 1319 Dorst, L. 438, 858 de Pillis, J. 760, 785 Dospinescu, G. 452 De Pillis, L. G. 519 Dou, Y.-N. 225, 762 de Seguins Pazzis, C. 1211 Doughty, S. P. 1207 De Silva, V. 945 Douglas, R. G. 714 de Souza, E. 1312 De Souza, P. N. 314, 318, 355, 356, 738 Doyle, J. C. 1156, 1228 Doyle, P. G. xxi de Vries, H. L. 890 Drachman, B. C. 276 DeAlba, L. M. 695, 711, 1310, 1311 Dragan, M. 452 Debbah, M. 181, 773 Dragan, V. 1319 Debnath, L. 51, 128, 145, 225 Dragomir, S. S. 140, 212, 217, Decell, H. P. 631 220–224, 226, 228, 230, 271, 272, Defant, A. 843, 864 274, 275, 584, 852, 855, 856, 944 Deistler, M. 1319 Dragovic, V. 32 Del Buono, N. 428 Drazin, M. P. 522 Delbourgo, R. 438 Delgado, R. V. 133, 134, 137–139, 150, Drissi, D. 144, 728 Drivaliaris, D. 643 151, 155, 160, 162, 164, 165, 167–169, 173, 177, 183, 184, 193, Drnovsek, R. 419, 420

AUTHOR INDEX

Drukker, M. 252 Drury, S. 1104, 1210 Drury, S. W. 429, 729, 812, 888 Du, H. 225, 740, 762 Du, H.-K. 751, 761, 902 Du, K. 908 Duarte, A. L. 777 Dubeau, F. 37 Duc, P. H. 160, 173 Duffin, R. J. 819 Duistermaat, J. J. 938, 942, 1212 Dukic, D. 116, 139, 164, 167, 177, 201, 253, 361, 362 Duleba, I. 1306 Dullerud, G. E. xxi Dumas, P. 953 Dummit, D. S. 370, 371, 434, 436, 616, 619 Dunbar, S. R. 169 Dunkl, C. F. 855 Duren, P. 27, 127, 271, 386, 960, 962, 971, 979, 980, 996, 1030, 1039, 1048, 1057, 1058, 1060, 1082, 1121 Dutkay, D. E. 1000 Dym, H. 275, 330, 333, 335, 494, 544, 568, 594, 595, 667, 714, 738, 799, 803, 816, 883, 902, 937, 940, 949, 1225, 1310 Dyson, F. J. 1065 Early, E. 966 Edelman, A. 727 Edelman, A. S. 896 Edwards, C. H. 926 Edwards, H. M. 997 Efthimiou, C. 1074 Efthimiou, C. J. 232, 972, 974 Egecioglu, O. 271, 272, 994 Egerstedt, M. 541 Eggleston, H. G. 362 Egozcue, J. J. 103 Ekhad, S. B. 334 Ekl, R. L. 34, 35 El-Haddad, M. 866 Elezovic, N. 967, 971 Elsner, L. 94, 352, 379, 536, 584, 694, 695, 697, 699, 733, 763, 823, 893, 941, 942, 1185, 1219, 1220 Embree, M. 1219 Engel, A. 31, 32, 67, 68, 100, 111, 121, 130, 133, 134, 137, 144, 150, 153, 164, 167, 168, 173, 177, 183, 193, 194, 198, 200, 206, 213, 216, 271, 272, 452, 472, 480, 484, 486, 489, 491, 1079 Engo, K. 1189, 1213 Epstein, M. 599 Epstein, M. P. 599 Erdelyi, T. 239 Erd˝os, P. 28 Erdmann, K. 440 Erdos, J. A. 609 Eriksson, K. 114, 616, 976, 1091 Eriksson, R. 497 Eschenbach, C. A. 540 Espinosa, O. 86, 1070, 1083, 1104 Evans, S. N. 1060 Evard, J.-C. 1247 Even, S. 104 Everest, G. 996 Everitt, W. N. 783 Fabrykowski, J. 169, 486 Fairgrieve, S. 100

1437

AUTHOR INDEX

Falcon, S. 75 Fallat, S. 733 Fan, K. 734, 774 Fang, M. 699 Fang, Y. 593 Farebrother, R. W. 178, 184, 440, 607, 674 Farenick, D. R. 615 Farhi, B. 55 Farkas, B. 957 Farmer, J. D. 968 Fassler, A. 440 Fay, T. H. 239, 959 Faynshteyn, O. 173, 446, 456, 472, 480 Feeman, T. G. 1095 Feil, T. 28, 370 Feiner, S. xxi, 396 Fekete, A. E. 386, 398 Fell, H. J. 1239 Feng, B. Q. 329, 863, 864, 875 Feng, X. 593 Feng, X. X. 592, 766 Feng, Z. 125, 137, 491 Fengming, D. 75 Fenn, R. 387, 393, 438, 497, 972 Fennessey, E. J. 67, 68, 75, 328, 1057, 1075 Fernandez, A. 140 Ferreira, P. G. 1319 Ferziger, J. H. xxi Fiedler, M. xxi, 188, 354, 427, 495, 519, 520, 539, 586, 614, 632, 665, 694, 728, 744, 750, 821, 823, 825, 1229 Field, M. J. 203, 1247 Fill, J. A. 636 Fillard, P. 1215 Fillmore, J. P. 1206 Fillmore, P. A. 571, 619, 714 Finch, S. R. 219 Fink, A. M. 123, 198, 222, 229, 271, 276, 536, 850, 851, 855, 856, 858 Fishkind, D. E. 636 Fitzgerald, C. H. 820 Flajolet, P. 974 Flanders, H. 599, 603, 740, 897 Fleischhack, C. 771 Fleming, W. 785 Flett, T. M. 1092 Florescu, L. 971 Floret, K. 843, 864 Foldes, S. 434 Foley, J. xxi, 396 Fonda, A. 494 Fong, C. K. 897 Fontijne, D. 438, 858 Foote, R. M. 370, 371, 434, 436, 616, 619 Formanek, E. 420, 429, 528 Foulds, L. R. xxi Foulis, D. L. 1213 Fraleigh, J. B. 436 Francis, B. A. xxi, 1156 Frank, R. L. 790, 791 Frankel, N. E. 1065 Franklin, J. xxii Frazho, A. E. xxi, 948 Frazier, M. xxi Frechet, M. 1211 Freedman, W. 204 Freese, R. W. 856 Freiling, G. xxi French, D. R. 92, 1113 Freudenberg, J. S. 1156

Friedland, S. 771, 895, 906, 911, 1220, 1223, 1234 Friswell, M. I. 387, 691, 694, 695 Frumosu, M. 50, 108, 111 Fuchs, D. 39, 55, 58, 100, 236, 523, 532, 1091 Fuhrmann, P. A. 390, 519–521, 544, 568, 724 Fujii, J. I. 794, 882 Fujii, M. 141, 206, 226, 271, 273, 742, 747, 748, 794, 814, 855, 856, 881, 882, 1215, 1236 Fuks, H. 54, 185 Fulci, R. 1189, 1213 Fulford, M. 123 Fuller, A. T. 687, 695, 1225 Fulton, W. 436 Funderlic, R. E. 319, 321, 430, 635, 638, 639, 662, 663, 737 Furdui, O. 75, 362, 957, 960, 961, 968, 971, 1000, 1024, 1026–1028, 1034, 1072–1074, 1087, 1089, 1098, 1131, 1134, 1141, 1142, 1150, 1154, 1174, 1176–1178 Furuichi, S. 140, 141, 144, 188, 766, 767, 770, 773 Furuta, T. 126, 411, 412, 715–717, 721, 735, 742, 743, 745–747, 749, 768, 794, 803, 812, 814, 849, 875, 882, 1213, 1215

Ger, R. 141 Gerdes, P. 615 Gerrard, A. xxi Gerrish, F. 578 Gessel, I. M. 86, 92, 105, 979, 980, 1008, 1079 Geveci, T. 1319 Ghalayini, B. 86 Ghausi, M. S. 1038, 1083 Gheondea, A. 604, 740 Ghouraba, F. A. A. 440 Gibergans-Baguena, J. 46, 92, 94, 197 Gidea, M. 178, 217, 858, 910 Gil’, M. 206, 804, 891, 897, 898, 1225 Gil’, M. I. 582, 583 Gilat, D. 517 Gilbert, A. C. 868 Gilbert, R. A. 115 Gillis, J. 104 Gilmore, R. 440 Girard, P. R. 387, 393, 394, 437, 691 Girgensohn, R. 50, 68, 94, 117, 210, 495, 540, 1070, 1087, 1097 Glaister, P. 258, 1097, 1158 Glasser, M. L. 1065, 1213 Gockenbach, M. S. 538, 619 Godsil, C. xxi Godunov, S. K. xxii, 416, 1219 Goebel, R. 958 Goh, C. J. 230 Gohberg, I. 502, 544, 595, 619, 941, 1302, 1303, 1315, 1319 Gaines, F. 429, 571 Golberg, M. A. 952, 1306 Gaitanas, K. 32, 966 Goldberg, M. 875, 897 Galantai, A. 314, 404, 412, 415, 440, Goller, H. 819 572–574, 585, 586, 590, 591, 595, Golub, G. H. xxi, 635, 662, 663, 737, 651, 736, 860, 940 762, 906, 911 Gallier, J. 394 Gong, M.-P. 810 Gamelin, T. W. 276, 928, 930, 1015, 1017, 1081–1083, 1104, 1108, 1115, Gonzalez, L. 39, 67, 68, 75, 86, 111 1121–1123, 1126, 1127, 1146, 1147 Gonzalez, N. C. 1210 Goodman, F. M. 436 Ganea, O. 383 Goodman, L. E. 386 Gangsong, L. 803 Gantmacher, F. R. xxii, 563, 576, 831, Goodson, G. R. 603, 604 Goodwin, G. C. 1156 1248 Gordon, G. 1027 Gao, F. 498 Gordon, N. 340 Gao, X. 763, 793 Gorkin, P. 496 Gao, Z. 193, 214, 861, 874, 1075 Goroncy, A. 195 Garcia-Caballero, E. M. 245, 1000, Gould, H. 58, 1008, 1049, 1083 1087 Gould, H. W. 67, 86, 92, 100, 276, 980, Garcia, S. R. 570, 606 1070 Garling, D. J. H. 135, 136, 206, 207, Govaerts, W. 695, 901, 902 213, 219, 361, 852, 864, 1108 Gover, E. 128 Garloff, J. 1225 Gow, R. 570, 605, 609, 610 Garnier, N. 100 Gradshteyn, I. S. 37, 43–45, 67, 75, 92, Garvey, S. D. 387, 691, 694, 695 132, 215, 239, 242–245, 257, 259, Gauthier, N. 68, 86, 103, 105, 246, 334, 1000, 1001, 1008, 1023, 1028, 1046, 980, 981 1047, 1056, 1084, 1087, 1106, 1108, Geerts, T. 1319 1109, 1118, 1119, 1121–1123, 1126, Gekeler, E. 907 1128, 1129, 1131, 1132, 1134–1136, Gelca, R. 29, 31, 32, 45–47, 49–52, 55, 1138, 1141–1144, 1146, 1147, 1149, 68, 75, 86, 100, 105, 122, 124, 132, 1150, 1152, 1154, 1155, 1160, 1161, 135, 144, 149–153, 155, 160, 161, 1165–1170, 1178 164, 167, 168, 180, 183, 190, 197, Graham, A. 701 210, 211, 216, 225, 239, 242, 244, Graham, R. L. 37, 48, 58, 67, 68, 75, 250, 252, 258, 327, 329, 355, 380, 86, 92, 106, 108, 111, 112, 118, 188, 386, 428, 429, 472, 484, 486, 491, 217, 980, 981, 983, 984, 1015, 1017 493, 495, 736, 944, 968, 971, 986, 1003, 1008, 1025, 1079, 1087, 1115, Granville, A. 44, 45, 1070 Grattan-Guinness, I. 118 1127, 1129, 1141, 1145, 1154, 1161–1163 Graybill, F. A. xxi Gelfand, I. M. 701 Grcar, J. 362 Genton, M. G. 725 Grcar, J. F. 846, 847 George, A. 604, 617 Green, W. L. 744, 819 Georgiev, S. 438 Greene, D. H. 67, 86, 92

1438 Gregg, J. 148 Greub, W. H. xxii, 701 Greubel, G. C. 1046 Greville, T. N. E. 415, 628, 631, 637, 642, 655, 656, 667, 670, 674, 679, 799 Gries, D. 853 Griffiths, H. B. 466 Griffiths, M. 100, 103, 966, 1077, 1079 Grigoriadis, K. xxi, 1247 Grimaldi, R. P. 28, 58, 94, 100, 103, 1077 Grinberg, D. 452 Grone, R. 379, 388, 584, 600, 650, 733 Gross, J. xxi, 387, 400, 402–404, 406, 407, 413, 415, 440, 591, 651, 656, 667, 673, 674, 814–816, 819 Gross, R. 436 Grossman, I. 436 Grosswald, E. 35, 39, 178 Grove, L. C. 436 Grover, P. 742 Gruber, P. M. 33, 116, 142, 307, 935, 936, 938 Grunbaum, B. 495 Guan, K. 144 Gudder, S. 740 Guggenheimer, H. W. 896 Guillemin, V. 691 Guillera, J. 1084, 1174–1176 Guler, O. 212, 278, 306, 307, 311, 312, 937, 940, 942 Gull, S. 387, 438, 691, 858 Gunther, M. 700 Guo, B.-N. 249, 250, 261, 264, 466, 484, 964, 1128 Guo, Q.-P. 699 Guo, R.-N. 76 Guo, V. J. W. 68, 86 Guobiao, Z. 803 Guorong, W. 400, 401, 404, 609, 632, 639, 667, 670, 910 Gupta, A. K. xxi Gupta, M. 196, 588 Gurlebeck, K. 438, 439, 1208 Gurvits, L. 1242 Gustafson, K. E. 788, 866 Gustafson, W. H. 609 Gutin, G. xxi Gutin, G. Z. 26 Guy, R. K. 40 Gwanyama, P. W. 205

Hampton, M. 456, 495 Han, J. H. 130, 1020 Handscomb, D. C. 986 Haneda, H. 1219 Hannan, E. J. 1319 Hansen, E. R. 258, 1044, 1061, 1062 Hansen, F. 142, 772 Hanson, A. J. 438, 439 Hardy, G. 215, 276 Hardy, G. H. 54, 130, 1031, 1037, 1049, 1061–1063, 1090, 1110, 1123, 1164, 1165, 1167, 1171 Hardy, M. 238 Hare, K. 1070 Harner, E. J. 590 Harper, J. D. 34, 130 Harries, J. 491 Harris, J. 436 Harris, J. M. xxi Harris, L. A. 354, 724, 758 Harris, W. A. 1206 Hart, G. W. xxi, 440 Hartfiel, D. J. xxi, 1242 Hartwig, R. 667 Hartwig, R. E. 320, 430, 431, 573, 580, 596, 619, 630, 631, 640, 643, 644, 667, 675, 815–817, 1312 Haruki, H. 1141 Haruki, S. 1141 Harvey, N. 856 Harville, D. A. xxi, 397, 400, 629, 634, 636, 638, 777, 1092, 1173 Hasan, M. A. 794–796, 800, 801 Hassani, M. 49, 104, 966, 1158 Hassen, A. 114, 115, 1008 Hattori, S. 1231 Hauke, J. 815 Haukkanen, P. 249 Hautus, M. L. J. xxi, 345, 1319 Havel, T. F. 1212 Havil, J. 35, 47, 48, 52, 54, 275, 332, 517, 957, 963, 966, 996, 1000, 1028, 1032, 1083 Hay, D. 245, 1084–1086 Hayajneh, S. 767, 873, 875 Haynes, T. 1240 Haynsworth, E. V. 661, 667, 721, 758, 831 He, C. 140, 747, 768, 809, 874, 875 He, G. 739 Heath, D. J. 232 Hecht, E. xxi Haddad, W. M. xxi, 362, 440, 604, 779, Heij, C. 1319 Heinig, G. 520 868, 912, 1232, 1233, 1240, 1287, Helmke, U. 520, 805 1313, 1319 Helton, B. W. 1247 Hadlock, C. R. 436 Henderson, H. V. 350, 362, 701 Hager, W. W. 362, 575 Heneghan, F. 981 Haghighi, M. H. Shirdareh 100, 103 Henrici, P. 92, 276 Hahn, R. 319, 413, 656 Herman, E. A. 784 Hahn, W. 1247 Herman, J. 23, 37, 48, 50, 58, 67, 68, Hairer, E. 1306 75, 86, 92, 94, 123–126, 134, 137, Hajja, M. 238, 472, 497 143, 150, 151, 155, 161, 164, 167, Halanay, A. 1319 168, 173, 177, 180, 183, 191, 193, Hale, J. K. 1195, 1307 197, 215, 484, 485 Hall, A. 745 Hershkowitz, D. 94, 396, 694, 759, Hall, B. C. 428, 1187, 1189, 1206, 1234 1191–1193, 1211, 1214, 1247 Hestenes, D. 387, 438, 691 Hall, F. J. 535 Hestenes, M. R. 319 Halliwell, G. T. 228, 231, 264 Halmos, P. R. 279, 315, 400, 543, 571, Hetmaniok, E. 131 576, 577, 599, 604, 608, 609, 632, Hiai, F. 264, 592, 657, 721, 725, 735, 657, 658, 731, 1211 741–744, 746, 749, 759, 766, 770, Hamermesh, M. 436 830, 865, 869–871, 873, 876, 882,

AUTHOR INDEX

883, 945, 1114, 1213, 1215, 1216, 1221, 1222 Higham, D. J. 362 Higham, N. J. xxi, 275, 362, 420, 427, 529, 584, 585, 587, 608, 723, 724, 856, 862–864, 866, 871, 897, 900, 907, 912, 932, 948, 1019, 1191, 1209, 1211, 1214, 1220–1222, 1227, 1228, 1247, 1317 Hile, G. N. 438 Hill, R. 1311 Hill, R. D. 345, 352, 427 Hillar, C.-J. 771 Hilliard, L. O. 544 Hindin, H. J. 103, 236, 262 Hinrichsen, D. 577, 582, 584, 843, 884, 945, 1219, 1220, 1222, 1227, 1234, 1235 Hiriart-Urruty, J.-B. 792, 797 Hirsch, M. W. xxi, 570 Hirschhorn, M. 179 Hirschhorn, M. D. 51, 53, 56, 86, 94, 114, 130, 228, 391, 964, 965, 967, 968, 981, 1020, 1030, 1087, 1088 Hirst, J. L. xxi Hirzallah, O. 140, 142, 145, 211, 271, 738, 747, 794, 855, 876, 877, 904, 905 Hjorungnes, A. 974 Hmamed, A. 831 Ho, C. 972, 973 Hoa, D. T. 139, 768 Hoagg, J. B. 424 Hoffman, A. J. 530, 536 Hoffman, K. xxii Hoffman, M. E. 994, 1054, 1161 Holbrook, J. 126, 577, 807, 1217 Holcombe, S. R. 1087 Holland, F. 820 Hollot, C. V. 796 Holmes, R. R. 436 Holton, D. A. 26 Holtz, O. 843, 1224, 1242 Holzel, M. 1239 Hong, Y. 609 Hong, Y. P. 899 Honsberger, R. 446 Hopkins, D. 1119 Horadam, K. J. 617 Horn, A. 803, 805 Horn, R. A. 325, 330, 341, 346, 347, 361, 384, 387, 390, 518, 536, 538, 539, 542–544, 546, 553, 568, 570, 572, 576, 578, 582, 584, 591, 603, 605, 606, 608, 609, 617, 663, 669, 685, 688, 694, 695, 699, 700, 711, 714, 722, 724, 727, 729, 742, 749, 753, 758, 776–778, 781, 782, 784, 788, 792–794, 797–799, 802–804, 807, 820–823, 827, 828, 831, 835, 838, 839, 841, 847–849, 860, 862, 866–868, 870, 872–874, 880, 882, 884, 894, 895, 897–899, 901, 903, 904, 907, 908, 953, 1092, 1187, 1188, 1191, 1218, 1220, 1221, 1229, 1233, 1306, 1312 Horne, B. G. 897 Horwitz, A. 496 Hou, H.-C. 751, 761, 902 Hou, Q.-H. 86 Hou, S.-H. 544 Householder, A. S. xxi, 520, 585, 635, 903, 937 Howard, J. 489

AUTHOR INDEX

Howe, R. 440, 1247 Howie, J. M. 276, 1126 Howland, R. A. xxi Hristova, S. 139, 214 Hsieh, P.-F. xxi, 570 Hu, G.-D. 1227 Hu, G.-H. 1227 Hu, X. 861 Hu-yun, S. 327, 811 Hua, Y. 249, 264 Huancas, F. 145, 161, 209 Huang, J. 869, 870 Huang, R. 694, 699, 700, 830, 897, 908, 909 Huang, T.-Z. 195, 539, 588, 700, 891, 896, 908 Huang, X. 145 Huang, Z. 699 Hubbard, B. B. 267, 861 Hubbard, J. H. 267, 861 Hughes, J. xxi, 396 Hughes, P. C. xxi, 1208 Hui, Q. 1240 Humphries, S. 333 Hung, C. H. 667 Hunter, J. J. 679 Huylebrouck, D. 1174 Hyde, T. 1090 Hyland, D. C. 440, 604, 700, 1232, 1233, 1319

Jia, G. 144, 228 Jiang, C.-C. 571 Jiang, W.-D. 446, 452, 466, 480 Jiang, Y. 144, 575, 744 Jiang, Y.-P. 1080 Jin, X.-Q. 884 Jocic, D. 872 Johnson, C. R. 325, 330, 333, 341, 346, 347, 361, 379, 383, 384, 387, 388, 390, 396, 424, 428, 434, 518, 536, 538, 539, 542–544, 546, 553, 566–570, 572, 576, 578, 582, 584, 600, 603, 605, 606, 608, 617, 650, 663, 685, 688, 694, 695, 698–700, 711, 714, 722, 724, 727, 729, 733, 753, 758, 774, 778, 782, 788, 792–794, 797–799, 802–804, 821–823, 827, 831, 835, 838, 839, 841, 847–849, 860, 862, 866–868, 870, 872, 880, 882, 894–899, 901, 903, 904, 907, 908, 912, 953, 1092, 1187, 1188, 1191, 1206, 1218, 1220, 1221, 1225, 1228, 1229, 1233, 1306, 1310–1312 Johnson, W. P. 108, 245, 246 Jolly, M. 1092 Jonas, P. 740 Jordan, T. F. 439 Jorswieck, E. A. xxi, 360 Jovanovic, M. 209, 210, 252, 264, 446, 472 Jovanovic, V. 209, 210, 252, 264, 446, Ibragimov, N. H. 1247 472 Ibrahim, A. 275 Joyner, D. 329, 370, 417, 436, 437, 440 Ibstedt, H. 39 Juhnke, F. 94 Ikebe, Y. 1092 Ikramov, K. D. 379, 584, 604, 617, 618, Jukna, S. 24 733, 761, 763, 783, 891, 1185, 1220 Jung, D. 141, 206 Junkins, J. L. 396, 1206 Inagaki, T. 1092 Jury, E. I. 695, 1230, 1234 Infante, E. F. 1207 Ionascu, E. J. 1128 Ionescu, V. xxi, 1319 Kaashoek, M. A. 1315 Ipsen, I. 314, 415, 585, 586, 595, 656 Kachi, Y. 961, 1070, 1087, 1094 Iserles, A. 386, 1207, 1212, 1306 Kaczkowski, S. 961, 967 Ito, T. 161, 173 Kaczor, W. J. 24, 47–49, 51, 94, 115, Ito, Y. 1231 125, 144, 156, 191, 197, 198, 208, Ivanov, N. V. 386 210–212, 225, 227, 229, 232, 249, Iwasaki, T. xxi, 1247 250, 253, 261, 264, 859, 955, 959, Izumino, S. 206, 223, 793 962, 963, 965–968, 1000, 1022, 1023, 1025, 1046, 1047, 1049, 1056, 1057, 1060–1062, 1070, 1071, 1073, Jacobson, D. H. 1319 1074, 1079–1081, 1083, 1084, 1087, Jagers, A. A. 249 1089, 1090, 1097, 1104, 1108, 1119, Jain, T. 952 1130, 1141, 1145, 1154 Jain, V. K. 942, 1235 Kadelburg, Z. 116, 139, 164, 167, 177, Jameson, A. 831 201, 253, 361, 362 Jameson, G. 257 Kadison, R. V. 740 Jameson, G. J. O. 52, 117, 145, 966 Kagan, A. 772 Jameson, N. 257 Kagstrom, J. B. 1219 Janezic, D. 440 Kailath, T. 504, 563, 612, 619, 1319 Janic, R. R. 452, 466, 484 Kalaba, R. E. xxi Janjic, M. 92, 425, 525–527 Kalantari, B. 949 Jank, G. xxi Kalman, D. 496, 518, 615 Janous, W. 472 Kamei, E. 742, 747, 814 Jaroma, J. H. 966 Kane, T. R. xxi Jazar, R. N. 237 Kanwar, R. K. 529 Jeffrey, A. 43, 67, 75, 76, 236, 237, Kanzo, T. 196, 859 262, 981, 984, 1019, 1046, 1109, 1114, 1120, 1123, 1154, 1161, 1163, Kapila, V. 1319 Kaplansky, I. xxii, 576 1166, 1168 Kapoor, G. 196, 588 Jeffries, C. 1234 Kapranov, M. M. 701 Jennings, A. xxi Karachik, V. 92 Jennings, G. A. 497 Karagiannis, N. 1049 Jensen, S. T. 195, 734 Karamzadeh, N. S. 577 Ji, J. 330 Jia, C. 76, 86, 92, 950 Karanasios, S. 643

1439 Karayannakis, D. 1025, 1032, 1050 Karcanias, N. 619, 1319 Karlin, S. 699 Karow, M. 843 Kasturiwale, N. N. 100 Kaszkurewicz, E. 1248 Kataria, K. K. 86, 189 Kato, M. 851, 852 Kato, T. 595, 651, 741, 893, 912, 941, 1220 Katsuura, H. 75, 183, 206 Katz, I. J. 643 Katz, S. M. 785 Kauderer, M. xxi Kauers, M. 107 Kayalar, S. 574 Kazakia, J. Y. 571 Kazarinoff, N. D. 141, 835 Keedwell, D. 39 Keel, L. 1224 Kelly, F. P. 1220 Kelly, J. J. 1038, 1083 Kendall, D. G. 115 Kendall, M. G. 498 Kenney, C. 584, 948, 1236 Keskin, R. 100, 103 Kestelman, H. 599 Keyfitz, N. xxi, 540 Khalil, W. xxi Khalili, P. 1053, 1143 Khan, N. A. 685, 701, 824 Khatri, C. G. 639, 651 Khattri, S. K. 128 Khoury, M. J. 45, 68, 197, 217, 963, 965, 966, 1022–1025, 1027, 1044, 1046, 1047, 1056, 1062, 1070, 1074, 1097 Khrushchev, S. 1097 Khuyrana, D. 327, 429, 528 Kik, P. 411, 638, 654 Kilic, E. 100, 103, 1079 Kilicman, A. 700, 821, 827, 873 Kim, H. 744 Kim, H. W. 75 Kim, P. 966 Kim, S. 744, 1213 Kimberling, C. 151, 184 Kindred, K. 76, 1070 King, C. 879, 889 King, R. B. 436 Kinyon, M. K. 257 Kirk, W. A. 850 Kirmaci, U. S. 205 Kisielewicz, A. P. 68 Kittaneh, F. 140–142, 145, 211, 271, 571, 584, 588, 738, 739, 760, 767, 768, 778, 794, 807–809, 812, 855, 866, 869–877, 879–882, 884, 885, 887–889, 897, 902–905, 1236, 1237 Klamkin, M. S. 29, 31, 32, 36, 41, 43, 50, 56, 119, 124, 131, 134, 146, 148, 149, 173, 178, 180, 183, 333, 480, 492 Klaus, A.-L. 875 Klee, V. 1234 Klosinski, L. F. 530 Klotz, L. 700 Klyachko, A. A. 805 Klyve, D. 966 Knapp, M. P. 242 Knill, O. 530 Knopp, K. 1021, 1022, 1044, 1046–1049, 1056, 1060, 1074, 1092 Knox, J. A. 127, 955

1440 Knuth, D. E. 37, 48, 58, 67, 68, 75, 86, 92, 106, 108, 111, 112, 118, 119, 188, 217, 980, 981, 983, 984, 1015, 1017 Knutson, A. 805 Ko, S.-L. 107 Koch, C. T. 1306 Koeber, M. 714 Koepf, W. A. 75 Koks, D. 438, 570 Koliha, J. J. 400, 402, 405, 406, 643, 669, 1210 Kolk, J. A. C. 938, 942, 1212 Kolman, B. 1247 Komaroff, N. 216, 592, 593, 809 Komornik, V. 1223 Konig, H. 892 Koning, R. H. 700 Konvalina, J. 531 Koornwinder, T. H. 68 Korevaar, J. 947 Korus, P. 50, 76, 189 Kosaki, H. 264, 725, 871, 872 Kosecka, J. xxi Koshy, T. xxi, 42, 52, 55, 58, 68, 75, 76, 86, 92, 94, 100, 103, 105, 131, 423, 974, 1008, 1049, 1070, 1071, 1075, 1097 Kouba, O. 143, 242, 249, 452, 466, 472, 480, 1026, 1027, 1034, 1044, 1046, 1100, 1120, 1130, 1136, 1142, 1143, 1154, 1155 Koumandos, S. 254 Kovac-Striko, J. 619 Kovacec, A. 249, 594 Kovalyov, M. 1083 Kowalenko, V. 242, 243 Kowalski, T. 232 Krafft, O. 819 Krantz, S. G. 276, 472, 946, 947 Krattenthaler, C. xxi, 333, 334, 341 Kratz, W. 1319 Krause, G. 893, 941, 942 Krauter, A. R. 762 Krebs, M. 76 Kreindler, E. 831 Kress, R. 890 Kressner, D. 597 Krook, M. 276, 983, 1015, 1017, 1032, 1040, 1047, 1062, 1104, 1105, 1109, 1161 Krupnik, M. 615 Krupnik, N. 615 Krusemeyer, M. 530, 953 Kubo, F. 226, 856, 1236 Kucera, R. 23, 37, 48, 50, 58, 67, 68, 75, 86, 92, 94, 123–126, 134, 137, 143, 150, 151, 155, 161, 164, 167, 168, 173, 177, 180, 183, 191, 193, 197, 215, 484, 485 Kucera, V. 1319 Kufner, A. 212, 219 Kuipers, J. B. xxi, 438, 439 Kulenovic, M. R. S. 1248 Kumar, M. U. 697 Kunze, R. xxii Kurepa, S. 1217 Kuriyama, K. 766 Kurt, V. 113, 976, 985, 992, 1010 Kurzweil, H. 370 Kwakernaak, K. xxi Kwapisz, M. 1239 Kwong, M. K. 254, 427, 738, 741, 821 Kwong, R. H. 1305, 1319

AUTHOR INDEX

Kyrchei, I. I. 330

Levinson, D. A. xxi Levrie, P. 1008 Lew, J. S. 528 Laberteaux, K. R. 380 Lewis, A. S. 228, 229, 231, 362, 543, Ladas, G. 1234, 1248 592, 593, 721, 741, 787, 935, 938, Laffey, T. J. 544, 577, 605, 610, 617 939 Lagarias, J. C. 36, 1242 Lewis, B. 103, 239, 245 Lai, H.-J. xxi Lewis, D. C. 698 Lakshminarayanan, S. 330 Li, C.-K. xxi, 331, 537, 585, 608, 609, Lam, T. Y. 173, 327, 429, 528 618, 717, 721, 740, 741, 744, 776, Lambrou, M. S. 578 777, 783, 805, 807, 822, 827, 831, Lampakis, E. 169, 202, 452, 466 853, 875, 888, 893, 903, 904 Lampret, V. 961 Li, C.-L. 212 Lancaster, P. xxi, 502, 520, 544, 563, Li, H. 193, 214, 780, 861, 874, 896, 578, 595, 599, 608, 619, 687, 797, 898, 900, 1216 847, 868, 912, 941, 1302, 1303, Li, H.-B. 699, 896 1310, 1311, 1319 Li, J. 145, 575, 1008, 1079 Landau, S. 53 Li, J.-L. 249, 250 Lander, L. J. 35 Li, Q. 740, 884 Lang, C. L. 100 Li, R.-C. 537 Lang, M. L. 100 Li, W. 881 Langer, H. 1211, 1242 Li, X. 667 Langholz, G. 1230 Larcombe, P. J. 67, 68, 75, 86, 92, 105, Li, Y. 225, 354, 415, 569, 762 Li, Y.-L. 249, 250 328, 1057, 1075, 1088, 1113 Larson, L. 28, 29, 31, 35, 43, 45, 47, 48, Li, Z. xxi, 540, 1247 Liao, J. 612 50, 75, 86, 94, 100, 116, 127, 128, Liao, W. 892 132, 138, 145, 164, 167, 173, 192, Lidskii, B. V. 805 198, 207, 236, 239, 244, 249, 250, Lieb, E. 859, 879 252, 261, 331, 433, 443, 472, 485, Lieb, E. H. 721, 758, 771, 790, 791, 489, 493, 1008, 1046, 1056, 1060, 810, 831, 859, 1216 1071, 1072, 1079, 1083, 1090, 1130 Ligh, S. 727 Larsson, L. 213 Ligouras, P. 452, 489 Lasaulce, S. 181, 773 Likins, P. W. xxi Lascu, M. 452 Lilov, L. 1207 Lasenby, A. 387, 438, 691, 858 Lim, J. S. 606 Lasenby, A. N. 387, 438, 691, 858 Lim, L.-H. 610 Lasenby, J. 387, 438, 691, 858 Lim, Y. 714, 744, 771, 1213, 1318 Lasserre, J. B. 592 Liman, A. 1234 Latulippe, J. 92 Lin, C.-S. 746, 795, 814, 855 Laub, A. J. xxii, 310, 563, 565, 584, Lin, M. 86, 226, 252, 326, 328, 350, 948, 1236, 1316, 1319 359, 360, 744, 745, 750–752, Laurie, C. 619 755–757, 759, 760, 764, 770, 773, Lavoie, J. L. 668 775–779, 781–785, 787, 792, 798, Lawson, C. L. 679 805, 806, 857, 858, 870, 872, 885, Lawson, J. D. 714, 744 886, 888, 905 Lax, P. D. 494, 522, 737, 893 Lin, P. Y. 1070 Lay, S. R. 307, 362, 940 Lin, T.-P. 141, 143 Lazarus, S. 577 Lin, W. 145 Le, T. 160, 164, 165, 177, 186 Lin, W.-W. 619 Leake, R. J. 383 Lin, Y. 773, 774, 781, 782, 811, 826 Leclerc, B. 330 Lin, Z. xxi, 311, 312, 315, 345, 582, LeCouteur, K. J. 810, 1212 592, 619 Lee, A. 427, 571, 632 Linden, H. 1236, 1237 Lee, C. 320 Lee, E.-Y. 759, 764, 771, 779, 805, 859, Lippert, R. A. 606 Lipschutz, S. 37, 130, 131, 979, 980, 869, 870, 882, 886, 888, 893 1162, 1170 Lee, J. M. 619 Lipsky, L. xxi Lee, S. H. 141, 206 Littlewood, J. E. 215, 276 Lee, W. Y. 615 Liu, B. xxi, 143 Lehmann, I. 34 Lehmer, D. H. 1008, 1063–1065, 1070 Liu, G. 1009 Liu, H. 128, 612 Lehnigk, S. H. 1247 Lei, T.-G. 733, 761, 772, 783, 784, 792, Liu, J. 37, 130, 131, 472, 694, 700, 721, 794, 798, 800 830, 831, 979, 980, 1162, 1170 Leite, F. S. 607, 1207, 1317 Liu, L.-M. 127, 214 Lemos, R. xxi, 1215–1217 Liu, Q. 699 Leonard, E. 1181 Liu, R.-W. 383 Leonard, I. E. 1119 Liu, S. 212, 700, 792, 821, 822, 825, Lesher, D. A. 974 828 Lesko, J. 1046, 1047 Liu, X. 430, 431, 440, 575, 632, 640, Lesniak, L. xxi 648, 649, 666, 670, 673, 674, 677, 1218, 1222 Lessmann, T. 35, 966 Liu, Y. 662 Letac, G. 725, 820 Leth, S. C. 968 Liz, E. 1181

AUTHOR INDEX

Loehr, N. 330 Loewy, R. 1310 Logofet, D. O. xxi Lokesha, V. 452 Longstaff, W. E. 578 Looze, D. P. 1156 Loparo, K. A. 593 Lopez-Aguayo, D. 68 Lopez, L. 428 Lopez, M. 94, 1097 Lopez-Valcarce, R. 1210 Lord, N. 239, 1000, 1037, 1038, 1119, 1152, 1163 Loss, M. 859 Lossers, O. P. 776 Louck, J. D. 245 Lounesto, P. 387, 438, 691 Lu, Z. 161 Lubich, C. 1306 Luca, F. 976 Luenberger, D. G. xxi, 756, 798 Lukic, M. 116, 139, 164, 167, 177, 201, 253, 361, 362 Lundquist, M. 324, 723, 782 Lupu, C. 173, 214, 226, 346, 383, 391, 420, 452, 466, 484, 856, 1158 Lupu, T. 173, 484, 1158 Luther, W. J. 1235 Lutkepohl, H. xxii Lutoborski, A. 532 Lynd, C. D. 972, 974 Lyubich, Y. I. 912

Mann, S. 438, 858 Manocha, H. L. 1092 Mansfield, L. E. xxii Mansour, M. 1224, 1230 Mansour, T. 68, 1008 Maor, E. 1095, 1119 Maradudin, A. A. 1306 Marcus, M. xxii, 204, 276, 337, 338, 440, 582, 592, 685, 694, 701, 730, 751, 760, 785, 824, 834, 891 Marden, M. 1237 Marengo, J. 94, 1097 Margaliot, M. 1230 Maric, A. 850–852 Marichev, O. I. 1178 Marinescu, D. S. 157, 160, 218, 220, 224, 225, 452, 486 Marion, C. 39 Markett, C. 1031 Markham, T. L. 354, 632, 661, 667, 721, 750, 823, 824, 830, 831 Markiewicz, A. 815 Markley, F. L. 386 Marquez, J.-B. R. 164, 229, 452, 1106 Marsaglia, G. 321, 362, 661, 663, 664 Marsden, J. E. xxi, 393 Marshall, A. W. 202, 216, 276, 360–362, 384, 466, 489, 582, 583, 593, 594, 694, 721, 722, 735, 752, 802, 805, 831, 904, 911, 912 Marsli, R. 535 Martensson, K. 1319 Martin, D. H. 1319 Martin, G. E. 1008 Ma, E.-C. 1312 Martinez, N. C. 76 Ma, Y. xxi Martinez, S. xxi, 541 Mabry, R. 492 Martinez-Serrano, M. F. 667, 675, 676, MacDuffee, C. C. 687, 692, 695 678 MacFarlane, A. G. J. 1319 Mascioni, V. 250, 252, 261 Machale, D. 436 Mason, J. C. 986 Maciejowski, J. M. 1240 Massey, J. Q. 332 Mackey, D. S. 440 Massey, W. S. 387, 691 Mackey, N. 440 Mastronardi, N. 325, 341 Mackiw, G. 37 Mateescu, C. 452 MacMillan, K. 32, 36, 37, 68 Matei, V. 452 Macys, J. J. 963 Matejicka, L. 145, 1237 Maddocks, J. H. 568, 798 Mathai, A. M. 1092 Maeda, H. 1231 Maestripieri, A. 399, 595, 650, 651, 656 Matharu, J. S. 128, 271, 274, 830, 855, 877, 880 Maftei, I. V. 452 Magnus, J. R. xxi, 630, 639, 660, 667, Mathes, B. 619 Mathias, R. 585, 589, 700, 717, 720, 679, 701, 763, 764, 769, 796, 1092 721, 728, 744, 804, 822, 828, 831, Magnus, W. 436, 1306 866, 873, 874, 884, 893, 897, 903, Mahon, B. J. 103, 1079 904, 907, 908, 1092, 1212 Mahon, B. J. M. 100 Matic, I. 116, 139, 164, 167, 177, 201, Majindar, K. N. 799 253, 361, 362 Malenfant, J. 106, 108, 980, 981 Maligranda, L. 138, 212, 213, 219, 843, Matic, M. 47, 223, 230, 231 Matousek, J. 24, 25, 860 850–852 Matson, J. B. 1319 Malik, S. B. 431, 640, 648, 649 Matsuda, T. 213 Malter, A. 119 Mattila, M. 249 Malyshev, A. N. 1227 Mavlo, D. 472, 480 Malzan, J. 607 Maybee, J. S. 598 Man, Y.-K. 43 Mazorchuk, V. 884 Manasrah, Y. 140, 141, 768, 778, 882 McCarthy, J. E. 729 Manfrino, R. B. 133, 134, 137–139, McCarthy, J. M. 1247 150, 151, 155, 160, 162, 164, 165, McClamroch, N. H. 387 167–169, 173, 177, 183, 184, 193, McCloskey, J. P. 418 196, 198, 199, 216, 218, 224, 226, McGrath, P. 1027 252, 361, 446, 452, 466, 472, 480, McKeown, J. J. xxi 485, 489, 492 McLaughlin, J. 531 Mangaldan, J. 111 McPhedran, R. 1071, 1151, 1152 Mangasarian, O. xxi Meehan, E. 544 Manjegani, S. M. 140, 764 Mann, H. B. 798 Meenakshi, A. R. 820

1441 Mehrabi, M. H. 164, 452 Mehta, C. L. 791 Melham, R. S. 92, 100, 103 Mellendorf, S. 535, 1230 Melman, A. 1239 Melnikov, Y. A. xxi Melville, J. 480 Melzak, Z. A. 68, 259, 968, 971, 1021, 1070, 1083, 1084, 1087, 1098, 1099, 1129, 1134, 1141, 1145, 1152, 1154, 1162, 1163, 1170, 1172 Mendez, M. A. 971 Menikhes, L. 92 Merca, M. 46, 67, 75, 86, 111, 242, 243, 252, 426, 980 Mercer, A. 249 Mercer, P. R. 228, 231, 264, 851, 855 Merikoski, J. K. 249, 431, 648, 897 Merino, D. I. 618 Merlini, D. 1079 Merris, R. 541, 701, 742, 753, 792, 825 Mertzios, B. G. 529 Mesbahi, M. 541 Meurant, G. 424 Meyer, C. 314, 415, 585, 586, 595, 656, 667 Meyer, C. D. 374–376, 417, 494, 508, 543, 604, 615, 632, 639, 651, 661, 667, 671, 675, 678, 679, 1241, 1247 Meyer, J. H. 189 Meyer, K. 1316 Mezerji, H. A. S. 1235 Miao, J.-M. 667 Miao, X. 780 Miao, Y. 127, 214 Michels, C. 843, 864, 884, 892 Migorski, S. 922, 923 Mihalyffy, L. 667 Mijalkovic, M. 207 Mijalkovic, Z. 207 Milicevic, A. 440 Miller, K. S. 518, 692, 1172, 1173 Milliken, G. A. 817 Milovanovic, G. V. 192, 941, 1234–1237 Milovanovic, I. Z. 192 Minamide, N. 635, 636 Minc, H. xxii, 204, 276, 440, 582, 834, 891 Minculete, N. 141, 188 Minculette, N. 472, 480 Minda, D. 496 Minton, G. 1119 Miranda, H. 831 Miranda, M. E. 594 Mirsky, L. xxii, 395, 861 Misra, P. 1315 Missinghoff, M. J. xxi Mitani, K.-I. 852 Mitchell, D. W. 446, 491, 492 Mitchell, J. D. 609 Mitra, S. K. 431, 639, 640, 648, 679, 701, 819, 831 Mitrinovic, D. S. 123, 144, 203, 205, 210, 215, 217, 222, 229, 271, 276, 446, 452, 456, 466, 472, 480, 484, 489, 536, 850, 851, 855, 856, 858, 1234, 1237 Mitroi, F.-C. 188 Mitter, S. K. 1312 Mityagin, B. 740 Miura, T. 196, 859 Mlynarski, M. 1219

1442 Moakher, M. 389, 724, 744, 952, 1207, 1217, 1220 Mohammadpour, A. 608 Moler, C. 1220, 1221, 1247 Molera, J. M. 606 Molinari, G. 1083 Moll, V. 1110 Moll, V. H. 31, 32, 35, 49, 50, 67, 68, 75, 86, 92, 100, 103, 105, 121, 239, 262, 968, 979, 980, 983, 989, 992, 993, 1008, 1024, 1032, 1036, 1047, 1061–1063, 1070, 1074, 1079, 1083, 1099, 1104, 1109, 1110, 1115, 1130, 1133, 1134, 1141, 1148, 1149, 1152, 1159, 1160, 1170, 1174, 1178 Mond, B. 737, 744, 793, 826–828 Monea, M. 157, 160, 218, 220, 224, 225, 452, 486 Monov, V. 694, 695, 697 Monov, V. V. 536 Moon, Y. S. 798 Moore, J. B. 745, 959 Morais, J. P. 438 Morawiec, A. 438, 1207 Moreland, T. 740 Moreno, S. G. 245, 1000, 1087 Morgan, F. 590 Mori, H. 223, 793 Mori, T. 831 Morley, T. D. 744, 819 Morozov, A. 701 Morris, H. C. 52 Morrison, K. E. 1060 Morsy, E. 1188 Mortari, D. 416 Mortici, C. 135, 151, 161, 162, 249, 971, 1040, 1090 Mortini, R. 252, 1081, 1189 Moschovakis, Y. 28 Moser, W. O. J. 29, 31, 32, 36, 41, 43, 50, 56, 119, 124, 131, 134, 146, 148, 149, 173, 178, 180, 183, 480, 492 Moses, P. 151, 184 Moslehian, M. S. 142, 211, 271, 274, 275, 742, 748, 850–852, 855, 856, 858, 870, 877, 878, 882 Mu, Y.-P. 86 Muckenhoupt, B. 571 Muir, T. 362 Muir, W. W. 721, 831 Mukherjea, K. 576 Munarini, E. 75, 86, 103, 113, 976, 985, 1009, 1079 Munkres, J. R. 691 Munn, W. D. 496 Munthe-Kaas, H. Z. 386, 1207, 1209, 1212, 1306 Murphy, I. S. 782 Murray, R. M. xxi, 1247 Murty, M. R. 108 Myland, J. 1058 Nagar, D. K. xxi Nahin, P. 996, 1102, 1104, 1106, 1107, 1109, 1114, 1116, 1121, 1122, 1124, 1126–1129, 1131, 1134, 1141, 1143, 1146, 1148, 1149, 1152, 1154, 1161–1163, 1175, 1177 Nahin, P. J. 21, 35, 178, 1088 Najafi, H. 742 Najfeld, I. 1212 Najman, B. 619 Nakamoto, R. 742, 747, 814, 881, 882, 1223

AUTHOR INDEX

Nakamura, S. 1079 Nakamura, Y. 619 Nandakumar, K. 330 Nanyes, O. 957 Narayan, D. A. 425 Narayan, J. Y. 425 Nataraj, S. xxi Nathanson, M. 879 Naylor, A. W. 118, 271, 849, 855, 917, 922, 1092 Needham, T. 276, 920, 1004, 1008 Nelsen, R. B. 30, 33, 133, 135, 149, 157, 161, 168, 177, 178, 212, 215, 222, 225, 249, 250, 252, 446, 452, 456, 466, 472, 480, 484–486, 489, 491, 495, 1095 Nelson, R. B. 126 Nemes, G. 971 Nemirovski, A. 308, 362 Nersesov, S. G. 440, 1233 Nett, C. N. 362 Neubauer, M. G. 775 Neudecker, H. xxi, 212, 630, 639, 660, 667, 679, 700, 701, 763, 764, 792, 796, 1092 Neuman, E. 249, 264, 266 Neumann, M. 424, 539, 604, 1232, 1233 Neuts, M. F. xxi Newcomb, R. W. 831 Newman, D. J. 94, 276, 1070, 1072, 1081, 1083 Newman, M. 333, 567, 1189, 1213 Nguyen, T. 325, 423 Nicholson, D. W. 811 Nicula, V. 452 Niculescu, C. 117, 143, 173, 204, 208, 209, 219, 252, 333, 772, 856, 944, 950 Niculescu, C. P. 117, 161, 178, 208, 217, 466, 858, 910, 1000, 1238 Nielsen, M. A. xxi Niezgoda, M. 195, 734 Nihei, M. 137, 164, 191, 244 Nihous, G. C. 86 Nijenhuis, A. 1000 Niknam, A. 870, 882 Nikolic, S. 440 Nikolov, N. 139, 214 Nimbran, A. S. 244, 1085 Nishio, K. 405 Niu, D.-W. 249, 250, 264, 466, 484, 1128 Noble, B. xxii, 362 Noel, J. 252, 1081 Nomakuchi, K. 667 Nordstrom, K. 349, 568, 738, 797, 817, 818 Norman, E. 1306 Norouzi, A. 140 Norsett, S. P. 386, 1207, 1212, 1306 Nowak, M. T. 24, 47–49, 51, 94, 115, 125, 144, 156, 191, 197, 198, 208, 210–212, 225, 227, 229, 232, 249, 250, 253, 261, 264, 859, 955, 959, 962, 963, 965–968, 1000, 1022, 1023, 1025, 1046, 1047, 1049, 1056, 1057, 1060–1062, 1070, 1071, 1073, 1074, 1079–1081, 1083, 1084, 1087, 1089, 1090, 1097, 1104, 1108, 1119, 1130, 1141, 1145, 1154 Nunemacher, J. 1247 Nyblom, M. 53, 1062 Nylen, P. 912

Oar, C. xxi, 1319 Odell, P. L. 226, 271, 272, 312, 314, 315, 319, 325, 326, 343, 379, 397, 407, 408, 411, 413, 530, 579, 591, 623, 628, 629, 633, 634, 636–640, 651, 656–658, 661, 666, 669, 670, 674, 675, 679, 736, 855–857, 910 Ogawa, H. 636 Ohtsuka, H. 100, 103, 122, 974, 1021, 1063, 1077, 1079 Ojeda, I. 700 Okubo, K. 592, 608, 766 Olah, C. 456, 466 Oldham, K. B. 1000, 1058 Oldknow, A. J. 466 Olesky, D. D. 598, 675, 678 Olkin, I. 202, 216, 276, 360–362, 384, 466, 489, 578, 582, 583, 593, 594, 694, 721, 722, 735, 737, 740, 752, 802, 805, 831, 904, 911, 912 Ollerton, R. L. 100 Olver, F. W. J. 245, 262, 264, 269 O’Meara, K. C. 576, 577, 619 Omidvar, M. E. 870, 882 O’Neill, S. T. 1057, 1075 Opincariu, M. 157, 160, 218, 220, 224, 225, 452, 486 Oppenheim, A. 484 Orrison, M. E. 37 Ortega, J. A. G. 133, 134, 137–139, 150, 151, 155, 160, 162, 164, 165, 167–169, 173, 177, 183, 184, 193, 196, 198, 199, 216, 218, 224, 226, 252, 361, 446, 452, 466, 472, 480, 485, 489, 492 Ortega, J. M. xxii Ortner, B. 762 Osaka, H. 139, 768 Osborn, R. 115 Osburn, S. L. 1212 Osler, T. J. 114, 115, 130, 1008, 1074, 1083, 1088 Ost, F. 699 Ostrowski, A. 1310 Ostrowski, A. M. 887 Oteo, J. A. 1306 Ouellette, D. V. 749, 750, 778, 831 Ovchinnikov, S. 258, 1061, 1062, 1079 Overdijk, D. A. 386 Ozdemir, M. E. 205 Paardekooper, M. H. C. 379 Pachter, M. 1319 Packard, E. 242 Paganini, F. xxi Paige, C. C. 350, 568, 574, 757, 780, 796 Paksoy, V. E. 193, 360, 361, 752, 770, 771, 781, 805, 806, 819, 886, 903, 906, 1221, 1222 Palanthandalam-Madapusi, H. 440, 789, 1306 Paliogiannis, F. C. 1211 Palka, B. P. 276 Pan, C.-T. 899 Pan, H. 980 Pan, W. 261 Pandeyend, K. P. 39 Pao, C. V. 1219 Papageorgiou, N. S. 922, 923 Papastavridis, J. G. xxi, 731 Pappacena, C. J. 578 Pappas, D. 643 Paris, R. B. 68

1443

AUTHOR INDEX

Parish, J. L. 496 Park, F. C. 1247 Park, P. 593 Park, P.-S. 54 Parker, D. F. 619 Parkin, T. R. 35 Parks, P. C. 1230 Parthasarathy, K. R. 727, 882, 883, 1223 Patashnik, O. 37, 48, 58, 67, 68, 75, 86, 92, 106, 108, 111, 112, 118, 188, 217, 980, 981, 983, 984, 1015, 1017 Patel, R. V. 751, 760, 1220, 1315, 1319 Patruno, G. N. 52 Paz, A. 578 Pearce, C. E. 223, 226, 228, 230 Pearce, C. E. M. 223, 230, 231 Pearcy, C. 576 Pearson, C. E. 276, 983, 1015, 1017, 1032, 1040, 1047, 1062, 1104, 1105, 1109, 1161 Pease, M. C. 440 Pecaric, J. 47, 205, 212, 213, 223, 226, 228, 230, 231, 748, 850–852 Pecaric, J. E. 118, 123, 222, 229, 271, 276, 362, 446, 452, 456, 466, 472, 480, 484, 489, 536, 737, 744, 745, 747, 793, 826–828, 850, 851, 855, 856, 858 Pemantle, R. 1073 Peng, Y. 771, 869, 870 Pennec, X. 1215 Pennisi, L. 21 Penson, K. A. 105, 1113 Peperko, A. 699 Peralta, J. 973 Perfetti, P. 1024 Peric, M. xxi Perlis, S. 362, 501, 502, 504, 619 Perrin, D. 607 Persson, L.-E. 117, 143, 173, 204, 208, 209, 212, 213, 219, 252, 333, 772, 856, 944, 950 Peter, T. 492 Petersen, I. R. 796 Petersen, K. 112, 975, 981 Peterson, A. C. 1319 Peterson, J. 75 Peterson, V. 320 Petkovic, L. D. 128, 210 Petkovic, M. S. 128, 210 Petkovsek, M. 43, 49, 67, 75, 76, 86, 92, 187, 968 Petz, D. 220, 657, 721, 735, 743, 744, 749, 770, 871, 1114, 1215, 1216 Phelps, S. 496 Phillips, D. 893 Piazza, G. 896 Piepmeyer, G. G. 1092 Pierce, S. 742, 753 Piezas, T. 34, 178, 189 Pillai, H. 1319 Ping, J. 436 Pinkus, A. 383 Pipes, L. A. 1205, 1206 Pippenger, N. 68, 1009 Pite, E. 763 Pittenger, A. O. xxi Piziak, R. 226, 271, 272, 312, 314, 315, 319, 325, 326, 343, 379, 397, 407, 408, 411, 413, 530, 579, 591, 623, 628, 629, 633, 634, 636–640, 651, 656–658, 661, 666, 669, 670, 674, 675, 736, 855–857, 910

Plaza, A. 67, 75 Plemmons, R. J. 375, 440, 542, 544, 1232 Plischke, E. 1227 Plouffe, S. 1011, 1049 Poghosyan, M. 86 Pohoata, C. 452 Polik, I. 789 Politi, T. 428, 896, 1206, 1208 Pollack, A. 691 Pollock, D. S. G. 1092 Polya, G. 215, 276 Polyak, B. T. 788, 789 Ponnusamy, S. xl, 21, 132, 244, 947, 1087, 1126 Poon, E. 618 Poon, Y.-T. 805 Poonen, B. 608, 728 Pop, O. T. 165, 177, 200, 217, 218, 271, 273, 452, 472, 480 Pop, V. 362, 1178 Popa, D. 856 Popescu, I. 86 Popescu, P. G. 92 Popescu-Zorica, A. 1170 Popov, V. M. xxi, 1247 Popovici, F. 117, 1000 Porta, H. 881 Porter, G. J. 396 Posamentier, A. S. 34 Pourciau, B. H. 940 Prach, A. 382 Pranesachar, C. R. 452 Prasad, S. 697 Prasolov, V. V. xxii, 134, 346, 347, 355, 361, 390, 407, 419, 429, 436, 438, 519, 520, 522, 533, 536, 539, 571, 578, 585, 590, 593, 599, 600, 604, 605, 608, 609, 618, 669, 694, 721, 758, 777, 779, 783, 799, 822, 861, 881, 893, 894, 900, 952, 953, 983, 1193 Prells, U. 387, 691, 694, 695 Pritchard, A. J. 577, 582, 584, 843, 884, 945, 1219, 1220, 1222, 1234, 1235 Pritchard, C. 492 Prodinger, H. 100, 103, 980, 1070, 1079 Proschan, F. 118, 362 Proulx, M. 966 Prudnikov, A. P. 1178 Pryce, J. D. 901, 902 Przemieniecki, J. S. xxi Psarrakos, P. J. 608 Ptak, V. 520, 744 Pugh, C. C. 951 Pukelsheim, F. 431, 701, 740, 814, 816, 817, 825 Pullman, N. J. 1247 Puntanen, S. 195, 662, 734, 793 Putcha, M. S. 580, 596, 619 Pye, W. C. 527 Qaisar, S. 747 Qi, F. 47, 52, 58, 67, 68, 76, 94, 127, 145, 214, 249, 250, 261, 264, 466, 484, 964, 981, 1019, 1128 Qi, H. 701 Qian, C. 145 Qian, R. X. 536 Qiao, S. 330, 398, 635, 679 Qin, H. 1131, 1141, 1174 Qiu, L. 698, 1228 Qiu, S.-L. 1080

Quaintance, J. 58, 92, 1008, 1049, 1083 Queiro, J. F. 594, 777 Quick, J. 606 Quinn, J. J. 23, 32, 37, 48, 50, 58, 67, 68, 75, 86, 92, 100, 103, 106–108, 111, 114, 115, 1009, 1079 Rabanovich, S. 884 Rabanovich, V. 619 Rabinowitz, S. 232, 953 Rachidi, M. 1206 Radjavi, H. 419, 420, 577, 607, 609, 617, 619, 655 Radulescu, M. 153 Radulescu, S. 153, 452 Radulescu, T.-L. T. 25, 47, 48, 50, 144, 160, 161, 194, 198, 207–209, 213, 217, 225, 227–229, 249, 250, 252, 261, 271, 472, 855, 934, 950, 951, 954, 955, 958–963, 967, 971, 1023, 1031, 1047, 1081, 1089, 1090, 1128 Radulescu, V. D. 25, 47, 48, 50, 144, 160, 161, 194, 198, 207–209, 213, 217, 225, 227–229, 249, 250, 252, 261, 271, 472, 855, 934, 950, 951, 954, 955, 958–963, 967, 971, 1023, 1031, 1047, 1081, 1089, 1090, 1128 Raghavan, T. E. S. 539 Rahman, Q. I. 891, 949 Rahmani, M. 112, 979, 980, 1009 Rajian, C. 820 Rajic, R. 748, 850–852 Rakocevic, V. 400, 402, 405, 406, 413–415, 595, 655 Ramare, O. 100 Ran, A. 1319 Ran, A. C. M. 1315, 1319 Rantzer, A. 578, 1228 Rao, C. R. xxi, 425, 679, 700, 819, 831 Rao, D. K. M. 788, 866 Rao, J. V. 667 Rao, M. B. xxi, 425, 700 Rasa, I. 856 Rashid, M. A. 341 Rassias, T. M. 220, 941, 1234–1237 Rather, N. A. 1237, 1238 Ratiu, R. S. 393 Ratiu, T. S. xxi Rauhala, U. A. 701 Raydan, M. 762 Recht, L. 881 Regalia, P. A. 701 Reinsch, M. W. 1189, 1213 Reitenbach, M. 242 Renaud, P. F. 593, 594, 911 Ressel, P. 153, 856 Reznick, B. 173 Richardson, T. J. 1305, 1319 Richeson, D. S. 26, 495 Richmond, A. N. 1214 Rickey, V. F. 1095 Riedel, K. S. 636 Ringrose, J. R. 912 Riordan, J. 92, 111 Rivlin, R. S. 528 Robbin, J. W. 118, 315, 362, 416, 440, 544, 545, 578, 606, 698, 952 Roberts, J. 33, 35, 178, 188 Robinson, D. W. 423 Robinson, P. 762 Robles, J. 667 Rockafellar, R. T. 362, 922, 935, 940 Rockett, A. M. 68

1444 Rodman, L. xxi, 396, 438, 502, 544, 563, 566, 569, 595, 610, 619, 740, 741, 797, 941, 1206, 1302, 1303, 1319 Rogers, G. S. 1092 Rogers, M. D. 1070 Rohan, R.-A. 394 Rohde, C. A. 667 Rohn, J. 864 Roitershtein, A. 245, 1084–1086 Rojo, O. 195, 196, 898 Rooin, J. 124, 209, 855 Room, T. G. 1207 Rooney, B. 36, 996, 997 Ros, J. 1306 Rosa, M. A. F. 994, 1130, 1144 Rose, D. J. 615 Rose, H. E. 36, 370, 432–434, 436, 437, 575 Rose, J. S. 437 Rose, N. J. 604, 667, 671, 678, 1227 Rosenbrock, H. H. 1283 Rosenfeld, M. xxi, 420 Rosenthal, J. 805 Rosenthal, P. 419, 420, 578, 579, 617 Rosoiu, A. 452 Rosser, J. B. 35 Rossmann, W. 440 Rothblum, U. G. 538, 539, 544, 675 Rotman, J. J. 432, 436, 616 Rottger, C. 996 Rowlinson, P. xxi Roy, R. 35, 86, 92, 114, 203, 208, 210, 238, 239, 245, 254, 262, 264, 269, 930, 943, 960, 968, 971, 989, 992, 996, 1000, 1003, 1039, 1044, 1049, 1053, 1056, 1058, 1060, 1121, 1157, 1161, 1162, 1176–1178 Royle, G. xxi Rozhkova, M. 484 Rozsa, P. 352 Rubin, M. H. xxi Rubinstein, Y. A. 446, 452 Rubio-Massegu, J. 127 Rugh, W. J. 516, 517, 1266, 1278, 1306, 1319 Rump, S. M. 1234 Rupp, R. 1189 Ruskai, M. B. 758, 831 Russell, A. M. 729 Russell, D. L. 1309 Rychlik, T. 195 Ryser, H. J. xxi, 326, 332, 792 Ryzhik, I. M. 37, 43–45, 67, 75, 92, 132, 215, 239, 242–245, 257, 259, 1000, 1001, 1008, 1023, 1028, 1046, 1047, 1056, 1084, 1087, 1106, 1108, 1109, 1118, 1119, 1121–1123, 1126, 1128, 1129, 1131, 1132, 1134–1136, 1138, 1141–1144, 1146, 1147, 1149, 1150, 1152, 1154, 1155, 1160, 1161, 1165–1170, 1178 Rzadkowski, G. 86, 236, 242, 244, 983 Sa, E. M. 379, 388, 584, 600, 650, 733 Saaty, T. L. 540 Saberi, A. xxi Sabourova, N. 843 Sadkane, M. 1227 Sain, M. K. 1319 Saito, K.-S. 748, 851, 852, 878 Salamon, D. A. 698 Salmond, D. 340

AUTHOR INDEX

Sandor, J. 46, 48, 51, 100, 117, 120, 121, 128, 132, 143, 209, 225, 249, 250, 261, 264, 266, 452, 466, 955 Sanfelice, R. G. 958 Sannuti, P. xxi Santos, J. P. O. 114, 1091 Sanzheng, Q. 400, 401, 404, 609, 632, 639, 667, 670, 910 Sarria, H. 897 Sastry, K. R. S. 333 Sastry, S. S. xxi, 1247 Satnoianu, R. 472 Satnoianu, R. A. 203, 452, 480 Sattinger, D. H. 355, 372, 1189, 1193, 1247 Saunders, M. 574 Savchev, S. 28, 29, 33, 35, 36, 47, 49, 52, 137, 151, 155, 173, 178, 202, 226 Sawhney, M. 164 Sayed, A. H. xxi Schafer, U. 714 Schaub, H. 396, 1206 Schep, A. R. 699 Scherer, C. W. 1302, 1319 Scherk, P. 607 Scheufens, E. E. 1032 Schiebold, C. 427, 692 Schilling, R. L. 153 Schleicher, D. 119 Schlosser, M. J. 68 Schmeisser, G. 891, 949 Schmidt, M. 1031 Schmoeger, C. 1185, 1211, 1212, 1217 Schneider, C. 1073 Schneider, H. 94, 330, 535, 694, 853, 1310 Schneider, R. 311 Schoenberg, I. J. 1092 Schoenfeld, L. 35 Scholkopf, B. 725 Scholz, D. 1214 Schott, J. R. xxi, 86, 313, 416, 422, 617, 632, 635–638, 666, 689, 711, 722, 776, 804, 805, 809–811, 821, 822, 824, 825, 839, 892, 910, 937, 1172, 1173 Schrader, C. B. 1319 Schreiber, M. 657, 658 Schreiner, R. 606 Schroder, B. S. W. 25, 118, 956 Schumacher, J. M. 1319 Schwartz, H. M. 704, 737 Schwarz, D. 226, 856 Schwenk, A. J. 698 Schwerdtfeger, H. 440 Scott, J. A. 127, 244, 456, 472, 1030, 1170 Scott, J. N. 77 Scott, L. L. 436 Searle, S. R. xxi, 350, 362, 701 Sebastian, P. 1092 Sebastiani, P. 952 Seber, G. A. F. 217, 225, 231, 312, 319, 341, 382, 398, 407, 410, 424, 440, 442, 443, 534, 536, 616, 636, 651, 669, 736, 778, 795, 816, 824, 899, 1232 Seberry, J. 700 Sedgewick, R. 974 Seiffert, H.-J. 103 Seiler, E. 774, 779 Seiringer, R. 771 Selfridge, J. L. 28, 35 Selig, J. M. xxi, 393, 438, 440

Sell, G. R. 118, 271, 849, 855, 917, 922, 1092 Semrl, P. 320, 853 Seo, Y. 141, 206, 223, 793, 1215, 1216 Seoud, M. A. 440 Seron, M. M. 1156 Serre, D. 372, 428, 528, 665 Serre, J.-P. 436 Servi, L. D. 232, 974 Seshadri, V. 1231 Seshu Aiyar, P. V. 54, 1031, 1037, 1049, 1061–1063, 1090, 1110, 1123, 1164, 1165, 1167, 1171 Shafroth, C. 341, 389 Shah, S. L. 330 Shah, W. M. 1234 Shaikhet, L. 142 Shakarchi, R. 31, 996 Shallit, J. 1070 Shamash, Y. xxi, 311, 312, 315, 345, 582, 592, 619 Shao, M. 612 Shapiro, H. 538, 576, 609, 617, 619, 775, 788 Shapiro, H. M. 698, 821 Sharma, R. 196, 588 Shattuck, M. 86 Shaw, R. 386 Shebrawi, K. 765, 766, 772, 865, 866, 870, 871, 876–879, 903 Shemesh, D. 617 Shen, S.-Q. 539, 700, 908 Sherali, H. D. 543, 922, 924 Sherif, N. 1188 Sherman, D. 606 Shetty, C. M. 543, 922, 924 Shi, H.-N. 190 Shi, S.-C. 480 Shilov, G. E. xxii Shirali, S. A. 249 Shiu, P. 1104, 1110 Shomron, N. 327, 429, 528 Shuster, M. D. 394, 438, 1206 Sibuya, Y. xxi, 570 Sierpinski, W. 35 Sijnave, B. 695 Siljak, D. D. xxi, 440, 1232 Silva, F. C. 806, 1311 Silva, J.-N. 314, 318, 355, 356, 738 Silverman, H. xl, 21, 132, 244, 947, 1087, 1126 Simic, S. xxi Simoes, R. 1311 Simon, B. 774, 779 Simons, C. S. 100 Simons, S. 121, 472, 1099 Simoson, A. 257 Simsa, J. 23, 37, 48, 50, 58, 67, 68, 75, 86, 92, 94, 123–126, 134, 137, 143, 150, 151, 155, 161, 164, 167, 168, 173, 177, 180, 183, 191, 193, 197, 215, 484, 485 Sinescu, V. 53 Singer, S. F. xxi Singh, D. 24, 118 Singh, J. N. 24, 118 Sinnamon, G. 787, 792 Sintamarian, A. 963, 964, 1051, 1087, 1142 Siong, C. C. 55 Sitaru, D. 452 Sittinger, B. D. 979, 1037 Sivan, R. xxi Sixdeniers, J.-M. 105, 1113

1445

AUTHOR INDEX

Skelton, R. E. xxi, 1247 Skubak, E. 496 Slota, D. 131 Smale, S. xxi, 570 Smarandache, L. 1238 Smiley, D. M. 274, 857 Smiley, M. F. 274, 850, 857 Smith, D. A. 178, 187, 436, 438 Smith, D. R. 1206 Smith, H. A. 1247 Smith, O. K. 532 Smith, P. J. 772 Smith, R. A. 1312 Smith, R. L. 383, 396, 1206 Smoktunowicz, A. 383, 667, 887 Smola, A. J. 725 Smotzer, T. 486 Snell, J. L. xxi Snieder, R. 630 Snyders, J. 1248, 1267, 1268, 1319 So, W. 379, 582, 763, 801, 805, 861, 1185, 1189, 1206, 1211–1213, 1217, 1218, 1221 Soatto, S. xxi Sobczyk, G. 387, 438, 691 Sofair, I. 472 Sofo, A. 48, 53, 92, 996, 1024, 1025, 1033, 1056, 1070, 1077, 1100, 1141, 1174, 1176 Soltan, V. 278, 306, 308, 309, 311, 312, 314, 315, 855, 935–938 Sommariva, A. M. 855 Sondow, J. 32, 36, 37, 68, 76, 996, 1079, 1084, 1174–1176 Song, M.-Y. 699 Song, Q. 452, 466 Song, R. 153 Sontag, E. D. xxi, 1231, 1309 Sorensen, D. C. 1310 Sourour, A. R. 609 Spanier, J. 1000, 1058 Specht, W. 206 Speed, R. P. 636 Spence, J. C. H. 1306 Spencer, J. 971 Spiegel, E. 619 Spiegel, M. R. 37, 130, 131, 979, 980, 1162, 1170 Spindelbock, K. 573, 630, 631, 644 Spitkovsky, I. M. 595, 737 Spivak, M. 438, 691, 923 Spivey, M. Z. 75, 92, 111, 112 Sprossig, W. 438, 439, 1208 Sprugnoli, R. 68, 75, 76, 86, 1070, 1079, 1102 Sra, S. 727, 785 Srivastava, H. M. 75, 76, 979, 980, 996, 1001, 1003, 1024, 1025, 1030, 1032–1036, 1050, 1070, 1071, 1074, 1092, 1130, 1132, 1146, 1151, 1152, 1154, 1166, 1170, 1177 Srivastava, R. 92, 1070 Stadler, A. 1087 Stan, A. I. 140 Stanciu, N. 183, 466, 480, 484 Stanford, W. 68 Stankovic, M. S. 964 Stanley, R. P. 23, 105 Starzhinskii, V. M. 1307 Steeb, W. H. 1220 Steele, J. M. 25, 124–126, 138, 160, 161, 164, 169, 177, 178, 180, 199–201, 203, 207, 208, 211–213,

215, 217, 221, 225, 226, 273, 274, 446, 856, 857, 944, 957, 959, 1172 Stein, E. M. 31, 996 Stellmacher, B. 370 Stengel, R. F. xxi Stenger, A. 1087 Stepniak, C. 816 Stern, R. J. 539, 1232, 1233 Stetter, H. J. xxi Stevanovic, M. R. 480 Stewart, G. W. xxi, xli, 362, 563, 565, 574, 595, 597, 617, 798, 831, 860, 866, 867, 893, 906, 907, 911, 912, 939–941 Stickel, E. U. 1231, 1247 Stiefel, E. 440 Stiller, L. xxi Stoer, J. 362, 868, 912, 940 Stofka, M. 1044 Stoica, G. 229, 272, 273, 961, 967 Stojanoff, D. 865, 882 Stolarsky, K. B. 143 Stone, M. G. 493 Stoorvogel, A. A. xxi, 345, 1319 Storey, C. xxi, 337, 619, 730, 1229 Strang, G. xxi, xxii, 325, 362, 423, 606, 727 Straskraba, I. 400, 402, 405, 406 Straub, A. 86, 1000, 1070, 1083, 1086, 1087, 1089, 1104 Strauss, M. J. 868 Strehl, V. 86 Strelitz, S. 1225 Strichartz, R. S. 1306 Stroe, M. 157, 160, 218, 220, 224, 225, 452, 486 Strom, T. 1219 Stuelpnagel, J. 1206 Styan, G. P. H. 195, 196, 212, 220, 320, 321, 324, 342, 350, 362, 398, 400–403, 405–407, 409, 414, 416–418, 429–431, 439, 440, 568, 573, 580, 588, 630, 640, 641, 661–664, 667, 668, 734, 738, 757, 780, 792, 793, 796, 816, 817, 825 Su, J. 780 Subramanian, R. 716, 741, 742, 1213 Suceava, B. 200, 201 Sugimoto, T. 424 Sullivan, P. 94 Sullivan, R. P. 609 Sumner, J. S. 249 Sun, H. 143 Sun, J. 563, 565, 574, 595, 597, 617, 798, 831, 860, 866, 867, 893, 906, 907, 912, 939–941 Sun, X.-J. 39, 40, 148 Sun, Y. 125, 137 Sun, Z.-W. 55, 86, 980, 1070 Sung, W. F. 1062, 1105 Sury, B. 67, 68, 75, 86, 103, 112, 1009, 1027, 1056, 1070, 1101 Sustik, M. A. 791 Swamy, K. N. 831 Swenson, C. 613 Switkes, J. 92 Sze, N. 783 Sze, N.-S. 608, 609 Szechtman, F. 570 Szekeres, P. 438 Szep, G. 617 Szirtes, T. xxi Szulc, T. 536, 662, 899

Tabachnikov, S. 39, 55, 58, 100, 236, 523, 532, 1091 Takagi, H. 196, 859 Takahashi, Y. 858 Takahasi, S.-E. 196, 858, 859 Takane, Y. 410, 667 Takeuchi, K. 410 Talvila, E. 1178 Tam, T. Y. 436 Tamura, T. 851, 852 Tan, J. 763, 793 Tannenbaum, A. R. 1156 Tao, T. 51, 717, 805, 899, 967, 1212, 1216 Tao, Y. 760, 807 Tapp, K. 394, 434, 436, 438, 943 Tarazaga, P. 729, 897 Tauraso, R. 122 Taussky, O. 362, 587, 619 Taylor, P. J. 111 Teel, A. R. 958 Teimoori, H. 523 Temesi, R. 744, 749 Tempelman, W. 386 ten Have, G. 608 Tener, J. E. 570 Teodorescu-Frumosu, A. 50, 108, 111 Terlaky, T. 789 Terrana, D. 100, 103 Terrell, R. E. 1213 Tetiva, M. 127, 177, 452, 466, 480 Thieu, U. 92 Thirring, W. E. 810 Thomas, J. A. xxi, 231, 721, 776, 782, 1172 Thome, N. 418 Thompson, A. C. 306 Thompson, C. J. 1221 Thompson, R. C. 268, 271, 362, 429, 582, 609, 619, 738, 774, 776, 779, 785, 801, 805, 831, 1189, 1213, 1217, 1218, 1306 Thuan, P. V. 94, 183 Thuong, L. Q. 166 Tian, J.-F. 222 Tian, Y. 317, 321, 324, 325, 342, 354, 357, 358, 377, 379, 397, 398, 400–407, 409, 411, 412, 414–418, 438–440, 567–570, 579, 630, 631, 636–638, 641, 643, 650–654, 657–668, 672, 679, 686, 688, 692, 735, 741, 745, 754, 816, 819, 820 Tie, L. 773, 774, 781, 782, 811, 826, 1238, 1239 Tigaeru, C. 271 Tingting, W. 993 Tismenetsky, M. 578, 599, 687, 847, 868, 912, 1310, 1311 Tisseur, F. 440 Toan, H. M. 139, 768 Toda, M. 751, 760, 1220 Todd, J. 587 Toffoli, T. 606 Tojo, F. A. F. 237, 262 Tominaga, M. 748, 812, 878, 1215 Tong, J. 944 Tong, Y. L. 118, 362 Tonge, A. 329, 863, 864, 875 Torabi-Dashti, M. 37 Torokhti, A. 906, 911 Toth, L. 963 Totolici, I. 200 Trainin, J. 1119 Trapp, G. E. 590, 744, 760, 819, 831

1446 Travaglia, M. V. 768 Trefethen, L. N. xxi, 1219, 1242 Trenkler, D. 386, 393, 394, 527, 528, 665, 740, 1208 Trenkler, G. 195, 217, 313, 314, 317–320, 329, 331, 376, 377, 379, 380, 383, 386, 387, 391, 393, 394, 397, 400, 402–413, 415, 418, 440, 527, 528, 573, 579, 589–591, 595, 609, 628, 630, 631, 633, 635, 641–644, 647, 648, 650–654, 656, 657, 664, 665, 667, 672–674, 677, 700, 733, 734, 736, 739, 740, 764, 772, 796, 815, 1208 Trentelman, H. L. xxi, 345, 1319 Treuenfels, P. 1317 Trickovic, S. B. 964 Trif, T. 68, 75, 1102 Trigiante, D. 613, 619, 727, 1205 Trimpe, S. 1223 Trinajstic, N. 440 Tromborg, B. 389 Tropp, J. A. 868 Troschke, S.-O. 387, 440, 815, 816 Trotter, T. 39 Trotter, W. T. 25, 118 Trudeau, R. J. 26 Trudgian, T. 35 Trustrum, G. B. 810 Tsatsomeros, M. 598 Tsatsomeros, M. J. 424, 733 Tsing, N.-K. 619, 797 Tsintsifas, G. A. 452 Tsiotras, P. 386, 396, 1206 Tsitsiklis, J. N. 1242 Tu, L. W. 691 Tuan, N. M. 166 Tuchinsky, P. M. 1095 Tuenter, H. J. H. 68, 86 Tuncbilek, Y. 145 Tung, S. H. 203 Turkington, D. A. 701 Turkmen, R. 193, 360, 361, 752, 762, 770, 771, 781, 805, 806, 819, 861, 874, 884, 886, 903, 906, 907, 1221, 1222 Turnbull, H. W. 362 Turner, J. D. 438 Tuynman, G. M. 1212 Tyan, F. 604, 1311 Tzermias, P. 961, 1070, 1087, 1094 Uchiyama, M. 813, 870, 1217, 1222 Udwadia, F. E. xxi Uhlig, F. 317, 607, 619, 797, 799, 1247 Ulukok, Z. 762, 861, 874, 884 Underwood, D. 30–32 Underwood, E. E. 345, 352 Upton, C. J. F. 729 Vajda, S. 100, 103, 493, 1079 Valean, C. I. 1134, 1141 Valentine, F. A. 362 Vamanamurthy, M. 123, 124, 227, 249 Van Barel, M. 325, 341, 520, 544 Van Brummelen, G. 497 Van-Brunt, A. 1213 van Dam, A. xxi, 396 van den Driessche, P. 598, 675, 678, 783, 888, 1234 van der Merwe, R. 1208 Van Dooren, P. 619 Van Huffel, S. 912

AUTHOR INDEX

Van Loan, C. F. xxi, 327, 615, 701, 906, 911, 1212, 1220, 1221, 1228, 1247 Van Overschee, P. 1319 Van Pelt, T. 789 van Schagen, F. 1319 Vandebril, R. 325, 341 Vandenberghe, L. xxi, 362, 756 Vandewalle, J. 912 Varadarajan, V. S. 440, 1189, 1212, 1247 Varah, J. M. 899 Vardi, I. 1003, 1143 Vardulakis, A. I. G. xxi, 544 Varga, R. S. xxi, 535, 539, 1232 Varosanec, S. 224 Vasic, P. M. 144, 203, 276, 452, 466, 484, 850 Vasudeva, H. L. 822, 828, 830 Vavrin, Z. 520 Vaz, J. 994, 1130, 1144 Vein, P. R. 362 Vein, R. xxi, 362 Veljan, D. 173, 177, 452, 466, 495 Venugopal, R. 440, 1306 Vepstas, L. 980, 1032, 1037 Vermeer, J. 605 Veselic, K. 586, 619, 1228 Vetter, W. J. 701 Vi, L. 183 Vidyasagar, M. 1219 Vignat, C. 86, 992 Villacorta, J. L. C. 1008 Villani, A. 1022 Villarino, M. B. 51, 228 Villet, C. M. 1220 Vinsonhaler, C. I. 576, 577, 619 Visick, G. 821, 825, 827, 828, 830 Visser, M. 1213 Volenec, V. 446, 452, 456, 466, 472, 480, 484, 489, 744, 745 Voll, N. G. 103 Vondracek, Z. 153 Vong, S.-W. 884 Vowe, M. 249 Vreugdenhil, R. 1319 Vuorinen, M. 123, 124, 227, 249

Wang, W. 76, 86, 92, 950 Wang, X. 1223 Wang, Y. 666, 1225, 1242 Wang, Y. W. 1218, 1313 Wang, Z.-K. 143 Wang, Z.-S. 452, 466 Wanner, G. 1306 Wansbeek, T. 700 Ward, A. J. B. 578 Ward, R. C. 1232 Ward, T. 996 Warga, J. 924, 925 Warner, W. H. 386 Wasserman, R. H. 691 Waterhouse, W. C. 618 Waterhouse, W. E. 337, 338 Waters, S. R. 427 Wathen, A. J. 762 Watkins, W. 730, 751, 760, 778, 785, 822, 826, 1079 Watson, G. S. 212, 220 Weatherby, C. 108 Weaver, J. R. 427, 572 Weaver, O. L. 355, 372, 1189, 1193, 1247 Webb, J. H. 489 Webster, R. 362 Wedin, P. A. 574 Wegert, E. 1242 Wegmann, R. 890 Wegner, S.-A. 957 Wei, C.-F. 981 Wei, F. 160, 166, 199 Wei, J. 1306 Wei, M. 574, 630, 637 Wei, S. 1052, 1056 Wei, Y. 330, 398, 635, 667, 670, 675, 679, 1210 Weinberg, D. A. 619 Weinert, H. L. 574 Weintraub, S. H. 691 Weirathmueller, A. 36, 996, 997 Weiss, G. 606 Weiss, G. H. 1306 Weiss, M. xxi, 1319 Weisstein, E. W. 39, 40, 68, 472, 1024, 1025, 1083, 1087 Welukar, R. M. 100 Wada, S. 220, 221, 858 Wen, P. 100, 1079 Wagner, D. G. 1225 Weng, K. 466, 484 Waldenstrom, S. 389 Wenpeng, Z. 993 Waldhauser, T. 86 Wenzel, D. 884 Walker, P. 497 Werman, M. 100 Walls, G. L. 239, 959 Wermuth, E. M. E. 1211, 1222 Walter, G. G. xxi, 1233 Werner, H.-J. 817 Wan, J. 1141 Wesseling, P. xxi Wang, A. Y. Z. 100, 1079 West, D. B. 118 Wang, B. 810, 811, 905 Westlake, J. R. xxi Wang, B.-Y. 390, 810, 818, 823, 830, Wets, R. J. B. 935 905, 907 Weyrauch, M. 1214 Wang, C.-C. 691 White, J. E. 615 Wang, C.-L. 212, 224 Wiegmann, N. A. 604 Wang, D. 715 Wiener, Z. 1241 Wang, F. 436 Wang, G. 330, 398, 635, 675, 679, 763, Wigner, E. P. 737 Wilcox, R. M. xxi, 1214 793 Wildberger, N. J. 24, 118 Wang, H. 630 Wildon, M. J. 440 Wang, J. 252, 721, 831 Wilf, H. S. 43, 49, 67, 68, 75, 76, 86, Wang, J.-H. 619 92, 187, 232, 947, 968, 980, 1008, Wang, L. 195, 588, 739, 874, 891 1014, 1021 Wang, L.-C. 212 Wilhelm, F. 1217 Wang, M.-K. 143, 1080 Wilker, J. B. 249 Wang, Q.-G. 536 Wang, T. 67, 68, 75, 1027, 1056, 1070, Wilkinson, J. H. xxi 1101 Willems, J. C. 1319

1447

AUTHOR INDEX

Zhang, F. 126, 193, 197, 216, 230, 271, 273, 314–316, 319, 320, 326, 327, 329–331, 336, 338, 340, 346, 347, 350, 359–362, 379, 380, 389, 390, 397, 407, 421, 427–429, 438, 440, 441, 531, 537, 540, 544, 566, 568, 571, 576, 577, 579, 580, 582–584, 591–593, 597, 600, 603–605, 610, 613, 615–617, 662, 669, 686, 691, 694, 699, 700, 713, 714, 722–725, 729–733, 735–738, 740–742, 745, 748–757, 760–766, 770–774, 776–785, 788, 792–794, 796–813, 815, 818, 819, 821–825, 827–831, 852, 853, 855, 856, 861, 865–867, 870–872, 874, 881, 886, 888, 889, 891, 892, 894–896, 898, 901, 903–908, 1079, 1221, 1222, 1242 Zhang, H. 466, 484 Zhang, L. 670 Zhang, P. 760, 784, 892 Zhang, R. 269 Zhang, S.-Q. 261 Zhang, X. 763, 793 Zhang, Y. 252, 699 Zhang, Z. 452, 466 Zhang, Z.-H. 161, 208, 452, 472, 480 Zhao, D. 193, 214, 861, 874, 898, 900, 1216 Zhao, F.-Z. 67, 68, 75, 1027, 1056, 1070, 1097, 1101, 1141 Zhao, K. 570 Zhao, T. 143 Zhao, Y. 701 Zheng, B. 320, 667 Zhong, J. 440, 649 Zhong, Q.-C. 399 Zacharias, J. 239 Zhou, H. 771 Zacharias, K. 1154 Zhou, K. xxi, 619, 899, 902, 1302 Zagier, D. 35, 119 Zhou, X. 222 Zakai, M. 1248, 1267, 1268, 1319 Zhu, H. 144 Zalcman, L. 947 Zhu, K. 466, 484 Zamfir, R. 610, 1237 Zhu, L. 128, 249, 250, 261, 264, 944 Zanna, A. 386, 1207, 1209, 1212, 1306 Zi-zong, Y. 742, 759, 781 Zassenhaus, H. 619 Ziegler, G. M. 35, 52 Zeilberger, D. 32, 33, 43, 49, 67, 68, 75, Zielke, G. 862, 909 76, 86, 92, 94, 100, 122, 187, 245, Zimmerman, S. 972, 973 362, 491, 617, 968, 1021, 1022 Zinn, J. 197 Zelevinsky, A. V. 701 Zireh, A. 942, 949, 1235 Xi, B.-Y. 818, 905, 907 Zemanek, J. 1220 Zitian, X. 128 Xia, W.-F. 143 Zhan, S. 594 Zizovic, M. R. 480 Xiang, J. X. 221 Zhan, X. 326, 350, 362, 443, 495, 535, Zlobec, S. 633 Xiao, X. 1052, 1056 539–544, 576, 578, 579, 583, 584, Zou, L. 140, 144, 744, 747, 768, 771, Xiao, Z.-G. 208 596, 618, 687, 694, 698, 711, 716, 869, 870, 875, 896 Xiaoxue, L. 1079 721, 728, 738, 777, 788, 802, 803, Zucker, I. J. 1070 Xie, Q. 700 805–808, 812, 819, 820, 825, 827, Zucker, J. 1071, 1151, 1152 Xin, L. 1079 828, 830, 839, 847, 849, 861, 865, Zuo, H. 271, 273, 747, 748, 855, 856 Xu, C. 86, 271, 350, 757, 766, 780, 855 867, 870, 873, 874, 879, 880, 882, Zuo, K. 400, 402 Xu, D. 394 883, 892–895, 899, 901, 904, 905, Zwart, H. J. 1219 Xu, H. 1185, 1228 907, 949, 950, 1214, 1223, 1235, Zwas, G. 897 Xu, M.-Z. 588 1237 Zwillinger, D. 106–108, 114, 446, 491, Xu, W.-w. 881 Zhang, B. 1225 497, 519, 616, 1009, 1097, 1108, Xu, X. 612, 809, 874 1130, 1134, 1168 Zhang, C.-E. 58, 67, 68, 94 Xu, Z. 271, 350, 757, 780, 855 Willems, J. L. 632 Williams, E. R. 636 Williams, J. P. 577, 607, 655, 714 Williams, K. S. 33, 119, 531, 795, 855, 1091, 1092 Wills, S. 820 Wilson, B. M. 54, 1031, 1037, 1049, 1061–1063, 1090, 1110, 1123, 1164, 1165, 1167, 1171 Wilson, D. A. 912 Wilson, P. M. H. 394, 497 Wilson, R. A. 437 Wimmer, H. K. 520, 568, 740, 876, 1303, 1310, 1312, 1319 Winkin, J. 632 Wirth, F. 1227 Witkowski, A. 123, 227 Wittenburg, J. 1207 Witula, R. 53, 131, 232 Witzgall, C. 362, 912, 940 Wolkowicz, H. 196, 350, 379, 388, 580, 584, 588, 600, 650, 733, 752, 770, 805, 806, 870, 872 Wolovich, W. A. 1319 Wonenburger, M. J. 609 Wong, C. S. 609, 765, 905 Wong, E. 725 Wonham, W. M. xxi, 525, 1181, 1308, 1319 Woo, C.-W. 733, 761, 772, 783, 784, 792, 794, 798, 800 Wright, E. M. 130 Wright, M. 100 Wrobel, I. 607 Wu, B. 200, 201, 1205 Wu, C.-F. 817 Wu, C. W. 530 Wu, D. W. 37 Wu, E. 1309 Wu, J. 892 Wu, L. 677 Wu, P. Y. 413, 608–610, 619, 831, 1228 Wu, S. 160, 166, 173, 177, 199, 214, 222, 452, 466, 472 Wu, S.-H. 480 Wu, Y.-D. 161, 452, 472, 480 Wu, Y.-L. 148, 452, 466

Yakubovich, V. A. 1307 Yamagami, S. 871 Yamamoto, Y. 328 Yamazaki, T. 206, 771, 814 Yan, K. 341, 342 Yan, Z.-z. 661 Yanagi, K. 766 Yanai, H. 410 Yanase, M. M. 737 Yandl, A. L. 613 Yang, B. 145, 220 Yang, H. 430, 431, 640, 666, 739, 874 Yang, J.-H. 1097, 1141 Yang, X. 128, 1224 Yang, X. M. 766 Yang, Z.-H. 143, 249, 266, 1170 Yang, Z. P. 592, 766 Yau, S. F. 824, 828 Yau, S. S.-T. 1206 Ye, K. 610 Ye, Q. 619 Yeadon, F. I. 386 Yeap, B. P. 242 Yellen, J. xxi Yeong, L. T. 75 Yi, H. 1084 Yibing, Z. 128 Yimin, W. 400, 401, 404, 609, 632, 639, 667, 670, 910 Yin, S. 869 Young, D. M. xxi Young, J. 1044 Young, N. J. 901 Young, P. M. 1228 Young, R. M. 1112, 1114

Index Symbols

GL

positive-semidefinite A ≤ B matrix square root generalized L¨owner 0 n×m definition, 714 partial ordering n × m zero matrix ˆT A definition, 815 definition, 283 c reverse transpose A≤B 1 n×m definition, 288 core partial ordering n × m ones matrix AT definition, 648 definition, 285 rs transpose 2 × 2 matrices A≤B definition, 286 commutator rank subtractivity partial A⊙α Fact 4.29.2, 428 ordering Schur power 2 × 2 matrix definition, 430 definition, 685 # discrete-time A(S1 ,S2 ) A≤B asymptotically stable submatrix sharp partial ordering matrix definition, 281 definition, 648 ∗ Fact 15.22.1, 1239 A[S1 ,S2 ] A ≤ B eigenvalue inequality submatrix star partial ordering Fact 10.21.1, 800 definition, 281 definition, 431 real square root A[i, j] i A←b Fact 3.15.34, 328 submatrix column replacement singular value definition, 281 definition, 280 Fact 7.12.35, 585 A⊥ 2 × 2 positive-semidefinite A ⊙ B complementary Schur product matrix idempotent matrix definition, 685 square root definition, 375 A ⊗ B Fact 10.10.8, 731 complementary projector Kronecker product 2 × 2 trace definition, 374 definition, 681 Fact 3.15.10, 326 B(p, q) A: B 3 × 3 matrices Bezout matrix parallel sum commutator definition, 519 definition, 818 Fact 4.29.3, 429 C(p) ˆ ∗ A 3 × 3 matrix equality companion matrix reverse complex conjugate trace definition, 546 transpose Fact 6.9.27, 529 C∗ definition, 288 3 × 3 symmetric matrix complex conjugate A# eigenvalue transpose group generalized inverse Fact 6.10.7, 531 definition, 288 definition, 627 4 × 4 matrices D|A A A commutator Schur complement adjugate Fact 4.29.4, 429 definition, 625 definition, 301 A⊕B Ei, j,n×m D A Kronecker sum n × m matrix with a single Drazin generalized inverse definition, 683 unit entry definition, 625 A#B definition, 285 AL geometric mean Ei, j left inverse definition, 743 matrix with a single unit definition, 294 A#α B entry R A generalized geometric definition, 285 right inverse mean Fσ set definition, 294 definition, 745 closed set A+ A−1 Fact 12.12.19, 939 generalized inverse inverse matrix Fact 12.12.19, 939 definition, 621 definition, 297 Gδ set A1/2

open set Fact 12.12.19, 939 H(g) Hankel matrix definition, 520 In identity matrix definition, 284 J[ ] 0 I −I 0

J 2n [

definition, 367 ] 0 I n

−In 0

definition, 367 K(x) cross-product matrix definition, 283 N standard nilpotent matrix definition, 284 Nn n × n standard nilpotent matrix definition, 284 PA,B pencil definition, 563 P n,m Kronecker permutation matrix definition, 683 V(λ1 , . . . , λ n) Vandermonde matrix definition, 613 [A, B] commutator definition, 283 arg z argument definition, 13 Bε (x) open ball Defn. 12.1.1, 913 C n×m n × m complex matrices definition, 280 F real or complex numbers definition, 14 F(s) rational functions definition, 513 F(s) n×m

1450

F(s)n×m prop

n × m rational transfer functions definition, 514 n×m F(s)prop n × m proper rational transfer functions definition, 514 F(s)prop proper rational functions definition, 513 F[s] polynomials with coefficients in F definition, 499 F[s] n×m polynomial matrices with coefficients in Fn×m definition, 501 F n×m n × m real or complex matrices definition, 280 N real numbers definition, 2 P real numbers definition, 2 Q real numbers definition, 2 R complex numbers definition, 13 real numbers definition, 2 R n×m n × m real matrices definition, 280 Sε (x) sphere Defn. 12.1.1, 913 Z real numbers definition, 2 Hn n × n Hermitian matrix definition, 703 Nn n × n positive-semidefinite matrix definition, 703 Pn n × n positive-definite matrix definition, 703 Im z imaginary part definition, 13 In A inertia definition, 511 Re z real part

definition, 13 C(A, B) controllable subspace definition, 1260 H Hamiltonian definition, 1297 H(G) Markov block-Hankel matrix definition, 1275 Hi, j,k (G) Markov block-Hankel matrix definition, 1274 J l (q) Jordan matrix definition, 549 real Jordan matrix definition, 551 K(A, B) controllability matrix definition, 1260 L{x(t)} Laplace transform definition, 1181 N(A) null space definition, 291 O(A, C) observability matrix definition, 1254 R(A) range definition, 290 S⊥ orthogonal complement definition, 289 Ss (A) asymptotically stable subspace definition, 1199 Su (A) unstable subspace definition, 1199 U(A, C) unobservable subspace definition, 1254 X∼ complement definition, 1 Y\X relative complement definition, 1 ∥A∥ p H¨older norm definition, 836 ∥A∥F Frobenius norm definition, 836 ∥A∥col column norm definition, 844 ∥A∥row row norm

definition, 844 ∥ A∥σ p Schatten norm definition, 837 ∥ A∥ p,q,F definition, 842 ∥ A∥ q,p H¨older-induced norm definition, 842 ∥x∥ p H¨older norm definition, 833 ∥y∥D dual norm definition, 859 adA adjoint operator definition, 283 affin S affine hull definition, 279 αmax (A) spectral abscissa definition, 510 αmax (p) root real abscissa definition, 500 bd S boundary Defn. 12.1.11, 915 bdS′ S relative boundary Defn. 12.1.11, 915 βmax (A) spectral imaginary abscissa definition, 511 βmax (p) root imaginary abscissa definition, 500 χA characteristic equation definition, 506 χ A,B characteristic polynomial definition, 564 cl S closure definition, 914 clS′ S relative closure definition, 914 conv S convex hull definition, 279 coco S convex conical hull definition, 279 coli (A) column definition, 280 cone S conical hull definition, 279 dcone S

dual cone definition, 290 def A defect definition, 292 deg p degree definition, 499 δ(A) spread definition, 510 det A determinant definition, 299 diag( A1 , . . . , A k ) block-diagonal matrix definition, 365 diag(a1 , . . . , a n) diagonal matrix definition, 365 dim S dimension of a set definition, 279 ecl S essential closure definition, 914 ℓ(A) lower bound definition, 845 ℓ p norm sequence Defn. 12.2.16, 918 Prop. 12.2.17, 918 ℓ q,p(A) H¨older-induced lower bound definition, 846 essglb(S) essential greatest lower bound definition, 8 esslub(S) essential least upper bound definition, 8 fcone(D, x0 ) feasible cone definition, 924 Iˆn reverse permutation matrix definition, 284 ind A index of a matrix definition, 375 ind A (λ) index of an eigenvalue definition, 558 inf(S) infimum definition, 8 int S interior definition, 913 intS′ S relative interior

Abel’s theorem definition, 913 iso S isolated subset definition, 914 λ( A) eigenvalue vector definition, 506 λ1 (A) maximum eigenvalue definition, 506 minimum eigenvalue definition, 506 λi (A) eigenvalue definition, 506 log( A) matrix logarithm definition, 1187 mroots(p) multiset of roots definition, 499 mspec( A) multispectrum definition, 506 µA minimal polynomial definition, 512 ν− (A), ν0 (A) inertia definition, 511 C complex conjugate definition, 287 π approximation Fact 1.15.7, 53 Archimedes Fact 12.18.57, 969 arithmetic-geometric mean Fact 12.18.57, 969 polar S dual cone definition, 290 rank A rank definition, 292 rank G normal rank for a rational transfer function definition, 514 rank P normal rank for a polynomial matrix definition, 502 relbdS relative boundary Defn. 12.1.11, 915 reldeg G relative degree definition, 514 relint S relative interior definition, 913 revdiag(a1 , . . . , a n)

reverse diagonal matrix definition, 365 ρmax (A) spectral radius definition, 510 ρmax (p) root radius definition, 500 d(A) vector of diagonal entries definition, 281 dmax ( A) maximum diagonal entry definition, 280 dmin (A) minimum diagonal entry definition, 280 di(A) diagonal entry definition, 280 roots(p) set of roots definition, 499 rowi (A) row definition, 280 sig A signature definition, 511 σ( A) singular value vector definition, 556 σmax (A) maximum singular value definition, 556 σmin (A) minimum singular value definition, 556 σi ( A) singular value definition, 555 sign x sign definition, 289 sign α sign definition, xxvi spec(A) spectrum definition, 506 sup(S) supremum definition, 8 tr A trace definition, 287 vec A column-stacking operator definition, 681 vec−1 A inverse vec operator definition, 681 |x| absolute value definition, 289

e

1451

amult A(λ) algebraic multiplicity definition, 506 circ(a0 , . . . , a n−1 ) circulant matrix definition, 614 eA esslim(S) matrix exponential essential limit definition, 1179 definition, 9 ei exp( A) ith column of the identity matrix exponential definition, 1179 matrix glb(S) definition, 285 greatest lower bound ei,n definition, 7 ith column of the n × n gmult A identity matrix geometric multiplicity definition, 285 definition, 511 f (k) (x0 ) lub(S) kth derivative least upper bound definition, 926 definition, 7 f ′(x0 ) msval( A) derivative singular value multiset definition, 925 definition, 556 kth derivative mult p(λ) definition, 926 multiplicity n-tuple definition, 499 definition, 2 sh(A, B) x >> 0 shorted operator positive vector definition, 819 definition, 277 (1)-inverse x ≥≥ 0 definition, 621 nonnegative vector idempotent matrix definition, 277 s Fact 8.4.28, 638 x≺y left inverse y strongly majorizes x Prop. 8.1.2, 622 definition, 305 right inverse slog x ≺ y Prop. 8.1.3, 622 y strongly log majorizes x (1,2)-inverse definition, 305 definition, 621 w (1,3)-inverse x≺y least squares y weakly majorizes x Fact 11.17.6, 910 definition, 305 limit Fact 12.18.23, 961 Fact 12.18.24, 961 series Fact 13.5.14, 1023

wlog

x ≺ y y weakly log majorizes x definition, 305 x↓ component decreasing ordering operator definition, 305 ↑ x component increasing ordering operator definition, 305 D+ f (x0 ; ξ) one-sided directional differential definition, 924 SO(3) logarithm Fact 15.16.11, 1220 SO(n) eigenvalue Fact 7.12.2, 579

A Abel

quintic polynomial Fact 4.31.16, 435 Abel identity binomial equality Fact 1.16.10, 59 Abel summability limit Fact 12.18.17, 959 Abel’s identity product equality Fact 2.12.14, 217 Abel’s inequality sum of products Fact 2.12.15, 217 Fact 2.12.16, 217 Abel’s test series Fact 12.18.13, 959 Abel’s theorem

1452

Abelian group

polynomial Fact 12.16.6, 949 Abelian group definition Defn. 4.4.1, 369 Abelian multiplication group definition Defn. 4.6.1, 371 equivalence relation Prop. 4.7.2, 373 absolute norm monotone norm Prop. 11.1.2, 833 absolute sum norm definition, 834 absolute value Frobenius norm Fact 11.15.13, 897 H¨older-induced norm Fact 11.9.30, 863 Fact 11.9.31, 863 inequality Fact 2.2.44, 138 Fact 5.2.24, 486 irreducible matrix Fact 4.25.9, 426 matrix, 289 maximum singular value Fact 11.15.12, 897 Niculescu’s inequality Fact 2.2.43, 138 reducible matrix Fact 4.25.9, 426 Schatten norm Fact 11.15.13, 897 spectral radius Fact 6.11.21, 543 Fact 6.11.22, 543 vector, 289 absolute-value matrix positive-semidefinite matrix Fact 10.10.2, 730 absolutely convergent series convergent series Prop. 12.3.2, 919 Prop. 12.3.4, 919 definition Defn. 12.3.1, 919 Defn. 12.3.3, 919 accretive-dissipative matrix determinant Fact 10.16.43, 783 accumulation point definition, 1092 acute triangle inequality Fact 5.2.13, 480 Ono’s inequality Fact 5.2.13, 480 acyclic graph definition

Defn. 1.4.4, 11

commutator Fact 3.23.4, 355 Fact 3.23.5, 355 norm inequality Fact 11.8.3, 853 adjugate quadratic inequality basic properties, 301 Fact 2.12.37, 221 characteristic polynomial Fact 2.12.38, 222 Fact 6.9.4, 525 compound matrix Fact 2.12.39, 222 Fact 9.5.19, 694 addition group cross product definition Fact 8.9.17, 664 Defn. 4.5.1, 371 defect addition subgroup Fact 3.19.2, 346 definition definition, 301 Defn. 4.5.1, 371 derivative additive compound Fact 12.16.19, 951 asymptotically stable Fact 12.16.22, 952 polynomial determinant Fact 15.18.16, 1225 Fact 3.17.31, 341 additive decomposition Fact 3.17.35, 341 diagonalizable matrix Fact 3.19.1, 345 Fact 7.10.4, 570 Fact 3.19.4, 346 Hermitian matrix Fact 3.19.6, 346 Fact 7.20.3, 618 Fact 3.21.3, 351 nilpotent matrix determinant of a Fact 7.10.4, 570 partitioned matrix orthogonal matrix Fact 3.19.10, 347 Fact 7.20.6, 618 Fact 3.19.12, 348 Fact 7.20.7, 618 Dodgson condensation Fact 7.20.8, 618 Fact 3.17.35, 341 skew-symmetric matrix eigenvalue Fact 7.20.4, 618 Fact 6.10.13, 532 unitary matrix eigenvector Fact 7.20.5, 618 Fact 7.15.30, 601 adjacency matrix elementary matrix definition Fact 3.21.1, 351 Defn. 4.2.1, 367 factor directed graph Fact 3.19.8, 346 Fact 4.26.3, 426 Frobenius norm graph of a matrix Fact 11.9.15, 861 Prop. 4.2.5, 368 inbound Laplacian matrix full-state feedback Fact 3.19.11, 347 Thm. 4.2.2, 368 generalized inverse Laplacian matrix Fact 8.3.21, 630 Thm. 4.2.2, 368 Fact 8.3.22, 630 Thm. 4.2.3, 368 Fact 8.9.17, 664 Fact 6.11.12, 541 Hermitian matrix outbound Laplacian Fact 4.10.9, 378 matrix iterated Thm. 4.2.2, 368 Fact 3.19.1, 345 outdegree matrix matrix powers Fact 4.26.3, 426 Fact 6.9.4, 525 row-stochastic matrix matrix product Fact 4.26.3, 426 Fact 3.19.9, 346 symmetric graph maximum singular value Fact 4.26.2, 426 Fact 10.21.23, 804 adjacent Fact 11.15.16, 897 Defn. 1.4.1, 10 Fact 11.15.17, 897 adjoint norm nilpotent matrix definition Fact 3.19.5, 346 Fact 11.9.7, 860 Fact 8.3.21, 630 dual norm null space Fact 11.9.7, 860 Fact 3.19.2, 346 H¨older-induced norm outer-product Fact 11.9.21, 862 perturbation adjoint operator Aczel’s inequality

Fact 3.21.2, 351 Fact 3.21.3, 351 partitioned matrix Fact 3.17.31, 341 positive-definite matrix Fact 10.21.23, 804 range Fact 3.19.2, 346 rank Fact 3.19.2, 346 Fact 3.19.3, 346 scalar factor Fact 3.19.1, 345 singular value Fact 7.12.39, 585 skew-Hermitian matrix Fact 4.10.9, 378 Fact 4.10.11, 379 skew-symmetric matrix Fact 6.9.19, 528 spectrum Fact 6.10.13, 532 submatrix Fact 3.19.7, 346 trace Fact 6.9.4, 525 transfer function Fact 16.24.15, 1315 transpose Fact 3.19.1, 345 affine closed half space closed half space Fact 3.11.13, 308 definition, 290 affine combination definition, 278 affine function definition, 282 left inverse Fact 3.18.8, 343 right inverse Fact 3.18.8, 343 affine hull closure Fact 12.11.12, 935 constructive characterization Fact 3.11.5, 306 Fact 3.11.6, 307 convex hull Fact 3.11.8, 308 convex set Thm. 12.4.6, 922 Fact 12.11.20, 935 definition, 279 linear mapping Fact 3.13.4, 315 matrix Fact 3.11.18, 309 Minkowski sum Fact 3.12.2, 310 subset Fact 3.12.1, 310 affine hyperplane affine subspace

antieigenvalue Fact 3.11.13, 308 definition, 279 determinant Fact 5.1.7, 441 affine mapping Hermitian matrix Fact 4.10.14, 380 normal matrix Fact 4.10.14, 380 affine open half space definition, 290 open half space Fact 3.11.13, 308 affine subspace affine hull Fact 3.11.4, 306 affine hull of image Fact 3.11.15, 309 affine hyperplane Fact 3.11.13, 308 Cartesian product Fact 3.12.18, 313 closed set Fact 12.11.11, 935 closure Fact 12.11.15, 935 definition, 278 dimension Prop. 3.1.4, 279 image under linear mapping Fact 3.11.15, 309 inclusion Prop. 3.1.4, 279 interior Fact 12.11.15, 935 intersection Prop. 3.1.2, 278 Fact 3.12.4, 311 left inverse Fact 3.11.15, 309 Minkowski sum Fact 3.11.12, 308 Fact 3.12.4, 311 open set Fact 12.11.11, 935 range Fact 3.14.7, 321 relative interior Fact 12.11.15, 935 scaling Fact 3.11.1, 306 span Fact 3.11.14, 309 Fact 5.1.8, 442 Fact 12.11.13, 935 subspace Fact 3.11.10, 308 Fact 3.11.11, 308 Fact 3.11.12, 308 Afriat spectrum of a product of projectors Fact 7.13.4, 590 Akers

elementary projectors Fact 11.16.10, 901 algebraic multiplicity block-triangular matrix Prop. 7.7.14, 560 definition Defn. 6.4.4, 506 Drazin inverse Fact 8.10.7, 670 geometric multiplicity Prop. 7.7.3, 558 index of an eigenvalue Prop. 7.7.6, 559 matrix power Fact 7.15.22, 600 orthogonal matrix Fact 7.12.2, 579 outer-product matrix Fact 7.15.5, 598 zero eigenvalue Fact 7.15.1, 597 Ali outer-product perturbations Fact 11.16.11, 901 positive-semidefinite matrix Fact 10.22.40, 813 Aljanaideh adjugate of a submatrix Fact 3.19.7, 346 determinant Fact 3.19.12, 348 almost nonnegative matrix asymptotically stable matrix Fact 15.20.5, 1232 definition Defn. 4.1.5, 367 group-invertible matrix Fact 15.20.4, 1232 irreducible matrix Fact 15.20.2, 1232 Lyapunov-stable matrix Fact 15.20.4, 1232 matrix exponential Fact 15.20.1, 1232 Fact 15.20.2, 1232 N-matrix Fact 15.20.3, 1232 Fact 15.20.5, 1232 nonnegative matrix Fact 15.20.1, 1232 positive matrix Fact 15.20.2, 1232 alternating group definition Prop. 4.6.6, 372 group Fact 4.31.16, 435 alternating series Bernoulli number Fact 13.5.93, 1052 Euler number Fact 13.5.93, 1052

rational function Fact 13.5.90, 1051 Fact 13.5.91, 1052 Fact 13.5.92, 1052 alternating sign matrix column and row sum Fact 7.18.17, 617 alternating signs determinant Fact 3.16.11, 330 alternating tensor Kronecker product Fact 9.4.41, 689 wedge product Fact 9.4.41, 689 Alzer’s inequality sum of integers Fact 1.12.42, 47 AM-GM inequality n variables Fact 2.11.81, 203 Akerberg’s refinement Fact 2.11.82, 203 alternative form Fact 2.11.119, 210 complex numbers Fact 2.21.18, 273 determinant Fact 10.16.48, 784 Fact 10.16.49, 784 difference Fact 2.11.115, 209 Heinz mean Fact 2.2.64, 143 interpolation Fact 2.2.65, 144 Fact 2.2.66, 144 Jensen’s inequality Fact 1.21.7, 117 Maclaurin’s inequality Fact 2.11.35, 195 main form Fact 2.11.114, 209 positive-definite matrix Fact 10.15.11, 775 power mean Fact 2.11.86, 204 Fact 2.11.89, 205 power-mean inequality Fact 2.2.58, 141 quartic equality Fact 2.4.12, 179 ratio Fact 2.11.115, 209 reverse inequality Fact 2.11.92, 205 Fact 2.11.93, 206 Fact 2.11.94, 206 Fact 2.11.95, 206 sextic equality Fact 2.6.1, 184 two variables Fact 2.2.15, 133 upper bound Fact 2.4.19, 180

1453

variation Fact 2.2.27, 135 weak form Fact 2.11.85, 203 weighted AM-GM inequality Fact 2.11.87, 204 Fact 2.11.118, 209 Amemiya’s inequality Schur product Fact 10.25.56, 828 analytic extension complex function Fact 12.14.11, 947 analytic function definition Defn. 12.6.4, 927 higher order derivatives Prop. 12.6.7, 928 Laurent series Prop. 12.6.9, 928 power series expansion Prop. 12.6.8, 928 vanishing function Prop. 12.6.5, 928 and definition, 2 Anderson trace inequality Fact 10.19.25, 798 Ando convex function Prop. 10.6.17, 831 angle coplanar vectors Fact 5.1.5, 441 definition, 285 determinant Fact 5.1.5, 441 positive-definite matrix Fact 10.18.26, 795 angle between complex vectors definition, 286 angle inequality Krein’s inequality Fact 11.8.9, 857 angular velocity vector quaternions Fact 15.12.16, 1208 Ansari block-Toeplitz matrix Fact 4.23.12, 422 anti-norm positive-definite matrix Fact 10.14.7, 764 reverse triangle inequality Fact 11.8.21, 859 Schatten norm Fact 11.10.53, 876 unitarily invariant norm Fact 11.10.4, 869 antieigenvalue definition Fact 11.9.54, 866

1454

antisymmetric graph

antisymmetric graph

Laplacian Fact 4.26.2, 426 antisymmetric relation definition Defn. 1.3.8, 7 one-sided cone induced by Prop. 3.1.7, 280 positive-semidefinite matrix Prop. 10.1.1, 703 aperiodic directed graph Defn. 1.4.3, 10 aperiodic graph Defn. 1.4.4, 11 nonnegative matrix Fact 6.11.5, 538 Apery’s constant series Fact 13.5.41, 1031 zeta function Fact 13.5.41, 1031 Araki positive-semidefinite matrix Fact 10.14.25, 767 Araki-Lieb-Thirring inequality positive-semidefinite matrix Fact 10.14.24, 767 arc length ellipse Fact 13.9.5, 1079 Archimedes approximations of π Fact 12.18.57, 969 arctan function sum Fact 2.18.4, 257 Fact 2.18.5, 258 area ellipse Fact 5.5.8, 496 Fact 5.5.9, 496 hypersphere Fact 5.5.13, 497 parallelogram Fact 5.4.4, 494 Fact 11.8.13, 857 pentagon Fact 5.3.12, 493 polygon Fact 5.3.10, 493 Fact 5.3.11, 493 Fact 5.3.13, 493 triangle Fact 5.2.3, 443 Fact 5.2.4, 443 Fact 5.2.5, 443 Fact 5.2.6, 443 areal coordinates triangle Fact 5.2.11, 466 argument

singular value Bode phase integral Fact 10.22.34, 812 Fact 14.7.14, 1156 definition, 4 arithmetic-mean– arithmetic mean harmonic-mean Carleman’s inequality inequality Fact 2.11.140, 213 scalar inequality geometric mean Fact 2.11.135, 212 Fact 2.2.63, 142 arrangement number Fact 2.11.97, 206 definition Fact 2.11.99, 206 Fact 1.18.1, 103 Fact 2.11.101, 207 permutation Fact 2.11.102, 207 Fact 1.18.1, 103 Fact 2.11.103, 207 associative equalities Fact 2.11.112, 208 definition, 283 Fact 2.11.113, 208 associativity Hermitian matrix composition Fact 10.11.54, 741 Prop. 1.6.1, 16 identric mean asymptotic sequence Fact 2.2.63, 142 big oh notation inequality Prop. 12.2.15, 918 Fact 2.3.49, 161 Defn. 12.2.14, 918 Fact 2.11.104, 207 little oh notation Fact 2.11.105, 207 Prop. 12.2.15, 918 Fact 2.11.110, 208 asymptotic solution Fact 2.11.111, 208 differential equation logarithmic mean Fact 16.20.1, 1305 Fact 2.11.112, 208 Lyapunov differential mixed equation arithmetic-geometric Fact 16.20.2, 1305 mean inequality asymptotic stability Fact 2.11.138, 213 eigenvalue positive-definite matrix Prop. 15.9.2, 1198 Fact 10.11.55, 741 input-to-state stability scalar inequality Fact 16.21.20, 1309 Fact 2.3.36, 155 linear dynamical system Fact 2.3.50, 161 Prop. 15.9.2, 1198 Fact 2.3.51, 161 Lyapunov equation Fact 2.3.60, 162 Cor. 15.10.1, 1201 arithmetic sequence matrix exponential rank Prop. 15.9.2, 1198 Fact 3.13.45, 320 nonlinear system arithmetic-geometric mean Thm. 15.8.5, 1196 π asymptotically stable Fact 12.18.57, 969 equilibrium complete elliptic integral definition of the first kind Defn. 15.8.3, 1196 Fact 12.18.55, 968 asymptotically stable limit matrix Fact 12.18.55, 968 2 × 2 matrix Fact 12.18.57, 969 Fact 15.19.35, 1231 arithmetic-mean almost nonnegative matrix inequality Fact 15.20.5, 1232 Fact 2.11.47, 197 asymptotically stable arithmetic-mean inequality polynomial harmonic mean Prop. 15.9.4, 1199 Fact 2.11.43, 197 Cayley transform scalar inequality Fact 15.22.10, 1240 Fact 2.11.129, 211 compartmental matrix arithmetic-mean– Fact 15.20.6, 1233 geometric mean controllability inequality Fact 16.21.6, 1308 polynomial controllability Gramian Fact 2.11.83, 203 Prop. 16.7.9, 1268 Siegel’s inequality Cor. 16.7.10, 1268 Fact 2.11.83, 203 controllable pair

Prop. 16.7.9, 1268 Cor. 16.7.10, 1268 controllably asymptotically stable Prop. 16.8.6, 1269 cyclic matrix Fact 15.19.25, 1229 definition Defn. 15.9.1, 1198 detectability Prop. 16.5.6, 1259 Cor. 16.5.7, 1259 diagonalizable over R Fact 15.18.14, 1225 discrete-time asymptotically stable matrix Fact 15.22.10, 1240 dissipative matrix Fact 15.19.21, 1228 Fact 15.19.37, 1231 factorization Fact 15.19.22, 1228 integral Lem. 15.10.2, 1201 inverse matrix Fact 15.19.15, 1227 Kronecker sum Fact 15.19.32, 1231 Fact 15.19.33, 1231 Fact 15.19.34, 1231 linear matrix equation Prop. 15.10.3, 1201 logarithmic derivative Fact 15.19.11, 1227 Lyapunov equation Prop. 15.10.5, 1202 Cor. 15.10.4, 1202 Cor. 15.10.7, 1203 Cor. 16.4.4, 1258 Cor. 16.5.7, 1259 Cor. 16.7.4, 1268 Cor. 16.8.7, 1270 Fact 16.22.7, 1311 Fact 16.22.17, 1312 matrix exponential Lem. 15.10.2, 1201 Fact 15.16.9, 1219 Fact 15.19.9, 1227 Fact 15.19.10, 1227 Fact 15.19.15, 1227 Fact 15.19.18, 1228 Fact 15.19.19, 1228 Fact 15.22.9, 1240 minimal realization Defn. 16.9.25, 1278 negative-definite matrix Fact 15.19.30, 1231 nonsingular N-matrix Fact 15.20.5, 1232 normal matrix Fact 15.19.37, 1231 observability Gramian Cor. 16.4.10, 1258 observable pair

Bertrand’s postulate Prop. 16.4.9, 1258 Cor. 16.4.10, 1258 observably asymptotically stable Prop. 16.5.6, 1259 perturbation Fact 15.19.16, 1227 positive-definite matrix Prop. 15.10.5, 1202 Prop. 16.4.9, 1258 Cor. 15.10.7, 1203 Fact 15.19.21, 1228 secant condition Fact 15.19.29, 1230 sign of entry Fact 15.20.5, 1233 sign stability Fact 15.20.5, 1233 similar matrices Fact 15.19.5, 1226 skew-Hermitian matrix Fact 15.19.30, 1231 spectrum Fact 15.19.13, 1227 square root Fact 15.19.36, 1231 stability radius Fact 15.19.17, 1228 stabilizability Prop. 16.8.6, 1269 Cor. 16.8.7, 1270 subdeterminant Fact 15.20.1, 1232 trace Fact 15.19.31, 1231 tridiagonal matrix Fact 15.19.24, 1229 Fact 15.19.25, 1229 Fact 15.19.26, 1229 Fact 15.19.27, 1230 Fact 15.19.28, 1230 asymptotically stable polynomial additive compound Fact 15.18.16, 1225 asymptotically stable matrix Prop. 15.9.4, 1199 definition Defn. 15.9.3, 1198 derivative Fact 15.18.1, 1223 even polynomial Fact 15.18.8, 1224 Hermite-Biehler theorem Fact 15.18.8, 1224 interlacing theorem Fact 15.18.8, 1224 Kharitonov’s theorem Fact 15.18.17, 1225 Kronecker sum Fact 15.18.15, 1225 odd polynomial Fact 15.18.8, 1224 polynomial coefficients

Fact 15.18.3, 1224 Bandila’s inequality Fact 15.18.5, 1224 triangle Fact 15.18.9, 1224 Fact 5.2.8, 446 Fact 15.18.10, 1224 Bankoff Fact 15.18.14, 1225 triangle Fact 15.18.15, 1225 Fact 5.2.8, 446 Fact 15.18.16, 1225 Barbalat’s lemma reciprocal argument Fact 12.17.14, 957 Fact 15.18.6, 1224 Fact 12.17.17, 957 Schur power Fact 12.17.16, 957 Fact 15.18.11, 1224 Barnett Schur product tridiagonal matrix Fact 15.18.12, 1225 Fact 15.19.24, 1229 subdeterminant Barnett factorization Fact 15.19.23, 1228 Bezout matrix asymptotically stable Fact 6.8.8, 519 barycenter subspace triangle definition, 1199 Fact 5.2.11, 466 asymptotically stable barycentric coordinates transfer function conjugate parameters minimal realization Fact 2.12.23, 219 Prop. 16.9.26, 1278 definition, 278 SISO entry triangle Prop. 16.9.27, 1278 Fact 5.2.11, 466 atan2 basis four-quadrant inverse definition, 279 tangent function Beckenbach Fact 2.18.7, 259 inequality Atkin Fact 2.12.49, 224 partition number Beckner’s two-point Fact 1.20.1, 113 inequality Autonne-Takagi powers factorization Fact 2.2.29, 135 complex-symmetric matrix Schatten norm Fact 7.10.22, 572 Fact 11.10.64, 879 average Bell number positive-semidefinite definition matrix Fact 1.19.6, 112 Fact 7.20.10, 619 determinant averaged limit Fact 6.9.9, 526 integral Dobinski’s formula Fact 12.13.12, 943 Fact 13.1.4, 976 series B Fact 13.1.4, 976 subset number Bagdasar Fact 1.19.6, 112 polygon upper Hessenberg matrix Fact 5.3.14, 493 Fact 6.9.9, 526 Baker-Campbell-Hausdorff Bell polynomial series properties matrix exponential Fact 13.2.4, 984 Prop. 15.4.12, 1189 Bellman Baker-Campbellquadratic form Hausdorff-Dynkin Fact 10.18.6, 792 expansion Bellman’s inequality time-varying dynamics quadratic inequality Fact 16.20.6, 1306 Fact 2.12.40, 222 balanced realization Ben-Israel definition generalized inverse Defn. 16.9.28, 1278 Fact 8.3.38, 632 minimal realization Bencze Prop. 16.9.29, 1278 arithmetic-mean– balancing transformation geometric-mean– existence Cor. 10.3.4, 708

1455

logarithmic-mean inequality Fact 2.11.112, 208 Bendixson’s theorem eigenvalue bound Fact 7.12.24, 582 Fact 11.13.7, 891 Beppo Levi inequality norm equality Fact 11.8.3, 853 Berezin trace of a convex function Fact 10.14.48, 771 Bergstrom determinant Fact 10.16.14, 778 Bergstrom’s inequality complex numbers Fact 2.21.20, 273 quadratic form Fact 10.12.5, 750 Fact 10.19.4, 795 sum of ratios Fact 2.12.19, 218 Fact 2.12.22, 218 Bernoulli formula sum of powers Fact 1.12.1, 36 Bernoulli matrix Vandermonde matrix Fact 7.18.5, 613 Bernoulli number alternating series Fact 13.5.93, 1052 definite integral Fact 14.6.45, 1149 Fact 14.8.16, 1160 definition Fact 13.1.6, 977 sum of powers of integers Fact 1.12.2, 37 Bernoulli polynomial series Fact 13.2.1, 981 Bernoulli’s inequality n variables Fact 2.11.11, 189 interpolation Fact 2.1.26, 124 scalar inequality Fact 2.1.21, 123 Fact 2.1.23, 123 Bernstein function inequality Fact 2.3.26, 153 Bernstein matrix Vandermonde matrix Fact 7.18.5, 613 Bernstein’s inequality matrix exponential Fact 15.16.4, 1218 Bernstein’s theorem derivative of a polynomial Fact 12.16.3, 949 Bertrand’s postulate

1456

Berwald

prime number Fact 1.11.42, 35 Berwald polynomial root bound Fact 15.21.18, 1237 Bessel’s inequality norm inequality Fact 11.8.10, 857 Bessis-Moussa-Villani trace conjecture derivative of a matrix exponential Fact 10.14.46, 771 power of a positive-semidefinite matrix Fact 10.14.45, 771 Bezout equation coprime polynomials Fact 6.8.7, 519 Bezout identity right coprime polynomial matrices Thm. 6.7.14, 516 Bezout matrix coprime polynomials Fact 6.8.8, 519 Fact 6.8.9, 520 Fact 6.8.10, 520 definition Fact 6.8.8, 519 distinct roots Fact 6.8.11, 521 factorization Fact 7.17.26, 608 polynomial roots Fact 6.8.11, 521 Bhat integral of a Gaussian density Fact 15.14.12, 1210 Bhatia Schatten norm inequality Fact 11.10.18, 871 bi-power series definition Defn. 12.3.5, 920 generating function Defn. 12.3.5, 920 inner radius of convergence Prop. 12.3.11, 921 outer radius of convergence Prop. 12.3.11, 921 radius of convergence Prop. 12.3.6, 921 bi-sequence definition Defn. 12.3.8, 920 bialternate product compound matrix Fact 9.5.20, 695 bidiagonal matrix

singular value Fact 7.12.51, 589 bidirectional implication definition, 4 biequivalence transformation definition, 374 biequivalent matrices congruent matrices Prop. 4.7.5, 374 definition Defn. 4.7.3, 373 Kronecker product Fact 9.4.12, 686 rank Prop. 7.1.2, 545 similar matrices Prop. 4.7.5, 374 Smith matrix Thm. 7.1.1, 545 unitarily similar matrices Prop. 4.7.5, 374 big oh notation sequence Defn. 12.2.13, 918 bijective function definition, 118 bilinear function definition, 926 Binet’s formula Fibonacci number Fact 1.17.1, 95 Lucas number Fact 1.17.2, 100 Binet-Cauchy formula determinant Fact 3.16.8, 330 Fact 3.16.9, 330 Fact 6.9.37, 530 Binet-Cauchy theorem compound of a matrix product Fact 9.5.18, 693 binomial coefficient binomial coefficient with repetition Fact 1.16.16, 94 Catalan number Fact 1.18.4, 104 congruence Fact 1.11.19, 32 definition, 15 determinant Fact 3.16.28, 333 Fact 3.16.29, 334 entropy Fact 12.18.49, 967 integer Fact 1.16.1, 54 integer pair Fact 1.12.17, 42 Lah number Fact 1.19.4, 108 least common multiple Fact 1.16.2, 55

limit Fact 12.18.47, 967 Fact 12.18.49, 967 Fact 12.18.50, 967 Fact 12.18.51, 968 Fact 12.18.52, 968 Fact 12.18.53, 968 lower triangular matrix Fact 4.25.3, 424 power Fact 1.16.4, 55 series Fact 13.7.1, 1063 Fact 13.7.2, 1063 Fact 13.7.3, 1064 Fact 13.7.4, 1064 Fact 13.7.5, 1064 Fact 13.7.6, 1064 Fact 13.7.8, 1065 Fact 13.7.9, 1070 Fact 13.7.11, 1071 binomial coefficient with repetition binomial coefficient Fact 1.16.16, 94 binomial expansion characteristic polynomial Fact 6.9.31, 529 binomial identity power equality Fact 1.16.10, 59 binomial inequality sum Fact 1.16.15, 92 binomial series complex numbers Fact 13.4.2, 1004 binomial transform identity Fact 1.16.7, 56 bipartite directed graph Defn. 1.4.3, 10 Birkhoff doubly stochastic matrix Fact 4.13.1, 387 bivector parallelogram Fact 11.8.13, 857 block definition, 281 block decomposition Hamiltonian Prop. 16.17.5, 1299 minimal realization Prop. 16.9.12, 1273 block Kronecker product Schur product Fact 9.6.25, 700 block-circulant matrix circulant matrix Fact 4.23.3, 420 Drazin generalized inverse Fact 8.12.5, 679 generalized inverse Fact 8.9.33, 668

inverse matrix Fact 3.22.7, 354 block-companion matrix characteristic polynomial Fact 7.13.33, 597 block-diagonal matrix companion matrix Prop. 7.3.7, 548 Lem. 7.3.2, 547 definition Defn. 4.1.3, 365 geometric multiplicity Prop. 7.7.14, 560 Hermitian matrix Fact 4.10.7, 378 least common multiple Lem. 7.3.6, 548 matrix exponential Prop. 15.2.8, 1184 maximum singular value Fact 7.12.37, 585 minimal polynomial Lem. 7.3.6, 548 normal matrix Fact 4.10.7, 378 shifted-unitary matrix Fact 4.13.10, 388 similar matrices Thm. 7.4.5, 550 singular value Fact 10.12.65, 760 Fact 10.22.16, 807 Fact 11.16.29, 904 Fact 11.16.34, 905 skew-Hermitian matrix Fact 4.10.7, 378 unitary matrix Fact 4.13.10, 388 block-Hankel matrix definition Defn. 4.1.3, 365 Hankel matrix Fact 4.23.3, 420 Markov block-Hankel matrix definition, 1274 block-Kronecker product Kronecker product, 700 block-Toeplitz matrix definition Defn. 4.1.3, 365 rank Fact 4.23.12, 422 Toeplitz matrix Fact 4.23.3, 420 block-triangular matrix algebraic multiplicity Prop. 7.7.14, 560 controllable dynamics Thm. 16.6.8, 1262 controllable subspace Prop. 16.6.9, 1263 Prop. 16.6.10, 1263 controllably asymptotically stable

Cartesian decomposition Prop. 16.7.3, 1266 detectability Prop. 16.5.4, 1259 determinant Fact 3.17.8, 335 Fact 10.16.47, 784 index of a matrix Fact 7.16.19, 604 inverse matrix Fact 3.22.1, 352 maximum singular value Fact 7.12.36, 585 minimal polynomial Fact 6.10.17, 534 observable dynamics Thm. 16.3.8, 1255 observably asymptotically stable Prop. 16.4.3, 1257 spectrum Prop. 7.7.14, 560 stabilizability Prop. 16.8.4, 1268 unobservable subspace Prop. 16.3.9, 1255 Prop. 16.3.10, 1255 blocking zero definition Defn. 6.7.10, 515 rational transfer function Defn. 6.7.4, 514 Smith-McMillan form Prop. 6.7.11, 515 Blundon triangle inequality Fact 5.2.8, 446 blunt cone definition, 278 Bode phase integral argument Fact 14.7.14, 1156 Bode sensitivity integral logarithm Fact 14.7.13, 1156 Bohr radius power series Fact 12.14.14, 947 Bohr’s inequality complex numbers Fact 2.21.8, 269 positive-semidefinite matrix Fact 10.11.83, 747 Bonami’s inequality powers Fact 2.2.32, 136 Fact 11.7.15, 852 Bonse’s inequality prime number Fact 1.11.45, 35 Borchers trace norm of a matrix difference Fact 11.10.19, 871 bordered matrix

factorization Fact 3.21.4, 351 inverse matrix Fact 3.21.5, 352 Borobia asymptotically stable polynomial Fact 15.18.10, 1224 boundary definition Defn. 12.1.11, 915 interior Fact 12.11.8, 935 Fact 12.11.9, 935 union Fact 12.12.2, 937 boundary relative to a set definition Defn. 12.1.11, 915 bounded derivative Hardy and Littlewood’s inequality Fact 12.16.13, 951 Landau’s inequality Fact 12.16.12, 950 bounded sequence definition, 917 bounded set definition Defn. 12.1.13, 915 image under linear mapping Fact 11.9.1, 860 open ball Fact 12.11.2, 935 Fact 12.11.3, 935 Bourbaki polynomial root bound Fact 15.21.5, 1235 Bourgeois matrix exponential Fact 15.15.1, 1211 Bourin spectral radius of a product Fact 10.22.33, 811 Boutin’s identity sum of powers Fact 2.11.9, 189 Brahmagupta’s formula quadrilateral Fact 5.3.1, 489 Brahmagupta’s identity five variables Fact 2.4.7, 178 Brauer spectrum bounds Fact 6.10.27, 536 Bretschneider’s formula quadrilateral Fact 5.3.1, 489 Brianchon’s theorem diagonals of a hexagon Fact 5.3.6, 492

Brouwer fixed-point theorem

1457

inclusion-exclusion principle image of a continuous Fact 1.8.4, 23 union function Fact 1.8.4, 23 Cor. 12.4.27, 924 cardinality of set Brown trace of a convex function intersections Fact 10.14.48, 771 sum Fact 1.12.16, 42 Browne’s theorem eigenvalue bound Carleman’s inequality Fact 7.12.24, 582 arithmetic mean Fact 2.11.140, 213 Fact 7.12.29, 583 Polya’s generalization Fact 11.13.6, 891 Fact 2.12.11, 216 Brownian motion positive-semidefinite Carlson inertia matrix Fact 16.22.4, 1310 Fact 10.9.4, 725 Carlson inequality Bruhat decomposition sum of powers upper triangular matrix Fact 2.11.141, 213 Fact 7.10.32, 575 Carlson’s inequality Burnside’s formula sum of powers factorial Fact 2.11.142, 213 Fact 12.18.58, 969 Carmichael Buzano’s inequality polynomial root bound Cauchy-Schwarz Fact 15.21.14, 1236 inequality Carnot’s theorem Fact 2.13.2, 226 triangle norm inequality Fact 5.2.11, 466 Fact 11.8.5, 856 Cartan orthogonal matrix C Fact 7.17.17, 607 Cartesian decomposition Callan determinant determinant Fact 10.16.2, 776 Fact 3.17.16, 337 eigenvalue Callebaut’s inequality Fact 7.12.24, 582 refined Cauchy-Schwarz Hermitian matrix inequality Fact 4.10.28, 381 Fact 2.12.33, 221 Fact 7.20.2, 618 Camion’s theorem Fact 7.20.3, 618 tournament normal matrix Fact 1.9.9, 26 Fact 7.20.2, 618 canonical form Fact 15.14.4, 1209 definition, 16 positive-definite matrix canonical mapping Fact 4.10.30, 381 definition, 16 positive-semidefinite Cantor intersection matrix theorem Fact 11.10.69, 880 intersection of compact rank sets Fact 7.20.1, 618 Fact 12.12.20, 939 Schatten norm Cao Fact 11.10.66, 879 group generalized inverse Fact 11.10.67, 879 Fact 8.12.3, 678 Fact 11.10.68, 880 Caratheodory’s theorem Fact 11.10.69, 880 convex set singular value Fact 3.11.5, 306 Fact 10.22.14, 807 Cardano’s trigonometric skew-Hermitian matrix solution Fact 4.10.28, 381 cubic polynomial Fact 7.20.2, 618 Fact 6.10.7, 531 Fact 7.20.3, 618 eigenvalue spectrum Fact 6.10.7, 531 Fact 7.12.24, 582 cardinality unitarily invariant norm definition, 16

1458

Cartesian product

Fact 11.14.17, 895

Cauchy interlacing theorem

Polya-Szego inequality Fact 2.12.44, 223 positive-semidefinite affine subspace Hermitian matrix Fact 3.12.18, 313 matrix eigenvalue cone Fact 10.12.33, 754 Lem. 10.4.4, 709 Fact 3.12.18, 313 Fact 10.12.35, 754 Cauchy matrix convex cone Fact 10.18.7, 792 determinant Fact 3.12.18, 313 vector case Fact 4.27.5, 427 convex set Fact 2.12.17, 217 Fact 4.27.6, 427 Fact 3.12.18, 313 Cayley positive-definite matrix definition, 2 determinant equality Fact 10.9.10, 727 partitioned matrix Fact 2.16.21, 246 Fact 10.9.20, 729 Fact 3.12.9, 312 Cayley diagram Fact 16.22.18, 1312 subspace definition, 440 positive-semidefinite Fact 3.12.18, 313 Cayley transform matrix Cartesian product asymptotically stable Fact 10.9.11, 728 representation matrix Fact 10.9.13, 728 definition, 278 Fact 15.22.10, 1240 Fact 16.22.19, 1312 cascade interconnection cross product quadratic form definition, 1289 Fact 4.14.9, 394 Fact 2.12.32, 221 transfer function cross-product matrix Cauchy product Prop. 16.13.2, 1289 Fact 4.12.1, 384 series cascaded systems definition Fact 13.5.8, 1022 geometric multiplicity Fact 4.13.24, 389 Cauchy’s estimate Fact 16.24.17, 1316 discrete-time polynomial root bound Catalan asymptotically stable Fact 15.21.9, 1235 infinite product Cauchy’s identity matrix Fact 13.10.3, 1081 product equality Fact 15.22.10, 1240 Catalan number Fact 2.14.1, 226 Hamiltonian matrix binomial coefficient Cauchy-Hadamard formula Fact 4.28.12, 428 Fact 1.18.4, 104 series Hermitian matrix definition Fact 12.18.9, 959 Fact 4.13.24, 389 Fact 1.18.4, 104 Cauchy-Khinchin orthogonal matrix determinant Fact 4.13.25, 390 inequality Fact 6.9.9, 526 Fact 4.13.26, 390 matrix Hankel matrix Fact 4.14.9, 394 Fact 3.15.39, 329 Fact 13.2.15, 994 positive-definite matrix Cauchy-Riemann integral Fact 10.10.35, 733 equations Fact 14.3.9, 1113 skew-Hermitian matrix definition, 927 series Fact 4.13.25, 390 derivative of a complex Fact 13.9.1, 1074 skew-symmetric matrix function upper Hessenberg matrix Fact 4.13.25, 390 Prop. 12.6.3, 927 Fact 6.9.9, 526 Fact 4.13.26, 390 Cauchy-Schwarz inequality Catalan numbers Fact 4.14.9, 394 Buzano’s inequality series symplectic matrix Fact 2.13.2, 226 Fact 13.5.18, 1025 Fact 4.28.12, 428 Callebaut’s inequality Catalan’s conjecture unitary matrix Fact 2.12.33, 221 integers Fact 4.13.25, 390 De Bruijn’s inequality Fact 1.11.41, 35 Cayley-Hamilton theorem Fact 2.12.43, 223 Catalan’s constant characteristic polynomial determinant integral Thm. 6.4.7, 509 Fact 10.16.25, 780 Fact 14.9.1, 1169 generalized version Frobenius norm series Fact 6.9.29, 529 Cor. 11.3.9, 840 Fact 13.5.86, 1050 polynomial matrix inner product bound Catalan’s identity Fact 6.9.30, 529 Cor. 11.1.7, 835 sum of reciprocals of Cayley-Menger interpolation integers determinant Fact 2.12.35, 221 Fact 1.12.44, 47 Heron’s formula McLaughlin’s inequality Cauchy Fact 5.4.7, 495 Fact 2.12.36, 221 polygonal numbers volume Milne’s inequality Fact 1.12.7, 40 Fact 5.4.7, 495 Fact 2.12.28, 220 polynomial disk inclusion center of mass Fact 2.12.29, 220 Fact 15.21.30, 1239 triangle Fact 2.12.30, 220 polynomial root bound Fact 5.2.11, 466 Ozeki’s inequality Fact 15.21.18, 1237 center subgroup Fact 2.12.45, 223 Cartesian product

commutator Fact 3.23.10, 355 central binomial coefficient asymptotic approximation Fact 12.18.48, 967 infinite product Fact 13.10.3, 1081 Fact 13.10.9, 1085 central trinomial coefficient geometric series Fact 13.4.2, 1004 centralizer commutator Fact 3.23.9, 355 Fact 9.5.3, 691 commuting matrices Fact 7.16.8, 603 Fact 7.16.9, 603 centrohermitian matrix complex conjugate transpose Fact 4.27.7, 427 definition Defn. 4.1.2, 364 generalized inverse Fact 8.3.35, 632 matrix product Fact 4.27.8, 427 centroid triangle Fact 5.2.11, 466 centrosymmetric matrix definition Defn. 4.1.2, 364 matrix product Fact 4.27.8, 427 matrix transpose Fact 4.27.7, 427 Cesaro limit cosine Fact 13.6.19, 1063 Cesaro summability limit Fact 12.18.17, 959 Cesaro summable discrete-time Lyapunov-stable matrix Fact 15.22.14, 1241 Cesaro’s lemma limit Fact 12.18.14, 959 chain definition Defn. 1.4.4, 11 chaotic ordering matrix logarithm Fact 10.23.1, 813 characteristic equation definition, 506 characteristic exponent periodic dynamics Fact 16.20.8, 1307 characteristic multiplier periodic dynamics

Cochran’s theorem Fact 6.9.13, 527 Fact 6.9.14, 527 Fact 6.9.16, 527 2 × 2 matrix Fact 6.9.17, 527 Fact 6.9.1, 524 Fact 6.9.32, 529 3 × 3 matrix Fact 6.9.33, 530 Fact 6.9.2, 524 similar matrices 4 × 4 matrix Fact 6.9.6, 525 Fact 6.9.3, 524 similarity invariant adjugate Prop. 6.4.2, 506 Fact 6.9.4, 525 Prop. 6.6.2, 513 binomial expansion skew-Hermitian matrix Fact 6.9.31, 529 Fact 6.9.12, 527 block-companion matrix skew-symmetric matrix Fact 7.13.33, 597 Fact 6.9.11, 527 Cayley-Hamilton theorem Fact 6.9.18, 527 Thm. 6.4.7, 509 Fact 6.9.19, 528 colleague matrix Fact 7.16.21, 605 Fact 7.18.2, 610 sum of derivatives companion matrix Fact 6.9.7, 525 Prop. 7.3.1, 546 upper block-triangular Cor. 7.3.4, 547 Cor. 7.3.5, 548 matrix comrade matrix Fact 6.10.16, 533 Fact 7.18.2, 610 upper triangular matrix confederate matrix Fact 6.10.14, 533 Fact 7.18.2, 610 Chebyshev congenial matrix polynomial Fact 7.18.2, 610 Fact 12.13.5, 942 cross-product matrix Chebyshev inequality Fact 6.9.18, 527 Schur product Fact 6.9.19, 528 Fact 10.25.69, 830 cyclic matrix Chebyshev polynomial of Prop. 7.7.15, 561 the first kind definition trigonometric functions Defn. 6.4.1, 506 Fact 13.2.6, 985 degree Chebyshev polynomial of Prop. 6.4.3, 506 the second kind derivative trigonometric functions Lem. 6.4.8, 509 Fact 13.2.7, 986 eigenvalue Chebyshev’s inequality Prop. 6.4.6, 508 rearrangement equalities Fact 2.12.7, 215 Prop. 6.4.5, 507 Chen matrix generalized inverse tridiagonal matrix Fact 8.3.28, 631 Fact 15.19.27, 1230 Hamiltonian matrix child Fact 6.9.20, 528 Defn. 1.4.1, 10 Fact 6.9.33, 530 Chirita inverse matrix inequality Fact 6.9.5, 525 Fact 2.12.57, 225 Leverrier’s algorithm Cholesky decomposition Prop. 6.4.9, 509 existence matrix product Fact 10.10.42, 734 Prop. 6.4.10, 509 circle Cor. 6.4.11, 510 complex numbers minimal polynomial Fact 5.5.2, 495 Fact 6.9.34, 530 plane partition monic Fact 5.5.1, 495 Prop. 6.4.3, 506 circulant matrix outer-product matrix block-circulant matrix Fact 6.9.15, 527 Fact 4.23.3, 420 Fact 6.9.17, 527 companion matrix output feedback Fact 7.18.13, 614 Fact 16.24.14, 1315 cyclic permutation matrix partitioned matrix Fact 7.18.13, 614 Fact 16.20.8, 1307

characteristic polynomial

Fourier matrix Fact 7.18.13, 614 primary circulant matrix Fact 7.18.13, 614 spectrum Fact 7.18.13, 614 CIUD closed inside unit disk definition, 14 Clarkson inequalities complex numbers Fact 2.21.8, 269 Schatten norm Fact 11.10.54, 876 Clausen’s integral series Fact 13.6.2, 1057 CLHP closed left half plane definition, 14 Clifford algebra real matrix representation Fact 4.32.1, 437 Cline generalized inverse of a matrix product Fact 8.4.23, 636 group generalized inverse Fact 8.11.1, 674 Cline’s formula Drazin generalized inverse Fact 8.11.6, 675 closed half space affine closed half space Fact 3.11.13, 308 cone Fact 3.11.7, 308 definition, 290 closed relative to a set continuous function Thm. 12.4.9, 922 closed set affine subspace Fact 12.11.11, 935 complement Fact 12.11.5, 935 continuous function Cor. 12.4.10, 922 convex set Fact 12.11.26, 936 definition Defn. 12.1.6, 914 intersection Fact 12.12.17, 939 Minkowski sum Fact 12.12.6, 937 Fact 12.12.7, 937 polar Fact 12.12.14, 938 positive-semidefinite matrix Fact 12.11.29, 936 quadratic form Fact 10.17.13, 789 subspace

1459

Fact 12.11.11, 935 union Fact 12.12.17, 939 closed-loop eigenvalues feedback interconnection Fact 16.21.22, 1309 closed-loop spectrum detectability Lem. 16.16.17, 1298 Hamiltonian Prop. 16.16.14, 1298 maximal solution of the Riccati equation Prop. 16.18.2, 1302 observability Lem. 16.16.17, 1298 observable eigenvalue Lem. 16.16.16, 1298 Riccati equation Prop. 16.16.14, 1298 Prop. 16.18.2, 1302 Prop. 16.18.3, 1302 Prop. 16.18.7, 1304 closest integer floor Fact 1.11.3, 28 closure affine hull Fact 12.11.12, 935 affine subspace Fact 12.11.15, 935 complement Fact 12.11.7, 935 cone Fact 12.11.15, 935 convergent sequence Prop. 12.2.6, 916 convex cone Fact 12.11.15, 935 convex hull Fact 12.11.21, 935 convex set Fact 12.11.15, 935 Fact 12.11.16, 935 Fact 12.12.9, 938 definition Defn. 12.1.6, 914 Defn. 12.1.9, 914 irrational number Fact 12.11.1, 934 limit point Prop. 12.1.8, 914 smallest closed set Fact 12.11.4, 935 subset Fact 12.12.1, 937 subspace Fact 12.11.15, 935 union Fact 12.12.2, 937 closure point definition Defn. 12.1.6, 914 Defn. 12.1.9, 914 Cochran’s theorem

1460

codomain

sum of projectors Fact 4.18.24, 416 codomain definition, 16 cofactor definition, 301 determinant expansion Prop. 3.8.5, 301 cogredient diagonalization commuting matrices Fact 10.20.2, 799 definition, 707 diagonalizable matrix Fact 10.20.3, 799 Fact 10.20.5, 799 positive-definite matrix Thm. 10.3.1, 707 Fact 10.20.7, 799 positive-semidefinite matrix Thm. 10.3.5, 708 unitary matrix Fact 10.20.2, 799 Weierstrass Fact 10.20.3, 799 cogredient transformation Hermitian matrix Fact 10.20.6, 799 Fact 10.20.8, 799 simultaneous diagonalization Fact 10.20.6, 799 Fact 10.20.8, 799 simultaneous triangularization Fact 7.19.10, 617 Cohn polynomial root bound Fact 15.21.18, 1237 colinear points determinant Fact 5.1.4, 441 Fact 5.1.9, 442 colleague matrix characteristic polynomial Fact 7.18.2, 610 collection of sets greatest lower bound Prop. 1.3.15, 8 Prop. 1.3.16, 8 least upper bound Prop. 1.3.15, 8 Prop. 1.3.17, 8 column definition, 280 column norm definition, 844 H¨older-induced norm Fact 11.9.25, 863 Fact 11.9.27, 863 Kronecker product Fact 11.10.93, 884 partitioned matrix Fact 11.9.22, 862 row norm

Fact 11.9.21, 862 spectral radius Cor. 11.4.10, 844 column vector definition, 277 column-stacking operator, see vec column-stochastic matrix definition Defn. 4.1.5, 367 common divisor definition, 501 common eigenvalues commuting matrices Fact 7.16.12, 604 common eigenvector commuting matrices Fact 7.16.2, 603 Fact 7.16.3, 603 norm equality Fact 11.10.52, 876 simultaneous triangularization Fact 7.19.2, 617 subspace Fact 7.16.15, 604 common multiple definition, 501 commutant commutator Fact 3.23.9, 355 Fact 9.5.3, 691 commutator 2 × 2 matrices Fact 4.29.2, 428 3 × 3 matrices Fact 4.29.3, 429 4 × 4 matrices Fact 4.29.4, 429 adjoint operator Fact 3.23.4, 355 Fact 3.23.5, 355 center subgroup Fact 3.23.10, 355 centralizer Fact 3.23.9, 355 Fact 9.5.3, 691 convergent sequence Fact 15.15.10, 1213 definition, 283 derivative of a matrix Fact 15.15.12, 1213 determinant Fact 3.17.30, 340 Fact 3.23.6, 355 Fact 4.29.3, 429 Fact 4.29.4, 429 dimension Fact 3.23.9, 355 Fact 3.23.10, 355 Fact 3.23.11, 356 Fact 9.5.3, 691 eigenvalue Fact 7.16.13, 604 Fact 7.16.14, 604

equalities Fact 3.15.25, 327 Fact 3.23.3, 354 factorization Fact 7.17.36, 609 Frobenius norm Fact 11.11.3, 884 Fact 11.11.4, 884 Hermitian matrix Fact 4.29.6, 429 Fact 4.29.8, 429 Fact 4.29.9, 429 Fact 7.16.13, 604 Fact 11.11.7, 885 idempotent matrix Fact 4.16.11, 402 Fact 4.16.15, 405 Fact 4.16.16, 405 Fact 4.22.10, 419 infinite product Fact 15.15.19, 1214 involutory matrix Fact 4.20.8, 417 lower triangular matrix Fact 4.22.13, 420 matrix exponential Fact 15.15.10, 1213 Fact 15.15.12, 1213 Fact 15.15.13, 1213 Fact 15.15.14, 1213 Fact 15.15.15, 1213 Fact 15.15.16, 1214 Fact 15.15.17, 1214 Fact 15.15.18, 1214 Fact 15.15.19, 1214 maximum eigenvalue Fact 11.11.7, 885 Fact 11.11.8, 885 maximum singular value Fact 11.11.6, 885 Fact 11.16.8, 900 nilpotent matrix Fact 4.22.10, 419 Fact 4.22.13, 420 Fact 4.22.14, 420 Fact 4.22.15, 420 Fact 7.16.4, 603 normal matrix Fact 4.29.12, 429 Fact 11.11.8, 885 power Fact 3.23.1, 354 powers Fact 3.23.2, 354 projector Fact 4.18.16, 413 Fact 4.18.18, 413 Fact 8.8.4, 652 Fact 11.11.2, 884 range Fact 8.8.4, 652 rank Fact 4.16.16, 405 Fact 4.18.18, 413 Fact 7.19.6, 617

Fact 8.5.2, 641 Schatten norm Fact 11.11.4, 884 series Fact 15.15.18, 1214 simultaneous triangularization Fact 7.19.6, 617 Fact 7.19.7, 617 skew-Hermitian matrix Fact 4.29.6, 429 Fact 4.29.10, 429 skew-symmetric matrix Fact 4.29.11, 429 spectrum Fact 7.11.30, 579 spread Fact 11.11.7, 885 Fact 11.11.8, 885 submultiplicative norm Fact 11.11.1, 884 subspace Fact 3.23.9, 355 Fact 3.23.10, 355 Fact 3.23.12, 356 sum Fact 3.23.12, 356 sum of matrices Fact 4.29.1, 428 trace Fact 3.23.1, 354 Fact 4.29.2, 428 Fact 4.29.3, 429 Fact 4.29.4, 429 Fact 4.29.5, 429 Fact 7.10.19, 571 triangularization Fact 7.19.6, 617 unitarily invariant norm Fact 11.11.6, 885 Fact 11.11.7, 885 Fact 11.11.8, 885 upper triangular matrix Fact 4.22.13, 420 zero diagonal Fact 4.29.7, 429 zero trace Fact 3.23.11, 356 commutator realization Shoda’s theorem Fact 7.10.19, 571 commuting matrices centralizer Fact 7.16.8, 603 Fact 7.16.9, 603 cogredient diagonalization Fact 10.20.2, 799 common eigenvalues Fact 7.16.12, 604 common eigenvector Fact 7.16.2, 603 Fact 7.16.3, 603 cyclic matrix Fact 7.16.6, 603 Fact 7.16.7, 603

completely solid set simultaneous Fact 7.16.8, 603 diagonalizable matrix diagonalization Fact 7.19.9, 617 Fact 10.20.2, 799 dimension simultaneous Fact 7.11.20, 577 triangularization Fact 7.11.21, 577 Fact 7.19.5, 617 Drazin generalized inverse spectral radius Fact 8.11.7, 675 Fact 7.13.23, 594 Fact 8.11.8, 675 spectrum eigenvector Fact 7.11.30, 579 Fact 7.16.1, 603 square root generalized Fact 10.8.1, 723 Cayley-Hamilton Fact 10.11.44, 740 star partial ordering theorem Fact 4.30.9, 431 Fact 6.9.29, 529 time-varying dynamics Hermitian matrix Fact 16.20.6, 1306 Fact 7.16.16, 604 triangularization Fact 10.20.2, 799 Fact 7.19.5, 617 idempotent matrix tripotent matrix Fact 4.16.2, 398 Fact 4.21.7, 418 Fact 4.21.7, 418 Kronecker sum compact set Fact 9.5.5, 691 continuous function linear combination Thm. 12.4.11, 922 Fact 3.23.7, 355 Thm. 12.4.20, 923 matrix exponential convergent subsequence Prop. 15.1.5, 1180 Thm. 12.2.9, 917 Cor. 15.1.6, 1180 definition Fact 15.15.1, 1211 Defn. 12.1.13, 915 Fact 15.15.6, 1212 intersection nilpotent matrix Fact 12.12.20, 939 Fact 4.22.11, 420 invariance of domain Fact 4.22.12, 420 Thm. 12.4.20, 923 normal matrix inverse image Fact 7.16.16, 604 Cor. 12.4.15, 922 Fact 7.19.8, 617 linear function Fact 7.20.2, 618 Cor. 12.4.13, 922 Fact 7.20.3, 618 minimizer Fact 15.15.6, 1212 Cor. 12.4.12, 922 polynomial representation Minkowski sum Fact 7.16.8, 603 Fact 12.12.8, 938 Fact 7.16.9, 603 one-to-one function Fact 7.16.10, 603 Cor. 12.4.14, 922 positive-definite matrix companion form matrix Fact 10.10.45, 734 discrete-time semistable positive-semidefinite matrix matrix Fact 15.22.25, 1242 Fact 10.11.1, 735 companion matrix Fact 10.11.47, 740 block-diagonal matrix Fact 10.23.4, 814 Prop. 7.3.7, 548 projector Lem. 7.3.2, 547 Fact 8.8.4, 652 bottom, right, top, left Fact 10.11.40, 739 Fact 7.18.3, 611 Fact 10.11.44, 740 characteristic polynomial range-Hermitian matrix Prop. 7.3.1, 546 Fact 8.5.9, 643 Cor. 7.3.4, 547 Fact 8.5.10, 643 Cor. 7.3.5, 548 rank subtractivity partial circulant matrix Fact 7.18.13, 614 ordering cyclic matrix Fact 10.23.4, 814 Fact 7.18.11, 614 semisimple matrix definition, 546 Fact 7.16.5, 603 elementary divisor simple matrix Thm. 7.3.8, 548 Fact 7.16.10, 603 example

Example 7.4.7, 551 Example 7.4.8, 552 generalized Frobenius companion matrix Fact 7.18.1, 610 hypercompanion matrix Lem. 7.4.4, 550 inverse matrix Fact 7.18.4, 612 Jordan matrix Lem. 7.4.1, 549 Leslie matrix Fact 7.18.1, 610 minimal polynomial Prop. 7.3.1, 546 Cor. 7.3.4, 547 Cor. 7.3.5, 548 multihypercompanion matrix Cor. 7.4.3, 550 nonnegative matrix Fact 6.11.18, 543 oscillator Fact 7.15.32, 601 similar matrices Fact 7.18.11, 614 singular value Fact 7.12.34, 584 Vandermonde matrix Fact 7.18.9, 613 compartmental matrix asymptotically stable matrix Fact 15.20.6, 1233 Lyapunov-stable matrix Fact 15.20.6, 1233 semistable matrix Fact 15.20.6, 1233 compatible norm induced norm Prop. 11.4.3, 842 compatible norms definition, 838 H¨older norm Prop. 11.3.5, 839 Schatten norm Prop. 11.3.6, 840 Cor. 11.3.7, 840 Cor. 11.3.8, 840 submultiplicative norm Prop. 11.3.1, 838 trace norm Cor. 11.3.8, 840 complement closed set Fact 12.11.5, 935 closure Fact 12.11.7, 935 definition, 1 interior Fact 12.11.7, 935 open set Fact 12.11.5, 935 relatively closed set Fact 12.11.6, 935

1461

relatively open set Fact 12.11.6, 935 complement of a directed graph Defn. 1.4.2, 10 complement of a relation definition Defn. 1.3.3, 6 complementary projector range Fact 4.17.2, 407 complementary projectors Defn. 4.8.5, 375 complementary relation relation Prop. 1.3.4, 6 complementary submatrices defect Fact 3.14.27, 325 Fact 3.14.28, 325 complementary subspaces complementary projectors Defn. 4.8.5, 375 complex conjugate transpose Fact 4.15.4, 396 definition, 279 group-invertible matrix Cor. 4.8.10, 375 Fact 4.9.2, 376 idempotent matrix Prop. 4.8.3, 374 Prop. 4.8.4, 375 Fact 4.15.4, 396 Fact 4.16.4, 399 Fact 8.7.9, 651 index of a matrix Prop. 4.8.9, 375 partitioned matrix Fact 4.16.4, 399 projector Prop. 4.8.6, 375 Fact 4.18.14, 413 Fact 4.18.19, 414 Fact 8.8.18, 657 range Fact 4.15.3, 396 simultaneous Fact 3.12.26, 315 Fact 3.12.27, 315 stable subspace Prop. 15.9.8, 1200 sum of dimensions Cor. 3.1.6, 279 unstable subspace Prop. 15.9.8, 1200 completely solid set convex set Fact 12.11.20, 935 definition Defn. 12.1.12, 915 open ball Fact 12.11.14, 935

1462

complex conjugate

partitioned matrix positive-semidefinite complex-symmetric matrix Fact 3.24.3, 356 Autonne-Takagi matrix Fact 3.24.4, 356 Fact 12.11.29, 936 factorization Fact 3.24.5, 356 solid set Fact 7.10.22, 572 Fact 3.24.7, 357 Fact 12.11.20, 935 T-congruence Fact 10.16.6, 777 Fact 7.10.22, 572 complex conjugate positive-definite matrix T-congruent determinant Fact 4.10.8, 378 Fact 3.24.8, 358 diagonalization positive-semidefinite Fact 10.15.15, 776 Fact 7.10.22, 572 matrix matrix exponential unitary matrix Fact 4.10.8, 378 Prop. 15.2.8, 1184 Fact 7.10.22, 572 rank partitioned matrix component Fact 3.24.6, 357 Fact 3.24.8, 358 definition, 277 complex number Fact 6.10.33, 536 composition convex combination similar matrices associativity Fact 2.21.14, 272 Fact 7.10.33, 575 Prop. 1.6.1, 16 complex numbers complex conjugate of a definition, 16 2 × 2 representation matrix one-to-one Fact 3.24.1, 356 definition, 287 Prop. 1.6.2, 16 Bohr’s inequality onto complex conjugate of a Fact 2.21.8, 269 Prop. 1.6.2, 16 vector circle composition of functions definition, 286 Fact 5.5.2, 495 one-to-one function complex exponential Clarkson inequalities Fact 1.10.6, 27 cyclotomic polynomial Fact 2.21.8, 269 onto function Fact 2.21.4, 268 cubic polynomial Fact 1.10.6, 27 product Fact 2.21.2, 266 compound matrix Fact 2.21.5, 269 Dunkl-Williams inequality adjugate Fact 2.21.6, 269 Fact 2.21.9, 271 Fact 9.5.19, 694 complex function equilateral triangle bialternate product analytic extension Fact 5.2.1, 443 Fact 9.5.20, 695 Fact 12.14.11, 947 exponential function matrix product conformal mapping Fact 2.21.27, 275 Fact 9.5.18, 693 Fact 12.14.6, 946 geometric series Fact 9.5.19, 694 de Branges’s theorem Fact 13.5.1, 1021 permanental bialternate Fact 12.14.12, 947 Hlawka’s inequality product Kobe function Fact 2.21.11, 272 Fact 9.5.21, 695 Fact 12.14.12, 947 logarithm function Schur product Liouville’s theorem Fact 2.21.29, 275 Fact 10.25.66, 830 Fact 12.14.8, 946 Maligranda inequality maximum modulus compound of a matrix Fact 2.21.9, 271 principle product Massera-Schaffer Fact 12.14.8, 946 Binet-Cauchy theorem inequality M¨obius transformation Fact 9.5.18, 693 Fact 2.21.9, 271 Fact 12.14.6, 946 comrade matrix parallellogram law open mapping theorem characteristic polynomial Fact 2.21.26, 274 Fact 12.14.7, 946 Fact 7.18.2, 610 parallelogram law Picard’s theorem concave function Fact 2.21.8, 269 Fact 12.14.9, 947 definition Poincar´e metric Riemann mapping Defn. 10.6.14, 717 Fact 2.21.30, 276 function composition theorem Fact 2.21.31, 276 Lem. 10.6.16, 718 Fact 12.14.10, 947 polarization identity integral Schlicht function Fact 2.21.8, 269 Fact 1.21.11, 118 Fact 12.14.12, 947 polygon nonincreasing function Schwarz lemma Fact 2.21.25, 274 Lem. 10.6.16, 718 Fact 12.14.4, 946 quadratic formula partitioned matrix Schwarz-Pick lemma Fact 2.21.1, 266 Fact 10.12.58, 759 Fact 12.14.5, 946 series positive-semidefinite complex matrix Fact 13.4.14, 1017 block 2 × 2 representation matrix triangle property Fact 3.24.6, 357 Fact 10.13.12, 762 Fact 5.2.2, 443 complex conjugate Fact 11.10.3, 869 complex symmetric Jordan trace Fact 3.24.3, 356 form determinant Fact 11.10.3, 869 similarity transformation conclusion Fact 3.24.6, 357 Fact 7.17.4, 605 Fact 3.24.10, 359 definition, 3

condition number

linear system solution Fact 11.17.1, 909 Fact 11.17.2, 909 Fact 11.17.3, 909 maximum singular value Fact 11.15.2, 896 positive-definite matrix Fact 10.21.6, 801 conditionally convergent series limit Fact 12.18.1, 957 cone blunt definition, 278 Cartesian product Fact 3.12.18, 313 closure Fact 12.11.15, 935 cone of image Fact 3.11.15, 309 conical hull Fact 3.11.4, 306 convex cone Fact 3.11.7, 308 Fact 3.11.10, 308 definition, 278 dictionary ordering Fact 3.11.23, 310 half space Fact 3.11.7, 308 image under linear mapping Fact 3.11.15, 309 interior Fact 12.11.15, 935 intersection Prop. 3.1.2, 278 Fact 3.12.4, 311 left inverse Fact 3.11.15, 309 lexicographic ordering Fact 3.11.23, 310 Minkowski sum Fact 3.12.4, 311 one-sided definition, 278 pointed definition, 278 quadratic form Fact 10.17.12, 789 Fact 10.17.16, 789 Fact 10.17.17, 789 relative interior Fact 12.11.15, 935 scaling Fact 3.11.1, 306 union Fact 3.12.13, 312 variational definition, 924 confederate matrix characteristic polynomial Fact 7.18.2, 610

controllability pencil conformal mapping

complex function Fact 12.14.6, 946 definition Defn. 12.6.6, 928 congenial matrix characteristic polynomial Fact 7.18.2, 610 congruence generalized inverse Fact 10.24.5, 816 modulo Fact 1.11.11, 30 Fact 1.11.12, 30 congruence transformation definition, 374 congruent matrices biequivalent matrices Prop. 4.7.5, 374 definition Defn. 4.7.4, 373 Hermitian matrix Thm. 7.5.7, 554 Prop. 4.7.5, 374 inertia Thm. 7.5.7, 554 Fact 7.9.26, 569 Kronecker product Fact 9.4.13, 686 matrix classes Prop. 4.7.5, 374 positive-definite matrix Prop. 4.7.5, 374 Cor. 10.1.3, 704 positive-semidefinite matrix Prop. 4.7.5, 374 Cor. 10.1.3, 704 range-Hermitian matrix Prop. 4.7.5, 374 Fact 7.10.7, 570 skew-Hermitian matrix Prop. 4.7.5, 374 skew-symmetric matrix Fact 4.10.37, 382 Fact 7.10.17, 571 Sylvester’s law of inertia, 554 symmetric matrix Fact 7.10.17, 571 unit imaginary matrix Fact 4.10.37, 382 congruent modulo integer Fact 1.11.9, 29 Fact 1.11.10, 29 congruent modulo k Defn. 1.5.2, 12 conical hull constructive characterization Fact 3.11.5, 306 Fact 3.11.6, 307 definition, 279

1463

matrix diagonal matrix Fact 12.13.9, 942 Fact 3.11.18, 309 Fact 16.21.13, 1308 limit Minkowski sum final state Prop. 12.4.5, 922 Fact 3.12.2, 310 Fact 16.21.5, 1307 linear function polyhedral cone geometric multiplicity Cor. 12.4.7, 922 Fact 3.11.19, 310 Fact 16.21.16, 1309 maximization subset Gramian Fact 12.13.8, 942 Fact 3.12.1, 310 Fact 16.21.19, 1309 minimizer input matrix Cor. 12.4.12, 922 conjecture Fact 16.21.17, 1309 open relative to a set definition, 2 PBH test Thm. 12.4.9, 922 conjugate exponents Thm. 16.6.19, 1265 open set image inequality positive-semidefinite Thm. 12.4.19, 923 Fact 2.2.51, 140 pathwise-connected set conjugate parameters matrix Fact 12.13.11, 943 barycentric coordinates Fact 16.21.7, 1308 rational function Fact 2.12.23, 219 positive-semidefinite Fact 12.13.11, 943 connected ordering weak majorization definition Fact 16.21.9, 1308 Fact 3.25.17, 362 Defn. 12.1.14, 915 range continuous-time control connected graph Fact 16.21.8, 1308 Defn. 1.4.4, 11 problem shift irreducible matrix LQG controller Fact 16.21.11, 1308 Fact 6.11.4, 538 Fact 16.25.8, 1318 shifted dynamics walk continuously differentiable Fact 16.21.10, 1308 Fact 6.11.3, 538 skew-symmetric matrix function connected set Fact 16.21.6, 1308 definition, 925 continuous function stabilization contractive matrix Thm. 12.4.25, 924 Fact 16.21.19, 1309 complex conjugate pathwise-connected set Sylvester’s equation transpose Prop. 12.4.24, 923 Fact 16.22.14, 1312 Fact 4.27.2, 427 constant polynomial transpose definition definition, 499 Fact 16.21.18, 1309 Defn. 4.1.2, 364 contains controllability Gramian determinant definition, 1 asymptotically stable Fact 10.16.22, 779 contingency matrix partitioned matrix definition, 5 Prop. 16.7.9, 1268 Fact 10.8.13, 724 continuity Cor. 16.7.10, 1268 positive-definite matrix spectrum controllably Fact 10.12.31, 753 Fact 12.13.21, 945 product asymptotically stable Fact 12.13.22, 945 Fact 4.27.3, 427 Prop. 16.7.3, 1266 continuity of roots unitarily invariant norm Prop. 16.7.4, 1267 coefficients Fact 11.10.21, 871 Prop. 16.7.5, 1267 polynomial contradiction Prop. 16.7.6, 1268 Fact 12.13.2, 941 definition, 4 Prop. 16.7.7, 1268 continuous function contragredient frequency domain closed relative to a set Cor. 16.11.5, 1286 diagonalization Thm. 12.4.9, 922 H2 norm definition, 707 compact set Cor. 16.11.4, 1286 positive-definite matrix Thm. 12.4.11, 922 Cor. 16.11.5, 1286 Thm. 10.3.2, 707 compact set image L2 norm Cor. 10.3.4, 708 Thm. 12.4.20, 923 Thm. 16.11.1, 1285 positive-semidefinite connected set controllability matrix matrix Thm. 12.4.25, 924 controllable pair Thm. 10.3.6, 708 convex function Thm. 16.6.18, 1265 Cor. 10.3.8, 708 Thm. 12.4.6, 922 definition, 1260 contraharmonic mean definition equality inequality Defn. 12.4.1, 921 Fact 16.21.23, 1309 Fact 2.2.15, 133 differentiable function rank contrapositive Prop. 12.5.4, 925 Cor. 16.6.3, 1260 definition, 4 distance Sylvester’s equation controllability Fact 12.13.4, 941 Fact 16.22.13, 1311 asymptotically stable fixed-point theorem controllability pencil matrix Thm. 12.4.26, 924 definition Fact 16.21.6, 1308 Cor. 12.4.27, 924 Defn. 16.6.12, 1263 cyclic matrix jointly continuous function Smith form Fact 16.21.14, 1308

1464

controllable canonical form

Prop. 16.6.15, 1264 Smith zeros Prop. 16.6.16, 1264 uncontrollable eigenvalue Prop. 16.6.13, 1263 uncontrollable spectrum Prop. 16.6.16, 1264 controllable canonical form definition, 1270 equivalent realizations Cor. 16.9.21, 1277 Cor. 16.9.23, 1278 realization Prop. 16.9.3, 1270 Prop. 16.9.20, 1277 controllable dynamics block-triangular matrix Thm. 16.6.8, 1262 orthogonal matrix Thm. 16.6.8, 1262 controllable eigenvalue controllable subspace Prop. 16.6.17, 1265 definition Defn. 16.6.11, 1263 controllable pair asymptotically stable matrix Prop. 16.7.9, 1268 Cor. 16.7.10, 1268 controllability matrix Thm. 16.6.18, 1265 cyclic matrix Fact 7.15.10, 599 eigenvalue placement Prop. 16.6.20, 1265 equivalent realizations Prop. 16.9.8, 1272 Prop. 16.9.11, 1273 Markov block-Hankel matrix Prop. 16.9.13, 1275 minimal realization Prop. 16.9.12, 1273 Prop. 16.9.18, 1276 positive-definite matrix Thm. 16.6.18, 1265 rank Fact 7.15.11, 599 similarity transformation Prop. 16.9.9, 1273 controllable subspace block-triangular matrix Prop. 16.6.9, 1263 Prop. 16.6.10, 1263 controllable eigenvalue Prop. 16.6.17, 1265 definition Defn. 16.6.1, 1259 equivalent expressions Lem. 16.6.2, 1260 final state Fact 16.21.4, 1307 full-state feedback

Fact 15.15.9, 1213 Prop. 16.6.5, 1261 inverse matrix identity-matrix shift Fact 3.20.22, 350 Lem. 16.6.7, 1262 limit invariant subspace Prop. 12.2.7, 917 Cor. 16.6.4, 1261 limit set nonsingular matrix Prop. 12.2.8, 917 Prop. 16.6.10, 1263 matrix exponential orthogonal matrix Prop. 15.1.3, 1180 Prop. 16.6.9, 1263 Fact 15.15.8, 1213 projector Fact 15.15.9, 1213 Lem. 16.6.6, 1261 Fact 15.15.10, 1213 controllably asymptotically Fact 15.22.18, 1241 stable matrix sign function asymptotically stable Fact 7.17.23, 608 matrix spectral radius Prop. 16.8.6, 1269 Fact 11.9.3, 860 block-triangular matrix square root Prop. 16.7.3, 1266 Fact 7.17.23, 608 controllability Gramian Fact 10.10.37, 734 Prop. 16.7.3, 1266 sum of limits Prop. 16.7.4, 1267 Prop. 12.4.8, 922 Prop. 16.7.5, 1267 unitary matrix Prop. 16.7.6, 1268 Fact 10.10.38, 734 Prop. 16.7.7, 1268 vectors definition Fact 12.18.4, 958 Defn. 16.7.1, 1266 convergent sequence of full-state feedback matrices Prop. 16.7.2, 1266 definition Lyapunov equation Defn. 12.2.4, 916 Prop. 16.7.3, 1266 convergent sequence of orthogonal matrix Prop. 16.7.3, 1266 scalars rank definition Prop. 16.7.4, 1267 Defn. 12.2.2, 916 Prop. 16.7.5, 1267 uniqueness stabilizability Prop. 12.2.1, 915 Prop. 16.8.6, 1269 convergent sequence of convergence test vectors Cauchy-Hadamard definition formula Defn. 12.2.4, 916 Fact 12.18.9, 959 convergent series ratio test absolutely convergent Fact 12.18.9, 959 series convergent product Prop. 12.3.2, 919 definition Prop. 12.3.4, 919 Defn. 12.7.1, 929 definition sequence Defn. 12.3.1, 919 Prop. 12.7.2, 930 Defn. 12.3.3, 919 convergent sequence inverse matrix closure Fact 6.10.11, 532 Prop. 12.2.6, 916 matrix exponential closure point Prop. 15.1.2, 1179 Prop. 12.2.5, 916 spectral radius commutator Fact 6.10.11, 532 Fact 15.15.10, 1213 convergent subsequence discrete-time semistable compact set matrix Thm. 12.2.9, 917 Fact 15.22.18, 1241 converse essential closure definition, 4 Prop. 12.2.6, 916 convex combination Prop. 12.2.8, 917 definition, 278 generalized inverse determinant Fact 8.3.38, 632 Fact 10.16.15, 778 Hermitian matrix norm inequality Fact 15.15.8, 1213 Fact 11.7.6, 850

positive-semidefinite matrix Fact 7.20.11, 619 Fact 10.16.15, 778 Fact 10.24.18, 817 convex cone Cartesian product Fact 3.12.18, 313 closure Fact 12.11.15, 935 cone Fact 3.11.10, 308 convex conincal hull Fact 3.11.4, 306 definition, 278 induced by transitive relation Prop. 3.1.7, 280 inner product Fact 12.12.22, 940 interior Fact 12.11.15, 935 intersection Prop. 3.1.2, 278 Fact 3.12.4, 311 Minkowski sum Fact 3.11.9, 308 Fact 3.12.4, 311 Fact 3.12.12, 312 polar Fact 12.12.14, 938 positive-semidefinite matrix, 703 quadratic form Fact 10.17.12, 789 Fact 10.17.16, 789 Fact 10.17.17, 789 relative interior Fact 12.11.15, 935 scaling Fact 3.11.1, 306 separation theorem Fact 12.12.22, 940 subspace Fact 3.11.9, 308 union Fact 3.12.13, 312 convex conical hull constructive characterization Fact 3.11.5, 306 Fact 3.11.6, 307 convex hull Fact 3.11.8, 308 definition, 279 dual cone Fact 3.11.8, 308 matrix Fact 3.11.18, 309 Minkowski sum Fact 3.12.2, 310 polyhedral cone Fact 3.11.18, 309 subset Fact 3.12.1, 310

coprime convex conjugate

Fact 10.17.18, 789 inverse function Niculescu’s inequality convex function Fact 3.11.17, 309 Fact 1.21.8, 117 Fact 10.17.7, 787 Fact 12.11.22, 936 nondecreasing function matrix convex function Lem. 10.6.16, 718 Fact 3.11.18, 309 constant function Fact 1.21.1, 116 Minkowski sum Fact 1.21.6, 117 one-sided directional Fact 3.12.2, 310 continuous function Fact 3.12.5, 311 Thm. 12.4.6, 922 differential Fact 12.12.12, 938 convex conjugate Prop. 12.5.1, 924 open set Fact 10.17.7, 787 Popoviciu’s inequality Fact 12.11.19, 935 definition Fact 1.21.9, 117 simplex Defn. 1.6.6, 21 positive-definite matrix Fact 5.1.8, 442 Defn. 10.6.14, 717 Fact 10.17.21, 790 solid set derivative Fact 10.17.22, 791 Fact 12.11.17, 935 Fact 12.16.10, 950 Fact 10.17.25, 791 spectrum determinant positive-semidefinite Fact 10.17.8, 787 Prop. 10.6.17, 718 matrix Fact 10.17.9, 788 Fact 3.16.27, 333 Fact 10.11.86, 748 strong majorization differential function Fact 10.17.18, 789 Fact 4.11.6, 384 Fact 12.13.23, 945 Fact 10.24.22, 819 subset directional differential reverse inequality Fact 3.12.1, 310 Fact 12.16.10, 950 Fact 10.11.20, 737 eigenvalue convex quadrilateral scalar inequality Cor. 10.6.19, 722 ellipse Fact 1.21.4, 116 Fact 10.22.11, 806 Fact 5.5.5, 496 Schur complement epigraph Fact 5.5.6, 496 Prop. 10.6.17, 718 Fact 12.13.25, 945 convex set Lem. 10.6.16, 718 function composition affine hull Schur product Lem. 10.6.16, 718 Thm. 12.4.6, 922 Fact 10.25.72, 830 Hermite-Hadamard Fact 12.11.20, 935 singular value Caratheodory’s theorem inequality Fact 15.16.12, 1220 Fact 3.11.5, 306 Fact 1.21.9, 117 strong log majorization Cartesian product Hermitian matrix Fact 3.25.9, 361 Fact 3.12.18, 313 Fact 10.14.47, 771 strong majorization closed set Fact 10.14.48, 771 Fact 3.25.8, 361 Fact 12.11.26, 936 Fact 15.15.26, 1215 Fact 3.25.12, 361 closure increasing function subdifferential Fact 12.11.15, 935 Thm. 10.6.15, 717 Fact 12.16.10, 950 Fact 12.11.16, 935 inequality sublevel set Fact 12.12.9, 938 Fact 1.21.2, 116 Fact 12.13.24, 945 completely solid set Fact 1.21.3, 116 trace Fact 12.11.20, 935 Jensen’s inequality Prop. 10.6.17, 718 convex cone Fact 1.21.7, 117 Fact 10.17.21, 790 Fact 3.11.10, 308 Fact 2.11.131, 211 transformation convex hull Kronecker product Fact 1.21.5, 117 Fact 3.11.4, 306 Prop. 10.6.17, 718 two-variable limit convexity of image log majorization Fact 12.17.18, 957 Fact 3.11.15, 309 Fact 3.25.13, 361 weak majorization definition, 278 logarithm Fact 3.25.8, 361 dimension Fact 15.16.12, 1220 Fact 3.25.9, 361 Fact 12.11.20, 935 logarithm of determinant Fact 3.25.10, 361 distance from a point Prop. 10.6.17, 718 Fact 3.25.12, 361 Fact 12.12.24, 940 logarithm of trace Fact 10.22.11, 806 Fact 12.12.25, 940 Prop. 10.6.17, 718 convex hull extreme point matrix exponential affine hull Fact 12.11.32, 937 Fact 10.17.23, 791 Fact 3.11.8, 308 image under linear Fact 10.17.24, 791 closure Fact 15.16.12, 1220 mapping Fact 12.11.21, 935 matrix functions Fact 3.11.15, 309 constructive Prop. 10.6.17, 718 interior characterization matrix logarithm Fact 12.11.15, 935 Fact 3.11.5, 306 Prop. 10.6.17, 718 Fact 12.11.16, 935 Fact 3.11.6, 307 midpoint convex Fact 12.11.18, 935 definition, 279 Fact 12.13.19, 944 Fact 12.12.4, 937 Hermitian matrix diagonal minimizer intersection Fact 10.21.11, 802

1465

Prop. 3.1.2, 278 Fact 3.12.4, 311 Fact 12.12.10, 938 left inverse Fact 3.11.15, 309 Minkowski sum Fact 3.11.2, 306 Fact 3.11.3, 306 Fact 3.12.4, 311 Fact 12.12.11, 938 norm Fact 11.7.1, 849 open ball Fact 12.11.14, 935 positive-semidefinite matrix Fact 10.17.2, 785 Fact 10.17.3, 785 Fact 10.17.4, 786 Fact 10.17.5, 786 Fact 10.17.6, 787 proper separation theorem Fact 12.12.23, 940 quadratic form Fact 10.17.2, 785 Fact 10.17.3, 785 Fact 10.17.4, 786 Fact 10.17.5, 786 Fact 10.17.6, 787 Fact 10.17.10, 788 Fact 10.17.12, 789 Fact 10.17.15, 789 Fact 10.17.16, 789 Fact 10.17.17, 789 relative interior Fact 12.11.15, 935 relatively boundary Fact 12.11.23, 936 scaling Fact 3.11.1, 306 set cancellation Fact 12.12.11, 938 solid set Fact 12.11.20, 935 sublevel set Fact 10.17.1, 785 union Fact 12.12.11, 938 convolution integral definition, 1251 coplanar determinant Fact 5.1.6, 441 coplanar vectors angle Fact 5.1.5, 441 copositive matrix nonnegative matrix Fact 10.19.24, 798 positive-semidefinite matrix Fact 10.19.24, 798 quadratic form Fact 10.19.24, 798 coprime

1466

coprime polynomials

least common multiple Fact 1.11.5, 28 polynomial Fact 6.8.5, 518 Fact 6.8.6, 518 coprime polynomials Bezout matrix Fact 6.8.8, 519 Fact 6.8.9, 520 Fact 6.8.10, 520 definition, 501 minimal realization Prop. 16.9.19, 1276 resultant Fact 6.8.6, 518 Smith-McMillan form Fact 6.8.17, 522 Sylvester matrix Fact 6.8.6, 518 coprime right polynomial fraction description Smith-McMillan form Prop. 6.7.16, 517 unimodular matrix Prop. 6.7.15, 517 Copson inequality sum of powers Fact 2.11.133, 212 Cordes inequality maximum singular value Fact 10.22.36, 812 core inverse group-invertible matrix Fact 8.5.15, 647 core partial ordering positive-semidefinite matrix Fact 8.5.17, 648 corollary definition, 5 correlation matrix positive-definite matrix Fact 10.25.36, 825 Schur product Fact 10.25.36, 825 cosine Cesaro limit Fact 13.6.19, 1063 Frobenius norm Fact 11.10.42, 874 series Fact 13.6.6, 1060 Fact 13.6.7, 1060 trigonometric equality Fact 2.16.7, 238 cosine law vector equality Fact 11.8.3, 853 cosine rule triangle Fact 5.2.7, 443 cotangent indefinite integral Fact 14.1.18, 1095 trigonometric equality

Fact 2.16.20, 245 COUD

closed outside unit disk definition, 14 Courant min-max theorem Fact 7.12.7, 579 covariance quadratic form Fact 14.12.4, 1172 Crabtree Schur complement of a partitioned matrix Fact 8.9.38, 669 Crabtree-Haynsworth quotient formula Schur complement of a partitioned matrix Fact 8.9.38, 669 Craig-Sakamoto theorem determinant Fact 3.16.14, 331 Cramer’s rule linear system solution Fact 3.16.12, 330 Crasta trigonometric sum Fact 12.18.28, 962 creation matrix upper triangular matrix Fact 15.12.4, 1205 CRHP closed right half plane definition, 14 Crimmins product of projectors Fact 8.8.10, 655 cross product n-dimensional space Fact 4.12.2, 387 adjugate Fact 8.9.17, 664 Cayley transform Fact 4.14.9, 394 equalities Fact 4.12.1, 384 generalized inverse Fact 8.9.18, 664 Jacobi identity Fact 4.12.1, 384 matrix exponential Fact 15.12.7, 1206 Fact 15.12.8, 1207 Fact 15.12.9, 1207 orthogonal matrix Fact 4.12.3, 387 Fact 4.12.4, 387 Fact 4.14.9, 394 outer-product matrix Fact 4.14.9, 394 parallelogram Fact 11.8.13, 857 triangle Fact 5.2.6, 443 cross-product matrix

characteristic polynomial Fact 6.9.18, 527 Fact 6.9.19, 528 diagonalization Fact 7.10.1, 569 Fact 7.10.2, 570 matrix exponential Fact 15.12.6, 1206 Fact 15.12.12, 1207 Fact 15.12.13, 1207 Fact 15.12.14, 1207 Fact 15.12.17, 1208 Fact 15.12.18, 1208 orthogonal matrix Fact 15.12.12, 1207 Fact 15.12.13, 1207 Fact 15.12.14, 1207 spectrum Fact 6.9.18, 527 CS decomposition unitary matrix Fact 7.10.29, 573 CT-dominant eigenvalue limit Fact 15.14.16, 1211 cube root equality Fact 3.15.35, 329 cubic equality Fact 2.2.8, 132 Maillet’s identity Fact 2.4.4, 178 scalar inequality Fact 2.3.31, 154 Fact 2.3.32, 154 Fact 2.3.33, 154 cubic inequality Nesbitt’s inequality Fact 2.3.60, 162 Fact 5.2.25, 486 Padoa’s inequality Fact 5.2.25, 486 Schur’s inequality Fact 2.3.60, 162 cubic polynomial Cardano’s trigonometric solution Fact 6.10.7, 531 complex numbers Fact 2.21.2, 266 root location Fact 15.18.4, 1224 curl derivative Fact 12.16.27, 953 Cusa-Huygens’s inequality trigonometric inequality Fact 2.17.1, 246 cycle definition Defn. 1.4.4, 11 symmetric graph Fact 1.9.7, 25 Fact 1.9.8, 26

cycle number

definition Fact 1.19.1, 105 equalities Fact 1.19.2, 106 limit Fact 12.18.54, 968 Vandermonde matrix Fact 7.18.7, 613 zeta function Fact 13.5.40, 1031 cyclic eigenvalue definition Defn. 7.7.4, 559 eigenvector Fact 7.15.4, 598 semisimple eigenvalue Prop. 7.7.5, 559 simple eigenvalue Prop. 7.7.5, 559 cyclic group definition Prop. 4.6.6, 372 group Fact 4.31.5, 433 Fact 4.31.16, 435 cyclic inequality scalar inequality Fact 2.3.61, 165 Fact 2.3.62, 166 Fact 2.3.63, 166 sums of positive numbers Fact 2.3.64, 166 Fact 2.3.65, 166 Fact 2.11.49, 198 Fact 2.11.57, 199 Fact 2.11.61, 200 trigonometric inequality Fact 2.17.10, 253 cyclic matrix asymptotically stable matrix Fact 15.19.25, 1229 campanion matrix Fact 7.18.11, 614 characteristic polynomial Prop. 7.7.15, 561 commuting matrices Fact 7.16.6, 603 Fact 7.16.7, 603 Fact 7.16.8, 603 controllability Fact 16.21.14, 1308 controllable pair Fact 7.15.10, 599 definition Defn. 7.7.4, 559 determinant Fact 7.15.10, 599 identity-matrix perturbation Fact 7.15.17, 600 linear independence Fact 7.15.10, 599 matrix power

derivative Fact 7.15.10, 599 minimal polynomial Prop. 7.7.15, 561 nonsingular matrix Fact 7.15.10, 599 rank Fact 7.12.1, 579 semisimple matrix Fact 7.15.13, 599 similar matrices Fact 7.18.11, 614 simple matrix Fact 7.15.13, 599 tridiagonal matrix Fact 15.19.25, 1229 cyclic multiplication group definition, 372 cyclic number totient function Fact 1.20.4, 115 cyclic permutation matrix definition, 284 determinant Fact 3.16.2, 329 involutory matrix Fact 4.20.7, 417 irreducible matrix Fact 7.18.15, 616 permutation matrix Fact 7.18.14, 616 primary circulant matrix Fact 7.18.13, 614 reverse permutation matrix Fact 4.20.7, 417 cyclic quadrilateral definition Fact 5.3.1, 489 cyclotomic polynomial complex exponential Fact 2.21.4, 268

orthogonal matrices and matrix exponentials Fact 15.12.13, 1207 de Branges’s theorem complex function Fact 12.14.12, 947 De Bruijn’s inequality refined Cauchy-Schwarz inequality Fact 2.12.43, 223 de Gua’s theorem tetrahedron Fact 2.3.11, 149 De Morgan’s laws logical equivalents Fact 1.7.1, 21 set equality Fact 1.8.1, 22 de Ruiter cross product singular value decomposition Fact 4.12.1, 384 SU(3) Fact 4.14.3, 392 Decell generalized inverse Fact 8.4.30, 638 decreasing function definition Defn. 10.6.12, 716 decreasing sequence definition, 917 defect adjugate Fact 3.19.2, 346 definition, 292 geometric multiplicity Prop. 6.5.2, 511 Fact 7.15.1, 597 group-invertible matrix Fact 4.9.2, 376 Hermitian matrix Fact 7.9.7, 566 D Fact 10.10.9, 731 Kronecker sum Dai Fact 9.5.3, 691 Laurent series partitioned matrix Fact 12.14.3, 946 Fact 3.14.10, 321 damped natural frequency Fact 3.14.15, 322 definition, 1187 Fact 3.14.17, 323 Fact 7.15.32, 601 Fact 8.9.2, 659 damped rigid body Fact 8.9.3, 659 dashpot power Fact 7.15.31, 601 Fact 3.13.27, 317 damping powers definition, 1187 Prop. 3.6.7, 293 damping matrix product partitioned matrix Prop. 3.7.3, 295 Fact 7.13.31, 596 product of matrices damping ratio Prop. 3.6.10, 294 definition, 1187 Fact 3.13.21, 316 Fact 7.15.32, 601 rank dashpot Cor. 3.6.5, 293 damped rigid body semisimple eigenvalue Fact 7.15.31, 601 Prop. 7.7.8, 559 Davenport

similar matrices Fact 7.11.7, 576 submatrix Fact 3.14.27, 325 Sylvester’s law of nullity Fact 3.13.22, 317 transpose Cor. 3.6.3, 293 zero eigenvalue Fact 7.15.1, 597 defective eigenvalue definition Defn. 7.7.4, 559 defective matrix definition Defn. 7.7.4, 559 identity-matrix perturbation Fact 7.15.17, 600 nilpotent matrix Fact 7.15.19, 600 outer-product matrix Fact 7.15.5, 598 deflating subspace pencil Fact 7.14.1, 597 Degen’s eight-square identity 16 variables Fact 2.10.1, 187 degree graph Defn. 1.4.4, 11 degree matrix definition Defn. 4.2.1, 367 symmetric graph Fact 4.26.2, 426 degree of a polynomial definition, 499 degree of a polynomial matrix definition, 501 Delrome inequality Fact 2.11.66, 201 derangement number definition Fact 1.18.2, 104 permutation Fact 1.18.2, 104 derivative adjugate Fact 12.16.19, 951 Fact 12.16.22, 952 adjugate matrix Fact 12.16.20, 951 asymptotically stable polynomial Fact 15.18.1, 1223 binomial coefficient Fact 12.16.8, 950 bounded derivative Fact 12.16.12, 950 Fact 12.16.13, 951

1467

convex function Fact 12.16.10, 950 curl Fact 12.16.27, 953 definition Defn. 12.5.3, 925 determinant Prop. 12.10.3, 933 Fact 12.16.19, 951 Fact 12.16.22, 952 Fact 12.16.23, 952 Fact 12.16.25, 953 Fact 12.16.26, 953 discrete-time asymptotically stable polynomial Fact 15.18.1, 1223 divergence Fact 12.16.27, 953 gradient Fact 12.16.27, 953 hyperbolic function Fact 2.19.6, 264 inverse hyperbolic function Fact 2.20.3, 266 inverse matrix Prop. 12.10.2, 933 Fact 12.16.18, 951 Fact 12.16.22, 952 inverse trigonometric function Fact 2.18.9, 261 logarithm of determinant Prop. 12.10.3, 933 matrix definition, 926 matrix exponential Fact 10.14.46, 771 Fact 15.15.3, 1212 Fact 15.15.4, 1212 Fact 15.15.5, 1212 Fact 15.15.11, 1213 Fact 15.16.2, 1218 matrix power Prop. 12.10.2, 933 maximum singular value Fact 15.16.2, 1218 product Fact 12.16.8, 950 second derivative of a matrix Fact 12.16.21, 952 squared matrix Fact 12.16.17, 951 trace Prop. 12.10.4, 934 Fact 15.15.4, 1212 transfer function Fact 16.24.6, 1314 trigonometric function Fact 2.16.22, 246 Fact 2.16.23, 246 uniqueness Prop. 12.5.2, 925

1468

derivative of a complex function

Wronskian Prop. 16.5.3, 1259 Fact 12.16.26, 953 PBH test Thm. 16.5.5, 1259 derivative of a complex Riccati equation function Cor. 16.17.3, 1299 Cauchy-Riemann Cor. 16.19.2, 1304 equations state convergence Prop. 12.6.3, 927 Fact 16.21.3, 1307 definition detectable Defn. 12.6.1, 926 definition Jacobian Defn. 16.5.1, 1259 Prop. 12.6.2, 926 determinant derivative of a matrix (1)-inverse commutator Fact 8.9.37, 669 Fact 15.15.12, 1213 accretive-dissipative matrix exponential matrix Fact 15.15.12, 1213 Fact 10.16.43, 783 matrix product adjugate Fact 15.14.3, 1209 Fact 3.17.31, 341 derivative of a matrix Fact 3.17.35, 341 exponential Fact 3.19.1, 345 Bessis-Moussa-Villani Fact 3.19.4, 346 trace conjecture Fact 3.21.3, 351 Fact 10.14.46, 771 affine hyperplane derivative of an integral Fact 5.1.7, 441 Leibniz’s rule alternating signs Fact 12.16.7, 950 Fact 3.16.11, 330 derivative polynomial AM-GM inequality trigonometric function Fact 10.16.48, 784 Fact 13.2.14, 993 Fact 10.16.49, 784 derogatory eigenvalue angle definition Fact 5.1.5, 441 Defn. 7.7.4, 559 basic properties derogatory matrix Prop. 3.8.2, 299 definition Bell number Defn. 7.7.4, 559 Fact 6.9.9, 526 identity-matrix Binet-Cauchy formula perturbation Fact 3.16.8, 330 Fact 7.15.17, 600 Fact 3.16.9, 330 Descartes rule of signs binomial coefficient polynomial Fact 3.16.28, 333 Fact 15.18.2, 1223 Fact 3.16.29, 334 Desnanot-Jacobi identity block-triangular matrix determinant Fact 3.17.8, 335 Fact 3.17.35, 341 Fact 10.16.47, 784 detectability Cartesian decomposition asymptotically stable Fact 10.16.2, 776 matrix Catalan number Prop. 16.5.6, 1259 Fact 6.9.9, 526 Cor. 16.5.7, 1259 Cauchy matrix block-triangular matrix Fact 4.27.5, 427 Prop. 16.5.4, 1259 Fact 4.27.6, 427 closed-loop spectrum Cauchy-Schwarz Lem. 16.16.17, 1298 inequality definition Fact 10.16.25, 780 Prop. 16.5.2, 1259 cofactor expansion Lyapunov equation Prop. 3.8.5, 301 Cor. 16.5.7, 1259 colinear points observably asymptotically Fact 5.1.4, 441 stable Fact 5.1.9, 442 Prop. 16.5.6, 1259 column interchange orthogonal matrix Prop. 3.8.2, 299 Prop. 16.5.4, 1259 commutator output convergence Fact 3.17.30, 340 Fact 16.21.3, 1307 Fact 3.23.6, 355 output injection

Fact 4.29.3, 429 Fact 4.29.4, 429 commuting matrices Fact 3.16.14, 331 complex conjugate Fact 3.24.8, 358 Fact 10.15.15, 776 complex conjugate transpose Prop. 3.8.1, 299 complex matrix Fact 3.24.6, 357 Fact 3.24.10, 359 contractive matrix Fact 10.16.22, 779 convex combination Fact 10.16.15, 778 convex function Prop. 10.6.17, 718 Fact 3.16.27, 333 coplanar Fact 5.1.6, 441 Craig-Sakamoto theorem Fact 3.16.14, 331 cyclic matrix Fact 7.15.10, 599 cyclic permutation matrix Fact 3.16.2, 329 definition, 299 derivative Prop. 12.10.3, 933 Fact 12.16.19, 951 Fact 12.16.22, 952 Fact 12.16.23, 952 Fact 12.16.25, 953 Fact 12.16.26, 953 Desnanot-Jacobi identity Fact 3.17.35, 341 diagonalizable matrix Fact 7.15.8, 599 difference of matrices Fact 6.9.23, 528 dissipative matrix Fact 10.15.2, 774 Fact 10.15.3, 774 Fact 10.16.20, 779 eigenvalue Fact 7.12.32, 584 Fact 7.12.33, 584 Fact 10.15.1, 774 elementary matrix Fact 3.21.1, 351 equality Fact 3.16.16, 331 Fact 3.16.17, 331 Fact 3.16.18, 331 Fact 3.16.19, 332 Fact 3.16.21, 332 Fact 3.16.22, 332 Fact 3.16.23, 333 Euclidean norm Fact 11.13.1, 890 factorial Fact 4.25.5, 425 factorization

Fact 7.17.9, 606 Fact 7.17.37, 609 falling factorial Fact 6.9.9, 526 Fibonacci number Fact 4.25.4, 425 Fischer’s inequality Fact 10.16.37, 782 Fact 10.16.39, 782 Frobenius norm Fact 11.9.56, 866 full-state feedback Fact 16.24.16, 1315 generalized Binet-Cauchy formula Fact 6.9.37, 530 generalized inverse Fact 8.9.35, 668 Fact 8.9.36, 668 Fact 8.9.37, 669 geometric mean Fact 10.11.68, 743 group Prop. 4.6.6, 372 Hadamard’s inequality Fact 10.15.7, 775 Fact 10.15.10, 775 Hankel matrix Fact 4.23.4, 420 Fact 13.2.15, 994 Hermitian matrix Cor. 10.4.10, 711 Fact 4.10.21, 381 Fact 10.16.5, 777 Hua’s inequalities Fact 10.16.28, 780 induced norm Fact 11.14.18, 895 inequality Fact 10.15.5, 774 Fact 10.16.27, 780 Fact 10.16.28, 780 Fact 10.16.29, 780 Fact 10.16.30, 781 Fact 10.16.31, 781 integral Fact 15.14.11, 1210 invariant zero Fact 16.24.16, 1315 inverse Fact 3.17.33, 341 inverse function theorem Thm. 12.5.5, 926 involutory matrix Fact 4.20.1, 417 Fact 4.20.2, 417 Fact 7.17.35, 609 Kronecker product Prop. 9.1.11, 683 Kronecker sum Fact 9.5.2, 691 Fact 9.5.12, 692 lower block-triangular matrix Prop. 3.8.1, 299

diagonal entry lower bound Fact 10.15.9, 775 lower reverse-triangular matrix Fact 3.16.15, 331 lower triangular matrix Fact 4.25.1, 424 Lucas number Fact 4.25.4, 425 majorization Fact 10.16.1, 776 matrix exponential Prop. 15.4.11, 1189 Cor. 15.2.4, 1183 Cor. 15.2.5, 1183 Fact 15.14.11, 1210 Fact 15.16.5, 1218 matrix logarithm Fact 10.15.11, 775 Fact 10.22.42, 813 Fact 11.9.56, 866 Fact 15.15.25, 1215 maximum singular value Fact 11.16.21, 903 Fact 11.16.22, 903 minimum singular value Fact 11.16.22, 903 multiplicative commutator Fact 7.17.37, 609 nilpotent matrix Fact 4.22.11, 420 nonsingular matrix Cor. 3.8.4, 301 Lem. 3.9.6, 303 normal matrix Fact 7.13.24, 594 ones matrix Fact 3.16.3, 329 ones matrix perturbation Fact 3.19.6, 346 orthogonal matrix Fact 4.13.18, 389 Fact 4.13.19, 389 Ostrowski-Taussky inequality Fact 10.15.3, 774 outer-product perturbation Fact 3.21.2, 351 Fact 3.21.3, 351 output feedback Fact 16.24.14, 1315 permutation matrix Fact 4.13.18, 389 ¨ Plucker identity Fact 3.17.36, 341 Fact 3.17.37, 342 polynomial Fact 6.9.10, 527 product Prop. 3.8.3, 300 rank-deficient matrix Fact 3.16.7, 330 reverse permutation matrix

Fact 3.16.1, 329 row interchange Prop. 3.8.2, 299 Schur complement Prop. 10.2.4, 705 semicontractive matrix Fact 10.16.19, 779 semidissipative matrix Fact 10.15.2, 774 Fact 10.15.4, 774 Fact 10.16.2, 776 singular value Fact 7.12.32, 584 Fact 7.12.33, 584 Fact 10.15.1, 774 Fact 11.15.26, 899 singular values Fact 7.13.26, 594 skew-Hermitian matrix Fact 4.10.11, 379 Fact 4.10.16, 380 Fact 10.16.3, 776 skew-symmetric matrix Fact 4.10.15, 380 Fact 4.10.35, 382 Fact 4.10.36, 382 Fact 6.8.16, 522 Fact 6.9.19, 528 Fact 6.10.8, 532 squared matrix Fact 3.16.5, 329 Fact 3.16.6, 329 strongly increasing function Prop. 10.6.13, 717 subdeterminant Fact 3.16.8, 330 Fact 3.16.9, 330 Fact 3.16.10, 330 Fact 3.17.12, 336 Fact 3.17.35, 341 Fact 6.9.37, 530 subdeterminant expansion Cor. 3.8.6, 302 submatrix Fact 3.16.4, 329 Fact 3.17.1, 334 sum of Kronecker product Fact 9.5.13, 692 Fact 9.5.14, 692 sum of matrices Fact 6.9.24, 529 Fact 6.9.25, 529 Fact 6.9.26, 529 Fact 7.13.24, 594 Fact 7.13.25, 594 Fact 10.16.9, 777 Fact 10.16.17, 779 Fact 10.16.18, 779 Fact 10.16.19, 779 Fact 11.16.22, 903 sum of orthogonal matrices Fact 4.13.19, 389

sum of positive-semidefinite matrices Fact 10.16.21, 779 Sylvester’s identity Fact 3.17.1, 334 symplectic matrix Fact 4.28.10, 428 Fact 4.28.11, 428 time-varying dynamics Fact 16.20.6, 1306 Toeplitz matrix Fact 3.16.19, 332 Fact 4.23.9, 421 Fact 4.25.6, 426 trace Prop. 10.4.14, 712 Cor. 15.2.4, 1183 Cor. 15.2.5, 1183 Fact 3.16.26, 333 Fact 10.13.8, 762 Fact 10.16.24, 779 Fact 15.15.21, 1214 transpose Prop. 3.8.1, 299 transposition matrix Fact 4.13.18, 389 tridiagonal matrix Fact 4.23.9, 421 Fact 4.24.3, 423 Fact 4.24.4, 423 Fact 4.24.6, 424 Fact 4.24.7, 424 Trudi’s formula Fact 4.25.6, 426 unimodular matrix Prop. 6.3.8, 504 unitary matrix Fact 4.13.17, 389 Fact 4.13.20, 389 Fact 4.13.21, 389 Fact 4.13.22, 389 Fact 4.13.23, 389 Fact 11.10.83, 882 upper bound Fact 3.16.24, 333 Fact 3.16.25, 333 Fact 10.15.6, 775 Fact 10.15.7, 775 Fact 10.15.9, 775 Fact 10.15.10, 775 upper Hessenberg matrix Fact 4.25.4, 425 Fact 4.25.5, 425 Fact 4.25.6, 426 Fact 6.9.8, 525 Fact 6.9.9, 526 Fact 6.9.10, 527 upper triangular matrix Fact 4.25.1, 424 Vandermonde matrix Fact 7.18.5, 613 Fact 7.18.6, 613 Fact 7.18.7, 613 Fact 7.18.8, 613

1469

Wronskian Fact 12.16.26, 953 determinant inequality Hua’s inequalities Fact 10.12.52, 757 determinant lower bound nonsingular matrix Fact 6.10.24, 535 determinant of a partitioned matrix adjugate Fact 3.19.10, 347 Fact 3.19.12, 348 Hadamard’s inequality Fact 8.9.35, 668 determinant of an outer-product perturbation Sherman-MorrisonWoodbury formula Fact 3.21.3, 351 determinantal compression partitioned matrix Fact 10.16.50, 784 diagonal eigenvalue Fact 10.13.14, 762 hexagon Fact 5.3.6, 492 positive-semidefinite matrix Fact 10.13.14, 762 quadrilateral Fact 5.3.5, 492 zero Fact 7.10.19, 571 diagonal dominance rank Fact 6.10.29, 536 diagonal dominance theorem nonsingular matrix Fact 6.10.23, 535 Fact 6.10.24, 535 diagonal entry definition, 280 eigenvalue Fact 10.21.11, 802 Fact 10.21.12, 802 Hermitian matrix Cor. 10.4.7, 710 Fact 10.21.11, 802 Fact 10.21.13, 803 Fact 10.21.19, 804 positive-definite matrix Fact 10.21.16, 803 positive-semidefinite matrix Fact 10.11.27, 738 Schur-Horn theorem Fact 4.13.16, 389 similar matrices

1470

diagonal matrix

Fact 7.10.14, 571 strong majorization Fact 10.21.11, 802 Fact 10.21.12, 802 unitarily similar matrices Fact 7.10.18, 571 Fact 7.10.20, 571 unitary matrix Fact 4.13.16, 389 Fact 10.21.14, 803 weak majorization Fact 7.12.26, 582 diagonal matrix controllability Fact 16.21.13, 1308 definition Defn. 4.1.3, 365 Hermitian matrix Cor. 7.5.5, 554 Lem. 10.5.1, 713 Kronecker product Fact 9.4.4, 686 matrix exponential Fact 15.14.13, 1210 orthogonally similar matrices Fact 7.10.16, 571 positive-semidefinite matrix Fact 10.10.48, 735 semisimple matrix Prop. 7.7.11, 560 unitary matrix Thm. 7.6.3, 557 Lem. 10.5.1, 713 diagonalizable matrix additive decomposition Fact 7.10.4, 570 cogredient diagonalization Fact 10.20.3, 799 Fact 10.20.5, 799 commuting matrices Fact 7.19.9, 617 determinant Fact 7.15.8, 599 eigenvector Fact 7.15.7, 599 example Example 7.7.18, 562 factorization Fact 7.17.29, 609 idempotent matrix Fact 7.15.21, 600 involutory matrix Fact 7.15.21, 600 Jordan-Chevalley decomposition Fact 7.10.4, 570 positive-definite matrix Cor. 10.3.3, 708 product of matrices Fact 10.20.4, 799 S-N decomposition Fact 7.10.4, 570

maximum singular value simultaneous Fact 10.22.15, 807 diagonalization Fact 11.10.22, 872 Fact 10.20.3, 799 projector Fact 10.20.5, 799 Fact 4.18.19, 414 trace Fact 7.13.27, 594 Fact 7.15.8, 599 Fact 8.8.12, 655 tripotent matrix rank Fact 7.15.21, 600 Fact 4.16.12, 402 diagonalizable over C Schatten norm definition, 560 Fact 11.10.17, 871 diagonalizable over F singular value identity-matrix Fact 10.12.65, 760 perturbation Fact 10.22.16, 807 Fact 7.15.17, 600 trace norm diagonalizable over R Fact 11.10.19, 871 asymptotically stable difference of matrices matrix determinant Fact 15.18.14, 1225 Fact 6.9.23, 528 definition, 560 idempotent matrix factorization Fact 4.16.6, 400 Prop. 7.7.13, 560 Fact 4.16.7, 400 Cor. 7.7.22, 562 Fact 4.16.8, 401 similar matrices Fact 4.16.9, 401 Prop. 7.7.13, 560 Fact 4.16.10, 402 Cor. 7.7.22, 562 Fact 4.16.11, 402 diagonalization difference of powers cross-product matrix equality Fact 7.10.1, 569 Fact 2.2.4, 131 Fact 7.10.2, 570 diagonally dominant matrix differentiable function continuous function nonsingular matrix Prop. 12.5.4, 925 Fact 6.10.23, 535 definition diagonally located block Defn. 12.5.3, 925 definition, 281 differential equation diagonals asymptotic solution pentagon Fact 16.20.1, 1305 Fact 5.3.7, 492 digamma function polygon Gamma function Fact 5.3.9, 492 Fact 13.3.3, 1000 Diaz-Goldman-Metcalf digraph inequality definition, 118 H¨older’s inequality dihedral group Fact 2.12.47, 223 definition dictionary ordering Prop. 4.6.6, 372 cone group Fact 3.11.23, 310 Fact 4.31.16, 435 total ordering Klein four-group Fact 1.8.16, 25 Fact 4.31.16, 435 Dieudonne´ dilogarithm function orthogonal matrix integral Fact 7.17.17, 607 Fact 13.3.5, 1002 difference series Drazin generalized inverse Fact 13.5.35, 1029 Fact 8.11.9, 675 dimension Fact 8.11.10, 675 commuting matrices Fact 8.11.11, 676 Fact 7.11.20, 577 Frobenius norm Fact 7.11.21, 577 Fact 11.10.14, 871 convex set generalized inverse Fact 12.11.20, 935 Fact 8.4.39, 639 feasible cone group generalized inverse Fact 12.11.30, 936 Fact 8.11.12, 676 open set idempotent matrix Fact 12.11.10, 935 Fact 4.16.12, 402 product of matrices Fact 7.13.29, 596

Fact 3.13.21, 316 rank inequality Fact 3.13.4, 315 solid set Fact 12.11.27, 936 subspace Fact 3.13.4, 315 subspace dimension theorem Thm. 3.1.3, 279 subspace inclusion Fact 3.12.23, 314 subspace intersection Fact 3.12.19, 313 Fact 3.12.20, 314 Fact 3.12.24, 314 zero trace Fact 3.23.11, 356 dimension of a set definition, 279 dimension of a subspace definition, 279 dimension of an affine subspace definition, 279 dimension theorem rank and defect Cor. 3.6.5, 293 Diophantus’s identity sum of squares Fact 2.4.7, 178 directed chain definition Defn. 1.4.3, 10 directed cut definition Defn. 1.4.3, 10 Defn. 1.4.4, 11 graph Fact 6.11.4, 538 directed cycle definition Defn. 1.4.3, 10 graph Fact 1.9.2, 25 Fact 1.9.4, 25 directed edge definition, 9 directed forest definition Defn. 1.4.3, 10 graph Fact 1.9.7, 25 directed graph adjacency matrix Fact 4.26.3, 426 antisymmetric graph Fact 4.26.2, 426 definition, 9 incidence matrix Fact 4.26.4, 426 Fact 4.26.5, 426 symmetric graph Fact 4.26.1, 426 directed graph of a matrix

discrete-time semistable polynomial definition Prop. 15.11.2, 1203 Defn. 4.2.4, 368 matrix exponential Prop. 15.11.2, 1203 directed Hamiltonian cycle definition discrete-time Defn. 1.4.3, 10 asymptotically stable graph matrix Fact 1.9.3, 25 2 × 2 matrix directed Hamiltonian path Fact 15.22.1, 1239 definition asymptotically stable Defn. 1.4.3, 10 matrix directed multigraph Fact 15.22.10, 1240 definition, 9 Cayley transform directed path Fact 15.22.10, 1240 definition definition Defn. 1.4.3, 10 Defn. 15.11.1, 1203 directed period discrete-time definition asymptotically stable Defn. 1.4.3, 10 polynomial directed subgraph Prop. 15.11.4, 1204 Defn. 1.4.3, 10 dissipative matrix directed trail Fact 15.22.5, 1240 definition Kronecker product Defn. 1.4.3, 10 Fact 15.22.7, 1240 directed tree Fact 15.22.8, 1240 definition Lyapunov equation Defn. 1.4.3, 10 Prop. 15.11.5, 1204 directed walk matrix exponential definition Fact 15.22.9, 1240 Defn. 1.4.3, 10 matrix limit direction cosines Fact 15.22.15, 1241 Euler parameters matrix power Fact 4.14.6, 392 Fact 11.13.11, 892 orthogonal matrix Fact 15.22.3, 1239 Fact 4.14.6, 392 Fact 15.22.16, 1241 directional differential nonnegative matrix convex function Fact 15.22.2, 1239 Fact 12.16.10, 950 normal matrix directionally acyclic graph Fact 15.22.5, 1240 definition partitioned matrix Defn. 1.4.3, 10 Fact 15.22.11, 1240 graph positive-definite matrix Fact 1.9.4, 25 Prop. 15.11.5, 1204 directionally connected Fact 15.22.11, 1240 graph Fact 15.22.19, 1241 Defn. 1.4.3, 10 Fact 15.22.21, 1242 Dirichlet S function similar matrices series Fact 15.19.5, 1226 Fact 13.3.6, 1003 discrete-time Dirichlet integral asymptotically stable trigonometric function polynomial Fact 14.4.14, 1119 definition Dirichlet’s test Defn. 15.11.3, 1204 series derivative Fact 12.18.12, 959 Fact 15.18.1, 1223 disconnected discrete-time definition asymptotically stable Defn. 12.1.14, 915 matrix discrete Fourier analysis Prop. 15.11.4, 1204 circulant matrix Jury criterion Fact 7.18.13, 614 Fact 15.21.2, 1234 discrete-time asymptotic polynomial coefficients stability Fact 15.21.2, 1234 eigenvalue Fact 15.21.3, 1234 Prop. 15.11.2, 1203 Fact 15.21.4, 1234 linear dynamical system

Schur-Cohn criterion Fact 15.21.2, 1234 unit circle Fact 15.21.1, 1234 discrete-time control problem LQG controller Fact 16.25.9, 1318 discrete-time dynamics matrix power Fact 15.22.4, 1239 discrete-time Lyapunov equation discrete-time asymptotically stable matrix Fact 15.22.19, 1241 Fact 15.22.21, 1242 discrete-time Lyapunov-stable matrix Prop. 15.11.6, 1204 solutions Fact 16.23.1, 1313 spectral radius Fact 15.22.20, 1241 Stein equation Fact 15.22.19, 1241 Fact 15.22.21, 1242 discrete-time Lyapunov stability eigenvalue Prop. 15.11.2, 1203 linear dynamical system Prop. 15.11.2, 1203 matrix exponential Prop. 15.11.2, 1203 discrete-time Lyapunov-stable matrix definition Defn. 15.11.1, 1203 discrete-time Lyapunov equation Prop. 15.11.6, 1204 discrete-time Lyapunov-stable polynomial Prop. 15.11.4, 1204 group generalized inverse Fact 15.22.14, 1241 Kreiss matrix theorem Fact 15.22.23, 1242 Kronecker product Fact 15.22.7, 1240 Fact 15.22.8, 1240 logarithm Fact 15.15.20, 1214 matrix exponential Fact 15.22.9, 1240 matrix limit Fact 15.22.14, 1241 matrix power Fact 15.22.3, 1239 Fact 15.22.13, 1241 maximum singular value

1471

Fact 15.22.23, 1242 normal matrix Fact 15.22.5, 1240 positive-definite matrix Prop. 15.11.6, 1204 positive-semidefinite matrix Fact 15.22.19, 1241 row-stochastic matrix Fact 15.22.12, 1240 semicontractive matrix Fact 15.22.5, 1240 semidissipative matrix Fact 15.22.5, 1240 similar matrices Fact 15.19.5, 1226 unitary matrix Fact 15.22.17, 1241 discrete-time Lyapunov-stable polynomial definition Defn. 15.11.3, 1204 discrete-time Lyapunov-stable matrix Prop. 15.11.4, 1204 discrete-time semistability eigenvalue Prop. 15.11.2, 1203 linear dynamical system Prop. 15.11.2, 1203 matrix exponential Prop. 15.11.2, 1203 discrete-time semistable matrix companion form matrix Fact 15.22.25, 1242 convergent sequence Fact 15.22.18, 1241 definition Defn. 15.11.1, 1203 discrete-time semistable polynomial Prop. 15.11.4, 1204 idempotent matrix Fact 15.22.13, 1241 Kronecker product Fact 15.22.6, 1240 Fact 15.22.7, 1240 Fact 15.22.8, 1240 limit Fact 15.22.13, 1241 matrix exponential Fact 15.22.9, 1240 Fact 15.22.18, 1241 row-stochastic matrix Fact 15.22.12, 1240 similar matrices Fact 15.19.5, 1226 discrete-time semistable polynomial definition Defn. 15.11.3, 1204

1472

discrete-time system

discrete-time semistable matrix Prop. 15.11.4, 1204 discrete-time system state convergence Fact 15.22.22, 1242 discrete-time unstable matrix definition Defn. 15.11.1, 1203 discrete-time unstable polynomial definition Defn. 15.11.3, 1204 discriminant compound matrix Fact 9.5.18, 693 Fact 9.5.19, 694 polynomial Fact 6.8.21, 523 resultant Fact 6.8.21, 523 disjoint definition, 2 disk inclusion polynomial Fact 15.21.30, 1239 dissipative matrix asymptotically stable matrix Fact 15.19.21, 1228 Fact 15.19.37, 1231 definition Defn. 4.1.1, 363 determinant Fact 10.15.2, 774 Fact 10.15.3, 774 Fact 10.16.20, 779 discrete-time asymptotically stable matrix Fact 15.22.5, 1240 Frobenius norm Fact 15.16.3, 1218 inertia Fact 7.9.13, 567 Kronecker sum Fact 9.5.9, 692 matrix exponential Fact 15.16.3, 1218 maximum singular value Fact 10.21.18, 803 nonsingular matrix Fact 4.27.1, 427 normal matrix Fact 15.19.37, 1231 positive-definite matrix Fact 10.21.18, 803 Fact 15.19.21, 1228 range-Hermitian matrix Fact 7.16.18, 604 semidissipative matrix Fact 10.16.20, 779 spectrum Fact 10.16.20, 779

strictly dissipative matrix Fact 10.10.36, 734 unitary matrix Fact 10.10.36, 734 distance from a point set Fact 12.12.24, 940 Fact 12.12.25, 940 distance to singularity nonsingular matrix Fact 11.16.6, 900 distinct eigenvalues eigenvector Prop. 6.5.4, 511 distinct roots Bezout matrix Fact 6.8.11, 521 distributive equalities definition, 283 divergence derivative Fact 12.16.27, 953 divides definition, 501 properties Fact 1.11.4, 28 division of polynomial matrices quotient and remainder Lem. 6.2.1, 502 divisor sum partition number Fact 1.20.2, 115 Dixon’s identity binomial equality Fact 1.16.14, 86 Djokovic elementary projectors Fact 11.16.10, 901 Kronecker product Fact 10.25.15, 822 Schur product Fact 10.25.18, 822 Djokovic inequality Euclidean norm Fact 11.8.16, 858 Dobinski’s formula Bell number Fact 13.1.4, 976 Dodgson condensation adjugate Fact 3.17.35, 341 domain definition, 16 domain of convergence of a bi-power series definition Defn. 12.3.5, 920 domain of convergence of a power series definition Defn. 12.3.5, 919 dominant eigenvalue limit Fact 15.14.16, 1211

Fact 15.22.24, 1242

positive-semidefinite matrix rotation matrix Fact 7.15.15, 599 Fact 15.12.14, 1207 Drazin generalized inverse Dormido block-circulant matrix asymptotically stable Fact 8.12.5, 679 Cline’s formula polynomial Fact 8.11.6, 675 Fact 15.18.10, 1224 commuting matrices double cover Fact 8.11.7, 675 orthogonal matrix Fact 8.11.8, 675 parameterization definition, 625 Fact 4.14.6, 392 difference spin group Fact 8.11.9, 675 Fact 4.14.6, 392 Fact 8.11.10, 675 double factorial Fact 8.11.11, 676 equalities equivalent Fact 1.13.15, 51 characterization infinite series Fact 8.10.1, 669 Fact 13.5.83, 1048 group generalized inverse limit Fact 8.10.9, 670 Fact 12.18.59, 971 idempotent matrix double integral Prop. 8.2.2, 626 golden ratio Fact 8.11.9, 675 Fact 14.13.6, 1176 Fact 8.11.10, 675 logarithm integral Fact 14.13.3, 1174 Fact 15.14.8, 1210 Fact 14.13.4, 1175 Fact 15.14.10, 1210 Fact 14.13.5, 1175 Kronecker product power Fact 9.4.37, 688 Fact 14.13.1, 1173 limit rational function Fact 8.10.6, 670 Fact 14.13.2, 1173 linear equation double sum Fact 8.10.8, 670 series matrix exponential Fact 13.8.17, 1074 Fact 15.14.8, 1210 Fact 13.8.18, 1074 Fact 15.14.10, 1210 Fact 13.8.19, 1074 matrix limit doublet Fact 8.10.14, 670 definition matrix product Fact 3.13.31, 318 Fact 8.11.6, 675 outer-product matrix Fact 8.11.7, 675 Fact 3.13.31, 318 matrix sum Fact 3.15.6, 326 Fact 8.11.8, 675 spectrum null space Fact 7.12.16, 581 Prop. 8.2.2, 626 doubly stochastic matrix partitioned matrix definition Fact 8.10.11, 670 Defn. 4.1.5, 367 Fact 8.12.1, 678 permutation matrix Fact 8.12.5, 679 Fact 4.11.6, 384 polynomial Fact 4.13.1, 387 Fact 8.10.8, 670 strong majorization positive-semidefinite Fact 4.11.6, 384 Douglas-Fillmore-Williams matrix Fact 10.24.2, 815 lemma projector factorization Fact 8.11.11, 676 Thm. 10.6.2, 714 range Fact 10.11.2, 735 Prop. 8.2.2, 626 Dragomir-Yang inequalities rank Euclidean norm Fact 8.10.6, 670 Fact 11.8.18, 859 sequence Fact 11.8.19, 859 Fact 8.10.6, 670 Drazin similar matrices commuting matrices Fact 8.11.5, 675 Fact 8.11.8, 675

Donkin’s theorem

eigenvalue sum Fact 8.11.9, 675 Fact 8.11.11, 676 Fact 8.12.5, 679 tripotent matrix Prop. 8.2.2, 626 uniqueness Thm. 8.2.1, 626 Drazin inverse algebraic multiplicity Fact 8.10.7, 670 geometric multiplicity Fact 8.10.7, 670 partitioned matrix Fact 8.12.4, 678 DT-dominant eigenvalue limit Fact 15.22.24, 1242 dual cone convex cone Fact 12.12.14, 938 convex conical hull Fact 3.11.8, 308 definition, 290 intersection Fact 12.12.15, 939 Minkowski sum Fact 12.12.15, 939 dual norm adjoint norm Fact 11.9.7, 860 definition Fact 11.8.24, 859 induced norm Fact 11.8.24, 859 quadratic form Fact 11.9.53, 866 dual pentagonal number equalities Fact 1.12.4, 39 generating function Fact 13.4.7, 1010 Duhamel formula matrix exponential Fact 15.15.3, 1212 Dunkl-Williams inequality complex numbers Fact 2.21.9, 271 norm Fact 11.7.11, 851 Fact 11.7.13, 852 Durfee square identity infinite product Fact 13.10.25, 1090 dynamic compensator LQG controller Fact 16.25.8, 1318 Fact 16.25.9, 1318 Dyson’s integral multiple integral Fact 14.13.14, 1178

Fact 11.16.39, 906 edge

Defn. 1.4.2, 10 Egyptian fraction

integer Fact 1.11.50, 36 eigensolution eigenvector Fact 16.20.4, 1305 Fact 16.20.5, 1305 eigenstructure intertwining matrices Fact 7.16.11, 603 eigenvalue, see spectrum, multispectrum SO(n) Fact 7.12.2, 579 adjugate Fact 6.10.13, 532 asymptotic spectrum Fact 6.10.35, 537 asymptotic stability Prop. 15.9.2, 1198 bound Fact 6.10.28, 536 Fact 7.12.29, 583 Fact 11.13.6, 891 bounds Fact 6.10.22, 535 Fact 6.10.26, 535 Cardano’s trigonometric solution Fact 6.10.7, 531 Cartesian decomposition Fact 7.12.24, 582 commutator Fact 7.16.13, 604 Fact 7.16.14, 604 convex function Cor. 10.6.19, 722 Fact 10.22.11, 806 definition, 506 determinant Fact 7.12.32, 584 Fact 7.12.33, 584 Fact 10.15.1, 774 diagonal entry Fact 10.13.14, 762 Fact 10.21.11, 802 Fact 10.21.12, 802 discrete-time asymptotic stability Prop. 15.11.2, 1203 discrete-time Lyapunov stability Prop. 15.11.2, 1203 discrete-time semistability Prop. 15.11.2, 1203 Frobenius norm Fact 11.13.2, 890 Fact 11.13.4, 891 Fact 11.14.4, 893 E Fact 11.14.5, 893 generalized eigenvector Eckart-Young theorem Fact 7.15.12, 599 fixed-rank approximation

generalized Schur inequality Fact 11.13.5, 891 geometric mean Fact 10.25.34, 824 Hermitian matrix Thm. 10.4.5, 710 Thm. 10.4.9, 710 Thm. 10.4.11, 711 Cor. 10.4.2, 709 Cor. 10.4.6, 710 Cor. 10.4.7, 710 Cor. 10.6.19, 722 Lem. 10.4.3, 709 Lem. 10.4.4, 709 Fact 7.12.6, 579 Fact 7.12.7, 579 Fact 10.11.13, 736 Fact 10.18.22, 794 Fact 10.19.6, 796 Fact 10.21.11, 802 Fact 10.21.13, 803 Fact 10.21.19, 804 Fact 10.21.20, 804 Fact 10.21.21, 804 Fact 10.22.4, 805 Fact 10.22.5, 805 Fact 10.22.7, 805 Fact 10.22.23, 809 Fact 10.22.24, 809 Fact 10.25.31, 824 Hermitian part Fact 7.12.27, 583 H¨older matrix norm Fact 11.13.5, 891 inner matrix Fact 10.21.20, 804 Fact 10.21.21, 804 Kronecker product Prop. 9.1.10, 683 Fact 9.4.17, 686 Fact 9.4.21, 687 Fact 9.4.27, 687 Fact 9.4.32, 688 Fact 9.4.40, 689 Kronecker sum Prop. 9.2.4, 684 Fact 9.5.6, 691 Fact 9.5.8, 692 Fact 9.5.17, 693 lower triangular matrix Fact 6.10.14, 533 Lyapunov stability Prop. 15.9.2, 1198 matrix logarithm Thm. 15.5.2, 1190 matrix sum Fact 7.13.8, 591 Fact 7.13.9, 592 maximum singular value Fact 7.12.33, 584 Fact 11.14.4, 893 Fact 11.14.5, 893 minimum singular value Fact 7.12.33, 584

1473

normal matrix Fact 7.12.30, 583 Fact 7.15.16, 599 Fact 10.13.2, 761 orthogonal matrix Fact 7.12.2, 579 partitioned matrix Prop. 7.6.5, 557 Fact 6.10.33, 536 Fact 7.13.30, 596 Fact 7.13.31, 596 Fact 7.13.32, 597 quadratic form Lem. 10.4.3, 709 Fact 7.12.6, 579 Fact 7.12.7, 579 Fact 10.19.6, 796 root locus Fact 6.10.35, 537 Schatten norm Fact 11.13.5, 891 Schur complement Fact 10.12.4, 749 Schur product Fact 10.25.29, 823 Fact 10.25.32, 824 Schur’s inequality Fact 10.21.10, 801 Fact 11.13.2, 890 semistability Prop. 15.9.2, 1198 singular value Fact 7.12.26, 582 Fact 7.12.33, 584 Fact 10.13.2, 761 Fact 10.21.3, 800 Fact 10.21.10, 801 Fact 10.21.17, 803 Fact 11.15.25, 899 skew-Hermitian matrix Fact 7.12.10, 580 skew-symmetric matrix Fact 6.10.8, 532 spectral abscissa Fact 7.12.27, 583 spread Fact 10.21.5, 801 Fact 11.13.10, 892 strong log majorization Fact 10.22.37, 812 Fact 10.22.38, 812 strong majorization Cor. 10.6.19, 722 Fact 10.21.11, 802 Fact 10.21.12, 802 Fact 10.22.37, 812 Fact 10.22.38, 812 Fact 10.22.41, 813 subscript convention, 506 symmetric matrix Fact 6.10.7, 531 Fact 10.21.5, 801 trace Prop. 10.4.13, 712 Fact 7.12.14, 581

1474

eigenvalue bound

Fact 10.21.5, 801 Fact 10.21.10, 801 Fact 10.22.25, 809 unitarily invariant norm Fact 11.14.4, 893 Fact 11.14.5, 893 upper triangular matrix Fact 6.10.14, 533 Vandermonde matrix Fact 7.18.10, 614 weak log majorization Fact 10.22.6, 805 Fact 10.22.13, 807 weak majorization Fact 7.12.26, 582 Fact 10.21.10, 801 Fact 10.22.11, 806 Fact 10.22.12, 806 eigenvalue bound Bendixson’s theorem Fact 7.12.24, 582 Fact 11.13.7, 891 Browne’s theorem Fact 7.12.24, 582 Frobenius norm Fact 11.14.3, 893 Henrici Fact 11.13.2, 890 Hermitian matrix Fact 11.14.3, 893 Hirsch’s theorem Fact 7.12.24, 582 Hirsch’s theorems Fact 11.13.7, 891 H¨older norm Fact 11.13.7, 891 mixed H¨older norm Fact 11.13.8, 891 trace Fact 7.12.48, 588 Fact 7.12.49, 588 eigenvalue bounds ovals of Cassini Fact 6.10.27, 536 eigenvalue characterization minimum principle Fact 10.21.20, 804 eigenvalue inclusion region Lyapunov equation Fact 16.22.20, 1313 eigenvalue inequality 2 × 2 matrix Fact 10.21.1, 800 Hermitian matrix Lem. 10.4.1, 709 Fact 10.22.2, 805 Fact 10.22.3, 805 Poincar´e separation theorem Fact 10.21.21, 804 eigenvalue interlacing Hermitian matrix Fact 10.22.1, 804

eigenvalue interlacing theorem

Prop. 9.1.10, 683 Fact 9.4.27, 687 Fact 9.4.40, 689 Hermitian matrix Kronecker sum eigenvalue Prop. 9.2.4, 684 Thm. 10.4.5, 710 Fact 9.5.17, 693 eigenvalue majorization M-matrix Hermitian matrix Fact 6.11.15, 542 Fact 10.22.8, 806 normal matrix Fact 10.22.9, 806 Prop. 6.5.4, 511 eigenvalue of Hermitian Lem. 6.5.3, 511 part partitioned matrix maximum singular value Fact 6.10.33, 536 Fact 7.12.25, 582 similarity transformation minimum singular value Fact 7.15.7, 599 Fact 7.12.25, 582 Fact 7.15.9, 599 singular value upper triangular matrix Fact 10.21.9, 801 Fact 7.19.2, 617 eigenvalue perturbation Elder’s theorem Frobenius norm partition number Fact 11.14.13, 894 Fact 1.20.1, 113 Fact 11.14.14, 894 element Fact 11.14.15, 895 definition, 1 Hermitian matrix elementary divisor Fact 6.10.34, 537 companion matrix maximum singular value Thm. 7.3.8, 548 Fact 11.14.6, 893 definition, 548 normal matrix factorization Fact 11.14.6, 893 Fact 7.17.40, 610 partitioned matrix hypercompanion matrix Fact 6.10.34, 537 Lem. 7.4.4, 550 unitarily invariant norm Jordan matrix Fact 11.14.16, 895 Lem. 7.4.1, 549 eigenvalue placement elementary matrix controllable pair definition Prop. 16.6.20, 1265 Defn. 4.1.2, 364 observable pair inverse matrix Prop. 16.3.20, 1257 Fact 4.10.20, 381 eigenvalue vector nonsingular matrix definition, 506 Fact 7.17.14, 606 eigenvalues properties and matrix strong majorization types Fact 10.22.3, 805 Fact 4.10.19, 380 totally nonnegative matrix reduced row echelon Fact 6.11.9, 540 matrix Fact 6.11.10, 540 Thm. 7.2.2, 546 eigenvector semisimple matrix adjugate Fact 7.15.18, 600 Fact 7.15.30, 601 spectrum commuting matrices Prop. 7.7.21, 562 Fact 7.16.1, 603 unitarily similar matrices cyclic eigenvalue Prop. 7.7.23, 562 Fact 7.15.4, 598 elementary definition, 511 multicompanion form diagonalizable matrix definition, 549 Fact 7.15.7, 599 elementary distinct eigenvalues Prop. 6.5.4, 511 multicompanion matrix eigensolution similarity invariant Fact 16.20.4, 1305 Cor. 7.3.9, 549 Fact 16.20.5, 1305 elementary polynomial generalized eigensolution matrix Fact 16.20.5, 1305 definition, 503 Hermitian matrix elementary projector Fact 11.13.12, 892 definition Kronecker product Defn. 4.1.1, 363

elementary reflector Fact 4.17.9, 408 Fact 4.19.3, 416 hyperplane Fact 4.17.10, 408 maximum singular value Fact 11.16.10, 901 reflector Fact 7.17.15, 606 spectrum Prop. 7.7.21, 562 trace Fact 7.9.10, 566 unitarily similar matrices Prop. 7.7.23, 562 elementary reflector definition Defn. 4.1.1, 363 elementary projector Fact 4.17.9, 408 Fact 4.19.3, 416 hyperplane Fact 4.19.5, 416 null space Fact 4.17.9, 408 orthogonal matrix Fact 7.17.17, 607 range Fact 4.17.9, 408 rank Fact 4.17.9, 408 reflection theorem Fact 4.19.4, 416 reflector Fact 7.17.16, 607 spectrum Prop. 7.7.21, 562 trace Fact 7.9.10, 566 unitarily similar matrices Prop. 7.7.23, 562 elementary symmetric function Schur-concave function Fact 2.11.96, 206 elementary symmetric mean Maclaurin’s inequality Fact 2.11.35, 195 Newton’s inequality Fact 2.11.35, 195 elementary symmetric polynomial Newton’s identities Fact 2.11.34, 194 Fact 6.8.4, 518 ellipse arc length Fact 13.9.5, 1079 area Fact 5.5.8, 496 Fact 5.5.9, 496 convex quadrilateral Fact 5.5.5, 496 Fact 5.5.6, 496

Euler constant parallelogram Fact 5.5.7, 496 perimeter Fact 13.9.5, 1079 polynomial Fact 5.5.3, 496 Steiner inellipse Fact 5.5.3, 496 triangle Fact 5.5.3, 496 Fact 5.5.4, 496 ellipsoid positive-definite matrix Fact 5.5.14, 498 volume Fact 5.5.14, 498 elliptic integral arithmetic-geometric mean Fact 12.18.55, 968 Embry commuting matrices Fact 7.11.30, 579 empty matrix definition, 283 empty set definition, 1 Enestrom-Kakeya theorem polynomial root locations Fact 15.21.4, 1234 Enflo inequality Fact 11.8.12, 857 entropy binomial coefficient Fact 12.18.49, 967 logarithm Fact 2.15.24, 229 Fact 2.15.25, 229 Fact 2.15.26, 230 Fact 2.15.27, 230 Fact 2.15.28, 230 Schur-concave function Fact 3.25.7, 361 strong majorization Fact 3.25.7, 361 entry definition, 280 EP matrix, see range-Hermitian matrix definition, 440 epigraph convex function Fact 12.13.25, 945 equi-induced norm definition Defn. 11.4.1, 841 normalized norm Thm. 11.4.2, 841 spectral radius Cor. 11.4.5, 842 submultiplicative norm Cor. 11.4.4, 842 Fact 11.9.63, 868

equi-induced self-adjoint norm

1475

characterization Prop. 16.9.11, 1273 Fact 11.7.5, 850 similar matrices convex combination Defn. 16.9.6, 1272 maximum singular value Fact 11.8.4, 856 ¨ Erdos-Debrunner Fact 11.15.7, 896 definition, 286 inequality equi-induced unitarily determinant triangle invariant norm Fact 11.13.1, 890 Fact 5.2.11, 466 maximum singular value Djokovic inequality ¨ Erdos-Ko-Rado theorem Fact 11.15.6, 896 Fact 11.8.16, 858 set intersection equilateral triangle Dragomir-Yang Fact 1.8.10, 24 complex numbers inequalities ¨ Erdos-Mordell theorem Fact 5.2.1, 443 Fact 11.8.18, 859 triangle equilibrium Fact 11.8.19, 859 Fact 5.2.11, 466 definition, 1196 generalized ¨ Erdos-Straus conjecture equivalence class Egyptian fraction Cauchy-Schwarz definition, 6 Fact 1.11.50, 36 equivalence relation inequality reciprocal Thm. 1.3.5, 6 Fact 11.8.6, 856 Fact 1.11.51, 36 equivalent matrices Fact 11.10.2, 869 ¨ Erdos-Szekeres theorem Fact 7.11.5, 576 generalized Hlawka monotonicity similar matrices inequality Fact 1.8.17, 25 Fact 7.11.5, 576 Fact 11.8.16, 858 unitarily similar matrices ergodic theorem generalized parallelogram unitary matrix limit Fact 7.11.5, 576 law Fact 8.3.37, 632 equivalence hull Fact 11.8.17, 858 essential boundary definition Fact 11.10.61, 877 definition Defn. 1.3.3, 6 Huygens-Leibniz identity Defn. 12.1.11, 915 relation Fact 11.8.15, 858 essential closure Prop. 1.3.4, 6 inequality convergent sequence equivalence relation Fact 11.8.3, 853 Prop. 12.2.6, 916 Abelian group Fact 11.8.10, 857 Prop. 12.2.8, 917 Prop. 4.7.2, 373 Fact 11.8.14, 858 definition definition Fact 11.8.18, 859 Defn. 12.1.9, 914 Defn. 1.3.1, 5 Fact 11.8.19, 859 essential closure point equivalence class Fact 11.8.20, 859 definition Thm. 1.3.5, 6 Lagrange’s second identity Defn. 12.1.9, 914 group Fact 11.8.15, 858 essential greatest lower Prop. 4.7.1, 373 maximum singular value Prop. 4.7.2, 373 bound Fact 11.15.3, 896 intersection sequence of sets minimum singular value Prop. 1.3.2, 6 Defn. 1.3.18, 8 Fact 11.15.3, 896 partition essential least upper polygon Thm. 1.3.6, 6 bound Fact 11.8.8, 856 representative set sequence of sets projector Defn. 1.3.7, 7 Defn. 1.3.18, 8 Fact 10.11.9, 736 equivalent matrices essential limit Fact 11.9.2, 860 equivalence class sequence of sets Fact 12.12.26, 940 Fact 7.11.5, 576 Defn. 1.3.20, 9 reverse triangle inequality equivalent norms essential singularity Fact 11.8.14, 858 equivalence definition, 929 weighted least square Thm. 11.1.8, 835 essentially nonnegative Fact 11.17.8, 910 norms Euler matrix Fact 11.9.23, 862 infinite product definition, 440 equivalent realizations Fact 13.10.23, 1090 Euclid controllable canonical pentagonal number prime numbers form Fact 12.18.41, 965 theorem Cor. 16.9.21, 1277 Euclidean distance matrix Fact 13.10.27, 1091 Cor. 16.9.23, 1278 negative-semidefinite zeta function controllable pair Fact 13.5.43, 1032 matrix Prop. 16.9.8, 1272 Euler characteristic Fact 11.9.17, 861 Prop. 16.9.11, 1273 Euler’s polyhedron Euclidean norm observable canonical form bounded sequence formula Cor. 16.9.22, 1277 Fact 11.8.29, 860 Fact 5.4.8, 495 Cor. 16.9.23, 1278 Cauchy-Schwarz planar graph observable pair Fact 1.9.10, 26 inequality Prop. 16.9.8, 1272 Euler constant Cor. 11.1.7, 835

1476

Euler number

logarithm exclusive or Fact 12.18.33, 963 definition, 2 symmetric difference Euler number Fact 1.8.3, 23 alternating series Fact 13.5.93, 1052 existence definition ordinary differential Fact 13.1.8, 980 equation Euler parameters Thm. 15.8.1, 1195 direction cosines Thm. 15.8.2, 1195 Fact 4.14.6, 392 existence of orthogonal matrix transformation Fact 4.14.6, 392 Hermitian matrix Fact 4.14.8, 394 Fact 4.11.2, 383 Rodrigues’s formulas orthogonal matrix Fact 4.14.8, 394 Fact 4.11.5, 384 Euler polynomial outer-product matrix series Fact 4.11.1, 383 Fact 13.2.3, 983 skew-Hermitian matrix Euler product formula Fact 4.11.4, 383 zeta function existential statement Fact 13.3.1, 994 definition, 3 Euler totient function logical equivalents limit Fact 1.7.6, 22 Fact 12.18.21, 960 exogenous input positive-semidefinite definition, 1291 matrix exponent Fact 10.9.8, 727 irrational number Euler’s inequality Fact 1.15.2, 52 triangle Fact 1.15.3, 52 Fact 5.2.8, 446 limit Euler’s polyhedron formula Fact 12.17.1, 954 Euler characteristic Fact 12.17.2, 954 Fact 5.4.8, 495 exponential, see matrix Euler’s theorem exponential quadrilateral geometric mean Fact 5.3.1, 489 Fact 2.11.103, 207 totient function inequality Fact 1.20.4, 115 Fact 2.1.51, 127 Eulerian number Fact 2.11.144, 214 permutation Fact 2.11.145, 214 Fact 1.19.5, 111 matrix logarithm Worpitzky’s identity Fact 15.15.28, 1215 Fact 1.19.5, 111 positive-definite matrix Eulerian numbers Fact 15.15.28, 1215 series power series Fact 13.1.2, 975 Fact 13.4.18, 1021 even integer exponential function prime complex numbers Fact 1.11.46, 35 Fact 2.21.27, 275 even permutation matrix convex function definition Fact 2.2.56, 141 Defn. 4.1.1, 363 inequality even polynomial Fact 2.2.57, 141 asymptotically stable limit polynomial Fact 12.17.6, 955 Fact 15.18.8, 1224 multiple integral definition, 500 Fact 14.13.14, 1178 exactly proper rational Fact 14.13.15, 1178 partial fractions function definition expansions Defn. 6.7.1, 513 Fact 13.4.10, 1014 exactly proper rational scalar inequalities Fact 2.1.54, 127 transfer function weak majorization definition Fact 3.25.11, 361 Defn. 6.7.2, 514

extended infinite interval

definition, xl extreme point convex set Fact 12.11.32, 937 Krein-Milman theorem Fact 12.11.32, 937

F fact

definition, 5 factorial

approximation Fact 12.18.58, 969 bounds Fact 1.13.9, 50 Fact 1.13.12, 50 Fact 1.13.14, 51 Burnside’s formula Fact 12.18.58, 969 determinant Fact 4.25.5, 425 divides Fact 1.13.16, 52 Fact 1.13.17, 52 harmonic number Fact 1.13.10, 50 inequality Fact 1.12.42, 47 Fact 1.13.11, 50 infinite product Fact 13.10.23, 1090 limit Fact 12.18.60, 971 Minc-Sathre inequality Fact 1.13.13, 51 prime Fact 1.11.13, 30 Fact 1.13.2, 49 prime number Fact 1.13.11, 50 product Fact 1.13.5, 50 Stirling’s formula Fact 12.18.58, 969 sum Fact 1.13.4, 49 Fact 1.13.6, 50 factorization asymptotically stable matrix Fact 15.19.22, 1228 Bezout matrix Fact 7.17.26, 608 bordered matrix Fact 3.21.4, 351 commutator Fact 7.17.36, 609 complex conjugate transpose Fact 7.17.25, 608 determinant Fact 7.17.9, 606 Fact 7.17.37, 609 diagonalizable matrix

Fact 7.17.29, 609 diagonalizable over R Prop. 7.7.13, 560 Cor. 7.7.22, 562 Douglas-FillmoreWilliams lemma Thm. 10.6.2, 714 Fact 10.11.2, 735 elementary divisor Fact 7.17.40, 610 generalized inverse Fact 8.9.34, 668 group generalized inverse Fact 8.11.1, 674 Hermitian matrix Fact 7.17.19, 607 Fact 7.17.27, 608 Fact 7.17.28, 608 Fact 7.17.43, 610 Fact 10.20.2, 799 idempotent matrix Fact 7.17.30, 609 Fact 7.17.33, 609 involutory matrix Fact 7.17.20, 607 Fact 7.17.34, 609 Fact 7.17.35, 609 Jordan form Fact 7.17.7, 606 lower triangular matrix Fact 7.17.12, 606 LULU decomposition Fact 7.17.13, 606 nilpotent matrix Fact 7.17.31, 609 nonsingular matrix Fact 7.17.14, 606 Fact 7.17.39, 609 orthogonal matrix Fact 7.17.17, 607 Fact 7.17.18, 607 Fact 7.17.34, 609 Fact 7.17.38, 609 partitioned matrix, 704 Prop. 3.9.3, 302 Prop. 3.9.4, 303 Fact 3.17.9, 335 Fact 3.17.11, 336 Fact 3.22.3, 352 Fact 3.22.4, 352 Fact 3.22.5, 353 Fact 3.22.6, 353 Fact 8.9.34, 668 Fact 10.12.53, 758 Fact 10.12.54, 758 positive-definite matrix Fact 7.17.28, 608 Fact 10.8.5, 724 Fact 10.8.6, 724 Fact 10.8.7, 724 Fact 10.8.9, 724 positive-semidefinite matrix Fact 7.17.24, 608

fraction Fact 7.17.28, 608 Fact 10.8.3, 723 Fact 10.8.4, 723 Fact 10.8.8, 724 Fact 10.10.41, 734 Fact 10.10.42, 734 projector Fact 7.17.15, 606 Fact 7.17.19, 607 Fact 8.8.10, 655 range Thm. 10.6.2, 714 Fact 10.11.2, 735 reflector Fact 7.17.16, 607 reverse-symmetric matrix Fact 7.10.13, 571 rotation-dilation Fact 3.24.2, 356 shear factor Fact 7.17.13, 606 similar matrices Fact 7.17.8, 606 skew-symmetric matrix Fact 7.17.40, 610 Fact 7.17.41, 610 symmetric matrix Cor. 7.4.10, 552 Fact 7.17.26, 608 ULU decomposition Fact 7.17.13, 606 unitary matrix Fact 7.17.11, 606 Fact 10.8.7, 724 upper triangular matrix Fact 7.17.11, 606 Fact 7.17.12, 606 factors sum Fact 1.20.3, 115 falling factorial definition, 15 determinant Fact 6.9.9, 526 upper Hessenberg matrix Fact 6.9.9, 526 Fan AM-GM inequality Fact 2.11.114, 209 convex function Prop. 10.6.17, 831 eigenvalue Fact 7.12.27, 583 Hermitian matrix product Fact 7.13.12, 592 orthogonal matrices Fact 7.13.21, 594 trace inequality Fact 7.13.14, 592 unitarily invariant norm Fact 11.10.8, 870 Fan constant definition Fact 10.11.77, 746 Fan dominance theorem

singular value Fact 11.16.23, 903 Fan trace minimization principle inequality Fact 7.13.19, 593 Fan-Lidskii theorem positive-semidefinite matrix Fact 11.14.5, 893 Farkas theorem linear system solution Fact 6.11.19, 543 fast Fourier transform circulant matrix Fact 7.18.13, 614 Faulhaber polynomial sum of powers of integers Fact 1.12.2, 37 feasible cone definition, 924 dimension Fact 12.11.30, 936 feedback interconnection closed-loop eigenvalues Fact 16.21.22, 1309 realization Prop. 16.13.4, 1290 Prop. 16.14.1, 1292 Fact 16.24.9, 1314 transfer function Fact 16.24.9, 1314 feedback signal definition, 1291 ´ Fejer polynomial root bound Fact 15.21.21, 1237 ´ Fejer’s theorem positive-semidefinite matrix Fact 10.25.19, 822 Fer expansion time-varying dynamics Fact 16.20.6, 1306 Fermat sum of squares of integers Fact 1.11.40, 35 Fermat prime prime Fact 1.11.13, 30 Fermat’s last theorem sum of powers Fact 1.11.39, 35 Fermat’s little theorem prime number Fact 1.11.15, 32 Fibonacci number Binet’s formula Fact 1.17.1, 95 Chebyshev polynomial of the second kind Fact 13.2.7, 986 determinant Fact 4.25.4, 425 Fibonacci polynomial

Fact 13.2.12, 992 generating function Fact 13.9.3, 1077 golden ratio Fact 13.9.2, 1075 Lucas number Fact 1.17.2, 100 nested radicals Fact 12.18.70, 974 recursion Fact 1.17.1, 95 root bound Fact 15.21.7, 1235 series Fact 13.9.3, 1077 Zeckendorf’s theorem Fact 1.17.1, 95 Fibonacci polynomial roots Fact 13.2.12, 992 field of values spectrum of convex hull Fact 10.17.8, 787 Fact 10.17.9, 788 final state controllability Fact 16.21.5, 1307 controllable subspace Fact 16.21.4, 1307 finite group group Fact 4.31.15, 434 Fact 4.31.16, 435 representation Fact 4.31.17, 436 finite interval definition, xl finite product equality Fact 1.14.1, 52 Fact 2.1.4, 119 multinomial coefficient Fact 2.21.24, 274 finite sequence monotonicity Fact 1.8.17, 25 finite set definition, 1 pigeonhole principle Fact 1.8.7, 24 finite sum maximum Fact 1.12.20, 43 minimum Fact 1.12.20, 43 finite-sum solution Lyapunov equation Fact 16.22.17, 1312 Finsler’s lemma positive-definite linear combination Fact 10.19.14, 796 Fact 10.19.15, 797 Fischer min-max theorem

1477

Fact 7.12.7, 579 Fischer’s inequality

determinant Fact 10.16.37, 782 Fact 10.16.39, 782 Fact 10.16.46, 784 five-parameter inequality scalar inequality Fact 2.5.6, 184 fixed point definition, 16 fixed-point theorem continuous function Thm. 12.4.26, 924 Cor. 12.4.27, 924 fixed-rank approximation Eckart-Young theorem Fact 11.16.39, 906 Frobenius norm Fact 11.16.39, 906 Fact 11.17.14, 911 least squares Fact 11.16.39, 906 Fact 11.17.14, 911 Schmidt-Mirsky theorem Fact 11.16.39, 906 singular value Fact 11.16.39, 906 Fact 11.17.14, 911 unitarily invariant norm Fact 11.16.39, 906 Fleck’s identity sextic Fact 2.4.14, 179 floor closest integer Fact 1.11.3, 28 floor function sum Fact 1.12.11, 41 Fact 1.12.12, 41 Fact 1.12.13, 41 forced response definition, 1251 forest definition Defn. 1.4.4, 11 symmetric graph Fact 1.9.8, 26 formal power series definition, 974 Fourier matrix circulant matrix Fact 7.18.13, 614 Vandermonde matrix Fact 7.18.13, 614 Fourier transform Parseval’s theorem Fact 16.24.1, 1313 fraction increasing function Fact 2.6.12, 185 scalar inequality Fact 2.4.1, 177 Fact 2.4.2, 177

1478

Frame

Fact 2.6.12, 185 Frame

inverse matrix Fact 3.20.21, 350 Franck distance to singularity Fact 11.16.5, 900 Franel number binomial equality Fact 1.16.13, 77 Franel’s inequality harmonic number Fact 12.18.34, 963 ´ Frechet characterization of the Euclidean norm Fact 11.7.5, 850 ´ Frechet derivative definition, 1092 free response definition, 1251 frequency domain controllability Gramian Cor. 16.11.5, 1286 frequency response imaginary part Fact 16.24.5, 1314 real part Fact 16.24.5, 1314 transfer function Fact 16.24.5, 1314 Friedland matrix exponential Fact 15.16.12, 1220 Frobenius similar to transpose Cor. 7.4.9, 552 singular value Cor. 11.6.9, 849 symmetric matrix factorization Fact 7.17.26, 608 Frobenius canonical form, see multicompanion form definition, 619 Frobenius inequality rank of partitioned matrix Fact 3.14.20, 324 Fact 8.9.16, 663 Frobenius matrix definition, 619 Frobenius norm absolute value Fact 11.15.13, 897 adjugate Fact 11.9.15, 861 Cauchy-Schwarz inequality Cor. 11.3.9, 840 commutator Fact 11.11.3, 884 Fact 11.11.4, 884 condition number Fact 11.9.14, 861

cosine Fact 11.10.42, 874 definition, 836 determinant Fact 11.9.56, 866 Fact 11.10.41, 874 dissipative matrix Fact 15.16.3, 1218 eigenvalue Fact 11.13.2, 890 Fact 11.13.4, 891 Fact 11.14.4, 893 Fact 11.14.5, 893 eigenvalue bound Fact 11.14.3, 893 eigenvalue perturbation Fact 11.14.13, 894 Fact 11.14.14, 894 Fact 11.14.15, 895 fixed-rank approximation Fact 11.16.39, 906 Fact 11.17.14, 911 Hermitian matrix Fact 11.9.8, 861 inequality Fact 11.10.14, 871 inverse Fact 11.9.13, 861 Kronecker product Fact 11.10.95, 884 matrix difference Fact 11.10.14, 871 matrix exponential Fact 15.15.36, 1217 Fact 15.16.3, 1218 maximum singular value bound Fact 11.15.15, 897 normal matrix Fact 11.9.18, 861 Fact 11.14.13, 894 Fact 11.14.14, 894 Fact 11.15.22, 898 Fact 11.15.23, 898 outer-product matrix Fact 11.8.25, 860 partitioned matrix Fact 11.12.11, 889 polar decomposition Fact 11.10.77, 881 positive-definite matrix Fact 11.10.40, 874 positive-semidefinite matrix Fact 11.9.56, 866 Fact 11.10.31, 872 Fact 11.10.36, 873 Fact 11.10.37, 873 Fact 11.10.44, 874 Fact 11.11.4, 884 rank Fact 11.9.12, 861 Fact 11.13.3, 891 Fact 11.16.39, 906 Fact 11.17.14, 911

Schatten norm, 838 Fact 11.9.20, 862 Schur product Fact 11.16.45, 907 Fact 11.16.47, 908 Schur’s inequality Fact 11.13.2, 890 skew-Hermitian matrix Fact 11.9.8, 861 spectral radius Fact 11.15.14, 897 Fact 11.15.22, 898 Fact 11.15.23, 898 sum of matrices Fact 11.10.7, 869 trace Fact 11.10.41, 874 Fact 11.13.2, 890 Fact 11.13.3, 891 Fact 11.13.4, 891 Fact 11.14.2, 892 trace norm Fact 11.10.35, 873 triangle inequality Fact 11.10.1, 868 unitarily invariant norm Fact 11.16.45, 907 unitary matrix Fact 11.10.77, 881 Fujii-Kubo polynomial root bound Fact 15.21.13, 1236 Fujiwara’s bound polynomial Fact 15.21.12, 1236 full column rank definition, 292 equivalent properties Thm. 3.7.1, 294 nonsingular equivalence Cor. 3.7.7, 297 full rank definition, 292 full row rank definition, 292 equivalent properties Thm. 3.7.1, 294 nonsingular equivalence Cor. 3.7.7, 297 full-rank factorization existence Prop. 7.6.6, 558 generalized inverse Prop. 8.1.12, 625 group-invertible matrix Prop. 4.8.11, 375 idempotent matrix Fact 4.15.19, 398 Fact 7.17.32, 609 outer-product matrix Prop. 7.6.7, 558 full-state feedback controllable subspace Prop. 16.6.5, 1261

controllably asymptotically stable Prop. 16.7.2, 1266 determinant Fact 16.24.16, 1315 invariant zero Prop. 16.10.10, 1284 Fact 16.24.16, 1315 stabilizability Prop. 16.8.3, 1268 uncontrollable eigenvalue Prop. 16.6.14, 1264 unobservable eigenvalue Prop. 16.3.14, 1256 unobservable subspace Prop. 16.3.5, 1254 function definition, 16 graph Fact 1.9.1, 25 Fact 1.9.2, 25 intersection Fact 1.10.3, 26 Fact 1.10.4, 27 union Fact 1.10.3, 26 Fact 1.10.4, 27 function composition matrix multiplication Thm. 3.2.1, 282 fundamental theorem of algebra definition, 499 fundamental triangle inequality Rouch´e Fact 5.2.8, 446 Fact 5.2.11, 466 triangle Fact 5.2.8, 446 Furdui limit Fact 12.18.25, 961 Furuta inequality positive-definite matrix Fact 10.11.79, 746 positive-semidefinite matrix Prop. 10.6.7, 715 spectral ordering Fact 10.23.3, 814

G Gaffke

quadratic form inequality Fact 10.19.19, 797 Galois quintic polynomial Fact 4.31.16, 435 gamma harmonic number Fact 12.18.34, 963 Fact 12.18.35, 964 inequality

generalized inverse Fact 12.18.34, 963 integral Fact 14.9.2, 1170 logarithm Fact 12.18.33, 963 Gamma function exponential function Fact 14.8.6, 1158 inequality Fact 2.2.78, 145 integral Fact 13.3.2, 997 Kummer’s expansion Fact 13.3.2, 997 gamma function multiple integral Fact 14.13.10, 1177 Fact 14.13.11, 1177 Fact 14.13.12, 1177 Fact 14.13.13, 1177 Gantmacher normal matrix Fact 7.16.16, 604 gap minimal principal angle Fact 12.12.28, 941 subspace Fact 12.12.28, 941 Gastinel distance to singularity Fact 11.16.6, 900 ˆ Gateaux differential definition, 1092 Gauss’s lemma polynomial Fact 6.8.2, 517 Gauss-Lucas theorem roots of polynomial Fact 12.16.5, 949 Gaussian density integral Fact 14.12.1, 1172 Fact 14.12.2, 1172 Fact 14.12.3, 1172 Fact 14.12.4, 1172 Fact 14.12.5, 1173 Fact 15.14.12, 1210 positive-definite matrix Fact 14.12.6, 1173 gcd series Fact 13.8.8, 1072 Fact 13.8.20, 1074 GCD identity integers Fact 1.11.5, 28 generalized algebraic multiplicity definition, 564 generalized Cayley-Hamilton theorem commuting matrices Fact 6.9.29, 529

generalized derangement number

definition Fact 1.18.3, 104 permutation Fact 1.18.3, 104 generalized eigensolution eigenvector Fact 16.20.5, 1305 generalized eigenvalue definition, 563 pencil Prop. 7.8.3, 564 Prop. 7.8.4, 565 regular pencil Prop. 7.8.3, 564 Prop. 7.8.4, 565 singular pencil Prop. 7.8.3, 564 generalized eigenvector eigenvalue Fact 7.15.12, 599 generalized Frobenius companion matrix companion matrix Fact 7.18.1, 610 Leslie matrix Fact 7.18.1, 610 generalized Furuta inequality positive-definite matrix inequality Fact 10.11.81, 746 generalized geometric mean positive-definite matrix Fact 10.11.72, 745 generalized geometric multiplicity definition, 564 generalized harmonic function integral Fact 13.3.4, 1001 Fact 14.2.7, 1099 Fact 14.6.10, 1136 Fact 14.6.11, 1137 Fact 14.6.12, 1137 ¨ generalized Holder inequality vector Fact 11.8.28, 860 generalized hypersphere volume Fact 5.5.15, 498 generalized inverse (1)-inverse Fact 8.3.1, 628 Fact 8.3.2, 628 Fact 8.3.3, 628 Fact 8.3.4, 628 Fact 8.3.5, 628 (1,2)-inverse Fact 8.3.10, 629

(1,3)-inverse Fact 8.3.6, 628 Fact 8.3.8, 628 (1,4)-inverse Fact 8.3.7, 628 Fact 8.3.8, 628 adjugate Fact 8.3.21, 630 Fact 8.3.22, 630 Fact 8.9.17, 664 basic properties Prop. 8.1.7, 622 block-circulant matrix Fact 8.9.33, 668 centrohermitian matrix Fact 8.3.35, 632 characteristic polynomial Fact 8.3.28, 631 characterization Fact 8.4.6, 633 commuting matrices Fact 8.4.5, 633 complex conjugate transpose Fact 8.3.26, 631 Fact 8.3.32, 631 Fact 8.4.16, 636 Fact 8.4.17, 636 Fact 8.4.18, 636 Fact 8.6.1, 649 Fact 8.6.2, 650 Fact 8.7.2, 650 Fact 8.7.6, 651 Fact 8.10.16, 671 Fact 8.10.17, 672 Fact 8.10.18, 672 congruence Fact 10.24.5, 816 convergent sequence Fact 8.3.38, 632 cross product Fact 8.9.18, 664 definition, 621 determinant Fact 8.9.35, 668 Fact 8.9.36, 668 Fact 8.9.37, 669 difference Fact 8.8.4, 652 equality Fact 8.3.36, 632 factorization Fact 8.9.34, 668 full-rank factorization Prop. 8.1.12, 625 generalized projector Fact 8.10.22, 673 Fact 8.10.24, 673 Fact 8.10.26, 674 group generalized inverse Fact 8.10.12, 670 group-invertible matrix Fact 8.10.2, 669 Hermitian matrix Fact 8.3.29, 631

1479

Fact 8.3.30, 631 Fact 8.4.12, 635 Fact 8.5.5, 642 Fact 10.24.13, 817 hypergeneralized projector Fact 8.10.23, 673 Fact 8.10.25, 674 Fact 8.10.27, 674 idempotent matrix Fact 7.13.28, 595 Fact 8.3.2, 628 Fact 8.4.27, 637 Fact 8.7.1, 650 Fact 8.7.2, 650 Fact 8.7.3, 651 Fact 8.7.4, 651 Fact 8.7.5, 651 Fact 8.7.6, 651 Fact 8.7.7, 651 Fact 8.7.8, 651 Fact 8.7.9, 651 Fact 8.8.11, 655 Fact 8.8.12, 655 Fact 8.8.13, 656 Fact 8.8.14, 656 Fact 8.8.15, 656 Fact 8.10.19, 672 Fact 8.10.20, 673 Fact 8.10.21, 673 Fact 8.10.26, 674 induced lower bound Fact 11.9.61, 868 inertia Fact 8.3.29, 631 Fact 8.9.8, 661 Fact 10.24.13, 817 integral Fact 15.14.6, 1210 inverse image Prop. 8.1.10, 624 Prop. 8.1.11, 625 Jordan form Fact 8.10.13, 670 Kronecker product Fact 9.4.36, 688 least squares Fact 11.17.6, 910 Fact 11.17.7, 910 Fact 11.17.9, 910 Fact 11.17.10, 910 Fact 11.17.11, 911 Fact 11.17.12, 911 left inverse Cor. 8.1.4, 622 Fact 8.3.13, 629 Fact 8.3.14, 629 Fact 8.4.41, 639 Fact 8.4.42, 640 Fact 11.17.4, 909 left-inner matrix Fact 8.3.11, 629 left-invertible matrix Prop. 8.1.5, 622 Prop. 8.1.12, 625

1480

¨ generalized Lowner partial ordering

Fact 8.4.20, 636 limit Fact 8.3.39, 632 linear equation Prop. 8.1.8, 623 Prop. 8.1.9, 624 Prop. 8.1.11, 625 linear matrix equation Fact 8.4.40, 639 lower bound Fact 11.9.61, 868 matrix difference Fact 8.4.39, 639 matrix exponential Fact 15.14.6, 1210 matrix inversion lemma Fact 8.4.13, 635 matrix limit Fact 8.3.27, 631 matrix product Prop. 8.1.12, 625 Fact 8.4.14, 636 Fact 8.4.15, 636 Fact 8.4.19, 636 Fact 8.4.20, 636 Fact 8.4.21, 636 Fact 8.4.22, 636 Fact 8.4.23, 636 Fact 8.4.24, 637 Fact 8.4.26, 637 Fact 8.4.27, 637 Fact 8.4.29, 638 Fact 8.4.30, 638 Fact 8.8.3, 652 Fact 8.8.5, 653 Fact 8.8.23, 658 matrix sum Fact 8.4.33, 638 Fact 8.4.34, 638 Fact 8.4.35, 638 Fact 8.4.36, 638 Fact 8.4.37, 639 Fact 8.4.38, 639 maximum singular value Fact 11.16.7, 900 Fact 11.16.41, 907 Newton-Raphson algorithm Fact 8.3.38, 632 normal matrix Prop. 8.1.7, 622 Fact 8.5.7, 642 Fact 8.6.1, 649 Fact 8.6.2, 650 null space Prop. 8.1.7, 622 Fact 8.9.2, 659 Fact 8.9.3, 659 observability matrix Fact 16.21.21, 1309 outer-product matrix Fact 8.3.17, 629 outer-product perturbation Fact 8.4.10, 634

Fact 8.4.11, 635 partial isometry Fact 8.3.31, 631 Fact 8.6.3, 650 range Prop. 8.1.7, 622 Fact 8.4.1, 632 Fact 8.4.2, 632 Fact 8.4.3, 633 Fact 8.4.8, 634 Fact 8.4.9, 634 Fact 8.9.27, 667 range-disjoint matrix Fact 8.5.14, 644 range-Hermitian matrix Prop. 8.1.7, 622 Fact 8.5.1, 641 Fact 8.5.2, 641 Fact 8.5.3, 641 Fact 8.5.4, 641 Fact 8.5.6, 642 Fact 8.5.8, 643 Fact 8.5.9, 643 Fact 8.5.10, 643 Fact 8.5.11, 643 Fact 8.5.12, 643 Fact 8.6.1, 649 Fact 8.6.2, 650 range-spanning matrix Fact 8.5.14, 644 rank Fact 8.3.24, 630 Fact 8.3.39, 632 Fact 8.4.2, 632 Fact 8.4.7, 634 Fact 8.4.10, 634 Fact 8.4.16, 636 Fact 8.4.25, 637 Fact 8.4.32, 638 Fact 8.5.2, 641 Fact 8.7.2, 650 Fact 8.9.5, 659 Fact 8.9.7, 660 Fact 8.9.11, 662 Fact 8.9.14, 663 Fact 8.9.15, 663 rank subtractivity partial ordering Fact 8.9.39, 669 right inverse Cor. 8.1.4, 622 Fact 8.3.12, 629 Fact 8.3.15, 629 Fact 11.17.5, 909 right-inner matrix Fact 8.3.11, 629 right-invertible matrix Prop. 8.1.5, 622 Prop. 8.1.12, 625 Fact 8.4.21, 636 sequence Fact 8.3.39, 632 singular value Fact 8.3.33, 631

singular value decomposition Fact 8.3.23, 630 square root Fact 10.24.4, 815, 816 star partial ordering Fact 8.4.43, 640 Fact 8.4.44, 640 star-dagger matrix Fact 8.3.32, 631 sum Fact 8.9.32, 668 Fact 8.9.33, 668 symmetric matrix Fact 8.3.3, 628 trace Fact 8.7.2, 650 tripotent matrix Fact 8.5.4, 641 Fact 8.5.5, 642 uniqueness Thm. 8.1.1, 621 unitarily invariant norm Fact 11.10.74, 881 Fact 11.17.4, 909 Fact 11.17.5, 909 unitary matrix Fact 8.3.37, 632 zero product Fact 8.4.4, 633 ¨ generalized Lowner partial ordering definition Fact 10.23.8, 815 generalized multispectrum definition, 563 generalized parallelogram law Euclidean norm Fact 11.8.17, 858 Fact 11.10.61, 877 positive-semidefinite matrix Fact 10.11.84, 747 vector equality Fact 11.8.3, 853 generalized pentagonal number generating function Fact 13.4.7, 1010 properties Fact 1.12.5, 39 generalized pentagonal numbers Gauss identity Fact 13.10.27, 1091 infinite product Fact 13.10.27, 1091 generalized projector definition Defn. 4.1.1, 363 generalized inverse Fact 8.10.22, 673 Fact 8.10.24, 673 Fact 8.10.26, 674

generalized Schur inequality

eigenvalues Fact 11.13.5, 891 generalized spectrum definition, 563 generating function bi-power series Defn. 12.3.5, 920 dual pentagonal number Fact 13.4.7, 1010 Fibonacci number Fact 13.9.3, 1077 generalized pentagonal number Fact 13.4.7, 1010 harmonic number Fact 13.4.5, 1009 Lucas number Fact 13.9.3, 1077 minimum integer Fact 13.4.17, 1020 partition number Fact 13.10.25, 1090 Pell number Fact 13.9.3, 1077 pentagonal number Fact 13.4.7, 1010 power series Defn. 12.3.5, 919 series Fact 13.4.9, 1012 triangular number Fact 13.4.6, 1010 Genocchi number definite integral Fact 14.6.45, 1149 Genocchi numbers series Fact 13.1.7, 980 geometric mean arithmetic mean Fact 2.11.97, 206 Fact 2.11.99, 206 Fact 2.11.101, 207 Fact 2.11.102, 207 Fact 2.11.103, 207 Fact 2.11.112, 208 Fact 2.11.113, 208 determinant Fact 10.11.68, 743 eigenvalue Fact 10.25.34, 824 exponential Fact 2.11.103, 207 inequality Fact 2.3.49, 161 limit Fact 12.17.10, 955 Fact 12.17.11, 956 matrix exponential Fact 10.11.70, 744 Fact 10.11.71, 744 matrix logarithm Fact 10.11.69, 744

group Fact 15.15.43, 1217 nondecreasing function Fact 10.11.68, 743 Fact 10.11.71, 744 positive-definite matrix Fact 10.11.68, 743 Fact 10.11.73, 745 Fact 10.11.74, 745 Fact 10.25.71, 830 positive-semidefinite matrix Fact 10.11.68, 743 power inequality Fact 2.11.116, 209 Fact 2.11.117, 209 Riccati equation Fact 16.25.6, 1317 scalar inequality Fact 2.3.50, 161 Fact 2.3.51, 161 Schur product Fact 10.25.71, 830 trace Fact 10.14.41, 770 geometric multiplicity algebraic multiplicity Prop. 7.7.3, 558 block-diagonal matrix Prop. 7.7.14, 560 cascaded systems Fact 16.24.17, 1316 controllability Fact 16.21.16, 1309 defect Prop. 6.5.2, 511 Fact 7.15.1, 597 definition Defn. 6.5.1, 511 Drazin inverse Fact 8.10.7, 670 matrix power Fact 7.15.22, 600 partitioned matrix Prop. 7.7.14, 560 rank Prop. 6.5.2, 511 similar matrices Prop. 7.7.10, 560 zero eigenvalue Fact 7.15.1, 597 Fact 7.15.2, 597 geometric series complex numbers Fact 13.4.2, 1004 Fact 13.5.1, 1021 geometric-mean decomposition unitary matrix Fact 7.10.31, 575 Gergonne point triangle Fact 5.2.11, 466 Germain identity equality Fact 2.2.1, 129

greatest lower bound definition, 925 collection of sets derivative eigenvalue bounds Prop. 1.3.15, 8 Fact 12.16.27, 953 Fact 6.10.22, 535 Prop. 1.3.16, 8 Gram matrix Fact 6.10.26, 535 definition positive-semidefinite Gerstenhaber Defn. 1.3.11, 7 algebra generated by two matrix preuniqueness Fact 10.10.41, 734 commuting matrices Lem. 1.3.10, 7 Fact 7.11.26, 577 Gram-Schmidt projector Gibson orthonormalization Fact 8.8.19, 657 diagonal entries of similar upper triangular matrix Fact 8.8.21, 658 matrices factorization union of sets Fact 7.10.14, 571 Fact 7.17.10, 606 Lem. 1.3.13, 7 Giuga’s conjecture Gramian Gregory’s series prime number controllability series Fact 1.11.17, 32 Fact 16.21.19, 1309 Fact 13.4.14, 1017 Givens rotation stabilization Greville orthogonal matrix Fact 16.21.19, 1309 generalized inverse Fact 7.17.18, 607 Graph Fact 8.4.23, 636 global asymptotic stability definition, 16 Fact 8.4.26, 637 nonlinear system graph Fact 8.7.1, 650 Thm. 15.8.5, 1196 definition, 9 Fact 8.9.19, 665 globally asymptotically directed cut projector Fact 6.11.4, 538 stable equilibrium Fact 8.8.13, 656 directed cycle definition group Fact 1.9.2, 25 Defn. 15.8.3, 1196 alternating group Fact 1.9.4, 25 globally Lipschitz Fact 4.31.16, 435 directed forest definition cyclic group Fact 1.9.7, 25 Defn. 12.4.28, 924 Fact 4.31.5, 433 directed Hamiltonian cycle Gohberg-Semencul Fact 4.31.16, 435 Fact 1.9.3, 25 formulas definition directionally acyclic graph Bezout matrix Defn. 4.4.1, 369 Fact 1.9.4, 25 Fact 6.8.8, 519 Defn. 4.5.1, 371 function Goldbach dihedral group Fact 1.9.1, 25 sum Fact 4.31.16, 435 Fact 1.9.2, 25 Fact 13.8.21, 1074 equivalence relation Hamiltonian cycle Goldbach conjecture Prop. 4.7.1, 373 Fact 1.9.9, 26 prime Prop. 4.7.2, 373 Hasse diagram Fact 1.11.46, 35 finite group Fact 1.9.4, 25 golden mean Fact 4.31.15, 434 positive-definite solution of irreducible matrix Fact 4.31.16, 435 Fact 6.11.4, 538 a Riccati equation Fact 4.31.17, 436 Laplacian matrix Fact 16.25.6, 1317 icosahedral group Fact 10.18.1, 792 Riccati equation Fact 4.31.16, 435 leaf Fact 16.25.6, 1317 isomorphism Fact 1.9.3, 25 golden ratio Prop. 4.6.5, 372 root double integral Lagrange’s theorem Fact 1.9.3, 25 Fact 14.13.6, 1176 Thm. 4.4.4, 370 Fact 1.9.4, 25 Fibonacci number Lie group spanning path Fact 13.9.2, 1075 Defn. 15.6.1, 1191 Fact 1.9.9, 26 Lucas number Prop. 15.6.2, 1191 tournament Fact 13.9.2, 1075 matrix exponential Fact 1.9.9, 26 Riccati equation Prop. 15.6.7, 1192 tree Fact 16.25.6, 1317 octahedral group Fact 1.9.5, 25 Golden-Thompson Fact 4.31.16, 435 walk inequality order of element Fact 6.11.1, 537 matrix exponential Fact 4.31.5, 433 graph of a matrix Fact 15.15.30, 1216 orthogonal matrix adjacency matrix Fact 15.15.31, 1216 Fact 4.31.13, 434 Prop. 4.2.5, 368 Fact 15.17.4, 1221 pathwise connected greatest common divisor normal matrix Prop. 15.6.8, 1193 definition, 501 Fact 15.15.31, 1216 permutation group least common multiple Gordan’s theorem Fact 4.31.16, 435 Fact 1.11.6, 28 positive vector real numbers series of rational functions Fact 6.11.20, 543 Fact 4.31.4, 433 Fact 13.5.3, 1021 gradient Gershgorin circle theorem

1481

1482

group generalized inverse

Fact 4.31.6, 433 Fact 15.20.4, 1232 range subgroup normal matrix Prop. 8.2.3, 627 Thm. 4.4.4, 370 Fact 8.10.17, 672 rank Fact 4.31.16, 435 outer-product matrix Fact 8.10.4, 669 symmetric group Fact 7.15.5, 598 sequence Fact 4.31.16, 435 positive-definite matrix Fact 8.10.4, 669 symmetry groups Fact 10.11.23, 737 singular value Fact 4.31.16, 435 positive-semidefinite decomposition tetrahedral group matrix Fact 8.5.13, 643 Fact 4.31.16, 435 Fact 10.11.23, 737 sum transpose projector Fact 8.11.12, 676 Fact 4.31.7, 433 Fact 4.18.15, 413 trace unipotent matrix projector onto Fact 8.10.5, 670 Fact 4.31.11, 434 Fact 8.10.2, 669 tripotent matrix Fact 15.23.1, 1243 range Fact 8.11.12, 676 unit sphere Fact 7.15.6, 598 group-invertible matrix Fact 4.31.8, 433 almost nonnegative matrix range-Hermitian matrix upper triangular matrix Prop. 4.1.7, 367 Fact 15.20.4, 1232 Fact 4.31.11, 434 Fact 8.10.16, 671 complementary subspaces Fact 15.23.1, 1243 rank Cor. 4.8.10, 375 group generalized inverse Fact 7.9.5, 566 Fact 4.9.2, 376 definition, 627 Fact 7.15.6, 598 core inverse difference semistable matrix Fact 8.5.15, 647 Fact 8.11.12, 676 Fact 15.19.4, 1226 definition discrete-time similar matrices Defn. 4.1.1, 363 Prop. 4.7.5, 374 Lyapunov-stable matrix equivalent Fact 7.10.6, 570 Fact 15.22.14, 1241 characterizations spectrum Drazin generalized inverse Fact 4.9.2, 376 Prop. 7.7.21, 562 Fact 8.10.9, 670 full-rank factorization square root factorization Prop. 4.8.11, 375 Fact 7.17.22, 608 Fact 8.11.1, 674 generalized inverse stable subspace generalized inverse Fact 8.10.2, 669 Prop. 15.9.8, 1200 Fact 8.10.12, 670 Hermitian matrix tripotent matrix idempotent matrix Fact 8.10.18, 672 Prop. 4.1.7, 367 Prop. 8.2.3, 627 idempotent matrix unitarily similar matrices Fact 8.11.12, 676 Prop. 4.1.7, 367 Prop. 4.7.5, 374 integral Prop. 4.8.11, 375 Fact 15.14.9, 1210 Prop. 8.2.3, 627 H Fact 15.14.10, 1210 Fact 4.16.20, 407 irreducible matrix Fact 7.12.11, 580 H2 norm Fact 8.11.4, 674 Fact 7.15.29, 601 controllability Gramian Kronecker product Fact 8.10.2, 669 Cor. 16.11.4, 1286 Fact 9.4.37, 688 Fact 8.10.3, 669 Cor. 16.11.5, 1286 limit Fact 8.11.14, 677 definition Fact 8.10.4, 669 idempotent matrix onto Defn. 16.11.2, 1285 Fact 8.10.15, 671 Prop. 4.8.11, 375 observability Gramian matrix exponential index of a matrix Cor. 16.11.4, 1286 Fact 15.14.9, 1210 Prop. 4.8.8, 375 Parseval’s theorem Fact 15.14.10, 1210 Cor. 7.7.9, 559 Thm. 16.11.3, 1286 Fact 15.19.6, 1226 Fact 7.15.6, 598 partitioned transfer Fact 15.19.7, 1226 inertia function matrix product Fact 7.9.5, 566 Fact 16.24.18, 1316 Fact 8.11.13, 677 Jordan form Fact 16.24.19, 1316 null space Fact 8.10.13, 670 Fact 16.24.20, 1316 Prop. 8.2.3, 627 Kronecker product quadratic performance partitioned matrix Fact 9.4.22, 687 measure Fact 8.12.2, 678 Fact 9.4.37, 688 Prop. 16.15.1, 1294 Fact 8.12.3, 678 Lyapunov-stable matrix submultiplicative norm positive-semidefinite Fact 15.19.3, 1226 Fact 16.24.23, 1316 matrix exponential matrix sum of transfer functions Fact 15.19.14, 1227 Fact 10.24.1, 815 Prop. 16.11.6, 1287 matrix power product transfer function Fact 4.9.3, 376 Fact 8.11.2, 674 Fact 16.24.18, 1316 Fact 8.11.3, 674 projector Fact 16.24.19, 1316 N-matrix Fact 8.11.13, 677 Fact 16.24.20, 1316

Fact 16.24.21, 1316 Fact 16.24.22, 1316 Hadamard matrix orthogonal matrix Fact 7.18.16, 616 Hadamard product, see Schur product Hadamard sum Schur product Fact 10.25.51, 827 Hadamard’s inequality determinant Fact 10.15.7, 775 Fact 10.15.10, 775 Fact 10.21.15, 803 determinant bound Fact 11.13.1, 890 determinant of a partitioned matrix Fact 8.9.35, 668 Hadamard-Fischer inequality positive-semidefinite matrix Fact 10.16.39, 782 Hadwiger-Finsler inequality triangle Fact 5.2.8, 446 Hahn-Banach theorem inner product inequality Fact 12.12.21, 939 half space polytope Fact 3.11.19, 310 half-vectorization operator Kronecker product, 701 Halmos Reid’s inequality Fact 10.19.4, 795 Hamiltonian block decomposition Prop. 16.17.5, 1299 closed-loop spectrum Prop. 16.16.14, 1298 definition, 1297 Hamiltonian matrix Prop. 16.16.13, 1297 Jordan form Fact 16.25.1, 1316 Riccati equation Thm. 16.17.9, 1300 Prop. 16.16.14, 1298 Cor. 16.16.15, 1298 spectral factorization Prop. 16.16.13, 1297 spectrum Thm. 16.17.9, 1300 Prop. 16.16.13, 1297 Prop. 16.17.5, 1299 Prop. 16.17.7, 1300 Prop. 16.17.8, 1300 Lem. 16.17.4, 1299 Lem. 16.17.6, 1300 stabilizability

Hermite Fact 16.25.1, 1316 stabilizing solution Cor. 16.16.15, 1298 uncontrollable eigenvalue Prop. 16.17.7, 1300 Prop. 16.17.8, 1300 Lem. 16.17.4, 1299 Lem. 16.17.6, 1300 unobservable eigenvalue Prop. 16.17.7, 1300 Prop. 16.17.8, 1300 Lem. 16.17.4, 1299 Lem. 16.17.6, 1300 Hamiltonian cycle definition Defn. 1.4.4, 11 graph Fact 1.9.9, 26 tournament Fact 1.9.9, 26 Hamiltonian graph definition Defn. 1.4.4, 11 Hamiltonian matrix Cayley transform Fact 4.28.12, 428 characteristic polynomial Fact 6.9.20, 528 Fact 6.9.33, 530 definition Defn. 4.1.6, 367 equality Fact 4.28.1, 427 Hamiltonian Prop. 16.16.13, 1297 inverse matrix Fact 4.28.5, 428 matrix exponential Prop. 15.6.7, 1192 matrix logarithm Fact 15.15.20, 1214 matrix sum Fact 4.28.5, 428 orthogonal matrix Fact 4.28.13, 428 orthosymplectic matrix Fact 4.28.13, 428 partitioned matrix Prop. 4.1.8, 367 Fact 4.28.6, 428 Fact 4.28.8, 428 Fact 6.9.32, 529 Fact 7.13.31, 596 skew reflector Fact 4.28.3, 427 skew-involutory matrix Fact 4.28.2, 427 Fact 4.28.3, 427 skew-symmetric matrix Fact 4.10.37, 382 Fact 4.28.3, 427 Fact 4.28.8, 428 spectrum Prop. 7.7.21, 562 symplectic matrix

Fact 4.28.2, 427 Fact 4.28.12, 428 Fact 4.28.13, 428 symplectic similarity Fact 4.28.4, 428 trace Fact 4.28.7, 428 unit imaginary matrix Fact 4.28.3, 427 Hamiltonian path definition Defn. 1.4.4, 11 Hankel matrix block-Hankel matrix Fact 4.23.3, 420 Catalan number Fact 13.2.15, 994 definition Defn. 4.1.3, 365 determinant Fact 13.2.15, 994 factorization Fact 7.17.45, 610 Hankel transform Fact 13.2.15, 994 Hilbert matrix Fact 4.23.4, 420 Markov block-Hankel matrix definition, 1274 rank Fact 4.23.8, 421 rational function Fact 6.8.10, 520 symmetric matrix Fact 4.23.2, 420 Toeplitz matrix Fact 4.23.1, 420 trigonometric matrix Fact 4.23.8, 421 Hankel transform Hankel matrix Fact 13.2.15, 994 Hanner’s inequality H¨older norm Fact 11.8.23, 859 Schatten norm Fact 11.10.65, 879 Hansen trace of a convex function Fact 10.14.48, 771 Hardy H¨older-induced norm Fact 11.9.35, 864 Hardy and Littlewood’s inequality bounded derivative Fact 12.16.13, 951 Hardy discrete inequality sum of powers Fact 2.11.132, 212 Hardy inequality integral Fact 12.13.17, 944 Hardy’s theorem

zeta function Fact 13.3.1, 994 Hardy-Hilbert inequality sum of powers Fact 2.12.25, 219 Fact 2.12.26, 219 Fact 2.12.27, 220 Hardy-Littlewood rearrangement inequality sum of products Fact 2.12.8, 216 Hardy-Littlewood-Polya theorem doubly stochastic matrix Fact 4.11.6, 384 harmonic function integral Fact 13.3.4, 1001 harmonic mean arithmetic-mean inequality Fact 2.11.43, 197 inequality Fact 2.2.15, 133 Fact 2.11.38, 196 Milne’s inequality Fact 2.12.31, 220 harmonic number asymptotic approximation Fact 12.18.36, 964 factorial Fact 1.13.10, 50 Franel’s inequality Fact 12.18.34, 963 gamma Fact 12.18.34, 963 generating function Fact 13.4.5, 1009 harmonic polynomial Fact 13.2.11, 992 inequality Fact 1.11.47, 36 Fact 1.12.46, 48 Fact 1.12.47, 48 Fact 1.12.48, 48 Fact 1.12.49, 48 integer Fact 1.11.38, 35 logarithm Fact 12.18.34, 963 Riemann hypothesis Fact 1.11.47, 36 series Fact 13.5.16, 1023 Fact 13.5.25, 1027 Fact 13.8.12, 1073 Fact 13.8.13, 1073 Fact 13.8.14, 1073 Fact 13.8.15, 1073 sum Fact 1.12.45, 47 sum of reciprocals Fact 12.18.37, 965 zeta function

1483

Fact 13.5.40, 1031 harmonic numbers

integral Fact 14.4.9, 1117 series Fact 13.5.18, 1025 harmonic polynomial harmonic number Fact 13.2.11, 992 harmonic steady-state response linear system Thm. 16.12.1, 1288 harmonic-mean– geometric-mean inequality power mean Fact 2.11.86, 204 Fact 2.11.89, 205 Hartfiel determinant of a sum of matrices Fact 10.16.9, 777 Hartwig rank of an idempotent matrix Fact 4.16.8, 401 Hasse diagram graph Fact 1.9.4, 25 Hausdorff distance set Fact 12.12.27, 940 Haynsworth determinant of a sum of matrices Fact 10.16.9, 777 positive-semidefinite matrix Fact 7.15.15, 599 Schur complement of a partitioned matrix Fact 8.9.38, 669 Haynsworth inertia additivity formula Schur complement Fact 8.9.9, 662 Heinz inequality unitarily invariant norm Fact 11.10.82, 882 Heinz mean scalar inequality Fact 2.2.64, 143 Heisenberg group unipotent matrix Fact 4.31.11, 434 Fact 15.23.1, 1243 upper triangular matrix Fact 4.31.11, 434 Fact 15.23.1, 1243 Henrici eigenvalue bound Fact 11.13.2, 890 Hermite

1484

Hermite polynomial

cotangent equality Fact 2.16.20, 245 Hermite polynomial trigonometric functions Fact 13.2.10, 990 Hermite-Biehler theorem asymptotically stable polynomial Fact 15.18.8, 1224 Hermite-Hadamard inequality convex function Fact 1.21.9, 117 integral Fact 12.13.18, 944 Hermitian matrices star partial ordering Fact 10.23.6, 814 Hermitian matrix, see symmetric matrix additive decomposition Fact 7.20.3, 618 adjugate Fact 4.10.9, 378 affine mapping Fact 4.10.14, 380 arithmetic mean Fact 10.11.54, 741 block-diagonal matrix Fact 4.10.7, 378 Cartesian decomposition Fact 4.10.28, 381 Fact 7.20.2, 618 Fact 7.20.3, 618 cogredient transformation Fact 10.20.6, 799 Fact 10.20.8, 799 commutator Fact 4.29.6, 429 Fact 4.29.8, 429 Fact 4.29.9, 429 Fact 7.16.13, 604 Fact 11.11.7, 885 commuting matrices Fact 7.16.16, 604 Fact 10.20.2, 799 complex conjugate transpose Fact 4.10.13, 379 Fact 7.10.9, 570 Fact 8.10.18, 672 congruent matrices Thm. 7.5.7, 554 Prop. 4.7.5, 374 convergent sequence Fact 15.15.8, 1213 Fact 15.15.9, 1213 convex function Fact 10.14.47, 771 Fact 10.14.48, 771 Fact 15.15.26, 1215 convex hull Fact 10.21.11, 802 defect Fact 7.9.7, 566

Fact 10.10.9, 731 definition Defn. 4.1.1, 363 determinant Cor. 10.4.10, 711 Fact 4.10.21, 381 Fact 10.16.5, 777 diagonal Fact 10.21.11, 802 diagonal entry Cor. 10.4.7, 710 Fact 10.21.11, 802 Fact 10.21.13, 803 Fact 10.21.19, 804 diagonal matrix Cor. 7.5.5, 554 Lem. 10.5.1, 713 eigenvalue Thm. 10.4.5, 710 Thm. 10.4.9, 710 Thm. 10.4.11, 711 Cor. 10.4.2, 709 Cor. 10.4.6, 710 Cor. 10.4.7, 710 Cor. 10.6.19, 722 Lem. 10.4.3, 709 Lem. 10.4.4, 709 Fact 7.12.6, 579 Fact 7.12.7, 579 Fact 10.11.13, 736 Fact 10.18.22, 794 Fact 10.19.6, 796 Fact 10.21.11, 802 Fact 10.21.13, 803 Fact 10.21.20, 804 Fact 10.21.21, 804 Fact 10.22.4, 805 Fact 10.22.5, 805 Fact 10.22.7, 805 Fact 10.22.23, 809 Fact 10.22.24, 809 Fact 10.25.31, 824 eigenvalue bound Fact 11.14.3, 893 eigenvalue inequality Lem. 10.4.1, 709 Fact 10.22.2, 805 Fact 10.22.3, 805 eigenvalue interlacing Fact 10.22.1, 804 eigenvalue majorization Fact 10.22.8, 806 Fact 10.22.9, 806 eigenvalue perturbation Fact 6.10.34, 537 eigenvalues Fact 10.21.19, 804 eigenvector Fact 11.13.12, 892 existence of transformation Fact 4.11.2, 383 factorization Fact 7.17.19, 607 Fact 7.17.27, 608

Fact 7.17.28, 608 Fact 7.17.43, 610 Fact 10.20.2, 799 Frobenius norm Fact 11.9.8, 861 generalized inverse Fact 8.3.29, 631 Fact 8.3.30, 631 Fact 8.4.12, 635 Fact 8.5.5, 642 Fact 10.24.13, 817 group-invertible matrix Fact 8.10.18, 672 inequality Fact 10.10.16, 732 Fact 10.10.18, 732 Fact 10.10.23, 732 Fact 10.15.5, 774 Fact 10.16.29, 780 inertia Thm. 10.4.11, 711 Prop. 7.5.6, 554 Fact 7.9.6, 566 Fact 7.9.8, 566 Fact 7.9.13, 567 Fact 7.9.14, 567 Fact 7.9.15, 567 Fact 7.9.16, 567 Fact 7.9.17, 568 Fact 7.9.18, 568 Fact 7.9.19, 568 Fact 7.9.27, 569 Fact 7.13.7, 591 Fact 8.3.29, 631 Fact 10.11.26, 737 Fact 10.24.13, 817 Fact 10.24.15, 817 Fact 16.22.1, 1309 Fact 16.22.2, 1310 Fact 16.22.3, 1310 Fact 16.22.4, 1310 Fact 16.22.5, 1310 Fact 16.22.6, 1310 Fact 16.22.7, 1311 Fact 16.22.8, 1311 Fact 16.22.10, 1311 Fact 16.22.11, 1311 Fact 16.22.12, 1311 inner matrix Fact 10.21.20, 804 Fact 10.21.21, 804 Kronecker product Fact 9.4.22, 687 Fact 10.25.31, 824 Kronecker sum Fact 9.5.9, 692 least squares Fact 7.13.20, 593 Fact 11.17.15, 911 left inverse Fact 8.3.13, 629 Lidskii-Mirsky-Wielandt theorem Fact 11.14.4, 893 limit

Fact 10.11.3, 735 linear combination Fact 10.19.14, 796 Fact 10.19.15, 797 Fact 10.19.20, 798 linear combination of projectors Fact 7.20.15, 619 matrix exponential Prop. 15.2.8, 1184 Prop. 15.4.3, 1187 Prop. 15.4.10, 1189 Cor. 15.2.6, 1183 Fact 15.15.8, 1213 Fact 15.15.9, 1213 Fact 15.15.22, 1214 Fact 15.15.30, 1216 Fact 15.15.32, 1216 Fact 15.15.35, 1217 Fact 15.15.36, 1217 Fact 15.15.38, 1217 Fact 15.16.1, 1217 Fact 15.17.4, 1221 Fact 15.17.7, 1222 Fact 15.17.15, 1223 Fact 15.17.19, 1223 maximum eigenvalue Lem. 10.4.3, 709 Fact 7.12.9, 580 Fact 10.11.12, 736 maximum singular value Fact 7.12.9, 580 minimum eigenvalue Lem. 10.4.3, 709 Fact 10.11.12, 736 normal matrix Prop. 4.1.7, 367 outer-product matrix Fact 4.10.18, 380 Fact 4.11.2, 383 outer-product perturbation Fact 10.22.1, 804 partitioned matrix Fact 4.10.27, 381 Fact 6.10.34, 537 Fact 7.9.27, 569 Fact 7.13.7, 591 Fact 8.9.9, 662 Fact 10.12.61, 759 Fact 10.16.6, 777 positive-definite matrix Fact 7.17.43, 610 Fact 10.11.24, 737 positive-semidefinite matrix Fact 7.17.43, 610 Fact 10.10.1, 730 Fact 10.10.13, 732 Fact 10.10.14, 732 Fact 10.11.24, 737 principal determinant Fact 4.10.10, 378 product Example 7.7.19, 562

¨ Holder norm projector Fact 4.17.1, 407 Fact 7.17.19, 607 Fact 10.10.27, 733 properties of < and ≤ Prop. 10.1.2, 703 quadratic form Fact 4.10.5, 378 Fact 4.10.6, 378 Fact 7.12.6, 579 Fact 7.12.7, 579 Fact 7.12.8, 580 Fact 10.18.22, 794 Fact 10.19.14, 796 Fact 10.19.15, 797 Fact 10.19.20, 798 quadratic matrix equation Fact 7.12.4, 579 range Lem. 10.6.1, 714 Fact 4.10.31, 381 rank Fact 4.10.22, 381 Fact 4.10.32, 382 Fact 7.9.6, 566 Fact 7.9.7, 566 Fact 10.7.10, 723 Fact 10.10.9, 731 Rayleigh quotient Lem. 10.4.3, 709 reverse-Hermitian matrix Fact 4.10.38, 382 right inverse Fact 8.3.12, 629 Schatten norm Fact 11.10.68, 880 Fact 11.11.4, 884 Schur complement Fact 10.12.8, 750 Fact 10.12.61, 759 Schur decomposition Cor. 7.5.5, 554 Schur product Prop. 9.3.1, 685 Fact 10.25.27, 823 Fact 10.25.31, 824 Fact 10.25.52, 827 signature Fact 7.9.6, 566 Fact 7.9.7, 566 Fact 10.11.28, 738 similar matrices Prop. 7.7.13, 560 Fact 7.11.17, 577 simultaneous diagonalization Fact 10.20.2, 799 Fact 10.20.6, 799 Fact 10.20.8, 799 skew-Hermitian matrix Fact 4.10.8, 378 Fact 7.20.2, 618 skew-symmetric matrix Fact 4.10.8, 378 spectral abscissa

Fact 7.12.9, 580 spectral radius Fact 7.12.9, 580 spectral variation Fact 11.14.8, 893 Fact 11.14.10, 894 spectrum Prop. 7.7.21, 562 Lem. 10.4.8, 710 Fact 6.10.3, 531 Fact 6.10.4, 531 spread Fact 10.18.22, 794 strong majorization Fact 10.21.11, 802 Fact 10.22.3, 805 submatrix Cor. 10.4.6, 710 Lem. 10.4.4, 709 Fact 7.9.8, 566 symmetric matrix Fact 4.10.8, 378 trace Prop. 10.4.13, 712 Cor. 10.4.10, 711 Lem. 10.4.12, 711 Fact 4.10.13, 379 Fact 4.10.22, 381 Fact 4.10.24, 381 Fact 7.13.17, 593 Fact 10.13.7, 762 Fact 10.14.1, 763 Fact 10.14.20, 766 Fact 10.14.57, 773 Fact 10.14.58, 773 Fact 15.15.26, 1215 trace of a product Fact 7.13.11, 592 Fact 10.14.3, 764 Fact 10.14.16, 766 trace of product Fact 7.13.10, 592 Fact 7.13.12, 592 Fact 7.13.13, 592 Fact 10.22.25, 809 tripotent matrix Fact 4.21.3, 417 Fact 4.21.4, 418 Fact 4.21.7, 418 Fact 8.5.5, 642 unitarily invariant norm Fact 11.10.23, 872 Fact 11.10.30, 872 Fact 11.10.56, 876 Fact 11.10.72, 880 Fact 11.10.73, 880 Fact 11.10.76, 881 Fact 11.14.4, 893 Fact 15.17.15, 1223 unitarily similar matrices Prop. 4.7.5, 374 Prop. 7.7.23, 562 Cor. 7.5.5, 554 unitary matrix Fact 4.13.24, 389

Fact 10.20.2, 799 Fact 15.15.38, 1217 Hermitian matrix eigenvalue Cauchy interlacing theorem Lem. 10.4.4, 709 eigenvalue interlacing theorem Thm. 10.4.5, 710 monotonicity theorem Thm. 10.4.9, 710 Fact 10.11.13, 736 Weyl’s inequality Thm. 10.4.9, 710 Fact 10.11.13, 736 Hermitian matrix inertia equality Styan Fact 10.11.26, 737 Hermitian part eigenvalue Fact 7.12.27, 583 Heron mean logarithmic mean Fact 2.2.67, 144 Heron’s formula Cayley-Menger determinant Fact 5.4.7, 495 triangle Fact 5.2.7, 443 Hessenberg matrix lower or upper Defn. 4.1.3, 365 Hessian definition, 926 hexagon area Fact 5.3.8, 492 diagonal Fact 5.3.6, 492 hidden convexity quadratic form Fact 10.17.12, 789 higher order derivatives analytic function Prop. 12.6.7, 928 Hilbert matrix Hankel matrix Fact 4.23.4, 420 positive-definite matrix Fact 4.23.4, 420 Hille-Yosida theorem matrix exponential bound Fact 15.16.8, 1219 Hiroshima’s theorem Schatten norm of a partitioned matrix Fact 11.12.6, 887 Hirsch’s theorem eigenvalue bound Fact 7.12.24, 582 Fact 11.13.7, 891 Hirschhorn’s 3-7-5 identity

1485

equality Fact 2.2.10, 132 Fact 2.4.15, 179 Hlawka’s equality norm equality Fact 11.8.5, 856 Hlawka’s inequality complex numbers Fact 2.21.11, 272 Euclidean norm Fact 11.8.16, 858 norm inequality Fact 11.8.5, 856 Hoffman eigenvalue perturbation Fact 11.14.13, 894 unitarily invariant norm Fact 11.10.8, 870 Hoffman-Wielandt theorem eigenvalue perturbation Fact 11.14.13, 894 ¨ Holder norm compatible norms Prop. 11.3.5, 839 definition, 833 eigenvalue Fact 11.13.5, 891 eigenvalue bound Fact 11.13.7, 891 Hanner’s inequality Fact 11.8.23, 859 H¨older-induced norm Prop. 11.4.11, 844 Fact 11.8.27, 860 Fact 11.9.23, 862 Fact 11.9.34, 864 Fact 11.9.35, 864 Fact 11.9.36, 864 Fact 11.9.37, 864 Fact 11.9.43, 865 inequality Prop. 11.1.5, 834 Prop. 11.1.6, 835 Fact 11.8.7, 856 Fact 11.8.20, 859 Fact 11.8.23, 859 Kronecker product Fact 11.8.26, 860 Fact 11.10.93, 884 matrix definition, 836 Minkowski’s inequality Lem. 11.1.3, 833 monotonicity Prop. 11.1.5, 834 outer-product matrix Fact 11.8.26, 860 Schatten norm Fact 11.9.10, 861 Fact 11.13.5, 891 submultiplicative norm Fact 11.10.48, 875 vector Fact 11.8.28, 860 vector norm

1486

¨ Holder’s inequality

Prop. 11.1.4, 834 ¨ Holder’s inequality

Diaz-Goldman-Metcalf inequality Fact 2.12.47, 223 extension Fact 2.13.1, 226 positive-semidefinite matrix Fact 10.14.8, 764 positive-semidefinite matrix trace Fact 10.14.6, 764 Fact 10.14.10, 765 Fact 10.14.11, 765 Fact 10.14.12, 765 Fact 10.14.13, 765 Fact 10.14.49, 772 Fact 10.14.51, 772 reversal Fact 2.12.47, 223 scalar case Fact 2.12.23, 219 Fact 2.12.24, 219 unitarily invariant norm Fact 11.10.38, 873 vector inequality Prop. 11.1.6, 835 ¨ Holder-induced lower bound definition, 846 ¨ Holder-induced norm absolute value Fact 11.9.30, 863 Fact 11.9.31, 863 adjoint norm Fact 11.9.21, 862 column norm Fact 11.9.25, 863 Fact 11.9.27, 863 Fact 11.9.29, 863 complex conjugate Fact 11.9.32, 863 complex conjugate transpose Fact 11.9.33, 863 definition, 842 field Prop. 11.4.7, 842 formulas Prop. 11.4.9, 843 H¨older norm Prop. 11.4.11, 844 Fact 11.8.27, 860 Fact 11.9.23, 862 Fact 11.9.34, 864 Fact 11.9.35, 864 Fact 11.9.36, 864 Fact 11.9.37, 864 Fact 11.9.43, 865 inequality Fact 11.9.25, 863 Fact 11.9.26, 863 Kronecker product Fact 11.10.94, 884

maximum singular value Fact 11.9.25, 863 mixed H¨older norm Fact 11.9.39, 864 Fact 11.9.40, 864 monotonicity Prop. 11.4.6, 842 partitioned matrix Fact 11.9.22, 862 quadratic form Fact 11.9.41, 864 Fact 11.9.42, 864 row norm Fact 11.9.25, 863 Fact 11.9.27, 863 Fact 11.9.29, 863 ¨ Holder-McCarthy inequality quadratic form Fact 10.18.18, 794 holds definition, 2 homomorphism definition Defn. 4.4.6, 371 Defn. 4.5.3, 371 Hopf’s theorem eigenvalues of a positive matrix Fact 6.11.27, 544 Horn singular value Fact 11.16.36, 905 unitary matrix Fact 10.21.14, 803 Householder matrix, see elementary reflector definition, 440 Householder reflector, see elementary reflector definition, 440 Hsu orthogonal matrix Fact 4.13.26, 390 orthogonally similar matrices Fact 7.10.16, 571 Hua’s identity matrix inverse Fact 3.15.28, 328 Hua’s inequalities determinant Fact 10.16.28, 780 determinant inequality Fact 10.12.52, 757 positive-semidefinite matrix Fact 10.12.52, 757 Hua’s inequality scalar inequality Fact 2.11.39, 196 Hua’s matrix equality generalization Fact 3.20.17, 350 Fact 3.20.18, 350

positive-semidefinite Fact 8.10.25, 674 Fact 8.10.27, 674 matrix hyperplane Fact 10.12.52, 757 definition, 279 Schur complement elementary projector Fact 10.12.52, 757 Fact 4.17.10, 408 Hui elementary reflector Lyapunov equation Fact 4.19.5, 416 Fact 15.19.1, 1226 intersection Hurwitz Fact 3.13.1, 315 subdeterminant hypersphere Fact 15.19.23, 1228 area Hurwitz matrix, see Fact 5.5.13, 497 asymptotically stable volume matrix Fact 5.5.13, 497 Hurwitz polynomial, see hypothesis asymptotically stable definition, 3 polynomial asymptotically stable I polynomial icosahedral group Fact 15.19.23, 1228 group Huygens Fact 4.31.16, 435 polynomial bound idempotent matrix Fact 15.21.26, 1238 (1)-inverse Huygens’s inequality Fact 8.4.28, 638 trigonometric inequality commutator Fact 2.17.1, 246 Fact 4.16.11, 402 Huygens-Leibniz identity Fact 4.16.15, 405 Euclidean norm Fact 4.16.16, 405 Fact 11.8.15, 858 Fact 4.22.10, 419 hyperbolic equalities commuting matrices Fact 2.19.2, 262 Fact 4.16.2, 398 Fact 2.19.1, 261 Fact 4.21.7, 418 hyperbolic function complementary derivative idempotent matrix Fact 2.19.6, 264 Fact 4.15.14, 397 equality complementary subspaces Fact 2.20.1, 264 Prop. 4.8.3, 374 increasing function Prop. 4.8.4, 375 Fact 2.19.5, 264 Fact 4.15.4, 396 hyperbolic functions Fact 4.16.4, 399 partial fractions Fact 8.7.9, 651 expansions complex conjugate Fact 13.4.12, 1016 Fact 4.15.9, 397 series complex conjugate Fact 13.4.11, 1015 transpose hyperbolic inequality Fact 4.15.9, 397 scalar Fact 4.15.16, 397 Fact 2.19.4, 263 definition hypercompanion matrix Defn. 4.1.1, 363 companion matrix diagonalizable matrix Lem. 7.4.4, 550 Fact 7.15.21, 600 definition, 550 difference elementary divisor Fact 4.16.6, 400 Lem. 7.4.4, 550 Fact 4.16.7, 400 real Jordan form Fact 4.16.11, 402 Fact 7.11.2, 575 Fact 4.16.12, 402 similarity transformation Fact 7.13.29, 596 Fact 7.11.2, 575 difference of matrices hyperellipsoid Fact 4.16.8, 401 volume Fact 4.16.9, 401 Fact 5.5.14, 498 Fact 4.16.10, 402 hypergeneralized projector discrete-time semistable generalized inverse matrix Fact 8.10.23, 673 Fact 15.22.13, 1241

impulse function Drazin generalized inverse Prop. 8.2.2, 626 Fact 8.11.9, 675 Fact 8.11.10, 675 equalities Fact 4.15.20, 398 factorization Fact 7.17.30, 609 Fact 7.17.33, 609 full-rank factorization Fact 4.15.19, 398 Fact 7.17.32, 609 generalized inverse Fact 7.13.28, 595 Fact 8.3.2, 628 Fact 8.4.27, 637 Fact 8.7.1, 650 Fact 8.7.2, 650 Fact 8.7.3, 651 Fact 8.7.4, 651 Fact 8.7.5, 651 Fact 8.7.6, 651 Fact 8.7.7, 651 Fact 8.7.8, 651 Fact 8.7.9, 651 Fact 8.8.11, 655 Fact 8.8.12, 655 Fact 8.8.13, 656 Fact 8.8.14, 656 Fact 8.8.15, 656 Fact 8.10.19, 672 Fact 8.10.20, 673 Fact 8.10.21, 673 Fact 8.10.26, 674 group generalized inverse Prop. 8.2.3, 627 Fact 8.11.12, 676 group-invertible matrix Prop. 4.1.7, 367 Prop. 4.8.11, 375 Prop. 8.2.3, 627 Fact 4.16.20, 407 Fact 7.12.11, 580 Fact 7.15.29, 601 Fact 8.10.2, 669 Fact 8.11.14, 677 inertia Fact 7.9.1, 565 involutory matrix Fact 4.20.3, 417 Kronecker product Fact 9.4.22, 687 left inverse Fact 4.15.12, 397 left-inner matrix Fact 4.13.8, 388 linear combination Fact 4.16.9, 401 Fact 7.20.14, 619 matrix exponential Fact 15.12.1, 1204 Fact 15.17.14, 1223 matrix product Fact 4.15.18, 398 Fact 4.15.19, 398

matrix sum Fact 7.20.12, 619 Fact 7.20.13, 619 Fact 7.20.14, 619 maximum singular value Fact 7.12.41, 585 Fact 7.12.42, 586 Fact 7.13.28, 595 Fact 8.7.7, 651 nilpotent matrix Fact 4.22.10, 419 nonsingular matrix Fact 4.15.13, 397 Fact 4.16.9, 401 Fact 4.16.10, 402 norm Fact 15.17.14, 1223 normal matrix Fact 4.17.3, 407 null space Fact 4.15.5, 396 Fact 4.16.3, 398 onto a subspace along another subspace definition, 375 outer-product matrix Fact 4.10.18, 380 Fact 4.15.8, 397 partitioned matrix Fact 4.15.21, 398 Fact 4.15.22, 398 Fact 4.16.4, 399 Fact 7.11.27, 579 positive-definite matrix Fact 7.17.33, 609 positive-semidefinite matrix Fact 7.17.33, 609 power Fact 4.15.2, 396 Fact 4.15.15, 397 product Fact 4.16.1, 398 Fact 4.16.13, 403 Fact 4.16.14, 404 projector Fact 4.17.3, 407 Fact 4.18.19, 414 Fact 7.11.18, 577 Fact 7.13.28, 595 Fact 8.7.4, 651 Fact 8.8.12, 655 Fact 8.8.13, 656 Fact 8.8.14, 656 Fact 8.8.15, 656 range Fact 4.15.1, 396 Fact 4.15.5, 396 Fact 4.15.6, 397 Fact 4.16.3, 398 range-Hermitian matrix Fact 4.17.3, 407 Fact 8.7.5, 651 Fact 8.7.6, 651 rank

Fact 4.15.8, 397 Fact 4.15.11, 397 Fact 4.15.16, 397 Fact 4.15.17, 398 Fact 4.15.22, 398 Fact 4.16.2, 398 Fact 4.16.5, 399 Fact 4.16.6, 400 Fact 4.16.7, 400 Fact 4.16.8, 401 Fact 4.16.12, 402 Fact 4.16.16, 405 Fact 4.16.17, 405 Fact 4.16.18, 406 Fact 4.16.19, 406 Fact 7.9.1, 565 Fact 7.13.1, 589 right inverse Fact 4.15.12, 397 right-inner matrix Fact 4.13.8, 388 semisimple matrix Fact 7.15.28, 601 similar matrices Prop. 4.7.5, 374 Prop. 7.7.23, 562 Cor. 7.7.22, 562 Fact 7.11.12, 577 Fact 7.11.18, 577 Fact 7.11.19, 577 Fact 7.11.27, 579 singular value Fact 7.12.41, 585 skew-Hermitian matrix Fact 4.15.10, 397 skew-idempotent matrix Fact 4.15.7, 397 spectrum Prop. 7.7.21, 562 Fact 7.13.1, 589 stable subspace Prop. 15.9.8, 1200 submultiplicative norm Fact 11.9.5, 860 sum Fact 4.16.5, 399 Fact 4.16.21, 407 trace Fact 7.9.1, 565 Fact 7.13.1, 589 transpose Fact 4.15.9, 397 tripotent matrix Fact 4.21.1, 417 Fact 4.21.7, 418 unitarily similar matrices Prop. 4.7.5, 374 Fact 7.10.26, 572 Fact 7.10.28, 573 Fact 7.11.13, 577 Fact 7.11.16, 577 unstable subspace Prop. 15.9.8, 1200 idempotent matrix onto group-invertible matrix

1487

Prop. 4.8.11, 375 idempotent matrix onto a subspace along another subspace

definition, 375 idempotent number

series Fact 13.4.9, 1012 identity function definition, 16 identity matrix definition, 284 symplectic matrix Fact 4.28.3, 427 identity theorem matrix function Thm. 12.8.3, 932 identity-matrix perturbation cyclic matrix Fact 7.15.17, 600 defective matrix Fact 7.15.17, 600 derogatory matrix Fact 7.15.17, 600 diagonalizable over F Fact 7.15.17, 600 inverse matrix Fact 6.8.14, 522 semisimple matrix Fact 7.15.17, 600 simple matrix Fact 7.15.17, 600 spectrum Fact 6.10.18, 534 Fact 6.10.20, 534 identity-matrix shift controllable subspace Lem. 16.6.7, 1262 unobservable subspace Lem. 16.3.7, 1255 identric mean arithmetic mean Fact 2.2.63, 142 logarithmic mean Fact 2.2.63, 142 image definition, 16 imaginary part frequency response Fact 16.24.5, 1314 transfer function Fact 16.24.5, 1314 imaginary vector definition, 286 implication definition, 3 improper rational function definition Defn. 6.7.1, 513 improper rational transfer function definition Defn. 6.7.2, 514 impulse function

1488

impulse response

Prop. 4.8.9, 375 definition Defn. 4.8.7, 375 definition, 1251 impulse response function group-invertible matrix Prop. 4.8.8, 375 definition, 1251 Cor. 7.7.9, 559 inbound Laplacian matrix Fact 7.15.6, 598 adjacency matrix Kronecker product Thm. 4.2.2, 368 Fact 9.4.31, 688 definition outer-product matrix Defn. 4.2.1, 367 Fact 7.15.5, 598 incidence matrix partitioned matrix definition Fact 7.16.19, 604 Defn. 4.2.1, 367 range directed graph Fact 7.15.6, 598 Fact 4.26.4, 426 rank Fact 4.26.5, 426 Prop. 7.7.2, 558 Laplacian matrix index of an eigenvalue Thm. 4.2.2, 368 algebraic multiplicity Thm. 4.2.3, 368 Prop. 7.7.6, 559 symmetric graph definition Fact 4.26.1, 426 Defn. 7.7.1, 558 inclusion-exclusion Jordan block principle Prop. 7.7.3, 558 cardinality minimal polynomial Fact 1.8.4, 23 Prop. 7.7.15, 561 inclusive or rank definition, 2 Prop. 7.7.2, 558 increasing function semisimple eigenvalue convex function Prop. 7.7.8, 559 Thm. 10.6.15, 717 index of imprimitivity definition nonnegative matrix Defn. 10.6.12, 716 Fact 6.11.5, 538 fraction induced lower bound Fact 2.6.12, 185 definition logarithm Defn. 11.5.1, 845 Prop. 10.6.13, 717 Prop. 11.5.2, 845 matrix functions generalized inverse Prop. 10.6.13, 717 Fact 11.9.61, 868 polynomial lower bound Fact 6.8.20, 523 Fact 11.9.60, 868 positive-definite matrix Fact 11.9.61, 868 Fact 10.11.89, 748 maximum singular value Schur complement Cor. 11.5.5, 846 Prop. 10.6.13, 717 minimum singular value increasing sequence Cor. 11.5.5, 846 definition, 917 properties positive-semidefinite Prop. 11.5.2, 845 matrix Prop. 11.5.3, 846 Prop. 10.6.3, 715 singular value indecomposable matrix, Prop. 11.5.4, 846 see irreducible matrix supermultiplicativity definition, 440 Prop. 11.5.6, 847 indegree induced norm graph compatible norm Defn. 1.4.3, 10 Prop. 11.4.3, 842 indegree matrix definition definition Defn. 11.4.1, 841 Defn. 4.2.1, 367 determinant index Fact 11.14.18, 895 minimal polynomial dual norm Fact 6.9.35, 530 Fact 11.8.24, 859 index of a matrix field block-triangular matrix Example 11.4.8, 843 Fact 7.16.19, 604 complementary subspaces maximum singular value definition, 1250

impulse response

Fact 11.9.28, 863 norm Thm. 11.4.2, 841 quadratic form Fact 11.9.53, 866 spectral radius Cor. 11.4.5, 842 Cor. 11.4.10, 844 induced norms symmetry property Fact 11.9.24, 862 inertia congruent matrices Thm. 7.5.7, 554 Fact 7.9.26, 569 definition, 511 difference of projectors Fact 7.9.11, 566 Fact 7.9.12, 567 dissipative matrix Fact 7.9.13, 567 generalized inverse Fact 8.3.29, 631 Fact 8.9.8, 661 Fact 10.24.13, 817 group-invertible matrix Fact 7.9.5, 566 idempotent matrix Fact 7.9.1, 565 involutory matrix Fact 7.9.2, 565 Lyapunov equation Fact 16.22.1, 1309 Fact 16.22.2, 1310 Fact 16.22.3, 1310 Fact 16.22.4, 1310 Fact 16.22.5, 1310 Fact 16.22.6, 1310 Fact 16.22.7, 1311 Fact 16.22.8, 1311 Fact 16.22.9, 1311 Fact 16.22.10, 1311 Fact 16.22.11, 1311 Fact 16.22.12, 1311 nilpotent matrix Fact 7.9.4, 565 partitioned matrix Fact 7.9.21, 568 Fact 7.9.22, 568 Fact 7.9.23, 569 Fact 7.9.24, 569 Fact 7.9.25, 569 Fact 7.9.27, 569 Fact 7.13.7, 591 Fact 8.9.8, 661 Fact 8.9.9, 662 positive-semidefinite matrix Fact 7.9.9, 566 Fact 16.22.9, 1311 projector Fact 7.9.20, 568 Fact 7.9.25, 569 rank Fact 7.9.5, 566

Fact 7.9.19, 568 Riccati equation Lem. 16.16.18, 1298 Schur complement Fact 8.9.9, 662 skew-Hermitian matrix Fact 7.9.4, 565 skew-involutory matrix Fact 7.9.4, 565 submatrix Fact 7.9.8, 566 tripotent matrix Fact 7.9.3, 565 inertia matrix positive-definite matrix Fact 10.10.7, 731 rigid body Fact 10.10.7, 731 inference definition, 4 inferior sequence Defn. 12.2.11, 917 Prop. 12.2.10, 917 infimum sequence Fact 12.18.5, 958 Fact 12.18.6, 958 set Fact 1.8.6, 24 infinite interval definition, xl infinite limit definition Defn. 12.2.3, 916 infinite matrix product convergence Fact 15.22.22, 1242 infinite product Euler product formula Fact 13.3.1, 994 infinite series equality Fact 13.5.80, 1047 Fact 13.5.81, 1047 Fact 13.5.82, 1048 Fact 13.5.84, 1049 Fact 13.5.85, 1049 infinite set definition, 1 infinity norm definition, 834 Kronecker product Fact 11.10.93, 884 submultiplicative norm Fact 11.10.26, 872 Fact 11.10.27, 872 injective function definition, 118 inner matrix eigenvalue Fact 10.21.20, 804 Fact 10.21.21, 804 Hermitian matrix Fact 10.21.20, 804

invariant zero asymptotically stable Fact 10.21.21, 804 singular value matrix Fact 10.21.22, 804 Lem. 15.10.2, 1201 inner product averaged limit convex cone Fact 12.13.12, 943 Fact 12.12.22, 940 determinant definition, 285–288 Fact 15.14.11, 1210 inequality Drazin generalized inverse Fact 3.15.1, 326 Fact 15.14.8, 1210 light cone inequality Fact 15.14.10, 1210 Fact 2.12.58, 225 Gaussian density open ball Fact 15.14.12, 1210 Fact 11.7.18, 852 generalized inverse separation theorem Fact 15.14.6, 1210 Fact 12.12.22, 940 group generalized inverse Fact 12.12.23, 940 Fact 15.14.9, 1210 subspace Fact 15.14.10, 1210 Fact 12.12.21, 939 infinite product inner-product minimization Fact 13.10.22, 1090 positive-definite matrix inverse matrix Fact 10.18.13, 793 Fact 15.14.7, 1210 input matrix limit controllability Fact 12.13.13, 943 Fact 16.21.17, 1309 Fact 12.13.14, 944 stabilizability matrix Fact 16.21.17, 1309 definition, 926 input-to-state stability matrix exponential asymptotic stability Prop. 15.1.4, 1180 Fact 16.21.20, 1309 Lem. 15.10.2, 1201 integer Fact 15.14.6, 1210 congruent modulo Fact 15.14.7, 1210 Fact 1.11.9, 29 Fact 15.14.8, 1210 Fact 1.11.10, 29 Fact 15.14.9, 1210 cubes Fact 15.14.10, 1210 Fact 1.11.37, 35 Fact 15.14.11, 1210 divisors Fact 15.14.12, 1210 Fact 1.11.29, 33 Fact 15.15.2, 1212 Egyptian fraction Fact 15.19.6, 1226 Fact 1.11.50, 36 Fact 15.19.7, 1226 equality positive-semidefinite Fact 1.11.23, 32 matrix harmonic number Prop. 10.6.10, 716 Fact 1.11.38, 35 integral representation Liouville’s theorem Kronecker sum Fact 1.11.29, 33 Fact 15.19.34, 1231 modulo interior Prop. 1.5.1, 12 affine subspace number of partitions Fact 12.11.15, 935 Fact 7.18.14, 616 boundary square Fact 12.11.8, 935 Fact 1.11.1, 28 Fact 12.11.9, 935 Fact 1.11.2, 28 complement squares Fact 12.11.7, 935 Fact 1.11.36, 35 cone integer pair Fact 12.11.15, 935 binomial coefficient convex cone Fact 1.12.17, 42 Fact 12.11.15, 935 integer powers convex set series Fact 12.11.15, 935 Fact 13.8.21, 1074 Fact 12.11.16, 935 integers Fact 12.12.4, 937 equality definition Fact 1.11.30, 33 Defn. 12.1.2, 913 integral intersection Fact 12.12.2, 937

largest open set Fact 12.11.4, 935 simplex Fact 5.1.8, 442 subset Fact 12.12.1, 937 subspace Fact 12.11.15, 935 union Fact 12.12.2, 937 Fact 12.12.3, 937 interior point definition Defn. 12.1.2, 913 interior point relative to a set definition Defn. 12.1.3, 913 interior relative to a set definition Defn. 12.1.3, 913 interlacing theorem asymptotically stable polynomial Fact 15.18.8, 1224 intermediate node definition Defn. 1.4.3, 10 Defn. 1.4.4, 11 interpolation polynomial Fact 6.8.13, 522 intersection affine subspace Fact 3.12.4, 311 cardinality bound Fact 1.8.12, 24 Fact 1.8.13, 25 closed set Fact 12.12.17, 939 Fact 12.12.18, 939 compact set Fact 12.12.20, 939 cone Fact 3.12.4, 311 convex cone Fact 3.12.4, 311 convex set Fact 3.12.4, 311 Fact 12.12.11, 938 definition, 1 dual cone Fact 12.12.15, 939 equivalence relation Prop. 1.3.2, 6 Erd¨os-Ko-Rado theorem Fact 1.8.10, 24 hyperplane Fact 3.13.1, 315 interior Fact 12.12.2, 937 inverse image Fact 3.12.8, 311 oddtown theorem Fact 1.8.11, 24

1489

open set Fact 12.12.16, 939 Fact 12.12.19, 939 reflexive relation Prop. 1.3.2, 6 span Fact 3.12.11, 312 Sperner lemma Fact 1.8.9, 24 subspace Fact 3.12.4, 311 symmetric relation Prop. 1.3.2, 6 transitive relation Prop. 1.3.2, 6 intersection of compact sets Cantor intersection theorem Fact 12.12.20, 939 intersection of ranges projector Fact 8.8.19, 657 intersection of subspaces subspace dimension theorem Thm. 3.1.3, 279 intertwining matrices eigenstructure Fact 7.16.11, 603 interval definition, xl invariance of domain compact set image Thm. 12.4.20, 923 open set image Thm. 12.4.19, 923 invariant subspace controllable subspace Cor. 16.6.4, 1261 definition, 291 lower triangular matrix Fact 7.10.3, 570 matrix representation Fact 3.11.21, 310 stable subspace Prop. 15.9.8, 1200 unobservable subspace Cor. 16.3.4, 1254 unstable subspace Prop. 15.9.8, 1200 upper triangular matrix Fact 7.10.3, 570 invariant zero definition Defn. 16.10.1, 1278 determinant Fact 16.24.16, 1315 equivalent realizations Prop. 16.10.10, 1284 feedback Prop. 16.10.10, 1284 full actuation Defn. 16.10.2, 1279 full observation

1490

inverse

Defn. 16.10.2, 1279 inequality full-state feedback Fact 2.20.2, 265 Fact 16.24.16, 1315 inverse hyperbolic observable pair functions Cor. 16.10.12, 1285 series pencil Fact 13.4.13, 1017 Cor. 16.10.4, 1280 inverse hyperbolic tangent Cor. 16.10.5, 1280 Poincar´e metric Cor. 16.10.6, 1281 Fact 2.21.30, 276 regular pencil Fact 2.21.31, 276 Cor. 16.10.4, 1280 series Cor. 16.10.5, 1280 Fact 1.16.6, 1062 Cor. 16.10.6, 1281 inverse image transmission zero closed set Thm. 16.10.8, 1282 Cor. 12.4.10, 922 Thm. 16.10.9, 1282 definition, 17 uncontrollable spectrum generalized inverse Thm. 16.10.9, 1282 Prop. 8.1.10, 624 uncontrollableProp. 8.1.11, 625 unobservable intersection Fact 3.12.8, 311 spectrum left-invertible matrix Thm. 16.10.9, 1282 Prop. 3.7.10, 298 unobservable eigenvalue linear equation Prop. 16.10.11, 1284 Prop. 8.1.11, 625 unobservable spectrum one-to-one function Thm. 16.10.9, 1282 Fact 1.10.5, 27 inverse onto function determinant Fact 1.10.5, 27 Fact 3.17.33, 341 open set Frobenius norm Cor. 12.4.10, 922 Fact 11.9.13, 861 right-invertible matrix left-invertible matrix Prop. 3.7.10, 298 Prop. 3.7.6, 297 sum polynomial matrix Fact 3.12.8, 311 definition, 503 inverse matrix positive-definite matrix 2×2 Fact 10.12.22, 752 Fact 3.20.2, 348 rank 2 × 2 block triangular Fact 3.14.28, 325 Lem. 3.9.2, 302 Fact 3.14.29, 325 3×3 right-invertible matrix Fact 3.20.2, 348 Prop. 3.7.6, 297 asymptotically stable subdeterminant Fact 3.17.33, 341 matrix uniqueness Fact 15.19.15, 1227 Prop. 1.6.4, 17 block-circulant matrix inverse function Fact 3.22.7, 354 convex hull block-triangular matrix Fact 3.11.17, 309 Fact 3.22.1, 352 Fact 12.11.22, 936 bordered matrix definition, 17 Fact 3.21.5, 352 subspace characteristic polynomial Fact 3.11.16, 309 Fact 6.9.5, 525 uniqueness companion matrix Thm. 1.6.5, 17 Fact 7.18.4, 612 inverse function theorem convergent sequence determinant Fact 3.20.22, 350 Thm. 12.5.5, 926 convergent series existence of local inverse Fact 6.10.11, 532 Thm. 12.5.5, 926 definition, 297 inverse hyperbolic function derivative derivative Prop. 12.10.2, 933 Fact 2.20.3, 266 Fact 12.16.22, 952 equality elementary matrix Fact 2.20.1, 264 Fact 3.21.1, 351

Fact 4.10.20, 381 equality Fact 3.20.3, 348 Fact 3.20.4, 348 Fact 3.20.5, 348 Fact 3.20.7, 348 Fact 3.20.8, 348 Fact 3.20.9, 349 Fact 3.20.10, 349 Fact 3.20.11, 349 Fact 3.20.12, 349 Fact 3.20.13, 349 Fact 3.20.14, 349 Fact 3.20.15, 349 Fact 3.20.16, 350 Fact 3.20.19, 350 Fact 3.20.20, 350 finite sequence Fact 3.20.21, 350 Hamiltonian matrix Fact 4.28.5, 428 Hankel matrix Fact 4.23.4, 420 identity-matrix perturbation Fact 6.8.14, 522 integral Fact 15.14.7, 1210 Kronecker product Prop. 9.1.7, 682 lower bound Fact 10.10.20, 732 matrix exponential Prop. 15.2.8, 1184 Fact 15.14.7, 1210 matrix inversion lemma Cor. 3.9.8, 304 matrix sum Cor. 3.9.10, 305 maximum singular value Fact 11.16.7, 900 Newton-Raphson algorithm Fact 3.20.22, 350 normalized submultiplicative norm Fact 11.9.62, 868 Fact 11.10.88, 883 Fact 11.10.89, 883 Fact 11.10.90, 883 Fact 11.10.91, 883 outer-product perturbation Fact 3.21.3, 351 partitioned matrix Fact 3.22.2, 352 Fact 3.22.3, 352 Fact 3.22.4, 352 Fact 3.22.5, 353 Fact 3.22.6, 353 Fact 3.22.7, 354 Fact 3.22.9, 354 Fact 7.13.31, 596 perturbation Fact 11.10.92, 884

polynomial representation Fact 6.8.15, 522 positive-definite matrix Prop. 10.6.6, 715 Lem. 10.6.5, 715 Fact 10.10.20, 732 Fact 10.10.46, 735 positive-semidefinite matrix Fact 10.11.61, 742 product Prop. 3.7.12, 298 push-through identity Fact 3.20.6, 348 rank Fact 3.22.11, 354 Fact 8.9.12, 662 Fact 8.9.13, 662 series Prop. 11.3.10, 841 similar matrices Fact 7.17.34, 609 similarity transformation Fact 7.17.5, 605 spectral radius Prop. 11.3.10, 841 spectrum Fact 7.12.17, 581 sum Fact 3.22.7, 354 Toeplitz matrix Fact 4.23.10, 421 tridiagonal matrix Fact 4.24.6, 424 Fact 4.24.7, 424 Fact 4.24.8, 424 upper block-triangular matrix Fact 3.22.8, 354 Fact 3.22.10, 354 inverse operation iterated Fact 1.10.2, 26 inverse tangent equality Fact 2.18.6, 258 series Fact 13.5.79, 1047 Fact 13.6.8, 1060 Fact 13.6.9, 1061 Fact 13.6.10, 1061 Fact 13.6.12, 1061 inverse tangent function atan2 Fact 2.18.7, 259 inverse trigonometric function derivative Fact 2.18.9, 261 equality Fact 2.18.1, 254 Fact 2.18.2, 255 Fact 2.18.3, 255 inverse trigonometric functions

Kantorovich inequality series Fact 13.4.9, 1012 inverse trigonometric inequality scalar Fact 2.18.8, 260 inverse vec definition, 681 invertible function definition, 17 involutory matrix commutator Fact 4.20.8, 417 cyclic permutation matrix Fact 4.20.7, 417 definition Defn. 4.1.1, 363 determinant Fact 4.20.1, 417 Fact 4.20.2, 417 Fact 7.17.35, 609 diagonalizable matrix Fact 7.15.21, 600 factorization Fact 7.17.20, 607 Fact 7.17.34, 609 Fact 7.17.35, 609 idempotent matrix Fact 4.20.3, 417 inertia Fact 7.9.2, 565 Kronecker product Fact 9.4.22, 687 matrix exponential Fact 15.12.1, 1204 normal matrix Fact 7.10.10, 571 Fact 7.10.11, 571 null space Fact 4.20.8, 417 partitioned matrix Fact 4.20.10, 417 product Fact 4.20.4, 417 range Fact 4.20.8, 417 rank Fact 4.20.6, 417 Fact 4.20.9, 417 reverse permutation matrix Fact 4.20.5, 417 semisimple matrix Fact 7.15.20, 600 signature Fact 7.9.2, 565 similar matrices Prop. 4.7.5, 374 Cor. 7.7.22, 562 Fact 7.17.34, 609 spectrum Prop. 7.7.21, 562 symmetric matrix Fact 7.17.39, 609 trace

1491

Fact 7.9.2, 565 Defn. 4.6.4, 372 Fact 8.10.13, 670 transpose group group-invertible matrix Fact 7.10.8, 570 Prop. 4.6.5, 372 Fact 8.10.13, 670 tripotent matrix Hamiltonian Fact 4.21.2, 417 Fact 16.25.1, 1316 J unitarily similar matrices minimal polynomial Prop. 4.7.5, 374 Prop. 7.7.15, 561 Jacobi identity normal matrix irrational number commutator Fact 7.11.8, 576 closure Fact 3.23.3, 354 real Jordan form Fact 12.11.1, 934 cross product Fact 7.11.3, 575 exponent Fact 4.12.1, 384 Schur decomposition Fact 1.15.2, 52 Jacobi theta function Fact 7.11.8, 576 Fact 1.15.3, 52 zeta function square root Kronecker’s theorem Fact 13.3.1, 994 Fact 7.17.21, 607 Fact 12.11.1, 934 Jacobi’s abstruse identity transfer function irreducible matrix infinite product Fact 16.24.11, 1315 absolute value Fact 13.10.28, 1091 Jordan matrix Fact 4.25.9, 426 Jacobi’s four-square companion matrix almost nonnegative matrix theorem Lem. 7.4.1, 549 Fact 15.20.2, 1232 sum of squares definition, 549 connected graph Fact 1.11.25, 32 elementary divisor Fact 6.11.4, 538 Jacobi’s identity Lem. 7.4.1, 549 cyclic permutation matrix determinant example Fact 7.18.15, 616 Fact 3.17.32, 341 Example 7.4.7, 551 definition Fact 3.17.34, 341 Example 7.4.8, 552 Defn. 4.1.1, 363 matrix differential graph Jordan structure equation Fact 6.11.4, 538 logarithm Fact 16.20.6, 1306 group generalized inverse Cor. 15.4.9, 1189 Jacobi’s triple product Fact 8.11.4, 674 matrix exponential identity M-matrix Cor. 15.4.9, 1189 infinite product Fact 6.11.15, 542 Jordan’s inequality Fact 13.10.26, 1090 permutation matrix trigonometric inequality Jacobian Fact 6.11.6, 540 Fact 2.17.1, 246 definition, 925 Fact 7.18.15, 616 Jordan-Chevalley Jacobson positive matrix decomposition nilpotent commutator Fact 6.11.5, 538 diagonalizable matrix Fact 7.16.4, 603 row-stochastic matrix Fact 7.10.4, 570 Janous Fact 15.22.12, 1240 nilpotent matrix inequality spectral radius convexity Fact 7.10.4, 570 Fact 2.6.9, 185 Fact 6.11.23, 544 Joyal Jensen spectral radius polynomial root bound convex function monotonicity Fact 15.21.11, 1235 Fact 12.13.19, 944 Fact 6.11.23, 543 Jung’s theorem Jensen’s inequality upper block-triangular open ball AM-GM inequality Fact 12.11.3, 935 matrix Fact 1.21.7, 117 Jury criterion Fact 6.11.6, 540 convex function discrete-time irreducible polynomial Fact 1.21.7, 117 definition, 501 asymptotically stable Fact 2.11.131, 211 isolated point polynomial JLL inequality definition Fact 15.21.2, 1234 trace of a matrix power Defn. 12.1.7, 914 Fact 6.11.29, 544 isolated singularity join operator K definition, 929 definition, 118 isolated subset jointly continuous function Kalman decomposition definition minimal realization continuous function Defn. 12.1.7, 914 Prop. 16.9.12, 1273 Fact 12.13.9, 942 isomorphic groups Kantorovich inequality Jordan block symplectic group and positive-definite matrix index of an eigenvalue unitary group Fact 10.11.43, 739 Prop. 7.7.3, 558 Fact 4.31.9, 434 positive-semidefinite Jordan form isomorphism definition, 550 matrix definition factorization Fact 10.11.42, 739 Defn. 4.4.6, 371 Fact 7.17.7, 606 quadratic form Defn. 4.5.3, 371 generalized inverse Fact 10.18.8, 792

1492

Karamata’s inequality

Fact 10.18.10, 793 quadratic form inequality scalar case Fact 10.19.19, 797 Fact 2.11.134, 212 Krein Schur product inertia of a Hermitian Fact 10.25.70, 830 matrix Karamata’s inequality Fact 16.22.5, 1310 weak majorization normal matrix Fact 3.25.17, 362 Fact 7.16.16, 604 Kato Krein’s inequality maximum singular value angle inequality of a matrix difference Fact 11.8.9, 857 Fact 11.10.22, 872 Krein-Krasnoselskiikernel function Milman positive-semidefinite equality matrix projector Fact 10.9.2, 725 Fact 7.13.27, 594 Fact 10.9.7, 726 Krein-Milman theorem Kharitonov’s theorem extreme points of a convex asymptotically stable set polynomial Fact 12.11.32, 937 Fact 15.18.17, 1225 Kreiss matrix theorem Khatri-Rao product maximum singular value Hadamard product, 700 Fact 15.22.23, 1242 Kittaneh Kristof Schatten norm inequality least squares Fact 11.10.18, 871 Fact 11.17.17, 911 KKT matrix Kronecker canonical form partitioned matrix pencil Fact 10.12.44, 756 Thm. 7.8.1, 563 Klamkin regular pencil scalar inequality Prop. 7.8.2, 563 Fact 2.3.69, 169 Kronecker permutation Klamkin’s inequality matrix triangle definition, 683 Fact 5.2.8, 446 Kronecker product Fact 5.2.10, 456 Fact 9.4.38, 688 Klein four-group orthogonal matrix dihedral group Fact 9.4.38, 688 Fact 4.31.16, 435 projector Klein’s inequality Fact 9.4.39, 689 trace of a matrix logarithm trace Fact 15.15.27, 1215 Fact 9.4.38, 688 Kleinman transpose stabilization Prop. 9.1.13, 683 Fact 16.21.19, 1309 vec Knopp’s inequality Fact 9.4.38, 688 reciprocals Kronecker product Fact 2.11.45, 197 biequivalent matrices Kobe function Fact 9.4.12, 686 complex function column norm Fact 12.14.12, 947 Fact 11.10.93, 884 Kojima’s bound complex conjugate polynomial transpose Fact 15.21.12, 1236 Prop. 9.1.3, 681 polynomial root bound congruent matrices Fact 15.21.5, 1235 Fact 9.4.13, 686 Kosaki convex function Schatten norm inequality Prop. 10.6.17, 718 Fact 11.10.18, 871 definition trace norm of a matrix Defn. 9.1.2, 681 difference determinant Fact 11.10.19, 871 Prop. 9.1.11, 683 trace of a convex function Fact 9.5.13, 692 Fact 10.14.48, 771 Fact 9.5.14, 692 Krafft diagonal matrix

Fact 9.4.4, 686 discrete-time asymptotically stable matrix Fact 15.22.7, 1240 Fact 15.22.8, 1240 discrete-time Lyapunov-stable matrix Fact 15.22.7, 1240 Fact 15.22.8, 1240 discrete-time semistable matrix Fact 15.22.6, 1240 Fact 15.22.7, 1240 Fact 15.22.8, 1240 Drazin generalized inverse Fact 9.4.37, 688 eigenvalue Prop. 9.1.10, 683 Fact 9.4.17, 686 Fact 9.4.21, 687 Fact 9.4.27, 687 Fact 9.4.32, 688 Fact 9.4.40, 689 eigenvector Prop. 9.1.10, 683 Fact 9.4.27, 687 Fact 9.4.40, 689 Frobenius norm Fact 11.10.95, 884 generalized inverse Fact 9.4.36, 688 group generalized inverse Fact 9.4.37, 688 group-invertible matrix Fact 9.4.22, 687 Fact 9.4.37, 688 Hermitian matrix Fact 9.4.22, 687 Fact 10.25.31, 824 H¨older norm Fact 11.8.26, 860 Fact 11.10.93, 884 H¨older-induced norm Fact 11.10.94, 884 idempotent matrix Fact 9.4.22, 687 index of a matrix Fact 9.4.31, 688 infinity norm Fact 11.10.93, 884 inverse matrix Prop. 9.1.7, 682 involutory matrix Fact 9.4.22, 687 Kronecker permutation matrix Fact 9.4.38, 688 Kronecker sum Fact 15.15.41, 1217 left inverse Fact 9.4.34, 688 left-equivalent matrices Fact 9.4.12, 686 lower triangular matrix

Fact 9.4.4, 686 matrix exponential Prop. 15.1.7, 1181 Fact 15.15.41, 1217 Fact 15.15.42, 1217 matrix multiplication Prop. 9.1.6, 682 matrix power Fact 9.4.5, 686 Fact 9.4.11, 686 Fact 9.4.27, 687 matrix sum Prop. 9.1.4, 682 maximum singular value Fact 11.10.95, 884 nilpotent matrix Fact 9.4.22, 687 normal matrix Fact 9.4.22, 687 orthogonal matrix Fact 9.4.22, 687 outer-product matrix Prop. 9.1.8, 682 partitioned matrix Fact 9.4.24, 687 Fact 9.4.25, 687 Fact 9.4.29, 687 positive-definite matrix Fact 9.4.22, 687 positive-semidefinite matrix Fact 9.4.22, 687 Fact 10.25.15, 822 Fact 10.25.37, 825 Fact 10.25.38, 825 Fact 10.25.39, 825 Fact 10.25.41, 825 Fact 10.25.42, 825 Fact 10.25.43, 826 Fact 10.25.44, 826 Fact 10.25.45, 826 Fact 10.25.48, 827 projector Fact 9.4.22, 687 range Fact 9.4.15, 686 Fact 9.4.16, 686 range-Hermitian matrix Fact 9.4.22, 687 rank Fact 9.4.20, 687 Fact 9.4.28, 687 Fact 9.4.29, 687 Fact 9.4.30, 688 Fact 10.25.15, 822 reflector Fact 9.4.22, 687 right inverse Fact 9.4.35, 688 right-equivalent matrices Fact 9.4.12, 686 row norm Fact 11.10.93, 884 Schatten norm Fact 11.10.95, 884

Laurent series Schur product Prop. 9.3.2, 685 semi-tensor product Fact 9.6.26, 700 semisimple matrix Fact 9.4.22, 687 similar matrices Fact 9.4.13, 686 singular matrix Fact 9.4.33, 688 singular value Fact 9.4.19, 687 skew-Hermitian matrix Fact 9.4.23, 687 spectral radius Fact 9.4.18, 686 square root Fact 10.25.26, 823 Fact 10.25.47, 826 submatrix Prop. 9.3.2, 685 tensor Fact 9.4.41, 689 trace Prop. 9.1.12, 683 Fact 10.25.25, 823 Fact 15.15.42, 1217 transpose Prop. 9.1.3, 681 triple product Prop. 9.1.5, 682 Fact 9.4.8, 686 tripotent matrix Fact 9.4.22, 687 unitarily similar matrices Fact 9.4.13, 686 unitary matrix Fact 9.4.22, 687 upper triangular matrix Fact 9.4.4, 686 vec Fact 9.4.6, 686 Fact 9.4.7, 686 Fact 9.4.9, 686 vector Fact 9.4.1, 685 Fact 9.4.2, 685 Fact 9.4.3, 686 Fact 9.4.26, 687 wedge product Fact 9.4.41, 689 Fact 9.4.42, 691 Kronecker sum associativity Prop. 9.2.3, 683 asymptotically stable matrix Fact 15.19.32, 1231 Fact 15.19.33, 1231 Fact 15.19.34, 1231 asymptotically stable polynomial Fact 15.18.15, 1225 commuting matrices Fact 9.5.5, 691

complex conjugate transpose Prop. 9.2.2, 683 defect Fact 9.5.3, 691 definition Defn. 9.2.1, 683 determinant Fact 9.5.2, 691 Fact 9.5.12, 692 dissipative matrix Fact 9.5.9, 692 eigenvalue Prop. 9.2.4, 684 Fact 9.5.6, 691 Fact 9.5.8, 692 Fact 9.5.17, 693 eigenvector Prop. 9.2.4, 684 Fact 9.5.17, 693 Hermitian matrix Fact 9.5.9, 692 integral representation Fact 15.19.34, 1231 Kronecker product Fact 15.15.41, 1217 linear matrix equation Prop. 15.10.3, 1201 linear system Fact 9.5.16, 693 Lyapunov equation Cor. 15.10.4, 1202 Lyapunov-stable matrix Fact 15.19.32, 1231 Fact 15.19.33, 1231 matrix exponential Prop. 15.1.7, 1181 Fact 15.15.40, 1217 Fact 15.15.41, 1217 matrix power Fact 9.5.1, 691 nilpotent matrix Fact 9.5.4, 691 Fact 9.5.9, 692 normal matrix Fact 9.5.9, 692 positive matrix Fact 9.5.9, 692 positive-semidefinite matrix Fact 9.5.9, 692 range-Hermitian matrix Fact 9.5.9, 692 rank Fact 9.5.3, 691 Fact 9.5.10, 692 Fact 9.5.11, 692 semidissipative matrix Fact 9.5.9, 692 semistable matrix Fact 15.19.32, 1231 Fact 15.19.33, 1231 similar matrices Fact 9.5.10, 692 skew-Hermitian matrix

Fact 9.5.9, 692 spectral radius Fact 9.5.7, 691 trace Fact 15.15.40, 1217 transpose Prop. 9.2.2, 683 Kronecker’s lemma limit Fact 12.18.15, 959 Fact 12.18.16, 959 positive-definite matrix Fact 10.11.76, 745 Kronecker’s theorem irrational number Fact 12.11.1, 934 Kummer’s expansion Gamma function Fact 13.3.2, 997 Ky Fan k-norm vector norm Fact 11.7.20, 853

1493

matrix function Thm. 12.8.2, 931 Laguerre polynomial trigonometric functions Fact 13.2.9, 989 Laguerre-Samuelson inequality mean Fact 2.11.37, 195 Fact 10.10.40, 734 Lah number binomial coefficient Fact 1.19.4, 108 Lalescu factorial limit Fact 12.18.60, 971 Lancaster’s formulas quadratic form integral Fact 14.12.3, 1172 Landau’s inequality bounded derivative Fact 12.16.12, 950 Laplace transform matrix exponential L Prop. 15.2.2, 1182 resolvent L2 norm Prop. 15.2.2, 1182 controllability Gramian Laplacian Thm. 16.11.1, 1285 symmetric graph definition, 1285 Fact 4.26.2, 426 observability Gramian Laplacian matrix Thm. 16.11.1, 1285 adjacency matrix ˆ l’Hopital’s rule Thm. 4.2.2, 368 limit Thm. 4.2.3, 368 Fact 12.17.13, 956 Fact 6.11.12, 541 ¨ Lowner-Heinz inequality definition positive-semidefinite Defn. 4.2.1, 367 matrix incidence matrix Cor. 10.6.11, 716 Thm. 4.2.2, 368 Labelle Thm. 4.2.3, 368 polynomial root bound quadratic form Fact 15.21.11, 1235 Fact 10.18.1, 792 Lagrange interpolation spectrum formula Fact 15.20.7, 1233 polynomial interpolation symmetric graph Fact 6.8.13, 522 Fact 10.18.1, 792 Lagrange’s four-square lattice theorem definition sum of powers Defn. 1.3.11, 7 Fact 1.11.26, 33 positive-semidefinite Lagrange’s identity matrix matrix equality Fact 10.11.51, 740 Fact 3.15.37, 329 Fact 10.11.52, 741 product equality Laurent series Fact 2.12.13, 217 analytic function sum of squares Prop. 12.6.9, 928 Fact 2.4.7, 178 Lagrange’s second identity definition, 928 infinite product Euclidean norm Fact 13.10.29, 1092 Fact 11.8.15, 858 rational function Lagrange’s theorem Fact 12.14.3, 946 subgroup square-summable Thm. 4.4.4, 370 coefficients Lagrange-Hermite Fact 12.14.15, 948 interpolation polynomial

1494

leading principal submatrix

leading principal submatrix

definition, 281 leaf

definition, 502 left equivalence transformation

definition, 374 Defn. 1.4.1, 10 left inverse graph (1)-inverse Fact 1.9.3, 25 Prop. 8.1.2, 622 least common multiple affine function binomial coefficient Fact 3.18.8, 343 Fact 1.16.2, 55 affine subspace block-diagonal matrix Fact 3.11.15, 309 Lem. 7.3.6, 548 complex conjugate coprime transpose Fact 1.11.5, 28 Fact 3.18.1, 342 definition, 501 Fact 3.18.2, 342 greatest common divisor Fact 3.18.5, 343 Fact 1.11.6, 28 cone sum Fact 3.11.15, 309 Fact 1.11.8, 29 convex set least squares Fact 3.11.15, 309 (1,3)-inverse definition, 17 Fact 8.3.6, 628 generalized inverse Fact 11.17.6, 910 Cor. 8.1.4, 622 (1,4)-inverse Fact 8.3.13, 629 Fact 11.17.7, 910 Fact 8.3.14, 629 fixed-rank approximation Fact 8.4.41, 639 Fact 11.16.39, 906 Fact 8.4.42, 640 Fact 11.17.14, 911 Fact 11.17.4, 909 generalized inverse Hermitian matrix Fact 11.17.6, 910 Fact 8.3.13, 629 Fact 11.17.7, 910 idempotent matrix Fact 11.17.9, 910 Fact 4.15.12, 397 Fact 11.17.10, 910 Kronecker product Fact 11.17.11, 911 Fact 9.4.34, 688 Fact 11.17.12, 911 left-inner matrix Hermitian matrix Fact 4.13.7, 388 Fact 7.13.20, 593 left-invertible matrix Fact 11.17.15, 911 Fact 3.18.3, 343 orthogonal Procrustes matrix product Fact 3.18.9, 343 problem Fact 3.18.11, 343 Fact 11.17.11, 911 Fact 3.18.13, 344 singular value Fact 3.18.16, 344 decomposition positive-definite matrix Fact 11.16.39, 906 Fact 4.10.25, 381 Fact 11.17.13, 911 range Fact 11.17.14, 911 Prop. 3.7.4, 295 Fact 11.17.16, 911 Fact 3.18.1, 342 Fact 11.17.17, 911 Fact 3.18.2, 342 least upper bound rank collection of sets Fact 3.18.1, 342 Prop. 1.3.15, 8 Fact 3.18.2, 342 Prop. 1.3.17, 8 representation definition Fact 3.18.6, 343 Defn. 1.3.11, 7 right inverse preuniqueness Prop. 1.6.3, 17 Lem. 1.3.10, 7 Fact 3.18.1, 342 projector Fact 3.18.2, 342 Fact 8.8.20, 657 subspace Fact 8.8.21, 658 Fact 3.11.15, 309 union of sets uniqueness Lem. 1.3.13, 7 Thm. 1.6.5, 17 Lebesgue’s identity unitarily invariant norm sum of squares Fact 11.17.4, 909 Fact 2.4.5, 178 left-equivalent matrices left divides

definition Defn. 4.7.3, 373 group-invertible matrix Fact 4.9.2, 376 Kronecker product Fact 9.4.12, 686 null space Prop. 7.1.2, 545 positive-semidefinite matrix Fact 7.11.24, 578 left-inner matrix definition Defn. 4.1.2, 364 generalized inverse Fact 8.3.11, 629 idempotent matrix Fact 4.13.8, 388 left inverse Fact 4.13.7, 388 trace Fact 7.13.19, 593 left-invertible function definition, 17 left-invertible matrix definition, 294 equivalent properties Thm. 3.7.1, 294 generalized inverse Prop. 8.1.5, 622 Prop. 8.1.12, 625 Fact 8.4.20, 636 inverse Prop. 3.7.6, 297 inverse image Prop. 3.7.10, 298 left inverse Fact 3.18.3, 343 nonsingular equivalence Cor. 3.7.7, 297 partitioned matrix Fact 3.18.20, 345 range Prop. 3.7.4, 295 unique left inverse Prop. 3.7.2, 295 upper block-triangular matrix Fact 3.18.20, 345 Legendre polynomial trigonometric functions Fact 13.2.8, 987 Legendre-Fenchel transform quadratic form Fact 10.17.7, 787 Lehmer matrix positive-semidefinite matrix Fact 10.9.8, 727 Lehmer mean power inequality Fact 2.2.62, 142 Leibniz’s inequality triangle

Fact 5.2.8, 446 Leibniz’s rule

derivative of an integral Fact 12.16.7, 950 Leibniz’s test series Fact 12.18.11, 959 lemma definition, 5 length definition Defn. 1.4.3, 10 Defn. 1.4.4, 11 Leslie matrix companion matrix Fact 7.18.1, 610 generalized Frobenius companion matrix Fact 7.18.1, 610 Leverrier’s algorithm characteristic polynomial Prop. 6.4.9, 509 lexicographic ordering cone Fact 3.11.23, 310 total ordering Fact 1.8.16, 25 Li eigenvalue inequality Fact 10.21.13, 803 Lidskii eigenvalues Fact 10.22.29, 811 Lidskii-Mirsky-Wielandt theorem Hermitian matrix Fact 11.14.4, 893 Lidskii-Wielandt inequalities eigenvalues of Hermitian matrices Cor. 10.6.19, 722 Lie algebra classical examples Prop. 4.3.2, 369 definition Defn. 4.3.1, 368 Lie group Prop. 15.6.4, 1192 Prop. 15.6.5, 1192 Prop. 15.6.6, 1192 matrix exponential Prop. 15.6.3, 1191 Prop. 15.6.7, 1192 strictly upper triangular matrix Fact 4.31.10, 434 Fact 15.23.1, 1243 upper triangular matrix Fact 4.31.10, 434 Fact 15.23.1, 1243 Lie group definition Defn. 15.6.1, 1191 group

linear matrix equation factorial Prop. 15.6.2, 1191 Fact 12.18.60, 971 Lie algebra generalized inverse Prop. 15.6.4, 1192 Fact 8.3.39, 632 Prop. 15.6.5, 1192 group generalized inverse Prop. 15.6.6, 1192 Fact 8.10.4, 669 Lie-Trotter formula Hermitian matrix matrix exponential Fact 10.11.3, 735 Cor. 15.4.13, 1190 infinite product Fact 15.15.8, 1213 Fact 13.10.18, 1088 Fact 15.17.2, 1220 Fact 13.10.20, 1089 Fact 15.17.3, 1221 Fact 13.10.21, 1089 Lieb concavity theorem, integral 831 Fact 12.13.13, 943 Lieb-Thirring inequality Fact 12.13.14, 944 positive-semidefinite Kronecker’s lemma matrix Fact 12.18.15, 959 Fact 10.14.18, 766 Fact 12.18.16, 959 Fact 10.22.27, 810 l’Hˆopital’s rule light cone inequality Fact 12.17.13, 956 inner product limit of derivative Fact 2.12.58, 225 Fact 12.17.12, 956 liminf matrix exponential sequence Fact 15.19.6, 1226 Prop. 12.2.12, 917 Fact 15.19.7, 1226 Fact 12.18.8, 958 Fact 15.19.8, 1226 limit matrix logarithm e Prop. 10.6.4, 715 Fact 12.18.23, 961 nondecreasing sequence of Fact 12.18.24, 961 sets Abel summability Prop. 1.3.21, 9 Fact 12.18.17, 959 nonincreasing sequence of arithmetic-geometric sets mean Prop. 1.3.21, 9 Fact 12.18.57, 969 positive-definite matrix binomial coefficient Fact 10.11.76, 745 Fact 12.18.49, 967 positive-semidefinite Fact 12.18.50, 967 matrix Fact 12.18.51, 968 Prop. 10.6.3, 715 Fact 12.18.52, 968 Fact 10.11.76, 745 Fact 12.18.53, 968 product of series Cesaro summability Fact 12.18.18, 960 Fact 12.18.17, 959 projector Cesaro’s lemma Fact 8.8.19, 657 Fact 12.18.14, 959 Fact 8.8.20, 657 convergent sequence rank Prop. 12.2.7, 917 Fact 12.13.26, 945 CT-dominant eigenvalue semistable matrix Fact 15.14.16, 1211 Fact 15.19.8, 1226 cycle number sequence Fact 12.18.54, 968 Fact 12.18.7, 958 definition series Defn. 12.2.2, 916 Fact 12.18.42, 966 discrete-time semistable Stolz-Cesaro lemma matrix Fact 12.18.19, 960 Fact 15.22.13, 1241 subset number double factorial Fact 12.18.54, 968 Fact 12.18.59, 971 Drazin generalized inverse sum Fact 13.5.19, 1025 Fact 8.10.6, 670 limit of a function Fact 8.10.14, 670 definition DT-dominant eigenvalue Defn. 12.4.2, 921 Fact 15.22.24, 1242 Defn. 12.4.3, 921 exponent Defn. 12.4.4, 922 Fact 12.17.1, 954 limit of derivative Fact 12.17.2, 954

limit Fact 12.17.12, 956 limit point closure Prop. 12.1.8, 914 definition, 1092 limit set convergent sequence Prop. 12.2.8, 917 limsup sequence Prop. 12.2.12, 917 Fact 12.18.8, 958 Lin outer-product perturbations Fact 11.16.11, 901 positive-semidefinite matrix Fact 10.22.40, 813 Linden polynomial root bound Fact 15.21.13, 1236 linear combination commuting matrices Fact 3.23.7, 355 definition, 278 Hermitian matrix Fact 10.19.14, 796 Fact 10.19.15, 797 Fact 10.19.20, 798 idempotent matrix Fact 7.20.14, 619 linear combination of matrices idempotent matrix Fact 4.16.5, 399 projector Fact 4.18.20, 415 linear combination of projectors Hermitian matrix Fact 7.20.15, 619 linear combination of two vectors definition, 277 linear constraint quadratic form Fact 10.17.11, 788 linear dependence absolute value Fact 11.8.2, 853 definition, 277, 279, 281 triangle inequality Fact 11.7.22, 853 linear dynamical system asymptotically stable Prop. 15.9.2, 1198 discrete-time asymptotically stable Prop. 15.11.2, 1203 discrete-time Lyapunov stable Prop. 15.11.2, 1203

1495

discrete-time semistable Prop. 15.11.2, 1203 Lyapunov stable Prop. 15.9.2, 1198 semistable Prop. 15.9.2, 1198 linear equation (1,4)-inverse Fact 8.3.7, 628 Drazin generalized inverse Fact 8.10.10, 670 generalized inverse Prop. 8.1.8, 623 Prop. 8.1.9, 624 Prop. 8.1.11, 625 inverse image Prop. 8.1.11, 625 rank Thm. 3.7.5, 296 right inverse Fact 8.3.16, 629 solutions Prop. 8.1.8, 623 Prop. 8.1.9, 624 linear function continuous function Cor. 12.4.7, 922 definition, 282 linear independence cyclic matrix Fact 7.15.10, 599 definition, 279 outer-product matrix Fact 3.15.9, 326 linear independence of rational functions definition, 515 linear interpolation strong majorization Fact 4.11.6, 384 linear matrix equation asymptotically stable matrix Prop. 15.10.3, 1201 existence of solutions Fact 7.11.25, 578 Fact 7.11.26, 578 generalized inverse Fact 8.4.40, 639 Kronecker sum Prop. 15.10.3, 1201 matrix exponential Prop. 15.10.3, 1201 rank Fact 3.13.23, 317 skew-symmetric matrix Fact 4.10.2, 377 solution Fact 8.4.40, 639 Sylvester’s equation Prop. 9.2.5, 684 Prop. 15.10.3, 1201 Fact 7.11.25, 578 Fact 7.11.26, 578 Fact 8.9.10, 662

1496

linear system

symmetric matrix Fact 4.10.2, 377 linear system harmonic steady-state response Thm. 16.12.1, 1288 Kronecker sum Fact 9.5.16, 693 solutions Fact 3.13.11, 315 linear system solution Cramer’s rule Fact 3.16.12, 330 nonnegative vector Fact 6.11.19, 543 norm Fact 11.17.1, 909 Fact 11.17.2, 909 Fact 11.17.3, 909 rank Cor. 3.7.8, 297 Cor. 3.7.9, 298 right-invertible matrix Fact 3.16.13, 330 total least squares Fact 11.17.1, 912 linear-quadratic control problem definition, 1293 Riccati equation Thm. 16.15.2, 1294 solution Thm. 16.15.2, 1294 lines plane Fact 5.1.2, 441 points Fact 5.1.1, 441 Liouville’s identity quartic Fact 2.4.4, 178 Liouville’s theorem complex function Fact 12.14.8, 946 integer Fact 1.11.29, 33 little oh notation sequence Defn. 12.2.13, 918 Littlewood H¨older-induced norm Fact 11.9.35, 864 Fact 11.9.36, 864 Ljance principal angle and subspaces Fact 7.12.42, 586 locally Lipschitz definition Defn. 12.4.28, 924 log majorization convex function Fact 3.25.13, 361 nondecreasing function Fact 3.25.13, 361

positive-semidefinite matrix Fact 10.12.19, 752 Fact 10.12.20, 752 logarithm, see matrix logarithm SO(3) Fact 15.16.11, 1220 Bode sensitivity integral Fact 14.7.13, 1156 convex function Fact 15.16.12, 1220 determinant Fact 10.15.11, 775 determinant and convex function Prop. 10.6.17, 718 double integral Fact 14.13.3, 1174 Fact 14.13.4, 1175 Fact 14.13.5, 1175 entropy Fact 2.15.24, 229 Fact 2.15.25, 229 Fact 2.15.26, 230 Fact 2.15.27, 230 Fact 2.15.28, 230 Euler constant Fact 12.18.33, 963 gamma Fact 12.18.33, 963 increasing function Prop. 10.6.13, 717 Jordan structure Cor. 15.4.9, 1189 limit Fact 12.17.3, 954 Fact 12.17.7, 955 orthogonal matrix Fact 15.16.11, 1220 rotation matrix Fact 15.16.11, 1220 series Fact 13.8.16, 1073 Shannon’s inequality Fact 2.15.28, 230 trace and convex function Prop. 10.6.17, 718 logarithm function complex numbers Fact 2.21.29, 275 principal branch Fact 2.21.29, 275 logarithmic derivative asymptotically stable matrix Fact 15.19.11, 1227 Lyapunov equation Fact 15.19.11, 1227 properties Fact 15.16.7, 1218 logarithmic mean arithmetic mean Fact 2.11.112, 208 Heron mean

Fact 2.2.67, 144 identric mean Fact 2.2.63, 142 integral Fact 14.2.17, 1102 limit Fact 12.18.56, 969 Napier’s inequality Fact 2.2.63, 142 Polya’s inequality Fact 2.2.63, 142 logical equivalents De Morgan’s laws Fact 1.7.1, 21 existential statement Fact 1.7.6, 22 implication Fact 1.7.1, 21 Fact 1.7.2, 22 Fact 1.7.3, 22 Fact 1.7.4, 22 sets Fact 1.7.5, 22 universal statement Fact 1.7.6, 22 lower bidiagonal matrix definition Defn. 4.1.3, 365 lower Hessenberg matrix Fact 4.24.1, 422 lower block-triangular matrix definition Defn. 4.1.3, 365 determinant Prop. 3.8.1, 299 right-invertible matrix Fact 3.18.20, 345 lower bound generalized inverse Fact 11.9.61, 868 induced lower bound Fact 11.9.60, 868 Fact 11.9.61, 868 minimum singular value Fact 11.15.24, 898 lower bound for a partial ordering definition Defn. 1.3.9, 7 lower Hessenberg matrix definition Defn. 4.1.3, 365 lower bidiagonal matrix Fact 4.24.1, 422 submatrix Fact 4.24.2, 423 tridiagonal matrix Fact 4.24.1, 422 lower reverse-triangular matrix definition Fact 3.16.15, 331 determinant Fact 3.16.15, 331

lower semicontinuity

rank Fact 12.13.26, 945 lower triangular matrix, see upper triangular matrix binomial coefficient Fact 4.25.3, 424 characteristic polynomial Fact 6.10.14, 533 commutator Fact 4.22.13, 420 definition Defn. 4.1.3, 365 determinant Fact 4.25.1, 424 eigenvalue Fact 6.10.14, 533 factorization Fact 7.17.12, 606 invariant subspace Fact 7.10.3, 570 Kronecker product Fact 9.4.4, 686 matrix exponential Fact 15.14.1, 1209 Fact 15.14.13, 1210 matrix power Fact 4.23.7, 421 matrix product Fact 4.25.2, 424 nilpotent matrix Fact 4.22.13, 420 similar matrices Fact 7.10.3, 570 spectrum Fact 6.10.14, 533 Toeplitz matrix Fact 4.23.7, 421 Fact 15.14.1, 1209 LQG controller continuous-time control problem Fact 16.25.8, 1318 discrete-time control problem Fact 16.25.9, 1318 dynamic compensator Fact 16.25.8, 1318 Fact 16.25.9, 1318 LU decomposition existence Fact 7.17.12, 606 Lucas number Binet’s formula Fact 1.17.2, 100 determinant Fact 4.25.4, 425 Fibonacci number Fact 1.17.2, 100 generating function Fact 13.9.3, 1077 golden ratio Fact 13.9.2, 1075 Lucas polynomial Fact 13.2.13, 993

mass-spring system recursion Prop. 15.9.4, 1199 Cor. 15.10.7, 1203 Fact 1.17.2, 100 matrix exponential matrix exponential Fact 15.19.6, 1226 Cor. 15.10.4, 1202 Lucas polynomial Fact 15.22.9, 1240 Fact 15.19.18, 1228 roots minimal realization Fact 15.19.19, 1228 Fact 13.2.13, 993 Defn. 16.9.25, 1278 null space Lukes N-matrix Fact 16.22.15, 1312 stabilization Fact 15.20.4, 1232 observability matrix Fact 16.21.19, 1309 normal matrix Fact 16.22.15, 1312 LULU decomposition Fact 15.19.37, 1231 observably asymptotically factorization positive-definite matrix Fact 7.17.13, 606 stable Prop. 15.10.6, 1202 Lyapunov differential Prop. 16.4.3, 1257 Cor. 15.10.7, 1203 positive-definite matrix equation semidissipative matrix Fact 16.22.16, 1312 asymptotic solution Fact 15.19.37, 1231 Fact 16.22.18, 1312 Fact 16.20.2, 1305 semistable matrix positive-semidefinite Lyapunov equation Fact 15.19.2, 1226 asymptotic stability matrix similar matrices Cor. 15.10.1, 1201 Fact 16.22.15, 1312 Fact 15.19.5, 1226 asymptotically stable Fact 16.22.19, 1312 step response Schur power matrix Fact 16.21.2, 1307 Fact 10.9.20, 729 Prop. 15.10.5, 1202 Lyapunov-stable semistability Cor. 15.10.4, 1202 polynomial Cor. 15.10.1, 1201 Cor. 15.10.7, 1203 definition semistable matrix Cor. 16.4.4, 1258 Defn. 15.9.3, 1198 Fact 15.19.1, 1226 Cor. 16.5.7, 1259 Lyapunov-stable matrix Fact 16.22.15, 1312 Cor. 16.7.4, 1268 Prop. 15.9.4, 1199 skew-Hermitian matrix Cor. 16.8.7, 1270 subdeterminant Fact 15.19.12, 1227 Fact 16.22.7, 1311 Fact 15.19.23, 1228 stabilizability Fact 16.22.17, 1312 Lyapunov-stable transfer Cor. 16.8.7, 1270 controllably Lyapunov stability function asymptotically stable eigenvalue minimal realization Prop. 16.7.3, 1266 Prop. 15.9.2, 1198 Prop. 16.9.26, 1278 detectability linear dynamical system SISO entry Cor. 16.5.7, 1259 Prop. 15.9.2, 1198 Prop. 16.9.27, 1278 discrete-time Lyapunov equation Fact 16.23.1, 1313 M Cor. 15.10.1, 1201 discrete-time matrix exponential asymptotically stable M-matrix Prop. 15.9.2, 1198 matrix definition nonlinear system Prop. 15.11.5, 1204 Fact 6.11.13, 541 Thm. 15.8.5, 1196 eigenvalue inclusion region determinant Lyapunov’s direct method Fact 16.22.20, 1313 Fact 6.11.16, 542 stability theory finite-sum solution eigenvector Thm. 15.8.5, 1196 Fact 16.22.17, 1312 Fact 6.11.15, 542 Lyapunov-stable inertia inverse equilibrium Fact 16.22.1, 1309 Fact 6.11.16, 542 definition Fact 16.22.2, 1310 irreducible matrix Defn. 15.8.3, 1196 Fact 16.22.3, 1310 Fact 6.11.15, 542 Lyapunov-stable matrix Fact 16.22.4, 1310 nonnegative matrix almost nonnegative matrix Fact 16.22.5, 1310 Fact 6.11.13, 541 Fact 15.20.4, 1232 Fact 16.22.6, 1310 rank compartmental matrix Fact 16.22.7, 1311 Fact 10.7.9, 723 Fact 15.20.6, 1233 Fact 16.22.8, 1311 Schur product definition Fact 16.22.9, 1311 Fact 9.6.22, 699 Defn. 15.9.1, 1198 Fact 16.22.10, 1311 submatrix group-invertible matrix Fact 16.22.11, 1311 Fact 6.11.14, 542 Fact 15.19.3, 1226 Fact 16.22.12, 1311 Z-matrix Kronecker sum Kronecker sum Fact 6.11.13, 541 Fact 15.19.32, 1231 Cor. 15.10.4, 1202 Fact 6.11.16, 542 Fact 15.19.33, 1231 logarithmic derivative Maclaurin’s inequality Lyapunov equation Fact 15.19.11, 1227 elementary symmetric Prop. 15.10.6, 1202 Lyapunov stability polynomial Cor. 15.10.7, 1203 Cor. 15.10.1, 1201 Fact 2.11.35, 195 Lyapunov-stable Lyapunov-stable matrix Magnus polynomial Prop. 15.10.6, 1202 determinant equalities

1497

Fact 3.16.26, 333 Magnus expansion

time-varying dynamics Fact 16.20.6, 1306 Maillet’s identity cubic Fact 2.4.4, 178 majorization, see strong majorization, weak majorization positive-semidefinite matrix Fact 10.12.21, 752 Makelainen idempotent matrices Fact 7.13.29, 596 Maligranda inequality complex numbers Fact 2.21.9, 271 norm Fact 11.7.11, 851 Mann positivity of a quadratic form on a subspace Fact 10.19.21, 798 Marcus quadratic form inequality Fact 10.19.5, 796 similar matrices and nonzero diagonal entry Fact 7.10.15, 571 Markov block-Hankel matrix controllable pair Prop. 16.9.13, 1275 definition, 1274, 1275 minimal realization Prop. 16.9.14, 1275 observable pair Prop. 16.9.13, 1275 rational transfer function Prop. 16.9.13, 1275 Prop. 16.9.14, 1275 Prop. 16.9.15, 1275 Markov parameter definition, 1253 rational transfer function Lem. 16.9.7, 1272 Martins’s inequality sum of integers Fact 1.12.42, 47 Mason polynomial root bound Fact 15.21.14, 1236 mass definition, 1187 undamped rigid body Fact 7.15.31, 601 mass matrix partitioned matrix Fact 7.13.31, 596 mass-spring system spectrum Fact 7.13.31, 596 stability

1498

Massera-Schaffer inequality

Fact 15.19.38, 1231

Prop. 15.9.2, 1198 asymptotically stable matrix Lem. 15.10.2, 1201 complex numbers Fact 15.16.9, 1219 Fact 2.21.9, 271 Fact 15.19.9, 1227 norm Fact 15.19.10, 1227 Fact 11.7.11, 851 Fact 15.19.15, 1227 matricial norm Fact 15.19.18, 1228 partitioned matrix Fact 15.19.19, 1228 Fact 11.12.4, 886 Fact 15.22.9, 1240 matrix block-diagonal matrix definition, 280 Prop. 15.2.8, 1184 matrix cosine commutator matrix exponential Fact 15.15.10, 1213 Fact 15.13.1, 1209 Fact 15.15.12, 1213 matrix sine Fact 15.15.13, 1213 Fact 15.13.1, 1209 Fact 15.15.14, 1213 matrix derivative Fact 15.15.15, 1213 definition, 933 Fact 15.15.16, 1214 matrix differential equation Fact 15.15.17, 1214 Jacobi’s identity Fact 15.15.18, 1214 Fact 16.20.6, 1306 Fact 15.15.19, 1214 matrix exponential commuting matrices Fact 16.20.3, 1305 Prop. 15.1.5, 1180 periodic dynamics Cor. 15.1.6, 1180 Fact 16.20.8, 1307 Fact 15.15.1, 1211 Fact 16.20.9, 1307 Fact 15.15.6, 1212 Riccati differential complex conjugate equation Prop. 15.2.8, 1184 Fact 16.25.7, 1318 complex conjugate time-varying dynamics transpose Fact 16.20.6, 1306 Prop. 15.2.8, 1184 Fact 16.20.7, 1306 Fact 15.16.4, 1218 matrix equality Fact 15.16.6, 1218 Lagrange’s identity convergence in time Fact 3.15.37, 329 Prop. 15.9.7, 1200 matrix exponential convergent sequence 2 × 2 matrix Prop. 15.1.3, 1180 Prop. 15.3.2, 1185 Fact 15.15.8, 1213 Cor. 15.3.3, 1185 Fact 15.15.9, 1213 Lem. 15.3.1, 1185 Fact 15.15.10, 1213 Example 15.3.4, 1186 Fact 15.22.18, 1241 Example 15.3.5, 1186 convergent series 3 × 3 matrix Prop. 15.1.2, 1179 Fact 15.12.5, 1206 convex function 3 × 3 orthogonal matrix Fact 10.17.23, 791 Fact 15.12.10, 1207 Fact 10.17.24, 791 Fact 15.12.11, 1207 Fact 15.16.12, 1220 3 × 3 skew-symmetric cross product matrix Fact 15.12.7, 1206 Fact 15.12.6, 1206 Fact 15.12.8, 1207 Fact 15.12.10, 1207 Fact 15.12.9, 1207 Fact 15.12.11, 1207 cross-product matrix 4 × 4 skew-symmetric Fact 15.12.6, 1206 matrix Fact 15.12.12, 1207 Fact 15.12.15, 1208 Fact 15.12.13, 1207 Fact 15.12.16, 1208 Fact 15.12.14, 1207 Fact 15.12.17, 1208 Fact 15.12.17, 1208 Fact 15.12.18, 1208 Fact 15.12.18, 1208 SO(n) definition Fact 15.12.3, 1205 Defn. 15.1.1, 1179 almost nonnegative matrix derivative Fact 15.20.1, 1232 Fact 10.14.46, 771 Fact 15.20.2, 1232 Fact 15.15.3, 1212 asymptotic stability Massera-Schaffer inequality

Fact 15.15.4, 1212 Fact 15.15.5, 1212 Fact 15.15.11, 1213 Fact 15.16.2, 1218 derivative of a matrix Fact 15.15.12, 1213 determinant Prop. 15.4.11, 1189 Cor. 15.2.4, 1183 Cor. 15.2.5, 1183 Fact 15.14.11, 1210 Fact 15.16.5, 1218 diagonal matrix Fact 15.14.13, 1210 discrete-time asymptotic stability Prop. 15.11.2, 1203 discrete-time asymptotically stable matrix Fact 15.22.9, 1240 discrete-time Lyapunov stability Prop. 15.11.2, 1203 discrete-time Lyapunov-stable matrix Fact 15.22.9, 1240 discrete-time semistability Prop. 15.11.2, 1203 discrete-time semistable matrix Fact 15.22.9, 1240 Fact 15.22.18, 1241 dissipative matrix Fact 15.16.3, 1218 Drazin generalized inverse Fact 15.14.8, 1210 Fact 15.14.10, 1210 Duhamel formula Fact 15.15.3, 1212 eigenstructure Prop. 15.2.7, 1183 Frobenius norm Fact 15.15.36, 1217 Fact 15.16.3, 1218 generalized inverse Fact 15.14.6, 1210 geometric mean Fact 10.11.70, 744 Fact 10.11.71, 744 Golden-Thompson inequality Fact 15.15.30, 1216 Fact 15.15.31, 1216 Fact 15.17.4, 1221 group Prop. 15.6.7, 1192 group generalized inverse Fact 15.14.9, 1210 Fact 15.14.10, 1210 Fact 15.19.6, 1226 Fact 15.19.7, 1226 group-invertible matrix Fact 15.19.14, 1227 Hamiltonian matrix

Prop. 15.6.7, 1192 Hermitian matrix Prop. 15.2.8, 1184 Prop. 15.4.3, 1187 Prop. 15.4.10, 1189 Cor. 15.2.6, 1183 Fact 15.15.8, 1213 Fact 15.15.9, 1213 Fact 15.15.22, 1214 Fact 15.15.30, 1216 Fact 15.15.32, 1216 Fact 15.15.35, 1217 Fact 15.15.36, 1217 Fact 15.15.38, 1217 Fact 15.16.1, 1217 Fact 15.17.4, 1221 Fact 15.17.7, 1222 Fact 15.17.15, 1223 Fact 15.17.19, 1223 idempotent matrix Fact 15.12.1, 1204 Fact 15.17.14, 1223 infinite product Fact 15.15.19, 1214 integral Prop. 15.1.4, 1180 Lem. 15.10.2, 1201 Fact 15.14.6, 1210 Fact 15.14.7, 1210 Fact 15.14.8, 1210 Fact 15.14.9, 1210 Fact 15.14.10, 1210 Fact 15.14.11, 1210 Fact 15.14.12, 1210 Fact 15.15.2, 1212 Fact 15.17.9, 1222 Fact 15.19.6, 1226 Fact 15.19.7, 1226 inverse matrix Prop. 15.2.8, 1184 Fact 15.14.7, 1210 involutory matrix Fact 15.12.1, 1204 Jordan structure Cor. 15.4.9, 1189 Kronecker product Prop. 15.1.7, 1181 Fact 15.15.41, 1217 Fact 15.15.42, 1217 Kronecker sum Prop. 15.1.7, 1181 Fact 15.15.40, 1217 Fact 15.15.41, 1217 Laplace transform Prop. 15.2.2, 1182 Lie algebra Prop. 15.6.7, 1192 Lie algebra of a Lie group Prop. 15.6.3, 1191 Lie-Trotter formula Fact 15.15.8, 1213 Lie-Trotter product formula Cor. 15.4.13, 1190 Fact 15.17.2, 1220

matrix logarithm Fact 15.17.3, 1221 limit Fact 15.19.6, 1226 Fact 15.19.7, 1226 Fact 15.19.8, 1226 linear matrix equation Prop. 15.10.3, 1201 logarithm Fact 15.15.22, 1214 lower triangular matrix Fact 15.14.1, 1209 Fact 15.14.13, 1210 Lyapunov equation Cor. 15.10.4, 1202 Fact 15.19.18, 1228 Fact 15.19.19, 1228 Lyapunov stability Prop. 15.9.2, 1198 Lyapunov-stable matrix Fact 15.19.6, 1226 Fact 15.22.9, 1240 matrix cosine Fact 15.13.1, 1209 matrix differential equation Fact 16.20.3, 1305 matrix logarithm Thm. 15.5.2, 1190 Prop. 15.4.2, 1187 Fact 15.14.14, 1210 Fact 15.14.15, 1211 Fact 15.15.35, 1217 matrix sine Fact 15.13.1, 1209 maximum eigenvalue Fact 15.17.4, 1221 maximum singular value Fact 15.16.1, 1217 Fact 15.16.2, 1218 Fact 15.16.5, 1218 Fact 15.17.8, 1222 Fact 15.17.12, 1222 nilpotent matrix Fact 15.12.1, 1204 Fact 15.14.14, 1210 nondecreasing function Fact 10.11.71, 744 norm Fact 15.17.11, 1222 Fact 15.17.13, 1222 Fact 15.17.14, 1223 norm bound Fact 15.19.10, 1227 normal matrix Prop. 15.2.8, 1184 Cor. 15.4.4, 1188 Fact 15.15.6, 1212 Fact 15.17.12, 1222 orthogonal matrix Prop. 15.4.8, 1188 Prop. 15.6.7, 1192 Fact 15.12.6, 1206 Fact 15.12.7, 1206 Fact 15.12.8, 1207 Fact 15.12.9, 1207

Fact 15.12.12, 1207 Fact 15.12.13, 1207 Fact 15.12.14, 1207 Fact 15.16.11, 1220 outer-product matrix Fact 15.12.1, 1204 partitioned matrix Fact 15.12.2, 1205 Fact 15.15.2, 1212 Fact 15.17.20, 1223 Peierls-Bogoliubov inequality Fact 15.15.32, 1216 polar decomposition Fact 15.14.4, 1209 polynomial matrix Prop. 15.2.1, 1181 positive-definite matrix Prop. 15.2.8, 1184 Prop. 15.4.3, 1187 Prop. 15.4.8, 1188 Fact 15.15.21, 1214 Fact 15.15.23, 1215 Fact 15.15.24, 1215 Fact 15.16.1, 1217 positive-semidefinite matrix Fact 10.13.13, 762 Fact 15.15.21, 1214 Fact 15.15.39, 1217 Fact 15.17.8, 1222 Fact 15.17.18, 1223 quadratic form Fact 10.17.23, 791 quaternions Fact 15.12.16, 1208 rank-two matrix Fact 15.12.19, 1209 resolvent Prop. 15.2.2, 1182 rotation matrix Fact 15.12.13, 1207 Fact 15.12.14, 1207 Schur product Fact 15.15.22, 1214 semisimple matrix Prop. 15.2.7, 1183 semistability Prop. 15.9.2, 1198 semistable matrix Fact 15.19.7, 1226 Fact 15.19.8, 1226 Fact 15.22.9, 1240 series Prop. 15.4.12, 1189 Fact 15.15.18, 1214 similar matrices Prop. 15.2.9, 1185 singular value Fact 15.16.5, 1218 Fact 15.16.12, 1220 Fact 15.17.5, 1222 Fact 15.17.6, 1222 skew-Hermitian matrix Prop. 15.2.8, 1184

Prop. 15.4.3, 1187 Fact 15.15.7, 1212 Fact 15.15.37, 1217 skew-involutory matrix Fact 15.12.1, 1204 skew-symmetric matrix Prop. 15.4.8, 1188 Example 15.3.6, 1186 Fact 15.12.3, 1205 Fact 15.12.6, 1206 Fact 15.12.7, 1206 Fact 15.12.8, 1207 Fact 15.12.9, 1207 Fact 15.12.16, 1208 Specht’s ratio Fact 15.15.30, 1216 spectral abscissa Fact 15.14.2, 1209 Fact 15.16.8, 1219 Fact 15.16.9, 1219 Fact 15.16.10, 1220 Fact 15.19.9, 1227 spectral radius Fact 15.14.2, 1209 spectrum Prop. 15.2.3, 1183 Cor. 15.2.6, 1183 stable subspace Prop. 15.9.8, 1200 state equation Prop. 16.1.1, 1249 strong log majorization Fact 15.17.4, 1221 submultiplicative norm Prop. 15.1.2, 1179 Fact 15.16.8, 1219 Fact 15.16.9, 1219 Fact 15.16.10, 1220 Fact 15.17.10, 1222 Fact 15.19.9, 1227 sum of integer powers Fact 15.12.4, 1205 symmetric matrix Prop. 15.4.8, 1188 symplectic matrix Prop. 15.6.7, 1192 thermodynamic inequality Fact 15.15.35, 1217 trace Cor. 15.2.4, 1183 Cor. 15.2.5, 1183 Fact 10.17.24, 791 Fact 15.12.6, 1206 Fact 15.15.4, 1212 Fact 15.15.11, 1213 Fact 15.15.30, 1216 Fact 15.15.32, 1216 Fact 15.15.33, 1216 Fact 15.15.34, 1216 Fact 15.15.35, 1217 Fact 15.15.40, 1217 Fact 15.15.42, 1217 Fact 15.16.4, 1218 Fact 15.16.5, 1218 Fact 15.17.1, 1220

1499

Fact 15.17.4, 1221 transpose Prop. 15.2.8, 1184 unipotent matrix Fact 15.14.14, 1210 unitarily invariant norm Fact 15.16.6, 1218 Fact 15.17.4, 1221 Fact 15.17.7, 1222 Fact 15.17.15, 1223 Fact 15.17.18, 1223 Fact 15.17.19, 1223 unitarily similar matrices Prop. 15.2.9, 1185 unitary matrix Prop. 15.2.8, 1184 Prop. 15.4.3, 1187 Prop. 15.6.7, 1192 Cor. 15.2.6, 1183 Fact 15.15.7, 1212 Fact 15.15.37, 1217 Fact 15.15.38, 1217 upper triangular matrix Fact 15.12.4, 1205 Fact 15.14.1, 1209 Fact 15.14.13, 1210 vibration equation Example 15.3.7, 1186 weak majorization Fact 15.17.4, 1221 Z-matrix Fact 15.20.1, 1232 Zassenhaus product formula Fact 15.15.19, 1214 matrix function definition, 931 identity theorem Thm. 12.8.3, 932 Lagrange-Hermite interpolation polynomial Thm. 12.8.2, 931 spectrum Cor. 12.8.4, 932 matrix function defined at a point definition Defn. 12.8.1, 931 matrix inequality matrix logarithm Prop. 10.6.4, 715 matrix inverse integral Fact 15.14.5, 1210 matrix inversion lemma generalization Fact 3.20.15, 349 generalized inverse Fact 8.4.13, 635 inverse matrix Cor. 3.9.8, 304 matrix logarithm chaotic ordering Fact 10.23.1, 813 complex matrix

1500

matrix measure

Defn. 15.4.1, 1187 convergent series Thm. 15.5.2, 1190 convex function Prop. 10.6.17, 718 definition, 714 determinant Fact 10.22.42, 813 Fact 11.9.56, 866 Fact 15.15.25, 1215 determinant and derivative Prop. 12.10.3, 933 discrete-time Lyapunov-stable matrix Fact 15.15.20, 1214 eigenvalues Thm. 15.5.2, 1190 exponential Fact 15.15.28, 1215 geometric mean Fact 15.15.43, 1217 Hamiltonian matrix Fact 15.15.20, 1214 Klein’s inequality Fact 15.15.27, 1215 limit Prop. 10.6.4, 715 matrix exponential Thm. 15.5.2, 1190 Prop. 15.4.2, 1187 Fact 15.14.14, 1210 Fact 15.14.15, 1211 Fact 15.15.22, 1214 Fact 15.15.35, 1217 matrix inequality Prop. 10.6.4, 715 maximum singular value Fact 10.22.42, 813 nonsingular matrix Prop. 15.4.2, 1187 norm Thm. 15.5.2, 1190 positive-definite matrix Prop. 10.6.4, 715 Prop. 15.4.10, 1189 Fact 10.10.50, 735 Fact 10.15.11, 775 Fact 10.22.41, 813 Fact 10.22.43, 813 Fact 10.23.1, 813 Fact 11.10.87, 883 Fact 15.15.25, 1215 Fact 15.15.27, 1215 Fact 15.15.28, 1215 Fact 15.15.29, 1215 positive-semidefinite matrix Fact 10.22.42, 813 Fact 11.10.86, 883 quadratic form Fact 10.18.19, 794 real matrix Prop. 15.4.5, 1188 Cor. 15.4.6, 1188

Cor. 15.4.7, 1188 Fact 15.15.20, 1214 relative entropy Fact 15.15.27, 1215 Schur product Fact 10.25.65, 829 spectrum Thm. 15.5.2, 1190 strong majorization Fact 10.22.43, 813 symplectic matrix Fact 15.15.20, 1214 trace Fact 15.15.25, 1215 Fact 15.15.27, 1215 Fact 15.15.29, 1215 Fact 15.15.35, 1217 twisted logarithm Fact 15.14.15, 1211 unitarily invariant norm Fact 11.10.86, 883 matrix measure properties Fact 15.16.7, 1218 matrix polynomial definition, 502 matrix power discrete-time asymptotically stable matrix Fact 11.13.11, 892 Fact 15.22.16, 1241 outer-product perturbation Fact 3.15.24, 327 positive-definite matrix Fact 10.23.2, 814 positive-definite matrix inequality Fact 10.11.81, 746 positive-semidefinite matrix Fact 10.14.45, 771 Fact 10.18.20, 794 spectral radius Fact 11.13.11, 892 Fact 15.22.16, 1241 submultiplicative norm Fact 11.13.11, 892 Fact 15.22.16, 1241 matrix product lower triangular matrix Fact 4.25.2, 424 normal matrix Fact 11.10.32, 872 strictly lower triangular matrix Fact 4.25.2, 424 strictly upper triangular matrix Fact 4.25.2, 424 unitarily invariant norm Fact 11.10.32, 872 upper triangular matrix Fact 4.25.2, 424

matrix sign function

convergent sequence Fact 7.17.23, 608 definition Defn. 12.9.2, 932 partitioned matrix Fact 12.15.3, 948 positive-definite matrix Fact 12.15.4, 948 properties Fact 12.15.2, 948 square root Fact 7.17.23, 608 matrix sine matrix cosine Fact 15.13.1, 1209 matrix exponential Fact 15.13.1, 1209 matrix sum rank Fact 3.13.40, 319 Fact 3.13.44, 320 maximal solution Riccati equation Defn. 16.16.12, 1297 Thm. 16.18.1, 1302 Thm. 16.18.4, 1302 Prop. 16.18.2, 1302 Prop. 16.18.7, 1304 maximal solution of the Riccati equation closed-loop spectrum Prop. 16.18.2, 1302 stabilizability Thm. 16.18.1, 1302 maximization continuous function Fact 12.13.8, 942 maximum finite sum Fact 1.12.20, 43 inequality Fact 2.6.6, 184 maximum eigenvalue commutator Fact 11.11.7, 885 Fact 11.11.8, 885 Hermitian matrix Lem. 10.4.3, 709 Fact 7.12.9, 580 Fact 10.11.12, 736 matrix exponential Fact 15.17.4, 1221 positive-semidefinite matrix Fact 10.22.17, 807 Fact 10.22.18, 808 quadratic form Lem. 10.4.3, 709 spectral abscissa Fact 7.12.9, 580 unitarily invariant norm Fact 11.11.7, 885 Fact 11.11.8, 885

maximum modulus principle

complex function Fact 12.14.8, 946 maximum singular value absolute value Fact 11.15.12, 897 adjugate Fact 10.21.23, 804 Fact 11.15.16, 897 Fact 11.15.17, 897 block-diagonal matrix Fact 7.12.37, 585 block-triangular matrix Fact 7.12.36, 585 bound Fact 10.10.24, 732 commutator Fact 11.11.6, 885 Fact 11.16.8, 900 complex conjugate transpose Fact 10.21.8, 801 Fact 10.22.17, 807 Fact 10.25.12, 822 condition number Fact 11.15.2, 896 Cordes inequality Fact 10.22.36, 812 derivative Fact 15.16.2, 1218 determinant Fact 11.16.21, 903 Fact 11.16.22, 903 discrete-time Lyapunov-stable matrix Fact 15.22.23, 1242 dissipative matrix Fact 10.21.18, 803 eigenvalue Fact 7.12.33, 584 Fact 11.14.4, 893 Fact 11.14.5, 893 eigenvalue of Hermitian part Fact 7.12.25, 582 eigenvalue perturbation Fact 11.14.6, 893 elementary projector Fact 11.16.10, 901 equi-induced self-adjoint norm Fact 11.15.7, 896 equi-induced unitarily invariant norm Fact 11.15.6, 896 Euclidean norm Fact 11.15.3, 896 generalized inverse Fact 11.16.7, 900 Fact 11.16.41, 907 Hermitian matrix Fact 7.12.9, 580 H¨older-induced norm Fact 11.9.25, 863

minimal polynomial idempotent matrix Fact 7.12.41, 585 Fact 7.12.42, 586 Fact 7.13.28, 595 Fact 8.7.7, 651 induced lower bound Cor. 11.5.5, 846 induced norm Fact 11.9.28, 863 inequality Prop. 11.2.2, 837 Cor. 11.6.5, 848 Cor. 11.6.11, 849 Fact 11.10.22, 872 Fact 11.16.20, 903 inverse matrix Fact 11.16.7, 900 Kreiss matrix theorem Fact 15.22.23, 1242 Kronecker product Fact 11.10.95, 884 matrix difference Fact 10.22.15, 807 Fact 11.10.22, 872 matrix exponential Fact 15.16.1, 1217 Fact 15.16.2, 1218 Fact 15.16.5, 1218 Fact 15.17.8, 1222 Fact 15.17.12, 1222 matrix logarithm Fact 10.22.42, 813 matrix power Fact 10.22.36, 812 Fact 11.15.9, 896 Fact 11.15.11, 897 normal matrix Fact 7.12.31, 584 Fact 7.15.16, 599 Fact 11.9.16, 861 Fact 11.14.6, 893 Fact 11.15.9, 896 Fact 11.15.10, 896 Fact 11.16.3, 900 Fact 11.16.4, 900 Fact 15.17.12, 1222 outer-product matrix Fact 7.12.19, 582 Fact 7.12.21, 582 Fact 11.8.25, 860 outer-product perturbation Fact 11.16.11, 901 Parrott’s theorem Fact 11.16.16, 902 partitioned matrix Fact 10.12.16, 751 Fact 10.12.17, 751 Fact 10.12.25, 753 Fact 10.12.30, 753 Fact 10.21.8, 801 Fact 11.12.4, 886 Fact 11.12.8, 888 Fact 11.12.9, 888 Fact 11.12.10, 889

Fact 11.16.15, 902 Fact 11.16.16, 902 Fact 11.16.17, 902 positive-definite matrix Fact 10.22.33, 811 positive-semidefinite matrix Fact 10.10.24, 732 Fact 10.11.32, 738 Fact 10.12.16, 751 Fact 10.12.17, 751 Fact 10.12.66, 760 Fact 10.22.15, 807 Fact 10.22.18, 808 Fact 10.22.19, 808 Fact 10.22.21, 808 Fact 10.22.33, 811 Fact 10.22.35, 812 Fact 10.22.36, 812 Fact 10.22.39, 813 Fact 10.22.42, 813 Fact 10.24.10, 816 Fact 15.17.8, 1222 power Fact 15.22.23, 1242 product Fact 11.16.1, 899 Fact 11.16.12, 901 projector Fact 7.12.41, 585 Fact 7.13.27, 594 Fact 7.13.28, 595 Fact 11.16.10, 901 Fact 11.16.41, 907 quadratic form Fact 11.15.1, 895 Fact 11.15.4, 896 resolvent Fact 11.15.16, 897 Fact 11.15.17, 897 Schur product Fact 10.25.12, 822 Fact 11.16.42, 907 Fact 11.16.43, 907 Fact 11.16.44, 907 Fact 11.16.47, 908 Fact 11.16.48, 908 Fact 11.16.49, 908 spectral abscissa Fact 7.12.28, 583 spectral radius Cor. 11.4.10, 844 Fact 7.12.9, 580 Fact 7.12.28, 583 Fact 10.22.33, 811 Fact 11.9.16, 861 Fact 11.15.11, 897 spectral variation Fact 11.14.7, 893 square root Fact 11.9.50, 865 Fact 11.16.18, 902 Fact 11.16.19, 903 sum of matrices Fact 11.16.18, 902

Fact 11.16.19, 903 trace Fact 7.13.15, 593 trace norm Cor. 11.3.8, 840 unitarily invariant norm Fact 11.10.34, 873 Fact 11.11.6, 885 maximum singular value bound Frobenius norm Fact 11.15.15, 897 minimum singular value bound Fact 11.15.18, 898 polynomial root Fact 11.15.18, 898 trace Fact 11.15.15, 897 McCarthy inequality positive-semidefinite matrix Fact 10.14.36, 769 Schatten norm Fact 11.10.53, 876 McIntosh’s inequality unitarily invariant norm Fact 11.10.72, 880 Fact 11.10.79, 881 McLaughlin’s inequality refined Cauchy-Schwarz inequality Fact 2.12.36, 221 McMillan degree Defn. 6.7.10, 515 minimal realization Thm. 16.9.24, 1278 Fact 16.21.15, 1308 mean Laguerre-Samuelson inequality Fact 2.11.37, 195 Fact 10.10.40, 734 quadratic form Fact 14.12.4, 1172 standard deviation Fact 2.11.38, 196 variance Fact 2.11.37, 195 Fact 10.10.40, 734 mean-value inequality product of means Fact 2.11.136, 212 Fact 2.11.137, 213 Fact 2.11.143, 214 mean-value theorem integral Fact 12.13.15, 944 mediant inequality Fact 2.12.6, 215 scalar inequality Fact 2.4.1, 177 meet operator definition, 118

1501

Mercator’s series

series Fact 13.4.14, 1017 Mersenne prime prime Fact 1.11.13, 30 Mertens’s theorem Cauchy product Fact 13.5.8, 1022 Metzler matrix definition, 440 ˘ Mihailescu Catalan’s conjecture Fact 1.11.41, 35 Mihet polynomial bound Fact 15.21.26, 1238 Milne’s inequality harmonic mean Fact 2.12.31, 220 refined Cauchy-Schwarz inequality Fact 2.12.29, 220 Fact 2.12.30, 220 Milnor simultaneous diagonalization Fact 10.20.8, 799 MIMO transfer function definition Defn. 16.9.1, 1270 min-max theorem eigenvalue Fact 7.12.7, 579 Minc-Sathre inequality factorial Fact 1.13.13, 51 minimal polynomial block-diagonal matrix Lem. 7.3.6, 548 block-triangular matrix Fact 6.10.17, 534 characteristic polynomial Fact 6.9.34, 530 companion matrix Prop. 7.3.1, 546 Cor. 7.3.4, 547 Cor. 7.3.5, 548 cyclic matrix Prop. 7.7.15, 561 definition, 512 existence Thm. 6.6.1, 512 index Fact 6.9.35, 530 index of an eigenvalue Prop. 7.7.15, 561 Jordan form Prop. 7.7.15, 561 null space Cor. 15.9.6, 1199 partitioned matrix Fact 6.10.17, 534 range Cor. 15.9.6, 1199

1502

minimal realization

similar matrices Prop. 6.6.3, 513 Fact 15.24.3, 1244 Fact 15.24.4, 1244 Fact 15.24.5, 1244 Fact 15.24.6, 1245 Fact 15.24.7, 1245 Fact 15.24.8, 1246 Fact 15.24.9, 1246 Fact 15.24.10, 1246 Fact 15.24.11, 1247 spectrum Fact 7.15.3, 597 stable subspace Prop. 15.9.5, 1199 Fact 15.24.1, 1243 Fact 15.24.2, 1243 upper block-triangular matrix Fact 6.10.17, 534 minimal realization asymptotically stable matrix Defn. 16.9.25, 1278 asymptotically stable transfer function Prop. 16.9.26, 1278 balanced realization Prop. 16.9.29, 1278 block decomposition Prop. 16.9.12, 1273 controllable pair Prop. 16.9.12, 1273 Prop. 16.9.18, 1276 coprime polynomials Prop. 16.9.19, 1276 definition Defn. 16.9.16, 1276 Kalman decomposition Prop. 16.9.12, 1273 Lyapunov-stable matrix Defn. 16.9.25, 1278 Lyapunov-stable transfer function Prop. 16.9.26, 1278 Markov block-Hankel matrix Prop. 16.9.14, 1275 McMillan degree Thm. 16.9.24, 1278 Fact 16.21.15, 1308 observable pair Prop. 16.9.12, 1273 Prop. 16.9.18, 1276 pole Fact 16.24.2, 1313 Fact 16.24.13, 1315 rational transfer function Prop. 16.9.17, 1276 Fact 16.24.13, 1315 semistable matrix Defn. 16.9.25, 1278 semistable transfer function Prop. 16.9.26, 1278

minimal-rank equality

partitioned matrix Fact 8.9.10, 662 minimizer compact set Cor. 12.4.12, 922 minimum finite sum Fact 1.12.20, 43 inequality Fact 2.12.1, 215 Fact 2.12.2, 215 Fact 2.12.3, 215 Fact 2.12.4, 215 Fact 2.12.5, 215 minimum eigenvalue Hermitian matrix Lem. 10.4.3, 709 Fact 10.11.12, 736 quadratic form Lem. 10.4.3, 709 minimum integer generating function Fact 13.4.17, 1020 minimum principle eigenvalue characterization Fact 10.21.20, 804 minimum root imaginary part definition, 500 minimum root modulus definition, 500 minimum root real part definition, 500 minimum singular value determinant Fact 11.16.22, 903 eigenvalue Fact 7.12.33, 584 eigenvalue of Hermitian part Fact 7.12.25, 582 Euclidean norm Fact 11.15.3, 896 induced lower bound Cor. 11.5.5, 846 inequality Cor. 11.6.6, 848 Cor. 11.6.7, 848 Cor. 11.6.8, 848 Fact 11.15.8, 896 lower bound Fact 11.15.24, 898 quadratic form Fact 11.15.1, 895 spectral abscissa Fact 7.12.28, 583 spectral radius Fact 7.12.28, 583 minimum singular value bound maximum singular value bound Fact 11.15.18, 898

subspace polynomial root Fact 3.12.4, 311 Fact 11.15.18, 898 Minkowski’s determinant minimum spectral theorem imaginary part positive-semidefinite definition, 510 minimum spectral modulus matrix definition, 510 Cor. 10.4.15, 712 minimum spectral real part Minkowski’s inequality definition, 510 H¨older norm nonnegative matrix Lem. 11.1.3, 833 Fact 6.11.17, 542 positive-semidefinite Z-matrix matrix Fact 6.11.17, 542 Fact 10.14.36, 769 Minkowski scalar case set-defined norm Fact 2.12.51, 224 Fact 12.11.31, 937 minor, see subdeterminant Minkowski difference Mircea’s inequality Minkowski sum triangle Fact 3.12.6, 311 Fact 5.2.8, 446 Minkowski inequality Mirsky unitarily invariant norm spread of a matrix Fact 11.10.11, 870 Fact 11.13.10, 892 Minkowski product Mirsky’s theorem definition, 283 singular value Minkowski sum perturbation affine hull Fact 11.16.40, 906 Fact 3.12.2, 310 MISO transfer function affine subspace definition Fact 3.11.12, 308 Defn. 16.9.1, 1270 Fact 3.12.4, 311 Mitrinovic closed set inequality Fact 12.12.6, 937 Fact 2.11.57, 199 Fact 12.12.7, 937 mixed compact set arithmetic-geometric Fact 12.12.8, 938 mean inequality cone arithmetic mean Fact 3.12.4, 311 Fact 2.11.138, 213 conical hull ¨ mixed Holder norm Fact 3.12.2, 310 definition, 836 convex cone eigenvalue bound Fact 3.11.9, 308 Fact 11.13.8, 891 Fact 3.12.4, 311 H¨older-induced norm Fact 3.12.12, 312 Fact 11.9.39, 864 convex conical hull Fact 11.9.40, 864 Fact 3.12.2, 310 mixed ratios convex hull inequality Fact 3.12.2, 310 Fact 2.12.20, 218 Fact 3.12.5, 311 ML-matrix Fact 12.12.12, 938 definition, 440 convex set ¨ Mobius transformation Fact 3.11.2, 306 complex function Fact 3.11.3, 306 Fact 12.14.6, 946 Fact 3.12.4, 311 modulo Fact 12.12.11, 938 congruence definition, 278 Fact 1.11.9, 29 dual cone Fact 1.11.10, 29 Fact 12.12.15, 939 Fact 1.11.11, 30 open set Fact 1.11.12, 30 Fact 12.12.5, 937 integer partitioned matrix Prop. 1.5.1, 12 Fact 3.12.9, 312 residue classes relative interior Fact 1.11.10, 29 Fact 12.12.12, 938 sum of digits span Fact 1.11.11, 30 Fact 3.12.2, 310 Moler

Newton-Raphson algorithm

1503

regular pencil existence asymptotically stable multirelation Fact 7.19.4, 617 Thm. 7.3.3, 547 definition, 5 matrix similar matrices Mollweide’s formula multiset Fact 15.19.30, 1231 Cor. 7.3.9, 549 triangle definition, 2 characteristic polynomial similarity invariant Fact 5.2.7, 443 multisubsets Fact 10.21.2, 800 Cor. 7.3.9, 549 Fact 1.8.5, 24 monic polynomial definition multihypercompanion form multispectrum, see definition, 499 Defn. 4.1.1, 363 definition, 550 monic polynomial matrix eigenvalue, spectrum negative-semidefinite existence definition, 501 definition matrix Thm. 7.4.5, 550 monodromy matrix Defn. 6.4.4, 506 characteristic polynomial multihypercompanion periodic dynamics properties Fact 10.21.2, 800 Fact 16.20.8, 1307 matrix Prop. 6.4.5, 507 definition monotone norm companion matrix mutual orthogonality of Defn. 4.1.1, 363 absolute norm Cor. 7.4.3, 550 Euclidean distance matrix complex matrices Prop. 11.1.2, 833 definition, 550 Fact 11.9.17, 861 definition, 288 definition, 833 example Nesbitt’s inequality mutual orthogonality of vector norm Example 7.4.7, 551 cubic inequality real matrices Fact 11.8.1, 853 Example 7.4.8, 552 Fact 2.3.60, 162 definition, 287 monotonic sequence multinomial Fact 5.2.25, 486 mutually orthogonal sets sum of powers equality scalar inequality definition, 289 Fact 1.12.43, 47 Fact 1.16.17, 94 Fact 2.3.61, 165 intersection monotonicity Fact 1.16.18, 94 Fact 2.3.62, 166 Prop. 3.3.5, 289 Erdo-Szekeres theorem multinomial coefficient Fact 2.3.63, 166 mutually orthogonal Fact 1.8.17, 25 definition, 15 Shapiro’s inequality vectors finite sequence finite product Fact 2.11.53, 198 definition, 285, 286 Fact 1.8.17, 25 Fact 2.21.24, 274 sums of positive numbers power inequality multinomial theorem Fact 2.3.64, 166 N Fact 2.2.31, 136 power of sum Fact 2.3.65, 166 power-mean inequality Fact 2.3.1, 146 Fact 2.11.57, 199 N-matrix Fact 2.2.58, 141 Fact 2.11.8, 189 almost nonnegative matrix nested radicals Fact 2.11.86, 204 multiple Fibonacci number Fact 15.20.3, 1232 Riccati equation definition, 501 Fact 12.18.70, 974 Fact 15.20.5, 1232 Prop. 16.18.5, 1303 multiple integral limit asymptotically stable Cor. 16.18.6, 1303 exponential function Fact 12.18.61, 972 matrix monotonicity theorem Fact 14.13.14, 1178 Fact 12.18.62, 972 Fact 15.20.5, 1232 Hermitian matrix Fact 14.13.15, 1178 Fact 12.18.63, 972 definition gamma function eigenvalue Fact 12.18.64, 972 Fact 15.20.3, 1232 Fact 14.13.10, 1177 Thm. 10.4.9, 710 Fact 12.18.65, 973 group-invertible matrix Fact 14.13.11, 1177 Fact 10.11.13, 736 Fact 12.18.66, 973 Fact 15.20.4, 1232 Fact 14.13.12, 1177 Moore-Penrose Fact 12.18.67, 973 Lyapunov-stable matrix Fact 14.13.13, 1177 Fact 12.18.69, 974 generalized inverse, Fact 15.20.4, 1232 Neuberg-Pedoe inequality see generalized inverse multiplication nonnegative matrix definition, 282 triangle Morley Fact 15.20.3, 1232 function composition Fact 5.2.22, 486 prime numbers Nagel point Thm. 3.2.1, 282 Newcomb Fact 1.16.3, 55 triangle Kronecker product simultaneous cogredient Morrie’s law Fact 5.2.11, 466 Prop. 9.1.6, 682 diagonalization, 831 trigonometric equality Nanjundiah multiplication group Newton’s identities Fact 2.16.17, 244 mixed definition elementary symmetric Muirhead’s theorem arithmetic-geometric Defn. 4.6.1, 371 strong majorization polynomial mean inequality order Fact 3.25.6, 360 Fact 2.11.34, 194 Fact 2.11.138, 213 Prop. 4.4.3, 370 multi-Jordan matrix Fact 6.8.4, 518 Napier’s inequality Prop. 4.6.3, 372 definition, 549 polynomial roots logarithmic mean rotation matrix similar matrices Fact 6.8.4, 518 Fact 2.2.63, 142 Prop. 4.6.6, 372 Thm. 7.4.2, 549 spectrum natural frequency multiplicative commutator multi-real-Jordan form Fact 6.10.12, 532 definition, 1187 realization definition, 551 Newton’s inequality Fact 7.15.32, 601 Fact 7.17.37, 609 similarity elementary symmetric necessary condition reflector realization Thm. 7.4.6, 551 polynomial definition, 3 Fact 7.17.38, 609 multi-real-Jordan matrix Fact 2.11.35, 195 necessity multiplicative perturbation definition, 551 Newton-Raphson definition, 4 small-gain theorem multicompanion form algorithm negative entries Fact 11.15.26, 899 definition, 547 generalized inverse nonnegative matrix multiplicity of a root multicompanion matrix Fact 8.3.38, 632 Fact 6.11.7, 540 definition, 499 definition, 547 inverse matrix negative-definite matrix

1504

Niculescu’s inequality

Fact 3.20.22, 350 square root Fact 7.17.23, 608 Niculescu’s inequality absolute value Fact 2.2.43, 138 convex function Fact 1.21.8, 117 square-root function Fact 2.2.48, 139 nilpotent matrix additive decomposition Fact 7.10.4, 570 adjugate Fact 3.19.5, 346 Fact 8.3.21, 630 commutator Fact 4.22.10, 419 Fact 4.22.13, 420 Fact 4.22.14, 420 Fact 4.22.15, 420 Fact 7.16.4, 603 commuting matrices Fact 4.22.11, 420 Fact 4.22.12, 420 defective matrix Fact 7.15.19, 600 definition Defn. 4.1.1, 363 determinant Fact 4.22.11, 420 example Example 7.7.17, 561 factorization Fact 7.17.31, 609 idempotent matrix Fact 4.22.10, 419 identity-matrix perturbation Fact 4.22.7, 419 Fact 4.22.8, 419 inertia Fact 7.9.4, 565 Jordan-Chevalley decomposition Fact 7.10.4, 570 Kronecker product Fact 9.4.22, 687 Kronecker sum Fact 9.5.4, 691 Fact 9.5.9, 692 lower triangular matrix Fact 4.22.13, 420 matrix exponential Fact 15.12.1, 1204 Fact 15.14.14, 1210 matrix sum Fact 4.22.12, 420 nonsingular matrix Fact 4.22.9, 419 null space Fact 4.22.1, 418 Fact 4.22.2, 419 Fact 4.22.3, 419 outer-product matrix

Fact 7.15.5, 598 partitioned matrix Fact 4.15.21, 398 Fact 7.11.28, 579 range Fact 4.22.1, 418 Fact 4.22.2, 419 Fact 4.22.3, 419 rank Fact 4.22.4, 419 Fact 4.22.5, 419 S-N decomposition Fact 7.10.4, 570 similar matrices Prop. 4.7.5, 374 Fact 7.11.28, 579 simultaneous triangularization Fact 7.19.7, 617 spectrum Prop. 7.7.21, 562 Toeplitz matrix Fact 4.23.6, 421 trace Fact 4.22.6, 419 triangular matrix Fact 7.19.7, 617 unitarily similar matrices Prop. 4.7.5, 374 upper triangular matrix Fact 4.22.13, 420 node definition, 9 nondecreasing function convex function Lem. 10.6.16, 718 Fact 1.21.1, 116 definition Defn. 10.6.12, 716 function composition Lem. 10.6.16, 718 geometric mean Fact 10.11.68, 743 Fact 10.11.71, 744 log majorization Fact 3.25.13, 361 matrix exponential Fact 10.11.71, 744 matrix functions Prop. 10.6.13, 717 Schur complement Prop. 10.6.13, 717 weak majorization Fact 3.25.10, 361 nondecreasing sequence definition, 917 nondecreasing sequence of sets definition, 9 limit Prop. 1.3.21, 9 nonderogatory eigenvalue definition Defn. 7.7.4, 559 nonderogatory matrix

definition Fact 11.13.9, 892 Defn. 7.7.4, 559 Fact 15.20.3, 1232 spectral radius convexity nonempty set Fact 6.11.26, 544 definition, 1 spectral radius nonincreasing function concave function monotonicity Lem. 10.6.16, 718 Fact 6.11.23, 543 definition submatrix Defn. 10.6.12, 716 Fact 6.11.24, 544 function composition trace Lem. 10.6.16, 718 Fact 6.11.29, 544 product of functions nonnegative matrix Fact 12.13.20, 945 eigenvalues sum of functions Perron-Frobenius theorem Fact 12.13.20, 945 Fact 6.11.5, 538 nonincreasing sequence nonnegative numbers definition, 917 inequality nonincreasing sequence of Fact 2.3.12, 149 sets nonnegative vector definition, 9 definition, 277 limit linear system solution Prop. 1.3.21, 9 Fact 6.11.19, 543 nonnegative matrix null space almost nonnegative matrix Fact 6.11.20, 543 Fact 15.20.1, 1232 nonsingular matrix aperiodic graph complex conjugate Fact 6.11.5, 538 Prop. 3.7.11, 298 companion matrix complex conjugate Fact 6.11.18, 543 transpose copositive matrix Prop. 3.7.11, 298 Fact 10.19.24, 798 Fact 3.20.23, 350 definition, 281 controllable subspace Defn. 4.1.5, 367 Prop. 16.6.10, 1263 discrete-time cyclic matrix asymptotically stable Fact 7.15.10, 599 definition, 297 matrix determinant Fact 15.22.2, 1239 Cor. 3.8.4, 301 eigenvalue Lem. 3.9.6, 303 Fact 6.11.5, 538 determinant lower bound index of imprimitivity Fact 6.10.24, 535 Fact 6.11.5, 538 diagonal dominance limit of matrix powers Fact 6.11.28, 544 theorem M-matrix Fact 6.10.23, 535 Fact 6.11.13, 541 Fact 6.10.24, 535 matrix power diagonally dominant Fact 6.11.29, 544 matrix minimum eigenvalue Fact 6.10.23, 535 Fact 6.11.17, 542 dissipative matrix N-matrix Fact 4.27.1, 427 Fact 15.20.3, 1232 distance to singularity negative entries Fact 11.16.6, 900 Fact 6.11.7, 540 elementary matrix product Fact 7.17.14, 606 Fact 3.15.36, 329 factorization Schur power Fact 7.17.14, 606 Fact 9.6.19, 698 Fact 7.17.39, 609 spectral radius group Fact 6.11.5, 538 Prop. 4.6.6, 372 Fact 6.11.13, 541 idempotent matrix Fact 6.11.21, 543 Fact 4.15.13, 397 Fact 6.11.22, 543 Fact 4.16.9, 401 Fact 6.11.24, 544 Fact 4.16.10, 402 Fact 6.11.25, 544 matrix logarithm Fact 9.6.18, 698 Prop. 15.4.2, 1187

normal matrix norm Fact 11.7.16, 852 open set Cor. 12.4.21, 923 partitioned matrix Fact 3.17.29, 340 perturbation Fact 11.16.5, 900 Fact 11.16.22, 903 positive-semidefinite matrix Fact 10.11.11, 736 quadratic form Fact 10.18.9, 792 Fact 10.18.10, 793 range-Hermitian matrix Prop. 4.1.7, 367 similar matrices Fact 7.11.14, 577 simplex Fact 5.1.8, 442 spectral radius Fact 6.10.36, 537 submultiplicative norm Fact 11.9.4, 860 Sylvester’s equation Fact 16.22.14, 1312 transpose Prop. 3.7.11, 298 unobservable subspace Prop. 16.3.10, 1255 weak diagonal dominance theorem Fact 6.10.25, 535 nonsingular matrix transformation Smith polynomial Prop. 6.3.9, 505 nonsingular polynomial matrix Defn. 6.2.5, 502 regular polynomial matrix Prop. 6.2.5, 502 nonzero diagonal entry similar matrices Fact 7.10.15, 571 norm absolute definition, 833 absolute sum definition, 834 column definition, 844 compatible definition, 838 complex conjugate transpose Fact 11.9.7, 860 convex set Fact 11.7.1, 849 Dunkl-Williams inequality Fact 11.7.11, 851 Fact 11.7.13, 852 equi-induced Defn. 11.4.1, 841

1505

spectral radius Fact 11.7.10, 850 equivalent Prop. 11.2.6, 838 Fact 11.7.12, 851 Thm. 11.1.8, 835 submultiplicative von Neumann–Jordan Euclidean definition, 838 definition, 286 inequality Euclidean-norm inequality trace Fact 11.7.4, 849 definition, 838 Fact 11.8.3, 853 norm monotonicity triangle inequality Fact 11.8.10, 857 power-sum inequality Defn. 11.1.1, 833 Fact 11.8.20, 859 Fact 2.2.59, 141 unitarily invariant Frobenius norm-compression definition, 837 definition, 836 inequality vector H¨older-norm inequality partitioned matrix Defn. 11.1.1, 833 Fact 11.8.20, 859 Fact 11.12.4, 886 weakly unitarily invariant idempotent matrix Fact 11.12.14, 890 definition, 837 Fact 15.17.14, 1223 positive-semidefinite induced norm bound matrix Defn. 11.4.1, 841 matrix exponential Fact 11.12.12, 889 induced norm Fact 15.19.10, 1227 normal matrix Thm. 11.4.2, 841 norm compression affine mapping infinity positive-semidefinite Fact 4.10.14, 380 definition, 834 matrix asymptotically stable linear combination of Fact 10.12.66, 760 matrix norms norm equality Fact 15.19.37, 1231 Fact 11.7.2, 849 common eigenvector block-diagonal matrix linear system solution Fact 11.10.52, 876 Fact 4.10.7, 378 Fact 11.17.1, 909 Hlawka’s equality Cartesian decomposition Fact 11.17.2, 909 Fact 11.8.5, 856 Fact 7.20.2, 618 Fact 11.17.3, 909 polarization identity Fact 15.14.4, 1209 Maligranda inequality Fact 11.8.3, 853 characterizations Fact 11.7.11, 851 Pythagorean theorem Fact 4.10.12, 379 Massera-Schaffer Fact 11.8.3, 853 commutator Schatten norm inequality Fact 4.29.12, 429 Fact 11.10.52, 876 Fact 11.7.11, 851 Fact 11.11.8, 885 norm inequality matrix commuting matrices Aczel’s inequality Defn. 11.2.1, 835 Fact 7.16.16, 604 Fact 11.8.3, 853 matrix exponential Fact 7.19.8, 617 Bessel’s inequality Fact 15.17.11, 1222 Fact 7.20.2, 618 Fact 11.8.10, 857 Fact 15.17.13, 1222 Fact 7.20.3, 618 Buzano’s inequality Fact 15.17.14, 1223 Fact 15.15.6, 1212 Fact 11.8.5, 856 matrix logarithm complex conjugate convex combination Thm. 15.5.2, 1190 transpose Fact 11.7.6, 850 monotone Fact 7.16.17, 604 Hlawka’s inequality definition, 833 Fact 8.6.2, 650 Fact 11.8.5, 856 nonsingular matrix Fact 8.10.17, 672 H¨older norm Fact 11.7.16, 852 definition Fact 11.8.23, 859 normalized Defn. 4.1.1, 363 orthogonal vectors definition, 837 determinant Fact 11.7.21, 853 partitioned matrix Fact 7.13.24, 594 Parseval’s inequality Fact 11.12.4, 886 discrete-time Fact 11.8.10, 857 Fact 11.12.5, 887 asymptotically stable polygonal inequalities Fact 11.12.6, 887 matrix Fact 11.8.5, 856 Fact 11.12.14, 890 Fact 15.22.5, 1240 quadrilateral inequality positive-definite matrix discrete-time Fact 11.8.5, 856 Fact 11.7.17, 852 Lyapunov-stable matrix Schatten norm quadratic form Fact 15.22.5, 1240 Fact 11.10.65, 879 Fact 11.7.17, 852 dissipative matrix Fact 11.10.66, 879 real and complex Fact 15.19.37, 1231 Fact 11.10.67, 879 Fact 11.9.38, 864 eigenvalue Selberg’s inequality row Fact 7.15.16, 599 Fact 11.8.11, 857 definition, 844 Fact 10.13.2, 761 unitarily invariant norm self-adjoint eigenvalue perturbation Fact 11.10.79, 881 definition, 837 Fact 11.14.6, 893 vector inequality set-defined eigenvector Fact 11.7.3, 849 Fact 12.11.31, 937 Prop. 6.5.4, 511 Fact 11.7.4, 849 spectral Lem. 6.5.3, 511 Fact 11.7.6, 850 definition, 838

1506

normal rank

example Example 7.7.17, 562 Frobenius norm Fact 11.9.18, 861 Fact 11.14.13, 894 Fact 11.14.14, 894 Fact 11.15.22, 898 Fact 11.15.23, 898 generalized inverse Prop. 8.1.7, 622 Fact 8.5.7, 642 Fact 8.6.1, 649 Fact 8.6.2, 650 Golden-Thompson inequality Fact 15.15.31, 1216 group-invertible matrix Fact 8.10.17, 672 Hermitian matrix Prop. 4.1.7, 367 idempotent matrix Fact 4.17.3, 407 involutory matrix Fact 7.10.10, 571 Fact 7.10.11, 571 Jordan form Fact 7.11.8, 576 Kronecker product Fact 9.4.22, 687 Kronecker sum Fact 9.5.9, 692 Lyapunov-stable matrix Fact 15.19.37, 1231 matrix exponential Prop. 15.2.8, 1184 Cor. 15.4.4, 1188 Fact 15.15.6, 1212 Fact 15.17.12, 1222 matrix power Fact 11.15.9, 896 maximum singular value Fact 7.12.31, 584 Fact 7.15.16, 599 Fact 11.9.16, 861 Fact 11.14.6, 893 Fact 11.15.9, 896 Fact 11.15.10, 896 Fact 11.16.3, 900 Fact 11.16.4, 900 Fact 15.17.12, 1222 orthogonal eigenvectors Cor. 7.5.8, 555 partial isometry Fact 8.6.1, 649 partitioned matrix Fact 4.15.21, 398 Fact 10.12.29, 753 polar decomposition Fact 15.14.4, 1209 positive-semidefinite matrix Fact 10.10.25, 732 Fact 10.11.22, 737 Fact 10.12.29, 753 Fact 10.13.2, 761

product Fact 11.10.32, 872 projector Fact 4.17.3, 407 Putnam-Fuglede theorem Fact 7.16.17, 604 quadratic form Fact 10.18.16, 793 range-Hermitian matrix Prop. 4.1.7, 367 rank subtractivity partial ordering Fact 4.30.17, 431 reflector Fact 7.10.10, 571 Fact 7.10.11, 571 Schatten norm Fact 11.11.4, 884 Fact 11.16.4, 900 Schur decomposition Cor. 7.5.4, 553 Fact 7.11.8, 576 Schur product Fact 11.10.9, 870 Schur’s inequality Fact 11.13.2, 890 semicontractive matrix Fact 15.19.37, 1231 semidissipative matrix Fact 15.19.37, 1231 semisimple matrix Prop. 7.7.12, 560 shifted-unitary matrix Fact 4.13.30, 391 similar matrices Prop. 7.7.12, 560 Fact 7.10.10, 571 Fact 7.10.11, 571 Fact 7.11.10, 576 similarity transformation Fact 7.17.4, 605 singular value Fact 7.15.16, 599 Fact 10.13.2, 761 skew-Hermitian matrix Prop. 4.1.7, 367 spectral radius Fact 7.15.16, 599 Fact 11.15.22, 898 Fact 11.15.23, 898 spectral variation Fact 11.14.8, 893 Fact 11.14.9, 894 Fact 11.14.11, 894 Fact 11.14.12, 894 spectrum Fact 6.10.30, 536 Fact 7.12.30, 583 Fact 10.17.8, 787 Fact 10.17.9, 788 square root Fact 10.10.31, 733 Fact 10.10.33, 733 trace Fact 4.10.12, 379

Fact 10.13.17, 763 Fact 10.14.2, 764 trace of product Fact 7.13.12, 592 transpose Fact 7.10.10, 571 Fact 7.10.11, 571 tripotent matrix Fact 4.21.4, 418 unitarily invariant norm Fact 11.10.32, 872 unitarily similar matrices Prop. 4.7.5, 374 Cor. 7.5.4, 553 Fact 7.11.8, 576 Fact 7.11.10, 576 unitary matrix Prop. 4.1.7, 367 Fact 4.13.6, 388 Fact 7.17.2, 605 normal rank definition for a polynomial matrix Defn. 6.2.4, 502 definition for a rational transfer function Defn. 6.7.4, 514 rank Prop. 6.3.6, 504 Prop. 6.3.7, 504 rational transfer function, 544 normal subgroup definition Defn. 4.4.1, 369 Defn. 4.6.1, 371 quotient group Thm. 4.4.5, 370 normalized norm definition, 837 equi-induced norm Thm. 11.4.2, 841 normalized submultiplicative norm inverse matrix Fact 11.9.62, 868 Fact 11.10.88, 883 Fact 11.10.89, 883 Fact 11.10.90, 883 Fact 11.10.91, 883 not definition, 2 nuclear norm trace norm, 912 null space adjugate Fact 3.19.2, 346 definition, 291 Drazin generalized inverse Prop. 8.2.2, 626 generalized inverse Prop. 8.1.7, 622 Fact 8.9.2, 659 Fact 8.9.3, 659 group generalized inverse

Prop. 8.2.3, 627 group-invertible matrix Fact 4.9.2, 376 idempotent matrix Fact 4.15.5, 396 inclusion Fact 3.13.10, 315 Fact 3.13.12, 315 inclusion for a matrix power Cor. 3.5.2, 291 inclusion for a matrix product Lem. 3.5.1, 291 Fact 3.13.3, 315 intersection Fact 3.13.14, 316 Fact 3.13.16, 316 involutory matrix Fact 4.20.8, 417 left-equivalent matrices Prop. 7.1.2, 545 Lyapunov equation Fact 16.22.15, 1312 matrix sum Fact 3.13.17, 316 minimal polynomial Cor. 15.9.6, 1199 nilpotent matrix Fact 4.22.1, 418 Fact 4.22.2, 419 Fact 4.22.3, 419 outer-product matrix Fact 3.13.18, 316 partitioned matrix Fact 3.14.5, 320 Fact 3.14.6, 320 Fact 3.14.10, 321 Fact 8.9.2, 659 Fact 8.9.3, 659 positive-semidefinite matrix Fact 10.7.3, 722 Fact 10.7.5, 722 Fact 10.18.2, 792 power Fact 3.13.27, 317 quadratic form Fact 10.18.2, 792 range Cor. 3.6.8, 293 Cor. 3.6.9, 293 Fact 3.13.2, 315 range inclusion Thm. 3.5.3, 291 range-Hermitian matrix Fact 4.9.6, 377 semisimple eigenvalue Prop. 7.7.8, 559 skew-Hermitian matrix Fact 10.7.3, 722 sum Fact 3.13.16, 316 Fact 3.13.34, 319 symmetric matrix

open annulus Fact 4.10.3, 377

Prop. 16.3.13, 1256 unobservable spectrum Prop. 16.3.16, 1256 observable canonical form defect of a submatrix definition, 1270 Fact 3.14.27, 325 equivalent realizations partitioned matrix Cor. 16.9.22, 1277 Fact 11.16.13, 901 Cor. 16.9.23, 1278 numerical radius weakly unitarily invariant realization Prop. 16.9.3, 1270 norm Prop. 16.9.20, 1277 Fact 11.9.55, 866 observable dynamics numerical range block-triangular matrix spectrum of convex hull Thm. 16.3.8, 1255 Fact 10.17.8, 787 orthogonal matrix Fact 10.17.9, 788 Thm. 16.3.8, 1255 observable eigenvalue O closed-loop spectrum Lem. 16.16.16, 1298 oblique projector, see definition idempotent matrix Defn. 16.3.11, 1255 observability observable subspace closed-loop spectrum Prop. 16.3.17, 1256 Lem. 16.16.17, 1298 observable pair PBH test asymptotically stable Thm. 16.3.19, 1257 matrix Riccati equation Prop. 16.4.9, 1258 Lem. 16.16.18, 1298 Cor. 16.4.10, 1258 Sylvester’s equation eigenvalue placement Fact 16.22.14, 1312 Prop. 16.3.20, 1257 observability Gramian equivalent realizations asymptotically stable Prop. 16.9.8, 1272 matrix Prop. 16.9.11, 1273 Cor. 16.4.10, 1258 invariant zero H2 norm Cor. 16.10.12, 1285 Cor. 16.11.4, 1286 Markov block-Hankel L2 norm matrix Thm. 16.11.1, 1285 Prop. 16.9.13, 1275 observably asymptotically minimal realization stable Prop. 16.9.12, 1273 Prop. 16.4.3, 1257 Prop. 16.9.18, 1276 Prop. 16.4.4, 1258 observability matrix Prop. 16.4.5, 1258 Thm. 16.3.18, 1256 Prop. 16.4.6, 1258 Fact 16.21.21, 1309 Prop. 16.4.7, 1258 positive-definite matrix observability matrix Thm. 16.3.18, 1256 definition, 1254 similarity transformation generalized inverse Prop. 16.9.10, 1273 Fact 16.21.21, 1309 observable subspace Lyapunov equation observable eigenvalue Fact 16.22.15, 1312 Prop. 16.3.17, 1256 observable pair observably asymptotically Thm. 16.3.18, 1256 stable Fact 16.21.21, 1309 asymptotically stable rank matrix Cor. 16.3.3, 1254 Prop. 16.5.6, 1259 Sylvester’s equation block-triangular matrix Fact 16.22.13, 1311 Prop. 16.4.3, 1257 observability pencil definition definition Defn. 16.4.1, 1257 Defn. 16.3.12, 1256 detectability Smith form Prop. 16.5.6, 1259 Prop. 16.3.15, 1256 Lyapunov equation Smith zeros Prop. 16.4.3, 1257 Prop. 16.3.16, 1256 observability Gramian unobservable eigenvalue nullity, see defect nullity theorem

Prop. 16.4.3, 1257 Prop. 16.4.4, 1258 Prop. 16.4.5, 1258 Prop. 16.4.6, 1258 Prop. 16.4.7, 1258 orthogonal matrix Prop. 16.4.3, 1257 output injection Prop. 16.4.2, 1257 rank Prop. 16.4.4, 1258 obtuse triangle inequality Fact 5.2.13, 480 octahedral group group Fact 4.31.16, 435 octonions inequality Fact 2.8.1, 187 real matrix representation Fact 4.32.1, 437 odd integers sum of integers Fact 1.12.18, 42 odd permutation matrix definition Defn. 4.1.1, 363 odd polynomial asymptotically stable polynomial Fact 15.18.8, 1224 definition, 500 oddtown theorem set intersection Fact 1.8.11, 24 off-diagonal entry definition, 280 off-diagonally located block definition, 281 OIUD open inside unit disk definition, 14 OLHP open left half plane definition, 14 one-sided cone definition, 278 induced by antisymmetric relation Prop. 3.1.7, 280 positive-semidefinite matrix, 703 quadratic form Fact 10.17.17, 789 one-sided directional differential convex function Prop. 12.5.1, 924 definition, 924 example Fact 12.16.14, 951 homogeneity Fact 12.16.11, 950

1507

one-to-one

composition Prop. 1.6.2, 16 definition, 16 inverse function Thm. 1.6.5, 17 one-to-one and onto function Schroeder-Bernstein theorem Fact 1.10.7, 27 one-to-one function compact set Cor. 12.4.14, 922 composition of functions Fact 1.10.6, 27 equivalent statements Fact 1.10.3, 26 Fact 1.10.4, 27 finite domain Fact 1.10.1, 26 inverse image Fact 1.10.5, 27 onto function Fact 1.10.7, 27 one-to-one matrix equivalent properties Thm. 3.7.1, 294 nonsingular equivalence Cor. 3.7.7, 297 ones matrix definition, 285 rank Fact 3.13.25, 317 Ono’s inequality acute triangle Fact 5.2.13, 480 triangle Fact 5.2.13, 480 onto composition Prop. 1.6.2, 16 inverse function Thm. 1.6.5, 17 onto function composition of functions Fact 1.10.6, 27 definition, 16 equivalent statements Fact 1.10.4, 27 finite domain Fact 1.10.1, 26 inverse image Fact 1.10.5, 27 one-to-one function Fact 1.10.7, 27 onto matrix equivalent properties Thm. 3.7.1, 294 nonsingular equivalence Cor. 3.7.7, 297 OOUD open outside unit disk definition, 14 open annulus

1508

open ball

definition, 928 open ball

bounded set Fact 12.11.2, 935 Fact 12.11.3, 935 completely solid set Fact 12.11.14, 935 convex set Fact 12.11.14, 935 inner product Fact 11.7.18, 852 Jung’s theorem Fact 12.11.3, 935 open ball of radius ε definition Defn. 12.1.1, 913 open half space affine open half space Fact 3.11.13, 308 cone Fact 3.11.7, 308 definition, 290 open mapping definition Defn. 12.4.16, 923 interior Thm. 12.4.17, 923 Thm. 12.4.18, 923 open set image Cor. 12.4.21, 923 open mapping theorem complex function Fact 12.14.7, 946 open set image Thm. 12.4.22, 923 open outer disk definition, 928 open relative to a set continuous function Thm. 12.4.9, 922 definition Defn. 12.1.3, 913 open set Gδ set Fact 12.12.19, 939 affine subspace Fact 12.11.11, 935 complement Fact 12.11.5, 935 continuous function Thm. 12.4.19, 923 Cor. 12.4.10, 922 convex hull Fact 12.11.19, 935 definition Defn. 12.1.2, 913 dimension Fact 12.11.10, 935 intersection Fact 12.12.16, 939 invariance of domain Thm. 12.4.19, 923 Minkowski sum Fact 12.12.5, 937 nonsingular matrix

Cor. 12.4.21, 923 right-invertible matrix Thm. 12.4.22, 923 subspace Fact 12.11.11, 935 union Fact 12.12.16, 939 Oppenheim triangle Fact 5.2.19, 485 Oppenheim’s geometric inequality triangle Fact 5.2.11, 466 Oppenheim’s inequality determinant inequality Fact 10.25.33, 824 optimal 2-uniform convexity powers Fact 2.2.29, 135 Schatten norm Fact 11.10.64, 879 or definition, 2 order definition Defn. 16.9.2, 1270 multiplication group Prop. 4.4.3, 370 Prop. 4.6.3, 372 order of an element of a group definition, 370 ordered Bell number series Fact 13.1.5, 976 subset number Fact 1.19.7, 112 ordered Bell polynomial properties Fact 13.2.5, 985 ordinary differential equation existence Thm. 15.8.1, 1195 Thm. 15.8.2, 1195 uniqueness Thm. 15.8.1, 1195 Thm. 15.8.2, 1195 ORHP open right half plane definition, 14 Orlicz H¨older-induced norm Fact 11.9.36, 864 orthogonal complement definition, 289 intersection Fact 3.12.15, 312 orthogonally complementary subspaces Prop. 3.3.6, 289 Cor. 3.3.7, 289

projector Prop. 4.8.2, 374 subspace Prop. 4.8.2, 374 Fact 3.11.22, 310 Fact 3.12.14, 312 Fact 3.18.19, 345 sum Fact 3.12.15, 312 orthogonal eigenvectors normal matrix Cor. 7.5.8, 555 orthogonal matrix, see unitary matrix 2 × 2 parameterization Fact 4.14.1, 391 3 × 3 skew-symmetric matrix Fact 15.12.10, 1207 Fact 15.12.11, 1207 additive decomposition Fact 7.20.6, 618 Fact 7.20.8, 618 algebraic multiplicity Fact 7.12.2, 579 Cayley transform Fact 4.13.25, 390 Fact 4.13.26, 390 Fact 4.14.9, 394 controllable dynamics Thm. 16.6.8, 1262 controllable subspace Prop. 16.6.9, 1263 controllably asymptotically stable Prop. 16.7.3, 1266 convex combination Fact 7.20.7, 618 cross product Fact 4.12.3, 387 Fact 4.12.4, 387 Fact 4.14.9, 394 cross-product matrix Fact 15.12.12, 1207 Fact 15.12.13, 1207 Fact 15.12.14, 1207 definition Defn. 4.1.1, 363 detectability Prop. 16.5.4, 1259 determinant Fact 4.13.18, 389 Fact 4.13.19, 389 direction cosines Fact 4.14.6, 392 eigenvalue Fact 7.12.2, 579 elementary reflector Fact 7.17.17, 607 Euler parameters Fact 4.14.6, 392 Fact 4.14.8, 394 existence of transformation Fact 4.11.5, 384

factorization Fact 7.17.17, 607 Fact 7.17.18, 607 Fact 7.17.34, 609 Fact 7.17.38, 609 group Prop. 4.6.6, 372 Hadamard matrix Fact 7.18.16, 616 Hamiltonian matrix Fact 4.28.13, 428 Kronecker permutation matrix Fact 9.4.38, 688 Kronecker product Fact 9.4.22, 687 logarithm Fact 15.16.11, 1220 matrix exponential Prop. 15.4.8, 1188 Prop. 15.6.7, 1192 Fact 15.12.6, 1206 Fact 15.12.7, 1206 Fact 15.12.8, 1207 Fact 15.12.9, 1207 Fact 15.12.10, 1207 Fact 15.12.11, 1207 Fact 15.12.12, 1207 Fact 15.12.13, 1207 Fact 15.12.14, 1207 Fact 15.16.11, 1220 observable dynamics Thm. 16.3.8, 1255 observably asymptotically stable Prop. 16.4.3, 1257 orthosymplectic matrix Fact 4.28.13, 428 parameterization Fact 4.14.5, 392 Fact 4.14.6, 392 Fact 4.14.7, 394 partitioned matrix Fact 4.13.12, 388 Fact 4.13.31, 391 permutation matrix Prop. 4.1.7, 367 quaternions Fact 4.14.6, 392 reflector Fact 4.14.5, 392 Fact 7.17.34, 609 Fact 7.17.38, 609 Rodrigues’s formulas Fact 4.14.8, 394 rotation matrix Fact 4.14.5, 392 Fact 4.14.6, 392 Fact 4.14.8, 394 Fact 4.14.9, 394 Fact 4.14.10, 396 skew-symmetric matrix Fact 4.13.25, 390 Fact 4.13.26, 390 SO(3)

Palanthadalam-Madapusi Fact 4.14.2, 392 spectrum Fact 4.13.32, 391 square root Fact 10.10.30, 733 stabilizability Prop. 16.8.4, 1268 subspace Fact 4.13.2, 387 Fact 4.13.3, 388 Fact 4.13.4, 388 trace Fact 4.13.14, 388 Fact 4.13.15, 388 Fact 4.13.33, 391 Fact 4.14.4, 392 Fact 7.13.18, 593 Fact 7.13.21, 594 unobservable subspace Prop. 16.3.9, 1255 orthogonal Procrustes problem least squares Fact 11.17.11, 911 orthogonal projector, see projector orthogonal vectors norm inequality Fact 11.7.21, 853 unitary matrix Fact 4.13.9, 388 vector sum and difference Fact 3.15.2, 326 orthogonally complementary subspaces definition, 289 orthogonal complement Prop. 3.3.6, 289 Cor. 3.3.7, 289 range-Hermitian matrix Fact 4.9.6, 377 subspace Fact 3.11.22, 310 orthogonally similar matrices definition Defn. 4.7.4, 373 diagonal matrix Fact 7.10.16, 571 skew-symmetric matrix Fact 7.16.20, 604 symmetric matrix Fact 7.10.16, 571 upper block-triangular matrix Cor. 7.5.2, 553 upper triangular matrix Cor. 7.5.3, 553 orthosymplectic matrix group Prop. 4.6.6, 372 Hamiltonian matrix Fact 4.28.13, 428 orthogonal matrix

Fact 4.28.13, 428 oscillator

companion matrix Fact 7.15.32, 601 definition, 1187 oscillatory matrix totally nonnegative matrix Fact 6.11.10, 540 Ostrowski inertia of a Hermitian matrix Fact 16.22.5, 1310 quantitative form of Sylvester’s law of inertia Fact 7.9.18, 568 Ostrowski-Taussky inequality determinant Fact 10.15.3, 774 outbound Laplacian matrix adjacency matrix Thm. 4.2.2, 368 definition Defn. 4.2.1, 367 outdegree graph Defn. 1.4.3, 10 outdegree matrix adjacency matrix Fact 4.26.3, 426 definition Defn. 4.2.1, 367 row-stochastic matrix Fact 4.26.3, 426 outer product Schur product Fact 9.6.4, 697 Fact 9.6.5, 697 Fact 9.6.6, 697 Fact 9.6.7, 697 Fact 9.6.8, 697 outer-product matrix algebraic multiplicity Fact 7.15.5, 598 characteristic polynomial Fact 6.9.15, 527 Fact 6.9.17, 527 cross product Fact 4.14.9, 394 defective matrix Fact 7.15.5, 598 definition, 287 Defn. 4.1.2, 364 doublet Fact 3.13.31, 318 Fact 3.15.6, 326 equality Fact 3.15.3, 326 Fact 3.15.5, 326 existence of transformation Fact 4.11.1, 383 Frobenius norm Fact 11.8.25, 860 full-rank factorization

Prop. 7.6.7, 558 generalized inverse Fact 8.3.17, 629 group-invertible matrix Fact 7.15.5, 598 Hermitian matrix Fact 4.10.18, 380 Fact 4.11.2, 383 H¨older norm Fact 11.8.26, 860 idempotent matrix Fact 4.10.18, 380 Fact 4.15.8, 397 index of a matrix Fact 7.15.5, 598 Kronecker product Prop. 9.1.8, 682 linear independence Fact 3.15.4, 326 Fact 3.15.9, 326 matrix exponential Fact 15.12.1, 1204 matrix power Fact 3.15.7, 326 maximum singular value Fact 7.12.19, 582 Fact 7.12.21, 582 Fact 11.8.25, 860 nilpotent matrix Fact 7.15.5, 598 null space Fact 3.13.18, 316 partitioned matrix Fact 6.9.17, 527 positive-definite matrix Fact 4.11.3, 383 positive-semidefinite matrix Fact 10.10.4, 731 Fact 10.10.5, 731 Fact 10.10.6, 731 Fact 10.18.3, 792 Fact 10.18.4, 792 quadratic form Fact 11.15.5, 896 range Fact 3.13.18, 316 rank Fact 3.13.26, 317 Fact 3.13.31, 318 Fact 4.10.17, 380 Fact 4.15.8, 397 semisimple matrix Fact 7.15.5, 598 singular value Fact 7.12.20, 582 skew-Hermitian matrix Fact 4.10.17, 380 Fact 4.11.4, 383 spectral abscissa Fact 7.12.16, 581 spectral radius Fact 7.12.16, 581 spectrum Fact 6.10.5, 531

1509

Fact 7.12.16, 581 sum Fact 3.13.31, 318 trace Fact 7.15.5, 598 unitarily invariant norm Fact 11.9.57, 866 outer-product perturbation adjugate Fact 3.21.2, 351 Fact 3.21.3, 351 determinant Fact 3.21.2, 351 Fact 3.21.3, 351 elementary matrix Fact 4.10.19, 380 generalized inverse Fact 8.4.10, 634 Fact 8.4.11, 635 Hermitian matrix Fact 10.22.1, 804 inverse matrix Fact 3.21.3, 351 matrix power Fact 3.15.24, 327 maximum singular value Fact 11.16.11, 901 rank Fact 3.13.32, 318 Fact 8.4.10, 634 unitary matrix Fact 4.13.23, 389 output convergence detectability Fact 16.21.3, 1307 output equation definition, 1251 output feedback characteristic polynomial Fact 16.24.14, 1315 determinant Fact 16.24.14, 1315 invariant zero Prop. 16.10.10, 1284 output injection detectability Prop. 16.5.3, 1259 observably asymptotically stable Prop. 16.4.2, 1257 ovals of Cassini spectrum bounds Fact 6.10.27, 536 Ozeki’s inequality reversed Cauchy-Schwarz inequality Fact 2.12.45, 223

P Padoa’s inequality

cubic inequality Fact 5.2.25, 486 triangle Fact 5.2.8, 446 Palanthandalam-Madapusi

1510

parallel affine subspaces

generalized inverse Fact 16.21.21, 1309 parallel affine subspaces definition, 278 parallel interconnection definition, 1289 transfer function Prop. 16.13.2, 1289 parallel sum definition Fact 10.24.20, 818 parallelepiped definition, 290 volume Fact 5.4.3, 494 Fact 5.4.4, 494 parallelogram area Fact 5.4.4, 494 Fact 11.8.13, 857 bivector Fact 11.8.13, 857 cross product Fact 11.8.13, 857 definition, 290 ellipse Fact 5.5.7, 496 parallelogram law complex numbers Fact 2.21.8, 269 Euclidean norm Fact 11.8.17, 858 Fact 11.10.61, 877 vector equality Fact 11.8.3, 853 parallelotope definition, 290 volume Fact 5.4.3, 494 parent Defn. 1.4.1, 10 Parker equal diagonal entries by unitary similarity Fact 7.10.18, 571 Parodi polynomial root bound Fact 15.21.13, 1236 Parrott’s theorem maximum singular value Fact 11.16.16, 902 Parseval’s inequality norm inequality Fact 11.8.10, 857 Parseval’s theorem Fourier transform Fact 16.24.1, 1313 H2 norm Thm. 16.11.3, 1286 partial derivative definition, 924 partial fractions equality Fact 2.11.3, 188 Fact 2.11.4, 188

Fact 13.1.3, 975 Fact 2.11.5, 188 series Fact 2.11.6, 189 Fact 13.1.3, 975 Fact 2.11.7, 189 partitioned matrix partial fractions adjugate expansions Fact 3.17.31, 341 exponential function Cartesian product Fact 13.4.10, 1014 Fact 3.12.9, 312 hyperbolic functions characteristic polynomial Fact 13.4.12, 1016 Fact 6.9.13, 527 trigonometric functions Fact 6.9.14, 527 Fact 13.4.10, 1014 Fact 6.9.16, 527 partial isometry Fact 6.9.17, 527 definition Fact 6.9.32, 529 Defn. 4.1.1, 363 Fact 6.9.33, 530 generalized inverse column norm Fact 8.3.31, 631 Fact 11.9.22, 862 Fact 8.6.3, 650 complementary subspaces normal matrix Fact 4.16.4, 399 Fact 8.6.1, 649 complex conjugate tripotent matrix Fact 3.24.8, 358 Fact 4.21.4, 418 Fact 6.10.33, 536 partial ordering complex conjugate definition transpose Defn. 1.3.8, 7 Prop. 3.9.1, 302 generalized L¨owner Fact 8.9.27, 667 ordering complex matrix Fact 10.23.8, 815 Fact 3.24.3, 356 planar case Fact 3.24.4, 356 Fact 1.8.15, 25 Fact 3.24.5, 356 positive-semidefinite Fact 3.24.7, 357 matrix Fact 10.16.6, 777 Prop. 10.1.1, 703 concave function rank subtractivity partial Fact 10.12.58, 759 ordering contractive matrix Fact 4.30.1, 430 Fact 10.8.13, 724 Fact 4.30.2, 430 defect Fact 4.30.3, 430 Fact 3.14.10, 321 Fact 4.30.4, 430 Fact 3.14.15, 322 Fact 4.30.5, 430 Fact 3.14.17, 323 star partial ordering Fact 8.9.2, 659 Fact 4.30.6, 430 Fact 8.9.3, 659 Fact 4.30.7, 430 definition, 281 Fact 4.30.8, 431 determinant Fact 4.30.9, 431 Prop. 10.2.4, 705 weak log majorization discrete-time Prop. 3.10.2, 306 asymptotically stable weak majorization matrix Prop. 3.10.2, 306 Fact 15.22.11, 1240 partially ordered set Drazin generalized inverse definition Fact 8.10.11, 670 Defn. 1.3.8, 7 Fact 8.12.1, 678 partition Fact 8.12.5, 679 definition, 2 Drazin inverse equivalence relation Fact 8.12.4, 678 Thm. 1.3.6, 6 eigenvector integer Fact 6.10.33, 536 Fact 7.18.14, 616 factorization, 704 partition number Fact 3.17.9, 335 definition sum Fact 3.17.11, 336 Fact 1.20.1, 113 Fact 3.22.3, 352 generating function Fact 3.22.4, 352 Fact 13.10.25, 1090 Fact 3.22.5, 353 infinite product Fact 3.22.6, 353 Fact 13.10.25, 1090 Fact 8.9.34, 668 Rademacher’s formula

Fact 10.12.53, 758 Fact 10.12.54, 758 factorization of block 2 × 2 Prop. 3.9.3, 302 Prop. 3.9.4, 303 Frobenius norm Fact 11.12.11, 889 geometric multiplicity Prop. 7.7.14, 560 group generalized inverse Fact 8.12.2, 678 Fact 8.12.3, 678 Hamiltonian matrix Prop. 4.1.8, 367 Fact 4.28.6, 428 Fact 4.28.8, 428 Fact 6.9.32, 529 Fact 7.13.31, 596 Hermitian matrix Fact 4.10.27, 381 Fact 6.10.34, 537 Fact 7.9.27, 569 Fact 7.13.7, 591 Fact 8.9.9, 662 Fact 10.12.61, 759 Fact 10.16.6, 777 H¨older-induced norm Fact 11.9.22, 862 idempotent matrix Fact 4.15.21, 398 Fact 4.15.22, 398 Fact 4.16.4, 399 Fact 7.11.27, 579 index of a matrix Fact 7.16.19, 604 inertia Fact 7.9.21, 568 Fact 7.9.22, 568 Fact 7.9.23, 569 Fact 7.9.24, 569 Fact 7.9.25, 569 Fact 7.9.27, 569 Fact 7.13.7, 591 Fact 8.9.8, 661 Fact 8.9.9, 662 inverse matrix Fact 3.22.2, 352 Fact 3.22.3, 352 Fact 3.22.4, 352 Fact 3.22.5, 353 Fact 3.22.6, 353 Fact 3.22.7, 354 Fact 3.22.9, 354 Fact 3.22.10, 354 Fact 7.13.31, 596 inverse of block 2 × 2 Prop. 3.9.7, 303 Cor. 3.9.9, 304 involutory matrix Fact 4.20.10, 417 Kronecker product Fact 9.4.24, 687 Fact 9.4.25, 687 Fact 9.4.29, 687 left-invertible matrix

pencil Fact 3.18.20, 345 matricial norm Fact 11.12.4, 886 matrix exponential Fact 15.12.2, 1205 Fact 15.15.2, 1212 Fact 15.17.20, 1223 matrix sign function Fact 12.15.3, 948 maximum eigenvalue Fact 7.13.30, 596 maximum singular value Fact 10.12.16, 751 Fact 10.12.17, 751 Fact 10.12.25, 753 Fact 10.12.30, 753 Fact 10.21.8, 801 Fact 11.12.4, 886 Fact 11.12.8, 888 Fact 11.12.9, 888 Fact 11.12.10, 889 Fact 11.16.15, 902 Fact 11.16.16, 902 Fact 11.16.17, 902 minimal polynomial Fact 6.10.17, 534 minimal-rank equality Fact 8.9.10, 662 minimum eigenvalue Fact 7.13.30, 596 Minkowski sum Fact 3.12.9, 312 multiplicative equalities, 283 nilpotent matrix Fact 4.15.21, 398 Fact 7.11.28, 579 nonsingular matrix Fact 3.17.29, 340 norm Fact 11.12.4, 886 Fact 11.12.5, 887 Fact 11.12.6, 887 Fact 11.12.14, 890 norm-compression inequality Fact 11.12.4, 886 Fact 11.12.14, 890 normal matrix Fact 4.15.21, 398 Fact 10.12.29, 753 null space Fact 3.14.5, 320 Fact 3.14.6, 320 Fact 3.14.10, 321 Fact 8.9.2, 659 Fact 8.9.3, 659 orthogonal matrix Fact 4.13.12, 388 Fact 4.13.31, 391 outer-product matrix Fact 6.9.17, 527 polynomial Fact 6.10.15, 533 power

Fact 3.15.30, 328 projector Fact 4.17.13, 409 Fact 4.18.18, 413 Fact 4.18.22, 415 Fact 4.18.23, 415 Fact 8.8.22, 658 Fact 8.9.14, 663 Fact 10.12.32, 754 quadratic form Fact 10.19.1, 795 Fact 10.19.2, 795 range Fact 3.14.2, 320 Fact 3.14.4, 320 Fact 3.14.8, 321 Fact 3.14.9, 321 Fact 3.14.14, 322 Fact 8.9.1, 659 Fact 8.9.2, 659 Fact 8.9.3, 659 Fact 8.9.4, 659 Fact 8.9.23, 666 Fact 8.9.27, 667 right-invertible matrix Fact 3.18.20, 345 row norm Fact 11.9.22, 862 Schatten norm Fact 11.12.5, 887 Fact 11.12.6, 887 Fact 11.12.8, 888 Fact 11.12.9, 888 Fact 11.12.10, 889 Fact 11.12.12, 889 Fact 11.12.13, 889 Fact 11.12.14, 890 Schur complement Fact 8.9.7, 660 Fact 8.9.9, 662 Fact 8.9.11, 662 Fact 8.9.38, 669 Fact 10.25.56, 828 Schur product Fact 10.25.7, 821 Fact 10.25.8, 821 Fact 10.25.56, 828 Fact 10.25.60, 829 semicontractive matrix Fact 10.8.11, 724 Fact 10.12.15, 751 similar matrices Fact 7.11.26, 578 Fact 7.11.27, 579 Fact 7.11.28, 579 Fact 7.11.29, 579 singular value Prop. 7.6.5, 557 Fact 11.16.13, 901 Fact 11.16.14, 901 Fact 11.16.33, 905 skew-Hermitian matrix Fact 4.10.27, 381 skew-symmetric matrix Fact 4.13.12, 388

spectral radius Fact 10.12.66, 760 spectrum Fact 3.24.6, 357 Fact 6.10.31, 536 Fact 6.10.32, 536 stability Fact 15.19.38, 1231 star partial ordering Fact 4.30.10, 431 Fact 4.30.11, 431 Fact 4.30.12, 431 Fact 4.30.13, 431 Fact 4.30.14, 431 Fact 4.30.15, 431 Fact 8.4.45, 640 Fact 8.4.46, 640 sum of matrices Fact 8.4.47, 640 Sylvester’s equation Fact 7.11.25, 578 Fact 7.11.26, 578 Fact 8.9.10, 662 symmetric matrix Fact 4.13.12, 388 symplectic matrix Fact 4.28.9, 428 trace Prop. 3.9.1, 302 Fact 10.12.11, 751 Fact 10.12.13, 751 Fact 10.12.14, 751 Fact 10.12.64, 760 Fact 10.14.54, 772 Fact 11.12.7, 888 transpose Prop. 3.9.1, 302 unitarily invariant norm Fact 11.9.52, 865 Fact 11.12.2, 885 unitarily similar matrices Fact 7.10.23, 572 unitary matrix Fact 4.13.11, 388 Fact 4.13.12, 388 Fact 4.13.21, 389 Fact 4.13.22, 389 Fact 4.13.31, 391 Fact 10.8.11, 724 Fact 10.8.12, 724 Fact 10.8.13, 724 Fact 10.12.58, 759 Fact 10.12.59, 759 Fact 10.12.60, 759 Fact 11.16.13, 901 partitioned transfer function H2 norm Fact 16.24.18, 1316 Fact 16.24.19, 1316 Fact 16.24.20, 1316 realization Prop. 16.13.3, 1290 Fact 16.24.8, 1314 transfer function

1511

Fact 16.24.8, 1314 Pascal matrix

positive-semidefinite matrix Fact 10.9.8, 727 Vandermonde matrix Fact 7.18.5, 613 Pascal’s identity binomial equality Fact 1.16.10, 59 path definition Defn. 1.4.4, 11 pathwise connected continuous function Fact 12.13.11, 943 definition Defn. 12.4.23, 923 group Prop. 15.6.8, 1193 pathwise-connected set connected set Prop. 12.4.24, 923 Pauli spin matrices quaternions Fact 4.32.6, 439 PBH test controllability Thm. 16.6.19, 1265 detectability Thm. 16.5.5, 1259 observability Thm. 16.3.19, 1257 stabilizability Thm. 16.8.5, 1269 Pecaric Euclidean norm inequality Fact 11.8.18, 859 Pedersen trace of a convex function Fact 10.14.48, 771 Peierls-Bogoliubov inequality matrix exponential Fact 15.15.32, 1216 Peixoto inequality Fact 2.11.57, 199 Pell number generating function Fact 13.9.3, 1077 recursion Fact 1.17.1, 103 pencil definition, 563 deflating subspace Fact 7.14.1, 597 generalized eigenvalue Prop. 7.8.3, 564 Prop. 7.8.4, 565 invariant zero Cor. 16.10.4, 1280 Cor. 16.10.5, 1280 Cor. 16.10.6, 1281 Kronecker canonical form

1512

Penrose

Thm. 7.8.1, 563 Weierstrass canonical form Prop. 7.8.3, 564 Penrose generalized inverse Fact 8.4.35, 638 pentagon area Fact 5.3.12, 493 diagonals Fact 5.3.7, 492 pentagonal number equalities Fact 1.12.4, 39 generating function Fact 13.4.7, 1010 pentagonal number theorem infinite product Fact 13.10.27, 1091 pentatopic numbers integers Fact 1.12.6, 40 perimeter ellipse Fact 13.9.5, 1079 polygon Fact 5.3.14, 493 period definition Defn. 1.4.4, 11 graph Defn. 1.4.4, 11 periodic dynamics characteristic exponent Fact 16.20.8, 1307 characteristic multiplier Fact 16.20.8, 1307 matrix differential equation Fact 16.20.8, 1307 Fact 16.20.9, 1307 monodromy matrix Fact 16.20.8, 1307 permanental bialternate product compound matrix Fact 9.5.21, 695 permutation arrangement number Fact 1.18.1, 103 definition, 21 derangement number Fact 1.18.2, 104 Eulerian number Fact 1.19.5, 111 generalized derangement number Fact 1.18.3, 104 permutation group definition Prop. 4.6.6, 372 group Fact 4.31.16, 435

permutation matrix

cyclic permutation matrix Fact 7.18.14, 616 definition Defn. 4.1.1, 363 determinant Fact 4.13.18, 389 doubly stochastic matrix Fact 4.11.6, 384 Fact 4.13.1, 387 irreducible matrix Fact 6.11.6, 540 Fact 7.18.15, 616 orthogonal matrix Prop. 4.1.7, 367 reducible matrix Fact 6.11.6, 540 spectrum Fact 7.18.14, 616 symmetric matrix Fact 7.17.44, 610 transposition matrix Fact 4.31.14, 434 vector norm Fact 11.7.19, 852 Perron-Frobenius theorem nonnegative matrix eigenvalues Fact 6.11.5, 538 perturbation asymptotically stable matrix Fact 15.19.16, 1227 inverse matrix Fact 11.10.92, 884 nonsingular matrix Fact 11.16.22, 903 perturbed matrix spectrum Fact 6.10.9, 532 Pesonen simultaneous diagonalization Fact 10.20.8, 799 Petrovich complex inequality Fact 2.21.21, 273 Pfaff’s theorem determinant of a skew-symmetric matrix Fact 6.8.16, 522 Pfaffian skew-symmetric matrix Fact 6.8.16, 522 phasor addition hyperbolic function equality Fact 2.19.3, 262 trigonometric equality Fact 2.16.5, 237 pi approximation Fact 1.15.7, 53 Picard theorem complex function

Fact 12.14.9, 947 Pick matrix

positive-semidefinite matrix Fact 10.9.21, 729 pigeonhole principle finite set Fact 1.8.7, 24 planar graph Euler characteristic Fact 1.9.10, 26 Planck’s integral exponential function Fact 14.8.15, 1160 plane 3-dimensional space Fact 5.1.3, 441 lines Fact 5.1.2, 441 plane partition circle Fact 5.5.1, 495 plane rotation orthogonal matrix Fact 7.17.18, 607 ¨ Plucker identity determinant Fact 3.17.36, 341 Fact 3.17.37, 342 Poincare´ separation theorem eigenvalue inequality Fact 10.21.21, 804 Poincare´ metric complex numbers Fact 2.21.30, 276 Fact 2.21.31, 276 pointed cone definition, 278 induced by reflexive relation Prop. 3.1.7, 280 positive-semidefinite matrix, 703 union Fact 3.12.13, 312 points lines Fact 5.1.1, 441 polar closed set Fact 12.12.14, 938 convex cone Fact 12.12.14, 938 definition, 290 polar cone definition, 362 polar decomposition existence Cor. 7.6.4, 557 Frobenius norm Fact 11.10.77, 881 matrix exponential Fact 15.14.4, 1209 normal matrix

Fact 10.8.9, 724 Fact 15.14.4, 1209 singular value Fact 11.16.26, 904 uniqueness Fact 10.8.3, 723 Fact 10.8.4, 723 Fact 10.8.5, 724 Fact 10.8.6, 724 Fact 10.8.7, 724 Fact 10.8.8, 724 unitarily invariant norm Fact 11.10.77, 881 unitary matrix Fact 10.8.9, 724 polarization identity complex numbers Fact 2.21.8, 269 norm equality Fact 11.8.3, 853 quadratic form Fact 3.15.8, 326 vector equality Fact 11.8.3, 853 polarized Cayley-Hamilton theorem trace Fact 6.9.21, 528 triple product equality Fact 6.9.22, 528 Fact 6.9.28, 529 pole definition, 929 minimal realization Fact 16.24.2, 1313 Fact 16.24.13, 1315 rational transfer function Defn. 6.7.4, 514 Smith-McMillan form Prop. 6.7.11, 515 pole of a rational function definition Defn. 6.7.1, 513 pole of a transfer function definition Defn. 6.7.10, 515 Polya’s generalization Carleman’s inequality Fact 2.12.11, 216 Polya’s inequality logarithmic mean Fact 2.2.63, 142 Polya-Szego inequality reversed Cauchy-Schwarz inequality Fact 2.12.44, 223 polygon area Fact 5.3.10, 493 Fact 5.3.11, 493 Fact 5.3.13, 493 definition, 290 diagonals Fact 5.3.9, 492 perimeter

positive matrix Fact 5.3.14, 493 polygonal inequalities

Euclidean norm Fact 11.8.5, 856 Fact 11.8.16, 858 polygonal numbers integers Fact 1.12.7, 40 polyhedral cone conical hull Fact 3.11.19, 310 convex conical hull Fact 3.11.18, 309 definition, 290 polyhedron definition, 290 polynomial Abel’s theorem Fact 12.16.6, 949 arithmetic-mean– geometric mean inequality Fact 2.11.83, 203 asymptotically stable Defn. 15.9.3, 1198 Bell polynomial Fact 13.2.4, 984 Bezout matrix Fact 6.8.8, 519 Fact 6.8.10, 520 bound Fact 12.13.7, 942 Fact 15.21.26, 1238 complex root Fact 15.21.28, 1238 Fact 15.21.29, 1239 continuity of roots Fact 12.13.2, 941 coprime Fact 6.8.5, 518 Fact 6.8.6, 518 Fact 6.8.7, 519 definition, 499 derivative Fact 12.16.2, 949 Fact 12.16.3, 949 Fact 12.16.4, 949 Fact 12.16.5, 949 Descartes rule of signs Fact 15.18.2, 1223 determinant Fact 6.9.10, 527 discrete-time asymptotically stable Defn. 15.11.3, 1204 discrete-time Lyapunov stable Defn. 15.11.3, 1204 discrete-time semistable Defn. 15.11.3, 1204 discrete-time unstable Defn. 15.11.3, 1204 discriminant Fact 6.8.21, 523 disk inclusion

Fact 15.21.30, 1239 distinct roots Fact 6.8.22, 523 Fact 6.8.23, 523 Drazin generalized inverse Fact 8.10.8, 670 ellipse Fact 5.5.3, 496 Fujiwara’s bound Fact 15.21.12, 1236 Gauss’s lemma Fact 6.8.2, 517 greatest common divisor Fact 6.8.7, 519 increasing function Fact 6.8.20, 523 interpolation Fact 6.8.13, 522 Kojima’s bound Fact 15.21.12, 1236 least common multiple Fact 6.8.5, 518 lower bound Fact 12.13.5, 942 Fact 12.13.6, 942 Lyapunov stable Defn. 15.9.3, 1198 maximum nonnegative root Fact 15.18.18, 1225 Fact 15.18.19, 1226 ordered Bell polynomial Fact 13.2.5, 985 partitioned matrix Fact 6.10.15, 533 root Fact 6.8.1, 517 Fact 6.8.2, 517 Fact 15.21.31, 1239 root bound Fact 15.21.5, 1235 Fact 15.21.6, 1235 Fact 15.21.7, 1235 Fact 15.21.8, 1235 Fact 15.21.9, 1235 Fact 15.21.10, 1235 Fact 15.21.11, 1235 Fact 15.21.12, 1236 Fact 15.21.13, 1236 Fact 15.21.14, 1236 Fact 15.21.15, 1237 Fact 15.21.16, 1237 Fact 15.21.17, 1237 Fact 15.21.21, 1237 Fact 15.21.22, 1238 Fact 15.21.23, 1238 Fact 15.21.24, 1238 Fact 15.21.25, 1238 Fact 15.21.27, 1238 root bounds Fact 15.21.18, 1237 Fact 15.21.19, 1237 Fact 15.21.20, 1237 root location Fact 15.18.4, 1224

root variation Fact 12.13.3, 941 roots Fact 6.8.3, 517 Fact 6.8.4, 518 Schur product Fact 15.18.13, 1225 semistable Defn. 15.9.3, 1198 sign Fact 12.13.1, 941 spectrum Fact 6.9.36, 530 Fact 6.10.15, 533 sum Fact 15.21.32, 1239 Fact 15.21.33, 1239 trinomial coefficient Fact 2.1.10, 121 unstable Defn. 15.9.3, 1198 Vandermonde matrix Fact 7.18.12, 614 polynomial coefficients asymptotically stable polynomial Fact 15.18.3, 1224 Fact 15.18.5, 1224 Fact 15.18.9, 1224 Fact 15.18.10, 1224 Fact 15.18.14, 1225 Fact 15.18.15, 1225 Fact 15.18.16, 1225 discrete-time asymptotically stable polynomial Fact 15.21.2, 1234 Fact 15.21.3, 1234 Fact 15.21.4, 1234 polynomial disk inclusion Cauchy Fact 15.21.30, 1239 polynomial division quotient and remainder Lem. 6.1.2, 501 polynomial matrix Cayley-Hamilton theorem Fact 6.9.30, 529 definition, 501 matrix exponential Prop. 15.2.1, 1181 Smith matrix Prop. 6.3.4, 504 polynomial matrix division linear divisor Cor. 6.2.3, 502 Lem. 6.2.2, 502 polynomial multiplication Toeplitz matrix Fact 6.8.12, 521 polynomial representation commuting matrices Fact 7.16.8, 603 Fact 7.16.9, 603 Fact 7.16.10, 603

1513

inverse matrix Fact 6.8.15, 522 polynomial root maximum singular value bound Fact 11.15.18, 898 minimum singular value bound Fact 11.15.18, 898 polynomial root bound Kojima’s bound Fact 15.21.5, 1235 Montel’s bound Fact 15.21.14, 1236 polynomial root locations Enestrom-Kakeya theorem Fact 15.21.4, 1234 polynomial roots Bezout matrix Fact 6.8.11, 521 Newton’s identities Fact 6.8.4, 518 polytope closed half space Fact 3.11.19, 310 definition, 290 volume Fact 5.4.7, 495 zonotope Fact 3.11.19, 310 Popoviciu AM-GM inequality Fact 2.11.115, 209 Popoviciu’s inequality convex function Fact 1.21.9, 117 quadratic inequality Fact 2.12.38, 222 Fact 2.12.39, 222 positive diagonal upper triangular matrix Fact 7.17.10, 606 positive integer sum of powers Fact 1.11.18, 32 sum of squares Fact 1.11.24, 32 Fact 1.11.25, 32 positive matrix almost nonnegative matrix Fact 15.20.2, 1232 definition, 281 Defn. 4.1.5, 367 eigenvalue Fact 6.11.27, 544 Kronecker sum Fact 9.5.9, 692 Schur product Fact 9.6.18, 698 Fact 9.6.20, 699 spectral radius Fact 9.6.20, 699 spectrum Fact 7.12.15, 581 unstable matrix

1514

positive vector

Fact 15.19.20, 1228 positive vector

definition, 277 null space Fact 6.11.20, 543 positive-definite matrix adjugate Fact 10.21.23, 804 AM-GM inequality Fact 10.15.11, 775 angle Fact 10.18.26, 795 anti-norm Fact 10.14.7, 764 arithmetic mean Fact 10.11.55, 741 asymptotically stable matrix Prop. 15.10.5, 1202 Prop. 16.4.9, 1258 Cor. 15.10.7, 1203 Fact 15.19.21, 1228 Cartesian decomposition Fact 4.10.30, 381 Cauchy matrix Fact 10.9.10, 727 Fact 10.9.20, 729 Fact 16.22.18, 1312 Cayley transform Fact 10.10.35, 733 cogredient diagonalization Thm. 10.3.1, 707 Fact 10.20.7, 799 commuting matrices Fact 10.10.45, 734 complex conjugate transpose Fact 10.10.44, 734 complex matrix Fact 4.10.8, 378 condition number Fact 10.21.6, 801 congruent matrices Prop. 4.7.5, 374 Cor. 10.1.3, 704 contractive matrix Fact 10.12.31, 753 contragredient diagonalization Thm. 10.3.2, 707 Cor. 10.3.4, 708 controllable pair Thm. 16.6.18, 1265 convex function Fact 10.17.21, 790 Fact 10.17.22, 791 Fact 10.17.25, 791 correlation matrix Fact 10.25.36, 825 definition Defn. 4.1.1, 363 diagonal entry Fact 10.21.16, 803 diagonalizable matrix Cor. 10.3.3, 708

discrete-time asymptotically stable matrix Prop. 15.11.5, 1204 Fact 15.22.11, 1240 Fact 15.22.19, 1241 Fact 15.22.21, 1242 discrete-time Lyapunov-stable matrix Prop. 15.11.6, 1204 dissipative matrix Fact 10.21.18, 803 Fact 15.19.21, 1228 eigenvalue Fact 10.11.41, 739 Fact 10.18.21, 794 Fact 10.18.23, 794 Fact 10.18.24, 795 Fact 10.18.25, 795 Fact 10.19.6, 796 Fact 10.19.8, 796 Fact 10.19.9, 796 Fact 10.19.23, 798 Fact 10.21.4, 800 Fact 10.22.41, 813 Fact 10.25.35, 825 eigenvalue inequality Fact 10.22.10, 806 ellipsoid Fact 5.5.14, 498 equality Fact 10.11.17, 737 Fact 10.11.18, 737 exponential Fact 15.15.28, 1215 factorization Fact 7.17.28, 608 Fact 10.8.5, 724 Fact 10.8.6, 724 Fact 10.8.7, 724 Fact 10.8.9, 724 Frobenius norm Fact 11.10.40, 874 Furuta inequality Fact 10.11.79, 746 Gaussian density Fact 14.12.6, 1173 generalized Furuta inequality Fact 10.11.81, 746 geometric mean Fact 10.11.68, 743 Fact 10.11.73, 745 Fact 10.11.74, 745 Fact 10.25.71, 830 group-invertible matrix Fact 10.11.23, 737 Hermitian matrix Fact 7.17.43, 610 Fact 10.11.24, 737 Hilbert matrix Fact 4.23.4, 420 idempotent matrix Fact 7.17.33, 609 increasing function

Fact 10.11.89, 748 inequality Fact 10.11.68, 743 Fact 10.11.72, 745 inertia matrix Fact 10.10.7, 731 inner-product minimization Fact 10.18.13, 793 integral Fact 14.12.1, 1172 Fact 14.12.2, 1172 Fact 14.12.3, 1172 Fact 14.12.4, 1172 Fact 14.12.5, 1173 inverse Fact 10.12.22, 752 inverse matrix Prop. 10.6.6, 715 Lem. 10.6.5, 715 Fact 10.10.20, 732 Fact 10.10.46, 735 Kantorovich inequality Fact 10.11.43, 739 Kronecker product Fact 9.4.22, 687 left inverse Fact 4.10.25, 381 limit Fact 10.11.76, 745 Lyapunov equation Fact 16.22.16, 1312 Fact 16.22.18, 1312 Lyapunov-stable matrix Prop. 15.10.6, 1202 Cor. 15.10.7, 1203 matrix exponential Prop. 15.2.8, 1184 Prop. 15.4.3, 1187 Prop. 15.4.8, 1188 Fact 15.15.21, 1214 Fact 15.15.23, 1215 Fact 15.15.24, 1215 Fact 15.16.1, 1217 matrix logarithm Prop. 10.6.4, 715 Prop. 15.4.10, 1189 Fact 10.10.50, 735 Fact 10.15.11, 775 Fact 10.22.41, 813 Fact 10.22.43, 813 Fact 10.23.1, 813 Fact 11.10.87, 883 Fact 15.15.25, 1215 Fact 15.15.27, 1215 Fact 15.15.28, 1215 Fact 15.15.29, 1215 matrix power Fact 10.11.66, 742 Fact 10.11.67, 743 matrix product Cor. 10.3.7, 708 matrix sign function Fact 12.15.4, 948 maximum singular value

Fact 10.22.15, 807 Fact 10.22.33, 811 norm Fact 11.7.17, 852 observable pair Thm. 16.3.18, 1256 outer-product matrix Fact 4.11.3, 383 power Fact 10.10.47, 735 Fact 10.11.62, 742 Fact 10.11.64, 742 Fact 10.11.77, 746 power inequality Fact 10.11.80, 746 Fact 10.11.82, 747 Fact 10.11.90, 749 product of Gaussian densities Fact 14.12.6, 1173 properties of < and ≤ Prop. 10.1.2, 703 regularized Tikhonov inverse Fact 10.10.45, 734 Riccati equation Fact 16.25.6, 1317 semicontractive matrix Fact 10.11.63, 742 simultaneous diagonalization Fact 10.20.7, 799 skew-Hermitian matrix Fact 10.16.3, 776 Fact 15.19.12, 1227 spectral ordering Fact 10.23.3, 814 spectral radius Fact 10.11.14, 736 Fact 10.22.33, 811 Fact 10.25.30, 823 spectrum Prop. 7.7.21, 562 strictly convex function Fact 10.17.18, 789 Fact 10.17.19, 790 strong majorization Fact 10.22.10, 806 Fact 10.22.43, 813 subdeterminant Prop. 10.2.9, 707 Fact 10.16.16, 779 submatrix Prop. 10.2.9, 707 Fact 10.12.56, 758 Toeplitz matrix Fact 10.9.23, 730 Fact 10.15.14, 776 tridiagonal matrix Fact 4.24.4, 423 Fact 4.24.5, 424 Fact 10.9.22, 730 unitarily invariant norm Fact 11.10.4, 869 unitarily similar matrices

positive-semidefinite matrix Prop. 4.7.5, 374 Prop. 7.7.23, 562 unitary matrix Fact 11.10.83, 882 upper bound Fact 10.11.50, 740 positive-definite solution Riccati equation Thm. 16.17.2, 1299 Prop. 16.19.3, 1305 Cor. 16.19.2, 1304 positive-semidefinite function positive-semidefinite matrix Fact 10.9.2, 725 positive-semidefinite matrix absolute-value matrix Fact 10.10.2, 730 antisymmetric relation Prop. 10.1.1, 703 Araki Fact 10.14.25, 767 Araki-Lieb-Thirring inequality Fact 10.14.24, 767 average Fact 7.20.10, 619 Bohr’s inequality Fact 10.11.83, 747 Brownian motion Fact 10.9.4, 725 Cartesian decomposition Fact 11.10.69, 880 Cauchy matrix Fact 10.9.11, 728 Fact 10.9.13, 728 Fact 16.22.19, 1312 Cauchy-Schwarz inequality Fact 10.12.33, 754 Fact 10.12.35, 754 Fact 10.18.7, 792 closed set Fact 12.11.29, 936 cogredient diagonalization Thm. 10.3.5, 708 commuting matrices Fact 10.11.1, 735 Fact 10.11.47, 740 Fact 10.23.4, 814 completely solid set Fact 12.11.29, 936 complex matrix Fact 4.10.8, 378 concave function Fact 10.13.12, 762 Fact 11.10.3, 869 congruent matrices Prop. 4.7.5, 374 Cor. 10.1.3, 704 contragredient diagonalization Thm. 10.3.6, 708

Cor. 10.3.8, 708 controllability Fact 16.21.7, 1308 convex combination Fact 7.20.11, 619 Fact 10.11.85, 748 Fact 10.16.15, 778 Fact 10.24.18, 817 convex cone, 703 convex function Fact 10.11.86, 748 Fact 10.17.18, 789 Fact 10.24.22, 819 convex set Fact 10.17.2, 785 Fact 10.17.3, 785 Fact 10.17.4, 786 Fact 10.17.5, 786 Fact 10.17.6, 787 copositive matrix Fact 10.19.24, 798 core partial ordering Fact 8.5.17, 648 definition Defn. 4.1.1, 363 diagonal entry Fact 10.10.10, 732 Fact 10.10.11, 732 Fact 10.11.27, 738 Fact 10.13.14, 762 diagonal matrix Fact 10.10.48, 735 discrete-time Lyapunov-stable matrix Fact 15.22.19, 1241 Drazin generalized inverse Fact 10.24.2, 815 eigenvalue Fact 10.13.14, 762 Fact 10.18.12, 793 Fact 10.19.10, 796 Fact 10.22.6, 805 Fact 10.22.12, 806 Fact 10.22.13, 807 Fact 10.22.22, 809 Fact 10.22.26, 809 Fact 10.22.27, 810 Fact 10.22.29, 811 Fact 10.22.30, 811 Fact 10.22.31, 811 Fact 10.22.32, 811 Fact 10.22.37, 812 Fact 10.22.38, 812 Fact 10.24.19, 818 Fact 10.25.29, 823 Fact 10.25.34, 824 equality Fact 10.12.36, 754 Euler totient function Fact 10.9.8, 727 factorization Fact 7.17.24, 608 Fact 7.17.28, 608 Fact 10.8.3, 723 Fact 10.8.4, 723

Fact 10.8.8, 724 Fact 10.10.41, 734 Fact 10.10.42, 734 Fan-Lidskii theorem Fact 11.14.5, 893 Fej´er’s theorem Fact 10.25.19, 822 Frobenius norm Fact 11.9.56, 866 Fact 11.10.31, 872 Fact 11.10.36, 873 Fact 11.10.37, 873 Fact 11.10.44, 874 Fact 11.11.4, 884 Furuta inequality Prop. 10.6.7, 715 geometric mean Fact 10.11.68, 743 group generalized inverse Fact 10.24.1, 815 group-invertible matrix Fact 10.11.23, 737 Hadamard-Fischer inequality Fact 10.16.39, 782 Hermitian matrix Fact 7.17.43, 610 Fact 10.10.1, 730 Fact 10.10.13, 732 Fact 10.10.14, 732 Fact 10.11.24, 737 H¨older’s inequality Fact 10.14.6, 764 Fact 10.14.8, 764 Fact 10.14.10, 765 Fact 10.14.11, 765 Fact 10.14.12, 765 Fact 10.14.13, 765 Fact 10.14.49, 772 Fact 10.14.51, 772 Hua’s inequalities Fact 10.12.52, 757 Hua’s matrix equality Fact 10.12.52, 757 idempotent matrix Fact 7.17.33, 609 increasing sequence Prop. 10.6.3, 715 inertia Fact 7.9.9, 566 Fact 16.22.9, 1311 integral Prop. 10.6.10, 716 inverse matrix Fact 10.11.61, 742 Kantorovich inequality Fact 10.11.42, 739 kernel function Fact 10.9.2, 725 Fact 10.9.7, 726 Kronecker product Fact 9.4.22, 687 Fact 10.25.15, 822 Fact 10.25.37, 825 Fact 10.25.38, 825

1515

Fact 10.25.39, 825 Fact 10.25.41, 825 Fact 10.25.42, 825 Fact 10.25.43, 826 Fact 10.25.44, 826 Fact 10.25.45, 826 Fact 10.25.48, 827 Kronecker sum Fact 9.5.9, 692 lattice Fact 10.11.51, 740 Fact 10.11.52, 741 left-equivalent matrices Fact 7.11.24, 578 Lehmer matrix Fact 10.9.8, 727 limit Prop. 10.6.3, 715 Fact 10.11.76, 745 log majorization Fact 10.12.19, 752 Fact 10.12.20, 752 Lyapunov equation Fact 16.22.15, 1312 Fact 16.22.19, 1312 majorization Fact 10.12.21, 752 matrix exponential Fact 10.13.13, 762 Fact 15.15.21, 1214 Fact 15.15.39, 1217 Fact 15.17.8, 1222 Fact 15.17.18, 1223 matrix logarithm Fact 10.22.42, 813 Fact 11.10.86, 883 matrix power Cor. 10.6.11, 716 Fact 10.10.17, 732 Fact 10.11.60, 742 Fact 10.11.78, 746 Fact 10.14.45, 771 Fact 10.18.17, 794 Fact 10.18.18, 794 Fact 10.18.19, 794 Fact 10.18.20, 794 Fact 11.10.50, 875 matrix product Cor. 10.3.7, 708 maximum singular value Fact 10.10.24, 732 Fact 10.11.32, 738 Fact 10.12.16, 751 Fact 10.12.17, 751 Fact 10.12.66, 760 Fact 10.22.17, 807 Fact 10.22.18, 808 Fact 10.22.19, 808 Fact 10.22.21, 808 Fact 10.22.33, 811 Fact 10.22.35, 812 Fact 10.22.36, 812 Fact 10.22.39, 813 Fact 10.22.42, 813 Fact 10.24.10, 816

1516

positive-semidefinite matrix root

Fact 15.17.8, 1222 McCarthy inequality Fact 10.14.36, 769 Minkowski’s determinant theorem Cor. 10.4.15, 712 Minkowski’s inequality Fact 10.14.36, 769 nonsingular matrix Fact 10.11.11, 736 norm compression Fact 10.12.66, 760 norm-compression inequality Fact 11.12.12, 889 normal matrix Fact 10.10.25, 732 Fact 10.11.22, 737 Fact 10.12.29, 753 Fact 10.13.2, 761 null space Fact 10.7.3, 722 Fact 10.7.5, 722 Fact 10.18.2, 792 Fact 10.19.13, 796 one-sided cone, 703 outer-product Fact 10.10.5, 731 outer-product matrix Fact 10.10.4, 731 Fact 10.10.6, 731 Fact 10.18.3, 792 Fact 10.18.4, 792 partial ordering Prop. 10.1.1, 703 Fact 10.23.7, 814 partitioned matrix Fact 11.12.1, 885 Fact 11.16.32, 905 Pascal matrix Fact 10.9.8, 727 Pick matrix Fact 10.9.21, 729 pointed cone, 703 positive-semidefinite function Fact 10.9.2, 725 power Fact 10.11.62, 742 Fact 10.11.64, 742 principal submatrix Fact 10.7.8, 723 Fact 10.10.3, 731 Fact 10.11.5, 735 Fact 10.11.6, 736 product Fact 10.11.15, 736 projector Fact 4.17.5, 408 properties of < and ≤ Prop. 10.1.2, 703 push-through identity Fact 10.11.15, 736 range Thm. 10.6.2, 714

Cor. 10.2.3, 705 Fact 10.7.1, 722 Fact 10.7.2, 722 Fact 10.7.3, 722 Fact 10.7.4, 722 Fact 10.7.5, 722 Fact 10.7.6, 723 Fact 10.11.2, 735 Fact 10.11.4, 735 Fact 10.24.8, 816 Fact 10.24.9, 816 Fact 10.24.11, 816 Fact 10.24.12, 817 range-Hermitian matrix Fact 10.24.23, 819 rank Fact 7.9.9, 566 Fact 10.7.1, 722 Fact 10.7.5, 722 Fact 10.7.7, 723 Fact 10.7.8, 723 Fact 10.7.9, 723 Fact 10.11.4, 735 Fact 10.11.25, 737 Fact 10.12.32, 754 Fact 10.24.12, 817 Fact 10.25.15, 822 rank subtractivity partial ordering Fact 10.23.4, 814 Fact 10.24.8, 816 Fact 10.24.9, 816 real eigenvalues Fact 7.15.15, 599 reflexive relation Prop. 10.1.1, 703 reproducing kernel space Fact 10.9.7, 726 reverse Fischer inequality Fact 10.16.46, 784 right inverse Fact 4.10.26, 381 Rotfel’d trace inequality Fact 11.10.3, 869 Schatten norm Fact 11.10.16, 871 Fact 11.10.53, 876 Fact 11.10.68, 880 Fact 11.10.69, 880 Fact 11.12.12, 889 Fact 11.12.13, 889 semicontractive matrix Fact 10.12.15, 751 Fact 10.12.31, 753 Fact 10.12.34, 754 semisimple matrix Cor. 10.3.7, 708 sharp partial ordering Fact 8.5.16, 648 shorted operator Fact 10.24.21, 819 signature Fact 7.9.9, 566 singular value Fact 10.11.32, 738

Fact 10.12.20, 752 Fact 10.22.14, 807 Fact 10.22.20, 808 Fact 11.16.36, 905 singular values Fact 10.12.19, 752 skew-Hermitian matrix Fact 10.10.15, 732 spectral ordering Fact 10.23.3, 814 spectral radius Fact 10.22.33, 811 Fact 10.24.9, 816 Fact 10.25.30, 823 spectrum Prop. 7.7.21, 562 Fact 10.24.17, 817 square Fact 10.7.2, 722 square root Fact 10.10.8, 731 Fact 10.10.32, 733 Fact 10.11.29, 738 Fact 10.11.35, 738 Fact 10.11.36, 738 Fact 10.11.37, 738 Fact 10.11.45, 740 Fact 10.11.48, 740 Fact 10.12.37, 755 Fact 11.9.50, 865 stabilizability Fact 16.21.7, 1308 star partial ordering Fact 10.23.5, 814 Fact 10.23.7, 814 Fact 10.24.9, 816 strong majorization Fact 10.22.6, 805 subdeterminant Prop. 10.2.8, 706 Fact 10.15.12, 776 submatrix Prop. 10.2.8, 706 Fact 10.7.9, 723 Fact 10.12.40, 756 Fact 10.16.39, 782 submultiplicative norm Fact 11.10.33, 872 Szasz’s inequality Fact 10.16.39, 782 transitive relation Prop. 10.1.1, 703 triangle inequality Fact 10.11.34, 738 tridiagonal matrix Fact 10.9.6, 726 unitarily invariant norm Fact 11.9.6, 860 Fact 11.9.51, 865 Fact 11.10.3, 869 Fact 11.10.5, 869 Fact 11.10.6, 869 Fact 11.10.8, 870 Fact 11.10.10, 870 Fact 11.10.12, 870

Fact 11.10.13, 870 Fact 11.10.15, 871 Fact 11.10.24, 872 Fact 11.10.33, 872 Fact 11.10.43, 874 Fact 11.10.44, 874 Fact 11.10.47, 875 Fact 11.10.49, 875 Fact 11.10.50, 875 Fact 11.10.78, 881 Fact 11.10.86, 883 Fact 11.11.4, 884 Fact 11.12.3, 886 Fact 11.14.5, 893 Fact 15.17.18, 1223 Fact 15.17.19, 1223 unitarily left-equivalent matrices Fact 7.11.23, 578 Fact 7.11.24, 578 unitarily right-equivalent matrices Fact 7.11.23, 578 unitarily similar matrices Prop. 4.7.5, 374 Prop. 7.7.23, 562 unitary matrix Fact 10.8.2, 723 Fact 10.12.58, 759 Fact 10.12.59, 759 Fact 10.12.60, 759 Fact 11.10.3, 869 upper bound Fact 10.11.57, 741, 742 Fact 10.11.58, 741 upper triangular matrix Fact 10.10.42, 734 weak log majorization Fact 10.22.13, 807 weak majorization Fact 10.22.12, 806 Young’s inequality Fact 10.14.8, 764 zero matrix Fact 10.11.21, 737 positive-semidefinite matrix root definition, 714 positive-semidefinite matrix square root definition, 714 positive-semidefinite solution Riccati equation Thm. 16.17.2, 1299 Thm. 16.18.4, 1302 Prop. 16.17.1, 1298 Prop. 16.19.1, 1304 Cor. 16.17.3, 1299 Cor. 16.18.8, 1304 Cor. 16.19.2, 1304 positive-semidefinite square root definition, 714 positivity

problem quadratic form on a subspace Fact 10.19.21, 798 Fact 10.19.22, 798 power series complex numbers Fact 13.4.3, 1008 definition Defn. 12.3.5, 919 domain of convergence Fact 13.4.1, 1004 exponential Fact 13.4.18, 1021 generating function Defn. 12.3.5, 919 radius of convergence Prop. 12.3.6, 920 Prop. 12.3.7, 920 Fact 12.16.1, 949 power series expansion analytic function Prop. 12.6.8, 928 Powers Schatten norm Fact 11.10.16, 871 Powers-Stormer’s inequality scalar inequality Fact 2.2.45, 139 Fact 10.14.32, 768 predicate definition, 3 primary circulant matrix circulant matrix Fact 7.18.13, 614 cyclic permutation matrix Fact 7.18.13, 614 prime even integer Fact 1.11.46, 35 factorial Fact 1.11.13, 30 Fact 1.13.2, 49 Fermat prime Fact 1.11.13, 30 Goldbach conjecture Fact 1.11.46, 35 Mersenne prime Fact 1.11.13, 30 prime number Bertrand’s postulate Fact 1.11.42, 35 Bonse’s inequality Fact 1.11.45, 35 bound Fact 1.11.43, 35 congruence Fact 1.11.16, 32 factorial Fact 1.13.11, 50 Fermat’s little theorem Fact 1.11.15, 32 Giuga’s conjecture Fact 1.11.17, 32 inequality

Fact 1.11.45, 35 product Fact 1.11.44, 35 sum of squares Fact 1.11.14, 32 Fact 1.11.40, 35 totient function Fact 1.20.4, 115 Vantieghem’s theorem Fact 1.11.21, 32 Wilson’s theorem Fact 1.11.20, 32 prime number theorem logarithm Fact 12.18.41, 965 prime numbers Euclid Fact 12.18.41, 965 sum of reciprocals Fact 12.18.41, 965 Wilson’s theorem Fact 1.11.22, 32 primitive matrix definition Fact 6.11.5, 538 row-stochastic matrix Fact 15.22.12, 1240 principal angle gap Fact 12.12.28, 941 subspace Fact 3.12.21, 314 Fact 7.12.42, 586 Fact 7.13.27, 594 Fact 12.12.28, 941 principal angle and subspaces Ljance Fact 7.12.42, 586 principal branch logarithm function Fact 2.21.29, 275 principal determinant Hermitian matrix Fact 4.10.10, 378 principal inverse definition, 19 principal logarithm definition Defn. 15.5.1, 1190 principal square root definition, 932 integral formula Fact 12.15.1, 948 square root Thm. 12.9.1, 932 principal submatrix definition, 281 positive-semidefinite matrix Fact 10.7.8, 723 Fact 10.10.3, 731 Fact 10.11.5, 735 Fact 10.11.6, 736 rank

Fact 10.7.8, 723 Schur product Fact 9.6.25, 700 principal-angle decomposition unitary matrix Fact 7.10.30, 574 problem adjoint norm Fact 11.9.7, 860 adjugate of a matrix Fact 3.19.5, 346 adjugate of an outer-product perturbation Fact 3.21.2, 351 Fact 3.21.3, 351 asymptotic stability of a compartmental matrix Fact 15.20.6, 1233 Cayley transform of a Lyapunov-stable matrix Fact 15.22.10, 1240 commutator realization Fact 4.29.7, 429 composition of left-invertible functions Fact 3.18.16, 344 constants for real and complex norms Fact 11.9.38, 864 convergent sequence for the generalized inverse Fact 8.3.38, 632 determinant lower bound Fact 10.16.20, 779 determinant of a partitioned matrix Fact 3.17.14, 337 diagonalization of the cross-product matrix Fact 7.10.2, 570 dimension of the centralizer Fact 9.5.3, 691 discrete-time asymptotic stability Fact 15.21.2, 1234 entries of an orthogonal matrix Fact 4.14.5, 392 equality in the triangle inequality Fact 11.7.22, 853 equilateral triangle Fact 5.2.11, 466 exponential representation of a discrete-time Lyapunov-stable matrix Fact 15.22.9, 1240 factorization of a nonsingular matrix by elementary matrices

1517

Fact 7.17.14, 606 factorization of a partitioned matrix Fact 8.9.34, 668 factorization of a unitary matrix Fact 7.17.18, 607 factorization of an orthogonal matrix Fact 7.17.17, 607 generalized inverse least-squares Fact 11.17.11, 911 generalized inverse of a partitioned matrix Fact 8.9.31, 667 geometric mean and generalized inverses Fact 10.11.68, 743 Hahn-Banach theorem interpretation Fact 12.12.21, 939 H¨older-induced norm inequality Fact 11.9.25, 863 Hurwitz stability test Fact 15.19.23, 1228 inverse matrix Fact 3.22.9, 354 Jordan form of a partitioned matrix Fact 7.13.32, 597 Kronecker product of positive-semidefinite matrices Fact 10.25.37, 825 least squares and unitary biequivalence Fact 11.17.17, 911 Lie algebra of upper triangular Lie groups Fact 15.23.1, 1243 Lyapunov-stable matrix and the matrix exponential Fact 15.19.37, 1231 matrix exponential Fact 15.22.5, 1240 matrix exponential and proper rotation Fact 15.12.7, 1206 Fact 15.12.8, 1207 Fact 15.12.9, 1207 matrix exponential formula Fact 15.15.38, 1217 maximum singular value of an idempotent matrix Fact 7.12.41, 585 nonsingularity of a partitioned matrix Fact 3.17.29, 340 orthogonal matrix

1518

product

product Fact 4.11.5, 384 adjugate Fact 7.17.18, 607 Fact 3.19.9, 346 polar decomposition of a characteristic polynomial matrix exponential Cor. 6.4.11, 510 Fact 15.14.4, 1209 Popoviciu’s inequality and compound matrix Fact 9.5.18, 693 Hlawka’s inequality Fact 9.5.19, 694 Fact 1.21.9, 117 contractive matrix positive-definite matrix Fact 4.27.3, 427 Fact 10.9.13, 728 derivative positive-semidefinite Fact 12.16.8, 950 matrix trace upper Drazin generalized inverse bound Fact 8.11.6, 675 Fact 10.14.24, 767 Fact 8.11.7, 675 power inequality equalities Fact 2.1.23, 123 Fact 3.15.25, 327 Fact 2.2.77, 145 equality product of matrix Fact 2.1.16, 122 exponentials group generalized inverse Fact 15.17.4, 1221 Fact 8.11.2, 674 quadratic form and angle Fact 8.11.13, 677 between vectors idempotent matrix Fact 10.18.26, 795 Fact 4.16.13, 403 rank of a matrix sum Fact 4.16.14, 404 Fact 3.13.40, 319 induced lower bound rank of a Prop. 11.5.3, 846 positive-semidefinite left inverse matrix Fact 3.18.9, 343 Fact 10.9.7, 726 Fact 3.18.11, 343 rank of the block Fact 3.18.13, 344 Kronecker product Fact 3.18.16, 344 Fact 9.6.25, 700 maximum singular value reflector Fact 11.16.12, 901 Fact 4.19.7, 416 positive-definite matrix reverse triangle inequality Cor. 10.3.7, 708 Fact 11.8.14, 858 positive-semidefinite right inverse matrix Fact 8.3.16, 629 Cor. 10.3.7, 708 simisimple imaginary Fact 10.11.15, 736 eigenvalues of a projector Fact 4.18.15, 413 partitioned matrix Fact 7.13.2, 589 Fact 7.13.31, 596 Fact 7.13.3, 589 singular value of a Fact 7.13.6, 591 partitioned matrix Fact 8.8.3, 652 Fact 11.16.17, 902 Fact 8.8.5, 653 special orthogonal group Fact 8.8.23, 658 and matrix exponentials Fact 8.11.13, 677 Fact 15.12.13, 1207 Fact 10.11.40, 739 spectrum of a partitioned quadruple matrix Fact 3.20.1, 348 Fact 7.13.32, 597 spectrum of a sum of outer range Fact 3.13.41, 319 products rank Fact 7.12.16, 581 Fact 3.13.42, 319 spectrum of the Laplacian Fact 4.10.32, 382 matrix Fact 8.9.4, 659 Fact 6.11.12, 541 reverse product sum of commutators Fact 3.15.29, 328 Fact 3.23.12, 356 right inverse sum of idempotent Fact 3.18.10, 343 matrices Fact 3.18.12, 344 Fact 4.18.24, 416 Fact 3.18.14, 344 zeros of a transfer function Fact 3.18.18, 344 Fact 16.24.15, 1315

semicontractive matrix Fact 4.27.3, 427 singular value Prop. 11.6.1, 847 Prop. 11.6.2, 847 Prop. 11.6.3, 847 Prop. 11.6.4, 848 Fact 10.22.28, 810 Fact 11.16.35, 905 singular value inequality Fact 10.13.5, 761 Fact 10.13.6, 761 skew-symmetric matrix Fact 7.17.40, 610 trace Fact 7.13.14, 592 Fact 7.13.15, 593 Fact 10.14.14, 765 Fact 10.14.15, 765 Fact 11.16.2, 899 vec Fact 9.4.7, 686 product equality Abel’s identity Fact 2.12.14, 217 Cauchy’s identity Fact 2.14.1, 226 Lagrange’s identity Fact 2.12.13, 217 product inequality 3n variables Fact 2.13.3, 226 product of Gaussian densities positive-definite matrix Fact 14.12.6, 1173 product of matrices definition, 282 diagonalizable matrix Fact 10.20.4, 799 Sylvester’s rank formula Fact 3.13.21, 316 trace Fact 3.15.17, 327 product of powers inequality Fact 2.12.54, 225 product of series limit Fact 12.18.18, 960 product sum rank Fact 3.13.42, 320 products of integers sum Fact 1.12.14, 42 Fact 1.12.15, 42 projection of a set into a subspace definition, 374 projector (2)-inverse Fact 8.3.9, 628 commutator Fact 4.18.16, 413

Fact 4.18.18, 413 Fact 8.8.4, 652 Fact 11.11.2, 884 commuting matrices Fact 8.8.4, 652 Fact 10.11.40, 739 Fact 10.11.44, 740 complementary subspaces Prop. 4.8.6, 375 Fact 4.18.14, 413 Fact 4.18.19, 414 Fact 8.8.18, 657 complex conjugate transpose Fact 4.17.4, 407 controllable subspace Lem. 16.6.6, 1261 definition Defn. 4.1.1, 363 difference Fact 4.18.10, 412 Fact 4.18.19, 414 Fact 7.13.27, 594 Fact 8.8.4, 652 Drazin generalized inverse Fact 8.11.11, 676 elementary reflector Fact 7.17.15, 606 equality Fact 4.17.11, 408 Euclidean norm Fact 10.11.9, 736 Fact 11.9.2, 860 Fact 12.12.26, 940 factorization Fact 7.17.15, 606 Fact 7.17.19, 607 Fact 8.8.10, 655 greatest lower bound Fact 8.8.19, 657 Fact 8.8.21, 658 group generalized inverse Fact 8.11.13, 677 group-invertible matrix Fact 4.18.15, 413 Fact 8.10.2, 669 Hermitian matrix Fact 4.17.1, 407 Fact 4.17.14, 409 Fact 7.17.19, 607 Fact 10.10.27, 733 idempotent matrix Fact 4.17.3, 407 Fact 4.17.14, 409 Fact 4.18.19, 414 Fact 7.11.18, 577 Fact 7.13.28, 595 Fact 8.7.4, 651 Fact 8.8.12, 655 Fact 8.8.13, 656 Fact 8.8.14, 656 Fact 8.8.15, 656 inequality Fact 10.10.26, 733 inertia

quadratic form Fact 7.9.11, 566 Fact 7.9.12, 567 Fact 7.9.20, 568 Fact 7.9.25, 569 intersection of ranges Fact 8.8.19, 657 Kronecker permutation matrix Fact 9.4.39, 689 Kronecker product Fact 9.4.22, 687 least upper bound Fact 8.8.20, 657 Fact 8.8.21, 658 linear combination Fact 8.8.8, 654 matrix difference Fact 4.18.19, 414 Fact 8.8.12, 655 matrix limit Fact 8.8.19, 657 Fact 8.8.20, 657 matrix product Fact 4.18.7, 410 Fact 4.18.15, 413 Fact 8.8.3, 652 Fact 8.8.5, 653 Fact 8.8.23, 658 Fact 8.11.13, 677 matrix sum Fact 7.20.9, 618 maximum singular value Fact 7.12.41, 585 Fact 7.13.27, 594 Fact 7.13.28, 595 Fact 11.16.10, 901 Fact 11.16.41, 907 normal matrix Fact 4.17.3, 407 onto a subspace definition, 374 orthogonal complement Prop. 4.8.2, 374 partitioned matrix Fact 4.17.13, 409 Fact 4.18.18, 413 Fact 4.18.22, 415 Fact 4.18.23, 415 Fact 8.8.22, 658 Fact 8.9.14, 663 Fact 10.12.32, 754 positive-semidefinite matrix Fact 4.17.5, 408 product Fact 4.18.8, 411 Fact 4.18.19, 414 Fact 7.13.2, 589 Fact 7.13.3, 589 Fact 7.13.5, 591 Fact 7.13.6, 591 Fact 8.8.13, 656 Fact 8.8.14, 656 Fact 10.11.40, 739 quadratic form

Fact 4.17.12, 408 range Prop. 4.8.1, 374 Fact 4.17.2, 407 Fact 4.17.7, 408 Fact 4.18.2, 409 Fact 4.18.3, 409 Fact 4.18.4, 409 Fact 4.18.5, 410 Fact 4.18.11, 412 Fact 4.18.12, 413 Fact 4.18.13, 413 Fact 8.8.19, 657 Fact 8.8.20, 657 range-Hermitian matrix Fact 4.17.3, 407 rank Fact 4.17.11, 408 Fact 4.17.13, 409 Fact 4.18.18, 413 Fact 4.18.22, 415 Fact 4.18.23, 415 Fact 7.13.27, 594 Fact 8.8.22, 658 Fact 10.12.32, 754 Fact 10.14.55, 772 reflector Fact 4.18.17, 413 Fact 4.19.1, 416 reverse diagonal Fact 4.17.6, 408 right inverse Fact 4.17.8, 408 similar matrices Cor. 7.7.22, 562 Fact 7.11.18, 577 simultaneous triangularization Fact 7.19.7, 617 skew-Hermitian matrix Fact 11.11.2, 884 spectrum Prop. 7.7.21, 562 Fact 7.13.4, 590 Fact 7.13.5, 591 square root Fact 10.11.44, 740 subspace Prop. 4.8.2, 374 Fact 4.18.1, 409 Fact 10.11.9, 736 Fact 12.12.26, 940 sum Fact 4.18.9, 412 Fact 4.18.18, 413 Fact 4.18.20, 415 Fact 4.18.24, 416 Fact 7.13.27, 594 Fact 8.8.6, 654 Fact 8.8.7, 654 trace Fact 4.17.6, 408 Fact 4.18.6, 410 Fact 4.18.21, 415 Fact 7.9.10, 566

Fact 10.14.55, 772 tripotent matrix Fact 8.8.4, 652 unitarily similar matrices Fact 7.11.15, 577 unobservable subspace Lem. 16.3.6, 1254 projector onto a subspace definition, 374 proof definition, 2 proper rational function definition Defn. 6.7.1, 513 proper rational transfer function definition Defn. 6.7.2, 514 realization Thm. 16.9.4, 1272 proper rotation matrix exponential Fact 15.12.7, 1206 Fact 15.12.8, 1207 Fact 15.12.9, 1207 proper separation theorem convex set Fact 12.12.23, 940 proper subset definition, 2 proposition definition, 5 Ptak maximum singular value Fact 11.15.11, 897 Ptolemy’s inequality complex numbers Fact 2.21.12, 272 quadrilateral Fact 5.3.1, 489 Ptolemy’s theorem quadrilateral Fact 5.3.1, 489 punctured open disk definition, 928 punctured plane definition, 928 Purves similar matrices and nonzero diagonal entry Fact 7.10.15, 571 push-through identity inverse matrix Fact 3.20.6, 348 positive-semidefinite matrix Fact 10.11.15, 736 Putnam-Fuglede theorem normal matrix Fact 7.16.17, 604 Pythagorean theorem norm equality Fact 11.8.3, 853 vector equality Fact 11.8.3, 853

1519

Pythagorean triples

quadratic equality Fact 1.11.23, 32

Q QR decomposition

existence Fact 7.17.11, 606 quadratic scalar inequality Fact 2.4.3, 178 quadratic form closed set Fact 10.17.13, 789 cone Fact 10.17.12, 789 Fact 10.17.16, 789 Fact 10.17.17, 789 convex cone Fact 10.17.12, 789 Fact 10.17.16, 789 Fact 10.17.17, 789 convex set Fact 10.17.2, 785 Fact 10.17.3, 785 Fact 10.17.4, 786 Fact 10.17.5, 786 Fact 10.17.6, 787 Fact 10.17.10, 788 Fact 10.17.12, 789 Fact 10.17.15, 789 Fact 10.17.16, 789 Fact 10.17.17, 789 copositive matrix Fact 10.19.24, 798 covariance Fact 14.12.4, 1172 definition, 364 dual norm Fact 11.9.53, 866 eigenvalue Lem. 10.4.3, 709 Fact 7.12.6, 579 Fact 7.12.7, 579 Fact 10.19.6, 796 field Fact 4.10.6, 378 Hermitian matrix Fact 4.10.5, 378 Fact 4.10.6, 378 Fact 7.12.6, 579 Fact 7.12.7, 579 Fact 7.12.8, 580 Fact 10.18.22, 794 Fact 10.19.14, 796 Fact 10.19.15, 797 Fact 10.19.20, 798 hidden convexity Fact 10.17.12, 789 H¨older-induced norm Fact 11.9.41, 864 Fact 11.9.42, 864 induced norm Fact 11.9.53, 866 inequality

1520

quadratic form on a subspace

Fact 10.18.6, 792 Fact 10.18.7, 792 Fact 10.18.17, 794 Fact 10.18.18, 794 Fact 10.18.19, 794 Fact 10.18.20, 794 Fact 10.19.4, 795 Fact 10.19.5, 796 Fact 10.19.11, 796 Fact 10.19.12, 796 integral Fact 14.12.3, 1172 Fact 14.12.4, 1172 Fact 14.12.5, 1173 Kantorovich inequality Fact 10.18.8, 792 Fact 10.18.10, 793 Laplacian matrix Fact 10.18.1, 792 Legendre-Fenchel transform Fact 10.17.7, 787 linear constraint Fact 10.17.11, 788 matrix exponential Fact 10.17.23, 791 matrix logarithm Fact 10.18.19, 794 maximum eigenvalue Lem. 10.4.3, 709 maximum singular value Fact 11.15.1, 895 Fact 11.15.4, 896 mean Fact 14.12.4, 1172 minimum eigenvalue Lem. 10.4.3, 709 minimum singular value Fact 11.15.1, 895 nonsingular matrix Fact 10.18.9, 792 Fact 10.18.10, 793 norm Fact 11.7.17, 852 normal matrix Fact 10.18.16, 793 null space Fact 10.18.2, 792 Fact 10.19.13, 796 one-sided cone Fact 10.17.17, 789 outer-product matrix Fact 11.15.5, 896 partitioned matrix Fact 10.19.1, 795 Fact 10.19.2, 795 polarization identity Fact 3.15.8, 326 projector Fact 4.17.12, 408 quadratic minimization lemma Fact 10.17.18, 789 Rayleigh quotient Lem. 10.4.3, 709

Reid’s inequality Fact 10.19.4, 795 S-lemma Fact 10.17.14, 789 Schur product Fact 10.25.13, 822 Fact 10.25.59, 828 skew-Hermitian matrix Fact 4.10.5, 378 skew-symmetric matrix Fact 4.10.4, 377 spectral condition number Fact 10.18.9, 792 spectrum Fact 10.17.8, 787 Fact 10.17.9, 788 subspace Fact 10.19.21, 798 Fact 10.19.22, 798 symmetric matrix Fact 4.10.4, 377 trace Fact 10.19.25, 798 vector derivative Prop. 12.10.1, 933 quadratic form on a subspace positivity Fact 10.19.21, 798 Fact 10.19.22, 798 quadratic formula complex numbers Fact 2.21.1, 266 quadratic inequality Aczel’s inequality Fact 2.12.37, 221 Fact 2.12.38, 222 Fact 2.12.39, 222 Bellman’s inequality Fact 2.12.40, 222 Popoviciu’s inequality Fact 2.12.38, 222 Fact 2.12.39, 222 ratio Fact 2.11.42, 197 sum Fact 2.2.39, 138 Fact 2.2.40, 138 sum of squares Fact 2.8.1, 187 Wagner’s inequality Fact 2.12.41, 222 quadratic matrix equation rank Fact 7.12.5, 579 solution Fact 16.25.3, 1317 spectrum Fact 7.12.3, 579 Fact 7.12.4, 579 quadratic minimization lemma quadratic form Fact 10.17.18, 789

quadratic performance measure

matrix exponential Fact 15.12.16, 1208 orthogonal matrix Fact 4.14.6, 392 Pauli spin matrices Fact 4.32.6, 439 real matrix representation Fact 4.32.1, 437 Fact 4.32.8, 440 Rodrigues’s formulas Fact 4.14.8, 394 unitary matrix Fact 4.32.9, 440 quintic equality Fact 2.2.9, 132 quintic inequality five variables Fact 2.5.5, 183 quintic polynomial Abel Fact 4.31.16, 435 Galois Fact 4.31.16, 435 quotient definition, 501 quotient group definition, 371 normal subgroup Thm. 4.4.5, 370 quotient group of a multiplication group definition Defn. 4.6.2, 372 quotient group of an addition group definition Defn. 4.5.2, 371

definition, 1293 H2 norm Prop. 16.15.1, 1294 quadrilateral Brahmagupta’s formula Fact 5.3.1, 489 Bretschneider’s formula Fact 5.3.1, 489 diagonal Fact 5.3.5, 492 Euler’s theorem Fact 5.3.1, 489 Ptolemy’s inequality Fact 5.3.1, 489 Ptolemy’s theorem Fact 5.3.1, 489 semiperimeter Fact 5.3.1, 489 sides Fact 5.3.2, 492 Fact 5.3.3, 492 Fact 5.3.4, 492 quadrilateral inequality Euclidean norm Fact 11.8.5, 856 quadrinomial root bound Fact 15.21.22, 1238 Fact 15.21.23, 1238 Fact 15.21.24, 1238 Fact 15.21.25, 1238 quadruple product trace Fact 9.4.10, 686 vec Fact 9.4.10, 686 quantum information matrix logarithm R Fact 15.15.29, 1215 quartic Rademacher’s formula AM-GM inequality partition number Fact 2.4.12, 179 Fact 13.1.3, 975 equality Rado Fact 2.2.1, 129 AM-GM inequality Liouville’s identity Fact 2.11.115, 209 Fact 2.4.4, 178 convex hull interpretation quaternion group of strong majorization symplectic group Fact 4.11.6, 384 Fact 4.32.4, 438 Radon’s inequality quaternions 2 × 2 matrix representation sum of ratios Fact 2.12.21, 218 Fact 4.32.6, 439 4 × 4 matrix representation Radstrom set cancellation Fact 4.32.3, 438 Fact 12.12.11, 938 angular velocity vector Rahman Fact 15.12.16, 1208 polynomial root bound complex decomposition Fact 15.21.11, 1235 Fact 4.32.2, 438 Ramanujan complex matrix cubic equality representation Fact 13.4.16, 1020 Fact 4.32.7, 439 nested radicals inequality Fact 12.18.62, 972 Fact 2.8.1, 187 partition number

rank Fact 1.20.1, 113 series Fact 13.7.8, 1065 trigonometric equalities Fact 2.16.1, 231 Ramanujan’s 6-10-8 identity equality Fact 2.2.10, 132 Fact 2.4.15, 179 range adjugate Fact 3.19.2, 346 affine subspace Fact 3.14.7, 321 commutator Fact 8.8.4, 652 complementary projector Fact 4.17.2, 407 complementary subspaces Fact 4.15.3, 396 complex conjugate transpose Fact 8.9.27, 667 controllability Fact 16.21.8, 1308 Drazin generalized inverse Prop. 8.2.2, 626 equality Fact 3.13.13, 315 Fact 3.13.19, 316 factorization Thm. 10.6.2, 714 Fact 10.11.2, 735 group generalized inverse Prop. 8.2.3, 627 group-invertible matrix Fact 4.9.2, 376 Fact 7.15.6, 598 Hermitian matrix Lem. 10.6.1, 714 Fact 4.10.31, 381 idempotent matrix Fact 4.15.1, 396 Fact 4.15.5, 396 Fact 4.15.6, 397 inclusion Fact 3.13.12, 315 Fact 3.13.13, 315 inclusion for a matrix power Cor. 3.5.2, 291 inclusion for a matrix product Lem. 3.5.1, 291 Fact 3.13.3, 315 index of a matrix Fact 7.15.6, 598 intersection Fact 3.13.15, 316 involutory matrix Fact 4.20.8, 417 Kronecker product Fact 9.4.15, 686 Fact 9.4.16, 686

left inverse Prop. 3.7.4, 295 Fact 3.18.1, 342 Fact 3.18.2, 342 left-invertible matrix Prop. 3.7.4, 295 matrix product Fact 3.13.41, 319 minimal polynomial Cor. 15.9.6, 1199 nilpotent matrix Fact 4.22.1, 418 Fact 4.22.2, 419 Fact 4.22.3, 419 null space Cor. 3.6.8, 293 Cor. 3.6.9, 293 Fact 3.13.2, 315 null space inclusion Thm. 3.5.3, 291 outer-product matrix Fact 3.13.18, 316 partitioned matrix Fact 3.14.2, 320 Fact 3.14.4, 320 Fact 3.14.8, 321 Fact 3.14.9, 321 Fact 3.14.14, 322 Fact 8.9.1, 659 Fact 8.9.2, 659 Fact 8.9.3, 659 Fact 8.9.4, 659 Fact 8.9.23, 666 Fact 8.9.27, 667 positive-semidefinite matrix Thm. 10.6.2, 714 Cor. 10.2.3, 705 Fact 10.7.1, 722 Fact 10.7.2, 722 Fact 10.7.3, 722 Fact 10.7.4, 722 Fact 10.7.5, 722 Fact 10.7.6, 723 Fact 10.11.2, 735 Fact 10.11.4, 735 Fact 10.24.8, 816 Fact 10.24.9, 816 Fact 10.24.11, 816 Fact 10.24.12, 817 power Fact 3.13.27, 317 projector Prop. 4.8.1, 374 Fact 4.17.2, 407 Fact 4.18.2, 409 Fact 4.18.3, 409 Fact 4.18.4, 409 Fact 4.18.5, 410 Fact 4.18.11, 412 Fact 4.18.12, 413 Fact 4.18.13, 413 Fact 8.8.19, 657 Fact 8.8.20, 657 rank

1521

Fact 3.14.12, 322 Fact 8.7.6, 651 right inverse Kronecker product Fact 3.18.1, 342 Fact 9.4.22, 687 Fact 3.18.2, 342 Kronecker sum right-equivalent matrices Fact 9.5.9, 692 Prop. 7.1.2, 545 nonsingular matrix Schur product Prop. 4.1.7, 367 Fact 9.6.14, 698 normal matrix skew-Hermitian matrix Prop. 4.1.7, 367 Fact 10.7.3, 722 null space stabilizability Fact 4.9.6, 377 Fact 16.21.8, 1308 orthogonally subspace complementary Fact 3.11.20, 310 subspaces sum Fact 4.9.6, 377 Fact 3.13.15, 316 positive-semidefinite Fact 3.13.34, 319 matrix symmetric matrix Fact 10.24.23, 819 Fact 4.10.3, 377 product range of a function Fact 8.5.12, 643 definition, 16 projector range of a matrix Fact 4.17.3, 407 definition, 290 range range-disjoint matrix Fact 4.9.9, 377 definition Fact 4.9.10, 377 Defn. 4.1.1, 363 range inclusion generalized inverse Fact 4.9.7, 377 Fact 8.5.14, 644 range-disjoint matrix range-Hermitian matrix Fact 4.9.4, 376 Fact 4.9.4, 376 rank range-Hermitian matrix Fact 4.9.6, 377 commuting matrices Fact 4.9.8, 377 Fact 8.5.9, 643 right-equivalent matrices Fact 8.5.10, 643 Fact 4.9.6, 377 complex conjugate Schur decomposition transpose Cor. 7.5.4, 553 Fact 4.9.5, 377 tripotent matrix Fact 8.10.16, 671 Fact 8.3.32, 642 congruent matrices Fact 8.5.4, 641 Prop. 4.7.5, 374 unitarily similar matrices Fact 7.10.7, 570 Prop. 4.7.5, 374 definition Cor. 7.5.4, 553 Defn. 4.1.1, 363 range-spanning matrix dissipative matrix definition Fact 7.16.18, 604 Defn. 4.1.1, 363 generalized inverse generalized inverse Prop. 8.1.7, 622 Fact 8.5.14, 644 Fact 8.5.1, 641 range-symmetric matrix Fact 8.5.2, 641 definition Fact 8.5.3, 641 Defn. 4.1.1, 363 Fact 8.5.4, 641 rank Fact 8.5.6, 642 additivity Fact 8.5.8, 643 Fact 3.14.11, 321 Fact 8.5.9, 643 Fact 8.4.31, 638 Fact 8.5.10, 643 adjugate Fact 8.5.11, 643 Fact 3.19.2, 346 Fact 8.5.12, 643 Fact 3.19.3, 346 Fact 8.6.1, 649 arithmetic sequence Fact 8.6.2, 650 Fact 3.13.45, 320 group-invertible matrix biequivalent matrices Prop. 4.1.7, 367 Prop. 7.1.2, 545 Fact 8.10.16, 671 block-Toeplitz matrix idempotent matrix Fact 4.23.12, 422 Fact 4.17.3, 407 bound Fact 8.7.5, 651 Cor. 3.6.4, 293

1522

rank deficient

Cor. 3.6.12, 294 Lem. 3.6.2, 292 commutator Fact 4.16.16, 405 Fact 4.18.18, 413 Fact 7.19.6, 617 Fact 8.5.2, 641 complex conjugate transpose Fact 3.13.35, 319 complex matrix Fact 3.24.6, 357 controllability matrix Cor. 16.6.3, 1260 controllable pair Fact 7.15.11, 599 controllably asymptotically stable Prop. 16.7.4, 1267 Prop. 16.7.5, 1267 cyclic matrix Fact 7.12.1, 579 defect Cor. 3.6.5, 293 definition, 292 diagonal dominance Fact 6.10.29, 536 difference Fact 4.16.12, 402 Fact 4.30.2, 430 dimension inequality Fact 3.13.4, 315 Drazin generalized inverse Fact 8.10.6, 670 equalities with defect Cor. 3.6.1, 292 equality Fact 3.13.19, 316 Fact 3.13.20, 316 Fact 3.13.24, 317 Frobenius norm Fact 11.9.12, 861 Fact 11.13.3, 891 Fact 11.16.39, 906 Fact 11.17.14, 911 geometric multiplicity Prop. 6.5.2, 511 group generalized inverse Fact 8.10.4, 669 group-invertible matrix Fact 4.9.2, 376 Fact 7.9.5, 566 Fact 7.15.6, 598 Hankel matrix Fact 4.23.8, 421 Hermitian matrix Fact 4.10.22, 381 Fact 4.10.32, 382 Fact 7.9.6, 566 Fact 7.9.7, 566 Fact 10.10.9, 731 inertia Fact 7.9.5, 566 Fact 7.9.19, 568 inverse

Fact 3.14.28, 325 Fact 3.14.29, 325 inverse matrix Fact 3.22.11, 354 Fact 8.9.12, 662 Fact 8.9.13, 662 involutory matrix Fact 4.20.6, 417 Fact 4.20.9, 417 Kronecker product Fact 9.4.20, 687 Fact 9.4.28, 687 Fact 9.4.29, 687 Fact 9.4.30, 688 Fact 10.25.15, 822 Kronecker sum Fact 9.5.3, 691 Fact 9.5.10, 692 Fact 9.5.11, 692 left inverse Fact 3.18.1, 342 Fact 3.18.2, 342 limit Fact 12.13.26, 945 linear equation Thm. 3.7.5, 296 linear matrix equation Fact 3.13.23, 317 linear system solution Cor. 3.7.8, 297 Cor. 3.7.9, 298 lower bound for product Prop. 3.6.11, 294 lower semicontinuity Fact 12.13.26, 945 M-matrix Fact 10.7.9, 723 matrix difference Fact 3.13.36, 319 Fact 4.30.1, 430 matrix powers Cor. 3.6.6, 293 Fact 4.22.5, 419 matrix product Fact 3.13.42, 319, 320 matrix sum Fact 3.13.36, 319 Fact 3.13.37, 319 Fact 3.13.38, 319 Fact 3.13.39, 319 Fact 3.13.40, 319 Fact 3.13.44, 320 Fact 3.14.11, 321 nilpotent matrix Fact 4.22.4, 419 Fact 4.22.5, 419 nonsingular submatrices Prop. 3.8.7, 302 normal rank Prop. 6.3.6, 504 Prop. 6.3.7, 504 observability matrix Cor. 16.3.3, 1254 observably asymptotically stable

Prop. 16.4.4, 1258 Prop. 6.7.7, 515 ones matrix Prop. 6.7.8, 515 Fact 3.13.25, 317 submatrix outer-product matrix Prop. 6.7.7, 515 Fact 3.13.26, 317 Fact 3.14.1, 320 Fact 3.13.31, 318 Fact 3.14.25, 324 Fact 4.15.8, 397 Fact 3.14.26, 325 outer-product Fact 3.14.27, 325 Fact 3.14.28, 325 perturbation Fact 3.14.29, 325 Fact 3.13.32, 318 Fact 4.24.2, 423 Fact 8.4.10, 634 subspace dimension power Fact 3.13.27, 317 theorem Fact 3.13.28, 317 Fact 3.14.16, 323 Fact 3.13.29, 317 subtractivity Fact 3.13.30, 317 Fact 4.30.1, 430 powers Fact 4.30.2, 430 Prop. 3.6.7, 293 sum of outer-product principal submatrix matrices Fact 10.7.8, 723 Fact 7.17.1, 605 product Sylvester’s equation Prop. 3.7.3, 295 Fact 16.22.13, 1311 Cor. 3.6.13, 294 Sylvester’s rank formula Fact 4.10.32, 382 Prop. 3.6.10, 294 Fact 8.9.4, 659 totally positive matrix product of matrices Fact 10.7.9, 723 Prop. 3.6.10, 294 trace Fact 3.13.21, 316 Fact 7.12.13, 580 Fact 3.13.33, 318 Fact 11.13.3, 891 quadratic matrix equation transpose Fact 7.12.5, 579 Cor. 3.6.3, 293 range trigonometric matrix Fact 3.14.12, 322 Fact 4.23.8, 421 range-Hermitian matrix tripotent matrix Fact 4.9.6, 377 Fact 4.21.3, 417 Fact 4.9.8, 377 Fact 4.21.5, 418 rational transfer function unitarily invariant norm Defn. 6.7.4, 514 Fact 11.16.39, 906 recurrence Wedderburn Fact 3.13.45, 320 Fact 8.4.10, 634 Riccati equation Fact 8.9.13, 662 Prop. 16.19.4, 1305 rank deficient right inverse definition, 292 Fact 3.18.1, 342 rank of a polynomial matrix Fact 3.18.2, 342 definition Rosenbrock system matrix Defn. 6.2.4, 502 Prop. 16.10.3, 1280 Defn. 6.3.3, 504 Prop. 16.10.11, 1284 submatrix Schur complement Prop. 6.2.7, 503 Fact 8.9.7, 660 Prop. 6.3.5, 504 Fact 8.9.11, 662 rank of a rational function Fact 8.9.12, 662 linearly independent Schur product columns Fact 9.6.15, 698 Prop. 6.7.6, 515 Fact 10.25.15, 822 Prop. 6.7.9, 515 simple matrix rank subtractivity partial Fact 7.12.1, 579 ordering singular value commuting matrices Prop. 7.6.2, 556 Fact 10.23.4, 814 Fact 11.16.39, 906 definition Fact 11.17.14, 911 Fact 4.30.3, 430 skew-Hermitian matrix Fact 4.30.4, 430 Fact 4.10.17, 380 Fact 4.30.5, 430 Fact 4.10.32, 382 generalized inverse Smith-McMillan form Fact 8.9.39, 669

reflexive relation normal matrix Fact 4.30.17, 431 positive-semidefinite matrix Fact 10.23.4, 814 Fact 10.23.7, 814 Fact 10.24.8, 816 Fact 10.24.9, 816 transitivity Fact 4.30.2, 430 rank-deficient matrix determinant Fact 3.16.7, 330 rank-two matrix matrix exponential Fact 15.12.19, 1209 ratio limit Fact 12.18.19, 960 ratio inequality quadratic Fact 2.11.42, 197 ratio of powers scalar inequalities Fact 2.2.74, 145 ratio test series Fact 12.18.9, 959 Fact 12.18.10, 959 rational canonical form, see multicompanion form or elementary multicompanion form rational function complex conjugate Fact 6.8.19, 523 continuous function Fact 12.13.11, 943 definition Defn. 6.7.1, 513 Hankel matrix Fact 6.8.10, 520 imaginary part Fact 6.8.19, 523 Laurent series Fact 12.14.3, 946 series Fact 13.8.10, 1072 spectrum Fact 7.12.18, 582 Taylor series Fact 12.14.2, 946 rational transfer function blocking zero Defn. 6.7.4, 514 definition Defn. 6.7.2, 514 Markov block-Hankel matrix Prop. 16.9.13, 1275 Prop. 16.9.14, 1275 Prop. 16.9.15, 1275 Markov parameter Lem. 16.9.7, 1272 minimal realization

Prop. 16.9.17, 1276 Fact 16.24.13, 1315 normal rank Defn. 6.7.4, 514 poles Defn. 6.7.4, 514 rank Defn. 6.7.4, 514 realization Fact 16.24.12, 1315 Rayleigh quotient Hermitian matrix Lem. 10.4.3, 709 quadratic form Lem. 10.4.3, 709 real eigenvalues positive-semidefinite matrix Fact 7.15.15, 599 real Jordan form hypercompanion matrix Fact 7.11.2, 575 Jordan form Fact 7.11.3, 575 similarity transformation Fact 7.11.2, 575 Fact 7.11.3, 575 real Jordan matrix definition, 551 real normal form existence Cor. 7.5.9, 555 real part frequency response Fact 16.24.5, 1314 transfer function Fact 16.24.5, 1314 real Schur decomposition definition, 553 existence Cor. 7.5.2, 553 Cor. 7.5.3, 553 real symplectic group special orthogonal group Fact 4.32.5, 438 real vector definition, 286 realization controllable canonical form Prop. 16.9.3, 1270 Prop. 16.9.20, 1277 definition Defn. 16.9.2, 1270 feedback interconnection Prop. 16.13.4, 1290 Prop. 16.14.1, 1292 Fact 16.24.9, 1314 observable canonical form Prop. 16.9.3, 1270 Prop. 16.9.20, 1277 partitioned transfer function Prop. 16.13.3, 1290 Fact 16.24.8, 1314

proper rational transfer function Thm. 16.9.4, 1272 rational transfer function Fact 16.24.12, 1315 similar matrices Prop. 16.9.5, 1272 transfer function Prop. 16.13.1, 1289 Fact 16.24.3, 1313 Fact 16.24.4, 1314 Fact 16.24.7, 1314 Fact 16.24.8, 1314 Fact 16.24.9, 1314 rearrangement inequality Chebyshev’s inequality Fact 2.12.7, 215 product of sums Fact 2.12.8, 215 reverse inequality Fact 2.12.10, 216 sum of differences Fact 2.12.8, 216 sum of products Fact 2.12.8, 216 reciprocal sum Fact 1.12.34, 46 reciprocal argument transfer function Fact 16.24.4, 1314 reciprocal of binomial coefficient series Fact 13.7.7, 1064 reciprocal powers inequality Fact 2.12.52, 224 reciprocals Knopp’s inequality Fact 2.11.45, 197 Walker’s inequality Fact 5.2.25, 486 recursion Fibonacci number Fact 1.17.1, 95 Lucas numbers Fact 1.17.2, 100 Pell number Fact 1.17.1, 103 reduced row echelon form definition, 546 reduced row echelon matrix definition Defn. 7.2.1, 545 elementary matrix Thm. 7.2.2, 546 reducible matrix absolute value Fact 4.25.9, 426 definition Defn. 4.1.1, 363 permutation matrix Fact 6.11.6, 540

1523

upper block-triangular matrix Fact 6.11.6, 540 zero entry Fact 4.25.7, 426 Fact 4.25.8, 426 redundant assumptions definition, 4 reflection theorem elementary reflector Fact 4.19.4, 416 reflector definition Defn. 4.1.1, 363 elementary reflector Fact 7.17.16, 607 equality Fact 4.19.8, 416 factorization Fact 7.17.16, 607 Hermitian matrix Fact 4.19.2, 416 involutory matrix Fact 4.19.2, 416 Kronecker product Fact 9.4.22, 687 normal matrix Fact 7.10.10, 571 Fact 7.10.11, 571 orthogonal matrix Fact 4.14.5, 392 Fact 7.17.34, 609 Fact 7.17.38, 609 projector Fact 4.18.17, 413 Fact 4.19.1, 416 rotation matrix Fact 4.14.5, 392 similar matrices Cor. 7.7.22, 562 skew reflector Fact 4.19.7, 416 spectrum Prop. 7.7.21, 562 trace Fact 7.9.10, 566 tripotent matrix Prop. 4.1.7, 367 unitary matrix Fact 4.19.2, 416 reflexive hull definition Defn. 1.3.3, 6 relation Prop. 1.3.4, 6 reflexive relation definition Defn. 1.3.1, 5 directed graph Defn. 1.4.2, 10 intersection Prop. 1.3.2, 6 pointed cone induced by Prop. 3.1.7, 280

1524

reflexivity

positive-semidefinite matrix Prop. 10.1.1, 703 reflexivity star partial ordering Fact 4.30.6, 430 regular pencil definition, 563 generalized eigenvalue Prop. 7.8.3, 564 Prop. 7.8.4, 565 invariant zero Cor. 16.10.4, 1280 Cor. 16.10.5, 1280 Cor. 16.10.6, 1281 Kronecker canonical form Prop. 7.8.2, 563 simultaneous triangularization Fact 7.19.3, 617 upper Hessenberg matrix Fact 7.19.4, 617 upper triangular matrix Fact 7.19.4, 617 regular polynomial matrix definition, 501 nonsingular polynomial matrix Prop. 6.2.5, 502 regular polytope simplex Fact 5.4.6, 495 regularized Tikhonov inverse positive-definite matrix Fact 10.10.45, 734 Reid’s inequality quadratic form Fact 10.19.4, 795 relation definition, 5 relative closure closure Prop. 12.1.10, 914 definition Defn. 12.1.9, 914 relative complement definition, 1 relative degree definition Defn. 6.7.1, 513 Defn. 6.7.3, 514 relative entropy matrix logarithm Fact 15.15.27, 1215 relative gain array definition Fact 10.25.4, 820 relative interior affine subspace Fact 12.11.15, 935 cone Fact 12.11.15, 935 convex cone Fact 12.11.15, 935

convex set Fact 12.11.15, 935 definition Defn. 12.1.3, 913 image Fact 12.11.24, 936 Fact 12.11.25, 936 interior Prop. 12.1.5, 913 Minkowski sum Fact 12.12.12, 938 open set Prop. 12.1.4, 913 subspace Fact 12.11.15, 935 union Fact 12.12.13, 938 relatively boundary convex set Fact 12.11.23, 936 relatively closed set complement Fact 12.11.6, 935 definition Defn. 12.1.9, 914 relatively open set complement Fact 12.11.6, 935 definition Defn. 12.1.3, 913 open set Prop. 12.1.4, 913 remainder definition, 501 removable singularity definition, 929 representative set equivalence relation Defn. 1.3.7, 7 reproducing kernel space positive-semidefinite matrix Fact 10.9.7, 726 residue classes modulo Fact 1.11.10, 29 resolvent definition, 508 Laplace transform Prop. 15.2.2, 1182 matrix exponential Prop. 15.2.2, 1182 maximum singular value Fact 11.15.16, 897 Fact 11.15.17, 897 restricted relation definition, 5 restriction of a function definition, 16 resultant coprime polynomials Fact 6.8.6, 518 discriminant Fact 6.8.21, 523

reversal of a directed graph

Hermitian matrix Fact 4.10.38, 382 reverse identity matrix Defn. 1.4.2, 10 Fact 4.10.38, 382 reversal of a relation reverse-symmetric matrix definition definition Defn. 1.3.3, 6 Defn. 4.1.1, 363 reverse AM-GM inequality factorization Specht’s ratio Fact 7.10.13, 571 Fact 2.11.95, 206 reverse complex conjugate reverse identity matrix Fact 4.10.38, 382 transpose similar matrices definition, 288 Fact 7.10.12, 571 reverse diagonal symmetric matrix projector Fact 4.10.38, 382 Fact 4.17.6, 408 Toeplitz matrix reverse identity matrix Fact 4.23.5, 420 reverse-Hermitian matrix reversed relation Fact 4.10.38, 382 reverse-symmetric matrix relation Prop. 1.3.4, 6 Fact 4.10.38, 382 Riccati differential reverse inequality equation AM-GM inequality matrix differential Fact 2.11.92, 205 equation Fact 2.11.93, 206 Fact 16.25.7, 1318 Fact 2.11.94, 206 Euclidean norm triangle Riccati equation closed-loop spectrum inequality Prop. 16.16.14, 1298 Fact 11.8.14, 858 Prop. 16.18.2, 1302 Young inequality Prop. 16.18.3, 1302 Fact 2.2.54, 140 Prop. 16.18.7, 1304 reverse permutation matrix cyclic permutation matrix detectability Cor. 16.17.3, 1299 Fact 4.20.7, 417 Cor. 16.19.2, 1304 definition, 284 existence determinant Fact 16.25.4, 1317 Fact 3.16.1, 329 Fact 16.25.5, 1317 involutory matrix geometric mean Fact 4.20.5, 417 Fact 16.25.6, 1317 spectrum golden mean Fact 7.10.25, 572 Fact 16.25.6, 1317 symplectic matrix golden ratio Fact 4.28.3, 427 Fact 16.25.6, 1317 reverse power-sum Hamiltonian inequality Thm. 16.17.9, 1300 Fact 2.11.91, 205 Prop. 16.16.14, 1298 reverse product Cor. 16.16.15, 1298 product inertia Fact 3.15.29, 328 Lem. 16.16.18, 1298 reverse transpose linear-quadratic control definition, 288 problem similar matrices Thm. 16.15.2, 1294 Fact 7.10.12, 571 reverse triangle inequality maximal solution Defn. 16.16.12, 1297 anti-norm Thm. 16.18.1, 1302 Fact 11.8.21, 859 Thm. 16.18.4, 1302 reverse-diagonal entry Prop. 16.18.2, 1302 definition, 280 Prop. 16.18.7, 1304 reverse-diagonal matrix monotonicity definition Prop. 16.18.5, 1303 Defn. 4.1.3, 365 Cor. 16.18.6, 1303 semisimple matrix observability Fact 7.15.14, 599 Lem. 16.16.18, 1298 reverse-Hermitian matrix positive-definite matrix definition Fact 16.25.6, 1317 Defn. 4.1.1, 363

rotation positive-definite solution Hermitian matrix Thm. 16.17.2, 1299 Fact 8.3.12, 629 Prop. 16.19.3, 1305 idempotent matrix Cor. 16.19.2, 1304 Fact 4.15.12, 397 positive-semidefinite Kronecker product Fact 9.4.35, 688 solution left inverse Thm. 16.17.2, 1299 Prop. 1.6.3, 17 Thm. 16.18.4, 1302 Fact 3.18.1, 342 Prop. 16.17.1, 1298 Fact 3.18.2, 342 Prop. 16.19.1, 1304 linear equation Cor. 16.17.3, 1299 Fact 8.3.16, 629 Cor. 16.18.8, 1304 matrix product Cor. 16.19.2, 1304 Fact 3.18.10, 343 rank Fact 3.18.12, 344 Prop. 16.19.4, 1305 Fact 3.18.14, 344 solution Fact 3.18.18, 344 Defn. 16.16.12, 1297 positive-semidefinite Fact 16.25.2, 1316 stabilizability matrix Thm. 16.17.9, 1300 Fact 4.10.26, 381 Thm. 16.18.1, 1302 projector Cor. 16.19.2, 1304 Fact 4.17.8, 408 stabilizing solution range Defn. 16.16.12, 1297 Fact 3.18.1, 342 Thm. 16.17.2, 1299 Fact 3.18.2, 342 Thm. 16.17.9, 1300 rank Thm. 16.18.4, 1302 Fact 3.18.1, 342 Fact 3.18.2, 342 Prop. 16.17.1, 1298 representation Prop. 16.18.3, 1302 Fact 3.18.7, 343 Prop. 16.19.4, 1305 right-inner matrix Cor. 16.16.15, 1298 Fact 4.13.7, 388 Riemann right-invertible matrix conditionally convergent Fact 3.18.4, 343 series transfer function Fact 12.18.1, 957 Fact 16.24.10, 1314 Riemann hypothesis uniqueness harmonic number Thm. 1.6.5, 17 Fact 1.11.47, 36 unitarily invariant norm zeta function Fact 11.17.5, 909 Fact 13.3.1, 994 Riemann mapping theorem right triangle inequality complex function Fact 5.2.13, 480 Fact 12.14.10, 947 Riemann-Lebesgue lemma right-equivalent matrices definition Fact 12.17.14, 957 Defn. 4.7.3, 373 right divides group-invertible matrix definition, 502 Fact 4.9.2, 376 right coprime polynomial Kronecker product matrices Fact 9.4.12, 686 Bezout identity range Thm. 6.7.14, 516 Prop. 7.1.2, 545 right equivalence range-Hermitian matrix transformation Fact 4.9.6, 377 definition, 374 right-inner matrix right inverse definition (1)-inverse Defn. 4.1.2, 364 Prop. 8.1.3, 622 generalized inverse affine function Fact 8.3.11, 629 Fact 3.18.8, 343 idempotent matrix definition, 17 Fact 4.13.8, 388 generalized inverse right inverse Cor. 8.1.4, 622 Fact 4.13.7, 388 Fact 8.3.12, 629 right-invertible function Fact 8.3.15, 629 definition, 17 Fact 11.17.5, 909

right-invertible matrix

definition, 294 equivalent properties Thm. 3.7.1, 294 generalized inverse Prop. 8.1.5, 622 Prop. 8.1.12, 625 Fact 8.4.21, 636 inverse Prop. 3.7.6, 297 inverse image Prop. 3.7.10, 298 linear system solution Fact 3.16.13, 330 lower block-triangular matrix Fact 3.18.20, 345 nonsingular equivalence Cor. 3.7.7, 297 open set Thm. 12.4.22, 923 partitioned matrix Fact 3.18.20, 345 right inverse Fact 3.18.4, 343 unique right inverse Prop. 3.7.2, 295 rigid body inertia matrix Fact 10.10.7, 731 rigid-body rotation matrix exponential Fact 15.12.6, 1206 rising factorial definition, 15 Rodrigues orthogonal matrix Fact 4.14.6, 392 Rodrigues’s formulas Euler parameters Fact 4.14.8, 394 orthogonal matrix Fact 4.14.8, 394 quaternions Fact 4.14.8, 394 Rogers inequality power inequality Fact 2.12.56, 225 ¨ Rogers-Holder inequality scalar case Fact 2.12.24, 219 Rogers-Ramanujan identities rational function Fact 13.5.6, 1022 root Defn. 1.4.1, 10 graph Fact 1.9.3, 25 Fact 1.9.4, 25 polynomial Fact 6.8.1, 517 Fact 6.8.2, 517 Fact 6.8.3, 517 Fact 6.8.4, 518

1525

Fact 15.21.5, 1235 Fact 15.21.6, 1235 Fact 15.21.7, 1235 Fact 15.21.8, 1235 Fact 15.21.9, 1235 Fact 15.21.10, 1235 Fact 15.21.11, 1235 Fact 15.21.12, 1236 Fact 15.21.13, 1236 Fact 15.21.14, 1236 Fact 15.21.15, 1237 Fact 15.21.16, 1237 Fact 15.21.17, 1237 root bound Fibonacci number Fact 15.21.7, 1235 polynomial Fact 15.21.21, 1237 Fact 15.21.22, 1238 Fact 15.21.23, 1238 Fact 15.21.24, 1238 Fact 15.21.25, 1238 root bounds polynomial Fact 15.21.18, 1237 Fact 15.21.19, 1237 Fact 15.21.20, 1237 root imaginary abscissa definition, 500 root imaginary part definition, 500 root locus eigenvalue Fact 6.10.35, 537 root modulus definition, 500 root radius sum of polynomials Fact 15.21.32, 1239 Fact 15.21.33, 1239 root radius of a polynomial definition, 500 root real abscissa definition, 500 root real part definition, 500 root variation polynomial Fact 12.13.3, 941 roots of polynomial convex hull Fact 12.16.5, 949 Gauss-Lucas theorem Fact 12.16.5, 949 Rosenbrock system matrix definition Defn. 16.10.1, 1278 rank Prop. 16.10.3, 1280 Prop. 16.10.11, 1284 Rosenthal two-variable limit Fact 12.17.18, 957 rotation vector

1526

rotation matrix

Fact 4.14.11, 396 rotation matrix

definition, 372 logarithm Fact 15.16.11, 1220 matrix exponential Fact 15.12.13, 1207 Fact 15.12.14, 1207 multiplication group Prop. 4.6.6, 372 orthogonal matrix Fact 4.14.5, 392 Fact 4.14.6, 392 Fact 4.14.8, 394 Fact 4.14.9, 394 Fact 4.14.10, 396 reflector Fact 4.14.5, 392 trace Fact 4.13.14, 388 rotation-dilation factorization Fact 3.24.2, 356 Rotfel’d trace inequality positive-semidefinite matrix Fact 11.10.3, 869 Roth solutions of Sylvester’s equation Fact 7.11.25, 578 Sylvester’s equation Fact 7.11.26, 578 Rothe’s identity binomial equality Fact 1.16.13, 77 Rouche´ fundamental triangle inequality Fact 5.2.8, 446 Fact 5.2.11, 466 Roup positive-definite matrix Fact 10.9.16, 728 Routh criterion asymptotically stable polynomial Fact 15.18.3, 1224 Routh matrix tridiagonal matrix Fact 15.19.27, 1230 row definition, 280 row norm column norm Fact 11.9.21, 862 definition, 844 H¨older-induced norm Fact 11.9.25, 863 Fact 11.9.27, 863 Kronecker product Fact 11.10.93, 884 partitioned matrix Fact 11.9.22, 862 spectral radius

Cor. 11.4.10, 844

Fact 11.10.65, 879 Hermitian matrix Fact 11.10.68, 880 adjacency matrix Fact 11.11.4, 884 Fact 4.26.3, 426 H¨older matrix norm definition Fact 11.13.5, 891 Defn. 4.1.5, 367 H¨older norm discrete-time Fact 11.9.10, 861 Lyapunov-stable matrix inequality Fact 15.22.12, 1240 Fact 11.10.18, 871 discrete-time semistable Fact 11.10.54, 876 matrix Fact 11.10.55, 876 Fact 15.22.12, 1240 Fact 11.10.65, 879 irreducible matrix Fact 11.10.66, 879 Fact 15.22.12, 1240 Fact 11.10.67, 879 outdegree matrix Kronecker product Fact 4.26.3, 426 Fact 11.10.95, 884 primitive matrix matrix difference Fact 15.22.12, 1240 Fact 11.10.17, 871 spectral radius McCarthy inequality Fact 6.11.11, 541 Fact 11.10.53, 876 spectrum monotonicity Fact 6.10.6, 531 Prop. 11.2.5, 838 normal matrix S Fact 11.11.4, 884 S-lemma Fact 11.16.4, 900 quadratic form optimal 2-uniform Fact 10.17.14, 789 convexity S-N decomposition Fact 11.10.64, 879 diagonalizable matrix partitioned matrix Fact 7.10.4, 570 Fact 11.12.5, 887 nilpotent matrix Fact 11.12.6, 887 Fact 7.10.4, 570 Fact 11.12.8, 888 Satnoianu’s inequality Fact 11.12.9, 888 weak form Fact 11.12.10, 889 Fact 2.2.20, 134 Fact 11.12.12, 889 Fact 2.3.46, 160 Fact 11.12.13, 889 Schatten norm Fact 11.12.14, 890 absolute value positive-semidefinite Fact 11.15.13, 897 matrix anti-norm Fact 11.10.16, 871 Fact 11.10.53, 876 Fact 11.10.53, 876 Beckner’s two-point Fact 11.10.68, 880 inequality Fact 11.10.69, 880 Fact 11.10.64, 879 Fact 11.12.12, 889 Cartesian decomposition Fact 11.12.13, 889 Fact 11.10.66, 879 power Fact 11.10.68, 880 Fact 11.9.47, 865 Fact 11.10.69, 880 power-sum inequality Clarkson inequalities Fact 11.10.59, 877 Fact 11.10.54, 876 Schur product commutator Fact 11.16.45, 907 Fact 11.11.4, 884 trace compatible norms Fact 11.14.1, 892 Prop. 11.3.6, 840 unitarily invariant norm Cor. 11.3.7, 840 Fact 11.9.11, 861 Cor. 11.3.8, 840 Schauder fixed-point definition, 837 theorem Prop. 11.2.3, 837 image of a continuous eigenvalue function Fact 11.13.5, 891 Thm. 12.4.26, 924 equality Schinzel Fact 11.10.52, 876 determinant upper bound Frobenius norm Fact 3.16.25, 333 Fact 11.9.20, 862 Schlicht function Hanner’s inequality row-stochastic matrix

complex function Fact 12.14.12, 947 Schmidt-Mirsky theorem fixed-rank approximation Fact 11.16.39, 906 Schneider inertia Fact 16.22.4, 1310 inertia of a Hermitian matrix Fact 16.22.5, 1310 Schoenberg Euclidean distance matrix Fact 11.9.17, 861 Schott’s theorem Schur product Fact 10.25.16, 822 Schroeder-Bernstein theorem one-to-one and onto function Fact 1.10.7, 27 Schur algebra generated by commuting matrices Fact 7.11.20, 577 arithmetic mean inequality Fact 2.11.111, 208 Schur complement convex function Prop. 10.6.17, 718 Lem. 10.6.16, 718 definition Defn. 8.1.13, 625 determinant Prop. 10.2.4, 705 eigenvalue Fact 10.12.4, 749 Hermitian matrix Fact 10.12.8, 750 Fact 10.12.61, 759 increasing function Prop. 10.6.13, 717 inequality Fact 10.12.38, 755 inertia Fact 8.9.9, 662 nondecreasing function Prop. 10.6.13, 717 partitioned matrix Fact 8.9.7, 660 Fact 8.9.9, 662 Fact 8.9.11, 662 Fact 8.9.38, 669 Fact 10.25.56, 828 positive-definite matrix Fact 10.12.6, 750 positive-semidefinite matrix Cor. 10.6.18, 721 Fact 10.12.4, 749 Fact 10.12.5, 750 Fact 10.12.7, 750 Fact 10.12.39, 755 Fact 10.12.40, 755

self-conjugate set Fact 10.12.48, 756 Fact 10.12.49, 756 Fact 10.12.51, 757 Fact 10.12.55, 758 Fact 10.24.21, 819 Fact 10.25.14, 822 rank Prop. 10.2.4, 705 Fact 8.9.7, 660 Fact 8.9.11, 662 Fact 8.9.12, 662 Schur product Fact 10.25.14, 822 Schur decomposition Hermitian matrix Cor. 7.5.5, 554 Jordan form Fact 7.11.8, 576 normal matrix Cor. 7.5.4, 553 Fact 7.11.8, 576 range-Hermitian matrix Cor. 7.5.4, 553 Schur inverse positive-semidefinite matrix Fact 10.25.1, 820 Schur power definition, 685 Lyapunov equation Fact 10.9.20, 729 nonnegative matrix Fact 9.6.19, 698 positive-semidefinite matrix Fact 10.25.2, 820 Fact 10.25.3, 820 Fact 10.25.40, 825 Schur power of polynomials asymptotically stable polynomial Fact 15.18.11, 1224 Schur product associative equalities, 685 block Kronecker product Fact 9.6.25, 700 Chebyshev inequality Fact 10.25.69, 830 commutative equalities, 685 complex conjugate transpose Fact 10.25.11, 822 compound matrix Fact 10.25.66, 830 convex function Fact 10.25.72, 830 correlation matrix Fact 10.25.36, 825 definition, 685 distributive equalities, 685 eigenvalue Fact 10.25.29, 823 Fact 10.25.32, 824

Frobenius norm Fact 11.16.45, 907 Fact 11.16.47, 908 geometric mean Fact 10.25.71, 830 Hadamard sum Fact 10.25.51, 827 Hermitian matrix Prop. 9.3.1, 685 Fact 10.25.27, 823 Fact 10.25.31, 824 Fact 10.25.52, 827 Kantorovich inequality Fact 10.25.70, 830 Kronecker product Prop. 9.3.2, 685 lower bound Fact 10.25.21, 823 M-matrix Fact 9.6.22, 699 matrix equality Fact 9.6.3, 697 Fact 9.6.11, 697 Fact 9.6.13, 697 matrix exponential Fact 15.15.22, 1214 matrix logarithm Fact 10.25.65, 829 matrix power Fact 9.6.16, 698 maximum singular value Fact 10.25.12, 822 Fact 11.16.42, 907 Fact 11.16.43, 907 Fact 11.16.44, 907 Fact 11.16.47, 908 Fact 11.16.48, 908 Fact 11.16.49, 908 nonnegative matrix Fact 9.6.18, 698 normal matrix Fact 11.10.9, 870 outer product Fact 9.6.4, 697 Fact 9.6.5, 697 Fact 9.6.6, 697 Fact 9.6.7, 697 Fact 9.6.8, 697 partitioned matrix Fact 10.25.7, 821 Fact 10.25.8, 821 Fact 10.25.56, 828 Fact 10.25.60, 829 positive matrix Fact 9.6.20, 699 principal submatrix Fact 9.6.25, 700 quadratic form Fact 9.6.9, 697 Fact 9.6.10, 697 Fact 10.25.13, 822 Fact 10.25.59, 828 range Fact 9.6.14, 698 rank

1527

eigenvalue inequality Fact 9.6.15, 698 Fact 10.21.11, 802 Fact 10.25.15, 822 positive-semidefinite Schatten norm Fact 11.16.45, 907 matrix Schur complement Fact 10.25.16, 822 Fact 10.25.14, 822 Schur-Cohn criterion singular value discrete-time Fact 11.16.42, 907 asymptotically stable Fact 11.16.43, 907 polynomial Fact 11.16.44, 907 Fact 15.21.2, 1234 Fact 11.16.46, 908 Schur-concave function spectral radius definition Fact 9.6.18, 698 Defn. 3.10.3, 306 Fact 9.6.20, 699 elementary symmetric Fact 9.6.23, 700 function Fact 9.6.24, 700 Fact 2.11.96, 206 Fact 11.16.44, 907 entropy submultiplicative norm Fact 3.25.7, 361 Fact 11.9.58, 867 strong majorization trace Fact 3.25.7, 361 Fact 9.6.12, 697 Schur-convex function Fact 9.6.21, 699 definition Fact 10.25.23, 823 Defn. 3.10.3, 306 Fact 10.25.24, 823 strong majorization Fact 10.25.25, 823 Fact 3.25.5, 360 Fact 10.25.28, 823 Fact 3.25.6, 360 Fact 11.16.46, 908 symmetric gauge function trace norm Fact 11.9.59, 867 Fact 11.16.47, 908 vector norm transpose Fact 11.7.19, 852 Fact 9.6.17, 698 Schur-Horn theorem unitarily invariant norm diagonal entry Fact 11.9.58, 867 Fact 4.13.16, 389 Fact 11.10.9, 870 unitary matrix Fact 11.10.96, 884 Fact 10.21.14, 803 Fact 11.16.50, 909 Schwarz lemma vector equality complex function Fact 9.6.1, 697 Fact 12.14.4, 946 Fact 9.6.2, 697 Schwarz matrix weak majorization tridiagonal matrix Fact 11.16.42, 907 Fact 15.19.25, 1229 Schur product of Fact 15.19.26, 1229 polynomials Schwarz-Pick lemma asymptotically stable conformal mapping polynomial Fact 12.14.5, 946 Fact 15.18.12, 1225 Schweitzer’s inequality Fact 15.18.13, 1225 scalar inequality Schur’s formulas Fact 2.11.135, 212 determinant of partitioned secant matrix indefinite integral Fact 3.17.14, 337 Fact 14.1.17, 1095 Schur’s inequality secant condition cubic inequality asymptotically stable Fact 2.3.60, 162 matrix eigenvalue Fact 15.19.29, 1230 Fact 10.21.10, 801 second derivative eigenvalues and the definition, 925 Frobenius norm Selberg’s inequality Fact 11.13.2, 890 norm inequality normal matrix Fact 11.8.11, 857 Fact 11.13.2, 890 self edge scalar inequality definition, 9 Fact 2.3.28, 153 self-adjoint norm Fact 2.3.30, 154 definition, 837 Schur’s theorem self-conjugate set

1528

self-directed edge

Defn. 7.7.4, 559 index of an eigenvalue Prop. 7.7.8, 559 definition, 9 null space self-edge Prop. 7.7.8, 559 Defn. 1.4.2, 10 simple eigenvalue semi-tensor product Prop. 7.7.5, 559 Kronecker product semisimple matrix Fact 9.6.26, 700 commuting matrices semicontractive matrix Fact 7.16.5, 603 complex conjugate cyclic matrix transpose Fact 7.15.13, 599 Fact 4.27.2, 427 definition definition Defn. 7.7.4, 559 Defn. 4.1.2, 364 diagonal matrix discrete-time Prop. 7.7.11, 560 Lyapunov-stable matrix elementary matrix Fact 15.22.5, 1240 Fact 7.15.18, 600 Frobenius norm idempotent matrix Fact 11.9.19, 861 Fact 7.15.28, 601 normal matrix identity-matrix Fact 15.19.37, 1231 perturbation partitioned matrix Fact 7.15.17, 600 Fact 10.8.11, 724 involutory matrix Fact 10.12.15, 751 Fact 7.15.20, 600 positive-definite matrix Kronecker product Fact 10.11.63, 742 Fact 9.4.22, 687 positive-semidefinite matrix exponential matrix Prop. 15.2.7, 1183 Fact 10.12.15, 751 normal matrix Fact 10.12.31, 753 Prop. 7.7.12, 560 Fact 10.12.34, 754 outer-product matrix product Fact 7.15.5, 598 Fact 4.27.3, 427 positive-semidefinite unitary matrix matrix Fact 4.27.4, 427 Cor. 10.3.7, 708 Fact 10.8.11, 724 reverse-diagonal matrix semidissipative matrix Fact 7.15.14, 599 definition similar matrices Defn. 4.1.1, 363 Prop. 7.7.12, 560 determinant Fact 7.10.5, 570 Fact 10.15.2, 774 Fact 7.11.7, 576 Fact 10.15.4, 774 simple matrix Fact 10.16.2, 776 Fact 7.15.13, 599 discrete-time Lyapunov-stable matrix skew-involutory matrix Fact 7.15.20, 600 Fact 15.22.5, 1240 semistability dissipative matrix eigenvalue Fact 10.16.20, 779 Prop. 15.9.2, 1198 Kronecker sum linear dynamical system Fact 9.5.9, 692 Prop. 15.9.2, 1198 Lyapunov-stable matrix Lyapunov equation Fact 15.19.37, 1231 Cor. 15.10.1, 1201 normal matrix matrix exponential Fact 15.19.37, 1231 Prop. 15.9.2, 1198 semiperimeter semistable matrix quadrilateral compartmental matrix Fact 5.3.1, 489 Fact 15.20.6, 1233 triangle definition Fact 5.2.7, 443 Defn. 15.9.1, 1198 semisimple eigenvalue group-invertible matrix cyclic eigenvalue Fact 15.19.4, 1226 Prop. 7.7.5, 559 Kronecker sum defect Fact 15.19.32, 1231 Prop. 7.7.8, 559 Fact 15.19.33, 1231 definition definition, 499

self-directed edge

limit Prop. 12.2.12, 917 Fact 15.19.8, 1226 Fact 12.18.8, 958 Lyapunov equation limit Fact 15.19.1, 1226 Fact 12.18.7, 958 Fact 16.22.15, 1312 limsup Lyapunov-stable matrix Prop. 12.2.12, 917 Fact 15.19.2, 1226 Fact 12.18.8, 958 matrix exponential little oh notation Fact 15.19.7, 1226 Defn. 12.2.13, 918 Fact 15.19.8, 1226 superior Fact 15.22.9, 1240 Defn. 12.2.11, 917 minimal realization Prop. 12.2.10, 917 Defn. 16.9.25, 1278 supremum semistable polynomial Fact 12.18.5, 958 Prop. 15.9.4, 1199 Fact 12.18.6, 958 similar matrices sequence of sets Fact 15.19.5, 1226 essential greatest lower unstable subspace bound Prop. 15.9.8, 1200 Defn. 1.3.18, 8 semistable polynomial essential least upper bound definition Defn. 1.3.18, 8 Defn. 15.9.3, 1198 essential limit reciprocal argument Defn. 1.3.20, 9 Fact 15.18.7, 1224 series semistable matrix e Prop. 15.9.4, 1199 Fact 13.5.14, 1023 semistable transfer Abel’s test function Fact 12.18.13, 959 minimal realization Apery’s constant Prop. 16.9.26, 1278 Fact 13.5.41, 1031 SISO entry Catalan numbers Prop. 16.9.27, 1278 Fact 13.5.18, 1025 Catalan’s constant sensitivity function Fact 13.5.86, 1050 integral Cauchy product Fact 14.7.13, 1156 Fact 13.5.8, 1022 separation theorem Clausen’s integral convex cone Fact 13.6.2, 1057 Fact 12.12.22, 940 commutator inner product Fact 15.15.18, 1214 Fact 12.12.22, 940 complex numbers Fact 12.12.23, 940 Fact 13.4.14, 1017 septic definition equality Defn. 12.3.1, 919 Fact 2.2.9, 132 Defn. 12.3.3, 919 sequence Dirichlet’s test ℓ p norm Fact 12.18.12, 959 Defn. 12.2.16, 918 Eulerian numbers Prop. 12.2.17, 918 Fact 13.1.2, 975 big oh notation Fibonacci number Defn. 12.2.13, 918 Fact 13.9.3, 1077 definition generating function Defn. 1.1.1, 2 Fact 13.4.9, 1012 Drazin generalized inverse Genocchi numbers Fact 8.10.6, 670 Fact 13.1.7, 980 generalized inverse greatest common divisor Fact 8.3.39, 632 Fact 13.5.3, 1021 group generalized inverse Gregory’s series Fact 8.10.4, 669 Fact 13.4.14, 1017 inferior harmonic number Defn. 12.2.11, 917 Fact 13.5.16, 1023 Prop. 12.2.10, 917 Fact 13.5.25, 1027 infimum harmonic numbers Fact 12.18.5, 958 Fact 13.5.18, 1025 Fact 12.18.6, 958 hyperbolic functions liminf

similar matrices Fact 13.4.11, 1015 infinite limit Fact 13.5.10, 1022 inverse hyperbolic functions Fact 13.4.13, 1017 inverse matrix Prop. 11.3.10, 841 inverse trigonometric functions Fact 13.4.9, 1012 Leibniz’s test Fact 12.18.11, 959 limit Fact 12.18.9, 959 Fact 12.18.11, 959 Fact 12.18.12, 959 Fact 12.18.13, 959 matrix exponential Fact 15.15.18, 1214 Mercator’s series Fact 13.4.14, 1017 monotonic Fact 12.18.10, 959 ordered Bell number Fact 13.1.5, 976 product Fact 13.5.7, 1022 ratio of products Fact 1.12.29, 45 rational function Fact 13.5.4, 1021 Fact 13.5.5, 1021 spectral radius Fact 13.4.15, 1019 square root Fact 1.12.30, 45 Stirling number of the second kind Fact 13.4.4, 1009 sum of powers Fact 13.5.103, 1057 triangular number Fact 13.4.6, 1010 trigonometric function Fact 13.5.95, 1054 trigonometric functions Fact 13.4.8, 1011 up/down numbers Fact 13.1.9, 981 zeta function Fact 13.5.39, 1031 set countable set definition, 1 countably infinite set definition, 1 definition, 1 distance from a point Fact 12.12.24, 940 Fact 12.12.25, 940 finite set definition, 1 Hausdorff distance Fact 12.12.27, 940

infimum Fact 1.8.6, 24 infinite set definition, 1 intersection Fact 3.12.7, 311 sum Fact 3.12.7, 311 supremum Fact 1.8.6, 24 symmetric difference Fact 1.8.3, 23 set cancellation convex set Fact 12.12.11, 938 set equality De Morgan’s laws Fact 1.8.1, 22 intersection Fact 1.8.1, 22 Fact 1.8.2, 23 union Fact 1.8.1, 22 Fact 1.8.2, 23 set union cardinality Fact 1.8.8, 24 set-valued inverse definition, 17 sextic Fleck’s identity Fact 2.4.14, 179 sextic inequality six variables Fact 2.6.17, 186 Fact 2.6.18, 186 Shannon’s inequality logarithm Fact 2.15.28, 230 Shapiro commuting matrices Fact 7.16.5, 603 Fact 7.16.6, 603 Shapiro’s inequality Nesbitt’s inequality Fact 2.11.53, 198 sharp partial ordering positive-semidefinite matrix Fact 8.5.16, 648 shear factor factorization Fact 7.17.13, 606 Shemesh common eigenvector Fact 7.16.15, 604 Sherman-MorrisonWoodbury formula determinant of an outer-product perturbation Fact 3.21.3, 351 shift controllability

1529

Fact 16.21.11, 1308 Fact 7.9.6, 566 stabilizability Fact 7.9.7, 566 Fact 16.21.12, 1308 Fact 10.11.28, 738 involutory matrix shifted argument Fact 7.9.2, 565 transfer function positive-semidefinite Fact 16.24.3, 1313 shifted-orthogonal matrix matrix definition Fact 7.9.9, 566 Defn. 4.1.1, 363 tripotent matrix shifted-unitary matrix Fact 7.9.3, 565 block-diagonal matrix signed volume Fact 4.13.10, 388 simplex definition Fact 5.4.1, 493 Defn. 4.1.1, 363 similar matrices normal matrix asymptotically stable Fact 4.13.30, 391 matrix spectrum Fact 15.19.5, 1226 Prop. 7.7.21, 562 biequivalent matrices unitary matrix Prop. 4.7.5, 374 Fact 4.13.29, 391 block-diagonal matrix Shoda Thm. 7.4.5, 550 factorization campanion matrix Fact 7.17.9, 606 Fact 7.18.11, 614 Fact 7.17.37, 609 characteristic polynomial Shoda’s theorem Fact 6.9.6, 525 commutator realization complex conjugate Fact 7.10.19, 571 Fact 7.10.33, 575 zero trace cyclic matrix Fact 7.10.19, 571 Fact 7.18.11, 614 shortcut of a relation defect definition Fact 7.11.7, 576 Defn. 1.3.3, 6 definition shorted operator Defn. 4.7.4, 373 definition diagonal entry Fact 10.24.21, 819 Fact 7.10.14, 571 positive-semidefinite diagonalizable over R matrix Prop. 7.7.13, 560 Fact 10.24.21, 819 Cor. 7.7.22, 562 sides discrete-time quadrilateral asymptotically stable Fact 5.3.2, 492 matrix Fact 5.3.3, 492 Fact 15.19.5, 1226 Fact 5.3.4, 492 discrete-time Siebeck Lyapunov-stable matrix ellipse Fact 15.19.5, 1226 Fact 5.5.3, 496 discrete-time semistable Siegel’s inequality matrix arithmetic-mean– Fact 15.19.5, 1226 geometric mean Drazin generalized inverse inequality Fact 8.11.5, 675 Fact 2.11.83, 203 equivalence class sign Fact 7.11.5, 576 matrix, 289 equivalent realizations vector, 289 Defn. 16.9.6, 1272 sign of entry example asymptotically stable Example 7.7.20, 562 matrix factorization Fact 15.20.5, 1233 Fact 7.17.8, 606 sign stability geometric multiplicity asymptotically stable Prop. 7.7.10, 560 group-invertible matrix matrix Prop. 4.7.5, 374 Fact 15.20.5, 1233 Fact 7.10.6, 570 signature Hermitian matrix definition, 511 Prop. 7.7.13, 560 Hermitian matrix

1530

similar over F

Fact 7.11.17, 577 idempotent matrix Prop. 4.7.5, 374 Prop. 7.7.23, 562 Cor. 7.7.22, 562 Fact 7.11.12, 577 Fact 7.11.18, 577 Fact 7.11.19, 577 Fact 7.11.27, 579 inverse matrix Fact 7.17.34, 609 involutory matrix Prop. 4.7.5, 374 Cor. 7.7.22, 562 Fact 7.17.34, 609 Kronecker product Fact 9.4.13, 686 Kronecker sum Fact 9.5.10, 692 lower triangular matrix Fact 7.10.3, 570 Lyapunov-stable matrix Fact 15.19.5, 1226 matrix classes Prop. 4.7.5, 374 matrix exponential Prop. 15.2.9, 1185 matrix power Fact 7.15.23, 600 Fact 7.15.24, 600 Fact 7.15.25, 600 Fact 7.15.26, 600 Fact 7.15.27, 601 minimal polynomial Prop. 6.6.3, 513 Fact 15.24.3, 1244 Fact 15.24.4, 1244 Fact 15.24.5, 1244 Fact 15.24.6, 1245 Fact 15.24.7, 1245 Fact 15.24.8, 1246 Fact 15.24.9, 1246 Fact 15.24.10, 1246 Fact 15.24.11, 1247 multi-Jordan matrix Thm. 7.4.2, 549 multicompanion matrix Cor. 7.3.9, 549 nilpotent matrix Prop. 4.7.5, 374 Fact 7.11.28, 579 nonsingular matrix Fact 7.11.14, 577 nonzero diagonal entry Fact 7.10.15, 571 normal matrix Prop. 7.7.12, 560 Fact 7.10.10, 571 Fact 7.10.11, 571 Fact 7.11.10, 576 partitioned matrix Fact 7.11.26, 578 Fact 7.11.27, 579 Fact 7.11.28, 579 Fact 7.11.29, 579

projector Cor. 7.7.22, 562 Fact 7.11.18, 577 real matrices Fact 7.11.9, 576 realization Prop. 16.9.5, 1272 reflector Cor. 7.7.22, 562 reverse transpose Fact 7.10.12, 571 reverse-symmetric matrix Fact 7.10.12, 571 semisimple matrix Prop. 7.7.12, 560 Fact 7.10.5, 570 Fact 7.11.7, 576 semistable matrix Fact 15.19.5, 1226 similarity invariant Thm. 6.3.11, 505 Cor. 7.3.9, 549 simultaneous diagonalization Fact 7.19.9, 617 skew-Hermitian matrix Fact 7.10.5, 570 Fact 15.19.12, 1227 skew-idempotent matrix Cor. 7.7.22, 562 skew-involutory matrix Prop. 4.7.5, 374 skew-symmetric matrix Fact 7.17.42, 610 Sylvester’s equation Cor. 9.2.6, 684 Fact 9.5.15, 692 symmetric matrix Fact 7.17.42, 610 transpose Prop. 7.7.13, 560 Cor. 6.3.12, 505 Cor. 7.4.9, 552 Cor. 7.7.22, 562 Fact 7.10.10, 571 Fact 7.10.11, 571 tripotent matrix Prop. 4.7.5, 374 Cor. 7.7.22, 562 unitarily invariant norm Fact 11.9.45, 865 unitarily similar matrices Fact 7.11.10, 576 upper triangular matrix Fact 7.10.3, 570 Vandermonde matrix Fact 7.18.11, 614 similar over F definition, 560 similar product eequality Fact 9.4.14, 686 similarity multi-real-Jordan form Thm. 7.4.6, 551

similarity invariant

characteristic polynomial Prop. 6.4.2, 506 Prop. 6.6.2, 513 definition Defn. 6.3.10, 505 elementary multicompanion matrix Cor. 7.3.9, 549 multicompanion matrix Cor. 7.3.9, 549 similar matrices Thm. 6.3.11, 505 Cor. 7.3.9, 549 similarity transformation complex conjugate transpose Fact 7.10.9, 570 Fact 7.17.5, 605 complex symmetric Jordan form Fact 7.17.4, 605 definition, 374 eigenvector Fact 7.15.7, 599 Fact 7.15.9, 599 hypercompanion matrix Fact 7.11.2, 575 inverse matrix Fact 7.17.5, 605 normal matrix Fact 7.17.4, 605 real Jordan form Fact 7.11.2, 575 Fact 7.11.3, 575 symmetric matrix Fact 7.17.3, 605 Fact 7.17.4, 605 SIMO transfer function definition Defn. 16.9.1, 1270 Simon Schatten norm Fact 11.16.4, 900 simple eigenvalue cyclic eigenvalue Prop. 7.7.5, 559 definition Defn. 7.7.4, 559 semisimple eigenvalue Prop. 7.7.5, 559 simple group definition Defn. 4.4.1, 369 simple matrix commuting matrices Fact 7.16.10, 603 cyclic matrix Fact 7.15.13, 599 definition Defn. 7.7.4, 559 identity-matrix perturbation Fact 7.15.17, 600 rank

Fact 7.12.1, 579 semisimple matrix Fact 7.15.13, 599 simple pole definition, 929 simplex convex hull Fact 5.1.8, 442 definition, 290 interior Fact 5.1.8, 442 nonsingular matrix Fact 5.1.8, 442 regular polytope Fact 5.4.6, 495 signed volume Fact 5.4.1, 493 volume Fact 5.4.6, 495 simultaneous block diagonalization unitary matrix Fact 10.20.1, 799 simultaneous diagonalization cogredient transformation Fact 10.20.6, 799 Fact 10.20.8, 799 commuting matrices Fact 10.20.2, 799 definition, 707 diagonalizable matrix Fact 10.20.3, 799 Fact 10.20.5, 799 Hermitian matrix Fact 10.20.2, 799 Fact 10.20.6, 799 Fact 10.20.8, 799 Milnor Fact 10.20.8, 799 Pesonen Fact 10.20.8, 799 positive-definite matrix Fact 10.20.7, 799 similar matrices Fact 7.19.9, 617 unitarily similar matrices Fact 7.19.8, 617 unitary matrix Fact 10.20.2, 799 simultaneous orthogonal biequivalence transformation upper Hessenberg matrix Fact 7.19.4, 617 upper triangular matrix Fact 7.19.4, 617 simultaneous triangularization cogredient transformation Fact 7.19.10, 617 common eigenvector Fact 7.19.2, 617 commutator Fact 7.19.6, 617

skew reflector Fact 7.19.7, 617 commuting matrices Fact 7.19.5, 617 nilpotent matrix Fact 7.19.7, 617 projector Fact 7.19.7, 617 regular pencil Fact 7.19.3, 617 simultaneous unitary biequivalence transformation Fact 7.19.3, 617 unitarily similar matrices Fact 7.19.5, 617 Fact 7.19.7, 617 unitary matrix Fact 7.19.1, 617 simultaneous unitary biequivalence transformation simultaneous triangularization Fact 7.19.3, 617 sine trigonometric equality Fact 2.16.8, 238 Fact 2.16.21, 246 sine rule triangle Fact 5.2.7, 443 singular matrix definition, 297 Kronecker product Fact 9.4.33, 688 spectrum Prop. 7.7.21, 562 singular pencil definition, 563 generalized eigenvalue Prop. 7.8.3, 564 singular polynomial matrix Defn. 6.2.5, 502 singular value 2 × 2 matrix Fact 7.12.35, 585 adjugate Fact 7.12.39, 585 arithmetic-mean– geometric mean inequality Fact 10.22.34, 812 bidiagonal matrix Fact 7.12.51, 589 block-diagonal matrix Fact 10.12.65, 760 Fact 10.22.16, 807 Fact 11.16.29, 904 Fact 11.16.34, 905 Cartesian decomposition Fact 10.22.14, 807 companion matrix Fact 7.12.34, 584

complex conjugate transpose Fact 7.12.23, 582 Fact 7.12.38, 585 convex function Fact 15.16.12, 1220 definition Defn. 7.6.1, 555 determinant Fact 7.12.32, 584 Fact 7.12.33, 584 Fact 7.13.26, 594 Fact 10.15.1, 774 Fact 11.15.26, 899 eigenvalue Fact 7.12.26, 582 Fact 7.12.33, 584 Fact 10.13.2, 761 Fact 10.21.3, 800 Fact 10.21.10, 801 Fact 10.21.17, 803 Fact 11.15.25, 899 eigenvalue of Hermitian part Fact 10.21.9, 801 Fan dominance theorem Fact 11.16.23, 903 fixed-rank approximation Fact 11.16.39, 906 Fact 11.17.14, 911 Frobenius Cor. 11.6.9, 849 generalized inverse Fact 8.3.33, 631 homogeneity Fact 7.12.22, 582 idempotent matrix Fact 7.12.41, 585 induced lower bound Prop. 11.5.4, 846 inequality Prop. 11.2.2, 837 Cor. 11.6.5, 848 Fact 11.16.31, 904 Fact 11.16.33, 905 inner matrix Fact 10.21.22, 804 Kronecker product Fact 9.4.19, 687 matrix difference Fact 10.12.65, 760 Fact 10.22.16, 807 matrix exponential Fact 15.16.5, 1218 Fact 15.16.12, 1220 Fact 15.17.5, 1222 Fact 15.17.6, 1222 matrix power Fact 11.15.19, 898 Fact 11.15.20, 898 Fact 11.15.21, 898 matrix product Prop. 11.6.1, 847 Prop. 11.6.2, 847 Prop. 11.6.3, 847

Prop. 11.6.4, 848 Fact 10.13.5, 761 Fact 10.13.6, 761 Fact 10.22.28, 810 Fact 11.16.35, 905 matrix sum Prop. 11.6.10, 849 Fact 11.16.24, 903 Fact 11.16.25, 903 Fact 11.16.29, 904 Fact 11.16.34, 905 normal matrix Fact 7.15.16, 599 Fact 10.13.2, 761 outer-product matrix Fact 7.12.20, 582 partitioned matrix Prop. 7.6.5, 557 Fact 11.16.13, 901 Fact 11.16.14, 901 Fact 11.16.33, 905 perturbation Fact 11.16.5, 900 polar decomposition Fact 11.16.26, 904 positive-semidefinite matrix Fact 10.11.32, 738 Fact 10.12.19, 752 Fact 10.12.20, 752 Fact 10.22.14, 807 Fact 10.22.20, 808 Fact 11.16.36, 905 product Fact 7.13.22, 594 Fact 11.16.1, 899 rank Prop. 7.6.2, 556 Fact 11.16.39, 906 Fact 11.17.14, 911 Schur product Fact 11.16.42, 907 Fact 11.16.43, 907 Fact 11.16.44, 907 Fact 11.16.46, 908 strong log majorization Fact 11.15.20, 898 Fact 11.15.21, 898 submatrix Fact 11.16.9, 901 trace Fact 7.13.14, 592 Fact 10.13.4, 761 Fact 10.21.7, 801 Fact 11.14.1, 892 Fact 11.16.2, 899 Fact 11.16.46, 908 unitarily biequivalent matrices Fact 7.11.23, 578 unitarily invariant norm Fact 11.9.48, 865 Fact 11.9.49, 865 Fact 11.16.39, 906 unitary matrix

1531

Fact 7.12.40, 585 Fact 11.16.13, 901 Fact 11.16.26, 904 Fact 11.16.27, 904 weak log majorization Prop. 11.6.2, 847 Fact 10.22.20, 808 weak majorization Prop. 11.2.2, 837 Prop. 11.6.3, 847 Fact 7.12.26, 582 Fact 10.13.5, 761 Fact 10.13.6, 761 Fact 10.21.10, 801 Fact 10.22.14, 807 Fact 10.22.28, 810 Fact 11.15.19, 898 Fact 11.16.23, 903 Fact 11.16.24, 903 Fact 11.16.25, 903 Fact 11.16.26, 904 Fact 11.16.27, 904 Fact 11.16.28, 904 Fact 11.16.37, 905 Fact 11.16.38, 906 Fact 11.16.42, 907 Fact 15.17.5, 1222 Fact 15.17.6, 1222 Weyl’s majorant theorem Fact 11.15.19, 898 singular value decomposition existence Thm. 7.6.3, 557 generalized inverse Fact 8.3.23, 630 group generalized inverse Fact 8.5.13, 643 least squares Fact 11.16.39, 906 Fact 11.17.13, 911 Fact 11.17.14, 911 Fact 11.17.16, 911 Fact 11.17.17, 911 unitary similarity Fact 7.10.27, 572 Fact 8.3.23, 630 Fact 8.5.13, 643 weak majorization Fact 11.16.27, 904 singular value multiset definition, 556 singular value perturbation unitarily invariant norm Fact 11.16.40, 906 singular value vector definition, 556 SISO transfer function definition Defn. 16.9.1, 1270 size definition, 280 skew reflector Hamiltonian matrix Fact 4.28.3, 427

1532

skew-Hermitian matrix

reflector Fact 10.10.17, 732 similar matrices Fact 4.19.7, 416 normal matrix Prop. 4.7.5, 374 skew-Hermitian matrix Prop. 4.1.7, 367 size Fact 4.19.6, 416 null space Fact 4.20.11, 417 skew-involutory matrix Fact 10.7.3, 722 skew-symmetric matrix Fact 4.19.6, 416 outer product Fact 4.28.3, 427 spectrum Fact 4.10.39, 382 spectrum Prop. 7.7.21, 562 outer-product matrix Prop. 7.7.21, 562 unitary matrix Fact 4.10.17, 380 symplectic matrix Fact 4.19.6, 416 Fact 4.11.4, 383 Fact 4.28.2, 427 unitarily similar matrices skew-Hermitian matrix, see partitioned matrix Fact 4.10.27, 381 Prop. 4.7.5, 374 skew-symmetric matrix positive-definite matrix skew-symmetric matrix, adjugate Fact 10.16.3, 776 Fact 4.10.9, 378 see skew-Hermitian Fact 15.19.12, 1227 Fact 4.10.11, 379 matrix positive-semidefinite asymptotically stable additive decomposition matrix matrix Fact 7.20.4, 618 Fact 10.10.15, 732 Fact 15.19.30, 1231 adjugate projector block-diagonal matrix Fact 6.9.19, 528 Fact 11.11.2, 884 Fact 4.10.7, 378 Cayley transform quadratic form Cartesian decomposition Fact 4.13.25, 390 Fact 4.10.5, 378 Fact 4.10.28, 381 Fact 4.13.26, 390 range Fact 7.20.2, 618 Fact 4.14.9, 394 Fact 10.7.3, 722 Fact 7.20.3, 618 characteristic polynomial rank Cayley transform Fact 6.9.11, 527 Fact 4.10.17, 380 Fact 4.13.25, 390 Fact 6.9.18, 527 Fact 4.10.32, 382 characteristic polynomial Fact 6.9.19, 528 similar matrices Fact 6.9.12, 527 Fact 7.16.21, 605 Fact 7.10.5, 570 commutator commutator Fact 15.19.12, 1227 Fact 4.29.6, 429 Fact 4.29.11, 429 skew-symmetric matrix Fact 4.29.10, 429 congruent matrices Fact 4.10.8, 378 complex conjugate Fact 4.10.37, 382 spectrum Fact 4.15.10, 397 Fact 7.10.17, 571 Prop. 7.7.21, 562 congruent matrices controllability symmetric matrix Prop. 4.7.5, 374 Fact 16.21.6, 1308 Fact 4.10.8, 378 definition definition trace Defn. 4.1.1, 363 Defn. 4.1.1, 363 Fact 4.10.24, 381 determinant determinant trace of a product Fact 4.10.11, 379 Fact 4.10.15, 380 Fact 7.13.11, 592 Fact 4.10.16, 380 Fact 4.10.35, 382 unitarily similar matrices Fact 10.16.3, 776 Fact 4.10.36, 382 Prop. 4.7.5, 374 eigenvalue Fact 4.10.40, 383 Prop. 7.7.23, 562 Fact 7.12.10, 580 Fact 6.8.16, 522 unitary matrix existence of Fact 6.9.19, 528 Fact 4.13.25, 390 Fact 6.10.8, 532 transformation Fact 15.15.37, 1217 eigenvalue Fact 4.11.4, 383 skew-idempotent matrix Fact 6.10.8, 532 Frobenius norm definition factorization Fact 11.9.8, 861 Defn. 4.1.1, 363 Fact 7.17.40, 610 Hermitian matrix idempotent matrix Fact 7.17.41, 610 Fact 4.10.8, 378 Fact 4.15.7, 397 Hamiltonian matrix Fact 7.20.2, 618 similar matrices Fact 4.10.37, 382 inertia Cor. 7.7.22, 562 Fact 4.28.3, 427 Fact 7.9.4, 565 skew-involutory matrix Fact 4.28.8, 428 Kronecker product definition Hermitian matrix Fact 9.4.23, 687 Defn. 4.1.1, 363 Fact 4.10.8, 378 Kronecker sum Hamiltonian matrix linear matrix equation Fact 9.5.9, 692 Fact 4.28.2, 427 Fact 4.10.2, 377 Lyapunov equation Fact 4.28.3, 427 matrix exponential Fact 15.19.12, 1227 inertia Prop. 15.4.8, 1188 matrix exponential Fact 7.9.4, 565 Example 15.3.6, 1186 Prop. 15.2.8, 1184 matrix exponential Fact 15.12.3, 1205 Prop. 15.4.3, 1187 Fact 15.12.1, 1204 Fact 15.12.6, 1206 Fact 15.15.7, 1212 semisimple matrix Fact 15.12.7, 1206 Fact 15.15.37, 1217 Fact 7.15.20, 600 Fact 15.12.8, 1207 matrix power

Fact 15.12.9, 1207 Fact 15.12.10, 1207 Fact 15.12.11, 1207 Fact 15.12.15, 1208 Fact 15.12.16, 1208 Fact 15.12.17, 1208 Fact 15.12.18, 1208 matrix product Fact 7.17.40, 610 orthogonal matrix Fact 4.13.25, 390 Fact 4.13.26, 390 Fact 15.12.10, 1207 Fact 15.12.11, 1207 orthogonally similar matrices Fact 7.16.20, 604 partitioned matrix Fact 4.13.12, 388 Pfaffian Fact 6.8.16, 522 quadratic form Fact 4.10.4, 377 similar matrices Fact 7.17.42, 610 skew-Hermitian matrix Fact 4.10.8, 378 skew-involutory matrix Fact 4.28.3, 427 spectrum Fact 6.9.19, 528 Fact 6.10.8, 532 Fact 7.16.20, 604 symmetric matrix Fact 7.10.17, 571 Fact 7.17.42, 610 trace Fact 4.10.23, 381 Fact 4.10.33, 382 unit imaginary matrix Fact 4.10.37, 382 small-gain theorem multiplicative perturbation Fact 11.15.26, 899 Smith form controllability pencil Prop. 16.6.15, 1264 definition, 504, 545 observability pencil Prop. 16.3.15, 1256 unimodular matrix Prop. 6.3.8, 504 Smith matrix biequivalent matrices Thm. 7.1.1, 545 definition, 504 existence Thm. 6.3.2, 504 polynomial matrix Prop. 6.3.4, 504 Smith polynomial nonsingular matrix transformation Prop. 6.3.9, 505

spectral radius Smith polynomials

definition Defn. 6.3.3, 504 Smith zeros controllability pencil Prop. 16.6.16, 1264 definition Defn. 6.3.3, 504 observability pencil Prop. 16.3.16, 1256 uncontrollable spectrum Prop. 16.6.16, 1264 unobservable spectrum Prop. 16.3.16, 1256 Smith’s method finite-sum solution of Lyapunov equation Fact 16.22.17, 1312 Smith-McMillan form blocking zero Prop. 6.7.11, 515 coprime polynomials Fact 6.8.17, 522 coprime right polynomial fraction description Prop. 6.7.16, 517 definition, 515 poles Prop. 6.7.11, 515 rank Prop. 6.7.7, 515 Prop. 6.7.8, 515 submatrix Prop. 6.7.7, 515 Smith-McMillan matrix definition, 515 existence Thm. 6.7.5, 514 SO(2) parameterization Fact 4.14.1, 391 SO(3) orthogonal matrix Fact 4.14.2, 392 solid angle circular cone Fact 5.5.11, 497 Fact 5.5.12, 497 cone Fact 5.5.10, 497 solid set completely solid set Fact 12.11.20, 935 convex hull Fact 12.11.17, 935 convex set Fact 12.11.20, 935 definition Defn. 12.1.12, 915 dimension Fact 12.11.27, 936 image Fact 12.11.28, 936 solution Riccati equation

Defn. 16.16.12, 1297

matrix exponential Fact 15.14.2, 1209 Fact 15.16.8, 1219 affine subspace Fact 15.16.9, 1219 Fact 3.11.14, 309 Fact 15.16.10, 1220 Fact 5.1.8, 442 Fact 15.19.9, 1227 Fact 12.11.13, 935 maximum eigenvalue constructive Fact 7.12.9, 580 characterization maximum singular value Fact 3.11.5, 306 Fact 7.12.28, 583 Fact 3.11.6, 307 minimum singular value convex conical hull Fact 7.12.28, 583 Fact 3.11.8, 308 outer-product matrix definition, 279 Fact 7.12.16, 581 intersection spectral radius Fact 3.12.11, 312 Fact 6.10.10, 532 Minkowski sum Fact 15.14.2, 1209 Fact 3.12.2, 310 spectral abscissas range definition, 510 Fact 3.11.18, 309 spectral condition number subset quadratic form Fact 3.12.1, 310 Fact 10.18.9, 792 subspace spectral factorization Fact 3.11.22, 310 definition, 500 union Hamiltonian Fact 3.12.11, 312 Prop. 16.16.13, 1297 spanning directed polynomial roots subgraph Prop. 6.1.1, 500 Defn. 1.4.3, 10 spectral imaginary spanning path abscissa graph definition, 511 Fact 1.9.9, 26 spectral imaginary tournament Fact 1.9.9, 26 abscissas spanning subgraph definition, 511 Defn. 1.4.4, 11 spectral imaginary part sparse matrix definition, 510 definition, 440 spectral moduli Specht definition, 510 reverse AM-GM inequality spectral modulus Fact 2.11.95, 206 definition, 510 Specht’s ratio spectral norm limit definition, 838 Fact 12.17.5, 955 spectral ordering matrix exponential positive-definite matrix Fact 15.15.30, 1216 Fact 10.23.3, 814 power of a positive-definite positive-semidefinite matrix matrix Fact 15.15.23, 1215 Fact 10.23.3, 814 Fact 15.15.24, 1215 spectral radius reverse AM-GM inequality bound Fact 2.11.95, 206 Fact 6.10.28, 536 reverse Young inequality column norm Fact 2.2.54, 140 Cor. 11.4.10, 844 Specht’s theorem commuting matrices unitarily similar matrices Fact 7.13.23, 594 Fact 7.11.11, 576 convergent sequence special orthogonal group Fact 11.9.3, 860 real symplectic group convergent series Fact 4.32.5, 438 Fact 6.10.11, 532 spectral abscissa convexity for nonnegative definition, 510 matrices eigenvalue Fact 6.11.26, 544 Fact 7.12.27, 583 definition, 510 Hermitian matrix discrete-time Lyapunov Fact 7.12.9, 580 equation span

1533

Fact 15.22.20, 1241 equi-induced norm Cor. 11.4.5, 842 Frobenius norm Fact 11.15.14, 897 Fact 11.15.22, 898 Fact 11.15.23, 898 Hermitian matrix Fact 7.12.9, 580 induced norm Cor. 11.4.5, 842 Cor. 11.4.10, 844 inverse matrix Prop. 11.3.10, 841 Kronecker product Fact 9.4.18, 686 Kronecker sum Fact 9.5.7, 691 lower bound Fact 11.15.14, 897 matrix exponential Fact 15.14.2, 1209 matrix power Fact 11.13.11, 892 Fact 15.22.16, 1241 matrix sum Fact 7.13.8, 591 Fact 7.13.9, 592 maximum singular value Cor. 11.4.10, 844 Fact 7.12.9, 580 Fact 7.12.28, 583 Fact 10.22.33, 811 Fact 11.9.16, 861 Fact 11.15.11, 897 minimum singular value Fact 7.12.28, 583 monotonicity for nonnegative matrices Fact 6.11.23, 543 nonnegative matrix Fact 6.11.13, 541 Fact 6.11.21, 543 Fact 6.11.22, 543 Fact 6.11.24, 544 Fact 6.11.25, 544 Fact 9.6.18, 698 Fact 11.13.9, 892 Fact 15.20.3, 1232 nonsingular matrix Fact 6.10.36, 537 norm Prop. 11.2.6, 838 normal matrix Fact 7.15.16, 599 Fact 11.15.22, 898 Fact 11.15.23, 898 outer-product matrix Fact 7.12.16, 581 partitioned matrix Fact 10.12.66, 760 perturbation Fact 11.16.5, 900 positive matrix Fact 9.6.20, 699

1534

spectral real part

positive-definite matrix Fact 10.11.14, 736 Fact 10.22.33, 811 Fact 10.25.30, 823 positive-semidefinite matrix Fact 10.22.33, 811 Fact 10.24.9, 816 Fact 10.25.30, 823 row norm Cor. 11.4.10, 844 row-stochastic matrix Fact 6.11.11, 541 Schur product Fact 9.6.18, 698 Fact 9.6.20, 699 Fact 9.6.23, 700 Fact 9.6.24, 700 Fact 11.16.44, 907 series Fact 13.4.15, 1019 spectral abscissa Fact 6.10.10, 532 Fact 15.14.2, 1209 submultiplicative norm Prop. 11.3.2, 839 Prop. 11.3.3, 839 Cor. 11.3.4, 839 Fact 11.9.3, 860 Fact 11.10.28, 872 trace Fact 6.10.28, 536 Fact 7.12.50, 588 Fact 11.15.14, 897 spectral real part definition, 510 spectral variation Hermitian matrix Fact 11.14.8, 893 Fact 11.14.10, 894 maximum singular value Fact 11.14.7, 893 normal matrix Fact 11.14.8, 893 Fact 11.14.9, 894 Fact 11.14.11, 894 Fact 11.14.12, 894 spectrum, see eigenvalue, multispectrum adjugate Fact 6.10.13, 532 asymptotic eigenvalue Fact 6.10.35, 537 asymptotically stable matrix Fact 15.19.13, 1227 block-triangular matrix Prop. 7.7.14, 560 bounds Fact 6.10.22, 535 Fact 6.10.26, 535 Fact 6.10.27, 536 Cartesian decomposition Fact 7.12.24, 582 circulant matrix

Fact 7.18.13, 614 commutator Fact 7.11.30, 579 commuting matrices Fact 7.11.30, 579 continuity Fact 12.13.21, 945 Fact 12.13.22, 945 convex hull Fact 10.17.8, 787 Fact 10.17.9, 788 cross-product matrix Fact 6.9.18, 527 definition Defn. 6.4.4, 506 dissipative matrix Fact 10.16.20, 779 doublet Fact 7.12.16, 581 elementary matrix Prop. 7.7.21, 562 elementary projector Prop. 7.7.21, 562 elementary reflector Prop. 7.7.21, 562 group-invertible matrix Prop. 7.7.21, 562 Hamiltonian Thm. 16.17.9, 1300 Prop. 16.16.13, 1297 Prop. 16.17.5, 1299 Prop. 16.17.7, 1300 Prop. 16.17.8, 1300 Lem. 16.17.4, 1299 Lem. 16.17.6, 1300 Hamiltonian matrix Prop. 7.7.21, 562 Hermitian matrix Prop. 7.7.21, 562 Lem. 10.4.8, 710 Fact 6.10.3, 531 Fact 6.10.4, 531 idempotent matrix Prop. 7.7.21, 562 Fact 7.13.1, 589 identity-matrix perturbation Fact 6.10.18, 534 Fact 6.10.20, 534 inverse matrix Fact 7.12.17, 581 involutory matrix Prop. 7.7.21, 562 Laplacian matrix Fact 15.20.7, 1233 lower triangular matrix Fact 6.10.14, 533 matrix exponential Prop. 15.2.3, 1183 Cor. 15.2.6, 1183 matrix function Cor. 12.8.4, 932 matrix logarithm Thm. 15.5.2, 1190 nilpotent matrix

Prop. 7.7.21, 562 normal matrix Fact 6.10.30, 536 Fact 7.12.30, 583 Fact 10.17.8, 787 Fact 10.17.9, 788 orthogonal matrix Fact 4.13.32, 391 outer-product matrix Fact 6.10.5, 531 Fact 7.12.16, 581 partitioned matrix Fact 3.24.6, 357 Fact 6.10.19, 534 Fact 6.10.31, 536 Fact 6.10.32, 536 permutation matrix Fact 7.18.14, 616 perturbed matrix Fact 6.10.9, 532 polynomial Fact 6.9.36, 530 Fact 6.10.15, 533 positive matrix Fact 7.12.15, 581 positive-definite matrix Prop. 7.7.21, 562 positive-semidefinite matrix Prop. 7.7.21, 562 Fact 10.24.17, 817 projector Prop. 7.7.21, 562 Fact 7.13.4, 590 Fact 7.13.5, 591 properties Prop. 6.4.5, 507 quadratic form Fact 10.17.8, 787 Fact 10.17.9, 788 quadratic matrix equation Fact 7.12.3, 579 Fact 7.12.4, 579 rational function Fact 7.12.18, 582 reflector Prop. 7.7.21, 562 reverse permutation matrix Fact 7.10.25, 572 row-stochastic matrix Fact 6.10.6, 531 shifted-unitary matrix Prop. 7.7.21, 562 singular matrix Prop. 7.7.21, 562 skew reflector Prop. 7.7.21, 562 skew-Hermitian matrix Prop. 7.7.21, 562 skew-involutory matrix Prop. 7.7.21, 562 skew-symmetric matrix Fact 6.9.19, 528 Fact 6.10.8, 532

Fact 7.16.20, 604 subspace decomposition Prop. 7.7.7, 559 Sylvester’s equation Cor. 9.2.6, 684 Fact 9.5.15, 692 symplectic matrix Prop. 7.7.21, 562 Toeplitz matrix Fact 6.10.21, 534 Fact 7.12.46, 587 Fact 7.12.47, 587 Fact 10.9.1, 724 Fact 10.9.23, 730 Fact 10.10.39, 734 trace Fact 6.10.12, 532 tridiagonal matrix Fact 7.12.43, 586 Fact 7.12.44, 586 Fact 7.12.45, 586 Fact 7.12.46, 587 Fact 7.12.47, 587 Fact 10.9.23, 730 tripotent matrix Prop. 7.7.21, 562 unipotent matrix Prop. 7.7.21, 562 unit imaginary matrix Fact 7.10.24, 572 unitary matrix Prop. 7.7.21, 562 upper triangular matrix Fact 6.10.14, 533 spectrum bounds ovals of Cassini Fact 6.10.27, 536 spectrum of convex hull field of values Fact 10.17.8, 787 Fact 10.17.9, 788 numerical range Fact 10.17.8, 787 Fact 10.17.9, 788 Sperner lemma set intersection Fact 1.8.9, 24 sphere of radius ε definition Defn. 12.1.1, 913 spin group double cover Fact 4.14.6, 392 spread commutator Fact 11.11.7, 885 Fact 11.11.8, 885 definition, 510 eigenvalue Fact 10.21.5, 801 Fact 11.13.10, 892 Hermitian matrix Fact 10.18.22, 794 spread of a polynomial definition, 500

stiffness square

definition, 280 positive-semidefinite matrix Fact 10.7.2, 722 trace Fact 10.13.3, 761 square root 2 × 2 matrix Fact 3.15.34, 328 2 × 2 positive-semidefinite matrix Fact 10.10.8, 731 asymptotically stable matrix Fact 15.19.36, 1231 commuting matrices Fact 10.8.1, 723 Fact 10.11.44, 740 convergent sequence Fact 7.17.23, 608 Fact 10.10.37, 734 definition, 714 equality Fact 10.10.28, 733 Fact 10.10.29, 733 generalized inverse Fact 10.24.4, 815, 816 group-invertible matrix Fact 7.17.22, 608 inequality Fact 1.12.41, 47 Fact 2.11.125, 210 Jordan form Fact 7.17.21, 607 Kronecker product Fact 10.25.26, 823 Fact 10.25.47, 826 limit Fact 12.18.30, 962 Fact 12.18.31, 962 matrix inequality Fact 10.10.34, 733 matrix sign function Fact 7.17.23, 608 maximum singular value Fact 11.9.50, 865 Newton-Raphson algorithm Fact 7.17.23, 608 normal matrix Fact 10.10.31, 733 Fact 10.10.33, 733 orthogonal matrix Fact 10.10.30, 733 positive-semidefinite matrix Fact 10.10.32, 733 Fact 10.11.29, 738 Fact 10.11.35, 738 Fact 10.11.36, 738 Fact 10.11.37, 738 Fact 10.11.45, 740 Fact 10.11.48, 740 Fact 10.12.37, 755

Fact 11.9.50, 865 principal square root Thm. 12.9.1, 932 projector Fact 10.11.44, 740 submultiplicative norm Fact 11.9.50, 865 sum Fact 1.12.31, 46 Fact 1.12.32, 46 Fact 1.12.33, 46 Fact 1.12.40, 47 sum of squares Fact 3.23.8, 355 unitarily invariant norm Fact 11.10.45, 874 Fact 11.10.51, 875 unitary matrix Fact 10.8.2, 723 Fact 10.10.30, 733 Fact 10.10.34, 733 square-root function Niculescu’s inequality Fact 2.2.48, 139 stability mass-spring system Fact 15.19.38, 1231 partitioned matrix Fact 15.19.38, 1231 stability radius asymptotically stable matrix Fact 15.19.17, 1228 stabilizability asymptotically stable matrix Prop. 16.8.6, 1269 Cor. 16.8.7, 1270 block-triangular matrix Prop. 16.8.4, 1268 controllably asymptotically stable Prop. 16.8.6, 1269 definition Defn. 16.8.1, 1268 Prop. 16.8.2, 1268 full-state feedback Prop. 16.8.3, 1268 Hamiltonian Fact 16.25.1, 1316 input matrix Fact 16.21.17, 1309 Lyapunov equation Cor. 16.8.7, 1270 maximal solution of the Riccati equation Thm. 16.18.1, 1302 orthogonal matrix Prop. 16.8.4, 1268 PBH test Thm. 16.8.5, 1269 positive-semidefinite matrix Fact 16.21.7, 1308

positive-semidefinite ordering Fact 16.21.9, 1308 range Fact 16.21.8, 1308 Riccati equation Thm. 16.17.9, 1300 Thm. 16.18.1, 1302 Cor. 16.19.2, 1304 shift Fact 16.21.12, 1308 stabilization controllability Fact 16.21.19, 1309 Gramian Fact 16.21.19, 1309 stabilizing solution Hamiltonian Cor. 16.16.15, 1298 Riccati equation Defn. 16.16.12, 1297 Thm. 16.17.2, 1299 Thm. 16.17.9, 1300 Thm. 16.18.4, 1302 Prop. 16.17.1, 1298 Prop. 16.18.3, 1302 Prop. 16.19.4, 1305 Cor. 16.16.15, 1298 stable subspace complementary subspaces Prop. 15.9.8, 1200 group-invertible matrix Prop. 15.9.8, 1200 idempotent matrix Prop. 15.9.8, 1200 invariant subspace Prop. 15.9.8, 1200 matrix exponential Prop. 15.9.8, 1200 minimal polynomial Prop. 15.9.5, 1199 Fact 15.24.1, 1243 Fact 15.24.2, 1243 standard control problem definition, 1293 standard deviation mean Fact 2.11.38, 196 standard nilpotent matrix definition, 284 Stanley’s theorem partition number Fact 1.20.1, 113 star partial ordering commuting matrices Fact 4.30.9, 431 definition Fact 4.30.8, 431 generalized inverse Fact 8.4.43, 640 Fact 8.4.44, 640 Hermitian matrices Fact 10.23.6, 814 partitioned matrix Fact 4.30.10, 431

1535

Fact 4.30.11, 431 Fact 4.30.12, 431 Fact 4.30.13, 431 Fact 4.30.14, 431 Fact 4.30.15, 431 Fact 8.4.45, 640 Fact 8.4.46, 640 positive-semidefinite matrix Fact 10.23.5, 814 Fact 10.23.7, 814 Fact 10.24.9, 816 reflexivity Fact 4.30.6, 430 sum Fact 4.30.16, 431 transitivity Fact 4.30.7, 430 star-dagger matrix generalized inverse Fact 8.3.32, 631 state convergence detectability Fact 16.21.3, 1307 discrete-time system Fact 15.22.22, 1242 state equation definition, 1249 matrix exponential Prop. 16.1.1, 1249 variation of constants formula Prop. 16.1.1, 1249 state transition matrix time-varying dynamics Fact 16.20.7, 1306 statement definition, 2 Stein equation discrete-time Lyapunov equation Fact 15.22.19, 1241 Fact 15.22.21, 1242 Steiner inellipse polynomial Fact 5.5.3, 496 step function, 1250 step response definition, 1251 Lyapunov-stable matrix Fact 16.21.2, 1307 step-down matrix resultant Fact 6.8.6, 518 Stephanos Kronecker product Fact 9.4.27, 687 Stewart regular pencil Fact 7.19.4, 617 Stieltjes constant zeta function Fact 13.3.1, 994 stiffness definition, 1187

1536

stiffness matrix

stiffness matrix

partitioned matrix Fact 7.13.31, 596 Stirling matrix Vandermonde matrix Fact 7.18.5, 613 Stirling number of the first kind cycle numbers Fact 1.19.1, 105 equalities Fact 1.19.2, 106 Stirling number of the second kind Bell number Fact 1.19.6, 112 equalities Fact 1.19.4, 108 series Fact 13.4.4, 1009 subset number Fact 1.19.3, 107 sum of powers Fact 1.19.4, 108 Stirling’s formula factorial Fact 12.18.58, 969 Stolz-Cesaro lemma limit Fact 12.18.19, 960 Storey tridiagonal matrix Fact 15.19.24, 1229 Stormer Schatten norm Fact 11.10.16, 871 Stothers’ theorem polynomial Fact 6.8.23, 523 Stothers’s theorem polynomial Fact 6.8.22, 523 strengthening definition, 4 strictly concave function definition Defn. 10.6.14, 717 strictly convex function definition Defn. 10.6.14, 717 positive-definite matrix Fact 10.17.18, 789 Fact 10.17.19, 790 trace Fact 10.17.19, 790 transformation Fact 1.21.5, 117 strictly dissipative matrix dissipative matrix Fact 10.10.36, 734 strictly lower triangular matrix definition Defn. 4.1.3, 365 matrix power

Fact 4.23.7, 421 matrix product Fact 4.25.2, 424 strictly proper rational function definition Defn. 6.7.1, 513 strictly proper rational transfer function definition Defn. 6.7.2, 514 strictly upper triangular matrix definition Defn. 4.1.3, 365 Lie algebra Fact 4.31.10, 434 Fact 15.23.1, 1243 matrix power Fact 4.23.7, 421 matrix product Fact 4.25.2, 424 strong Kronecker product Kronecker product, 700 strong log majorization convex function Fact 3.25.9, 361 definition Defn. 3.10.1, 305 eigenvalue Fact 10.22.37, 812 Fact 10.22.38, 812 matrix exponential Fact 15.17.4, 1221 singular value inequality Fact 11.15.20, 898 Fact 11.15.21, 898 strong majorization Fact 3.25.3, 360 strong majorization convex function Fact 3.25.8, 361 Fact 3.25.12, 361 convex hull Fact 4.11.6, 384 definition Defn. 3.10.1, 305 diagonal entry Fact 10.21.11, 802 Fact 10.21.12, 802 doubly stochastic matrix Fact 4.11.6, 384 eigenvalue Cor. 10.6.19, 722 Fact 10.22.6, 805 Fact 10.22.37, 812 Fact 10.22.38, 812 Fact 10.22.41, 813 eigenvalues Fact 10.22.3, 805 entropy Fact 3.25.7, 361 Hermitian matrix Fact 10.21.11, 802 Fact 10.21.12, 802

Fact 10.22.3, 805 linear interpolation Fact 4.11.6, 384 matrix logarithm Fact 10.22.43, 813 Muirhead’s theorem Fact 3.25.6, 360 ones vector Fact 3.25.4, 360 positive-definite matrix Fact 10.22.10, 806 Fact 10.22.43, 813 positive-semidefinite matrix Fact 10.22.6, 805 Schur-concave function Fact 3.25.7, 361 Schur-convex function Fact 3.25.5, 360 Fact 3.25.6, 360 strong log majorization Fact 3.25.3, 360 weak majorization Fact 3.25.1, 359 Fact 3.25.2, 359 strongly connected graph definition, 118 strongly decreasing function definition Defn. 10.6.12, 716 strongly increasing function definition Defn. 10.6.12, 716 determinant Prop. 10.6.13, 717 matrix functions Prop. 10.6.13, 717 Styan Hermitian matrix inertia equality Fact 10.11.26, 737 idempotent matrices Fact 7.13.29, 596 rank of an idempotent matrix Fact 4.16.8, 401 SU(2) quaternions Fact 4.32.6, 439 SU(3) unitary matrix Fact 4.14.3, 392 subdeterminant asymptotically stable matrix Fact 15.20.1, 1232 asymptotically stable polynomial Fact 15.19.23, 1228 definition, 301 determinant Fact 3.16.10, 330 Fact 3.17.35, 341

inverse Fact 3.17.33, 341 Lyapunov-stable polynomial Fact 15.19.23, 1228 positive-definite matrix Prop. 10.2.9, 707 Fact 10.16.16, 779 positive-semidefinite matrix Prop. 10.2.8, 706 Fact 10.15.12, 776 subdiagonal definition, 280 subdifferential convex function Fact 12.16.10, 950 subgraph Defn. 1.4.4, 11 subgroup definition Defn. 4.4.1, 369 Defn. 4.6.1, 371 group Thm. 4.4.4, 370 Prop. 4.4.1, 370 Fact 4.31.16, 435 Lagrange’s theorem Thm. 4.4.4, 370 sublevel set convex function Fact 12.13.24, 945 convex set Fact 10.17.1, 785 submatrix adjugate Fact 3.19.7, 346 complementary Fact 3.14.27, 325 Fact 3.14.28, 325 defect Fact 3.14.27, 325 definition, 281 determinant Fact 3.16.4, 329 Fact 3.17.1, 334 Hermitian matrix Cor. 10.4.6, 710 Lem. 10.4.4, 709 Fact 7.9.8, 566 inertia Fact 7.9.8, 566 Kronecker product Prop. 9.3.2, 685 lower Hessenberg matrix Fact 4.24.2, 423 M-matrix Fact 6.11.14, 542 nonnegative matrix Fact 6.11.24, 544 positive-definite matrix Prop. 10.2.9, 707 positive-semidefinite matrix Prop. 10.2.8, 706

sum Fact 10.7.9, 723 Fact 10.10.3, 731 Fact 10.12.40, 756 Fact 10.16.39, 782 rank Prop. 6.7.7, 515 Fact 3.14.1, 320 Fact 3.14.25, 324 Fact 3.14.26, 325 Fact 3.14.27, 325 Fact 3.14.28, 325 Fact 3.14.29, 325 Fact 4.24.2, 423 rank of a polynomial matrix Prop. 6.3.5, 504 singular value Fact 11.16.9, 901 Smith-McMillan form Prop. 6.7.7, 515 tridiagonal Fact 4.24.2, 423 Z-matrix Fact 6.11.14, 542 submultiplicative norm commutator Fact 11.11.1, 884 compatible norms Prop. 11.3.1, 838 definition, 838 equi-induced norm Cor. 11.4.4, 842 Fact 11.9.63, 868 H2 norm Fact 16.24.23, 1316 H¨older norm Fact 11.10.48, 875 idempotent matrix Fact 11.9.5, 860 infinity norm Fact 11.10.26, 872 Fact 11.10.27, 872 matrix exponential Prop. 15.1.2, 1179 Fact 15.16.8, 1219 Fact 15.16.9, 1219 Fact 15.16.10, 1220 Fact 15.17.10, 1222 Fact 15.19.9, 1227 matrix norm Fact 11.10.29, 872 matrix power Fact 11.13.11, 892 Fact 15.22.16, 1241 nonsingular matrix Fact 11.9.4, 860 positive-semidefinite matrix Fact 11.10.33, 872 Schur product Fact 11.9.58, 867 spectral radius Prop. 11.3.2, 839 Prop. 11.3.3, 839 Cor. 11.3.4, 839

Fact 11.9.3, 860 Fact 11.10.28, 872 square root Fact 11.9.50, 865 unitarily invariant norm Fact 11.9.58, 867 Fact 11.10.25, 872 Fact 11.10.33, 872 subset affine hull Fact 3.12.1, 310 closure Fact 12.12.1, 937 conical hull Fact 3.12.1, 310 convex conical hull Fact 3.12.1, 310 convex hull Fact 3.12.1, 310 definition, 1 interior Fact 12.12.1, 937 span Fact 3.12.1, 310 subset number Bell number Fact 1.19.6, 112 definition Fact 1.19.3, 107 equalities Fact 1.19.4, 108 limit Fact 12.18.54, 968 ordered Bell number Fact 1.19.7, 112 Vandermonde matrix Fact 7.18.7, 613 subset numbers series Fact 13.1.1, 975 subset operation induced partial ordering Fact 1.8.14, 25 transitivity Fact 1.8.14, 25 subspace affine subspace Fact 3.11.10, 308 Fact 3.11.11, 308 Fact 3.11.12, 308 Cartesian product Fact 3.12.18, 313 closed set Fact 12.11.11, 935 closure Fact 12.11.15, 935 common eigenvector Fact 7.16.15, 604 complementary Fact 3.12.17, 313 Fact 3.12.22, 314 Fact 3.12.26, 315 Fact 3.12.27, 315 complex conjugate transpose

Fact 3.12.10, 312 convex cone Fact 3.11.9, 308 definition, 278 dimension Cor. 3.1.5, 279 Fact 3.12.19, 313 Fact 3.12.20, 314 Fact 3.12.23, 314 Fact 3.12.24, 314 dimension inequality Fact 3.13.4, 315 gap Fact 12.12.28, 941 image under linear mapping Fact 3.11.15, 309 inclusion Cor. 3.1.5, 279 Fact 3.12.10, 312 Fact 3.12.14, 312 Fact 3.12.16, 312 inner product Fact 12.12.21, 939 interior Fact 12.11.15, 935 intersection Prop. 3.1.2, 278 Fact 3.12.4, 311 Fact 3.12.25, 314 inverse function Fact 3.11.16, 309 left inverse Fact 3.11.15, 309 minimal principal angle Fact 7.12.42, 586 Fact 7.13.27, 594 Fact 12.12.28, 941 Minkowski sum Fact 3.12.4, 311 open set Fact 12.11.11, 935 orthogonal complement Prop. 4.8.2, 374 Fact 3.11.22, 310 Fact 3.12.14, 312 Fact 3.12.17, 313 Fact 3.18.19, 345 orthogonal matrix Fact 4.13.2, 387 Fact 4.13.3, 388 Fact 4.13.4, 388 orthogonally complementary subspaces Fact 3.11.22, 310 principal angle Fact 3.12.21, 314 projector Prop. 4.8.1, 374 Prop. 4.8.2, 374 Fact 4.18.1, 409 Fact 10.11.9, 736 Fact 12.12.26, 940 quadratic form

1537

Fact 10.19.21, 798 Fact 10.19.22, 798 range Prop. 4.8.1, 374 Fact 3.11.20, 310 relative interior Fact 12.11.15, 935 scaling Fact 3.11.1, 306 span Fact 3.11.4, 306 Fact 3.11.22, 310 span of image Fact 3.11.15, 309 sum Fact 3.12.12, 312 Fact 3.12.25, 314 union Fact 3.12.16, 312 unitary matrix Fact 4.13.2, 387 Fact 4.13.3, 388 Fact 4.13.4, 388 subspace decomposition spectrum Prop. 7.7.7, 559 subspace dimension theorem dimension Thm. 3.1.3, 279 rank Fact 3.14.16, 323 sufficiency definition, 4 sufficient condition definition, 3 sum cardinality of set intersections Fact 1.12.16, 42 Drazin generalized inverse Fact 8.11.8, 675 Fact 8.11.9, 675 Fact 8.11.11, 676 eigenvalue Fact 7.13.8, 591 Fact 7.13.9, 592 equality Fact 1.12.26, 44 floor function Fact 1.12.11, 41 Fact 1.12.12, 41 Fact 1.12.13, 41 generalized inverse Fact 8.4.33, 638 Fact 8.4.34, 638 Fact 8.4.35, 638 Fact 8.4.36, 638 Fact 8.4.37, 639 Fact 8.4.38, 639 group generalized inverse Fact 8.11.12, 676 Hamiltonian matrix Fact 4.28.5, 428 idempotent matrix

1538

sum of digits

Fact 4.16.21, 407 inequality Fact 2.11.12, 190 Fact 2.11.13, 190 Fact 2.11.14, 190 Fact 2.11.44, 197 Fact 2.11.46, 197 inverse image Fact 3.12.8, 311 least common multiple Fact 1.11.8, 29 outer-product matrix Fact 3.13.31, 318 products of integers Fact 1.12.14, 42 Fact 1.12.15, 42 projector Fact 4.18.24, 416 Fact 7.13.27, 594 singular value Fact 11.16.24, 903 Fact 11.16.25, 903 Fact 11.16.29, 904 Fact 11.16.34, 905 spectral radius Fact 7.13.8, 591 Fact 7.13.9, 592 star partial ordering Fact 4.30.16, 431 sum of digits modulo Fact 1.11.11, 30 sum of integer powers inequality Fact 1.12.42, 47 matrix exponential Fact 15.12.4, 1205 sum of integers odd integers Fact 1.12.18, 42 sum of limits convergent sequence Prop. 12.4.8, 922 sum of matrices determinant Fact 4.13.19, 389 Fact 7.13.24, 594 Fact 7.13.25, 594 Fact 10.16.21, 779 Fact 11.16.22, 903 idempotent matrix Fact 4.16.5, 399 Fact 7.20.12, 619 Fact 7.20.13, 619 Fact 7.20.14, 619 inverse matrix Cor. 3.9.10, 305 Kronecker product Prop. 9.1.4, 682 nilpotent matrix Fact 4.22.12, 420 projector Fact 4.18.18, 413 Fact 4.18.20, 415 Fact 7.20.9, 618

Hardy-Littlewood rank definition, 118 Fact 7.17.1, 605 Sylvester matrix rearrangement sum of polynomials coprime polynomials inequality root radius Fact 6.8.6, 518 Fact 2.12.8, 216 Fact 15.21.32, 1239 Sylvester’s equation sum of projectors Fact 15.21.33, 1239 controllability Cochran’s theorem sum of powers Fact 16.22.14, 1312 Fact 4.18.24, 416 Bernoulli formula controllability matrix sum of ratios Fact 1.12.1, 36 Fact 16.22.13, 1311 Bergstrom’s inequality Boutin’s identity linear matrix equation Fact 2.12.19, 218 Fact 2.11.9, 189 Prop. 9.2.5, 684 Fact 2.12.22, 218 Carlson inequality Prop. 15.10.3, 1201 Radon’s inequality Fact 2.11.141, 213 Fact 7.11.25, 578 Fact 2.12.21, 218 Carlson’s inequality Fact 7.11.26, 578 sum of squares Fact 2.11.142, 213 Fact 8.9.10, 662 Diophantus’s identity Copson inequality nonsingular matrix Fact 2.4.7, 178 Fact 2.11.133, 212 Fact 16.22.14, 1312 Lagrange’s identity equality observability Fact 2.4.7, 178 Fact 1.12.25, 44 Fact 16.22.14, 1312 Lebesgue’s identity Fact 1.12.27, 44 observability matrix Fact 2.4.5, 178 Fact 1.12.28, 45 Fact 16.22.13, 1311 positive integer Fact 1.13.3, 49 partitioned matrix Fact 1.11.24, 32 Fact 2.1.13, 121 Fact 7.11.25, 578 Fact 1.11.27, 33 Hardy discrete inequality Fact 7.11.26, 578 Fact 10.18.27, 795 Fact 2.11.132, 212 Fact 8.9.10, 662 prime number Lagrange’s four-square rank Fact 1.11.40, 35 Fact 16.22.13, 1311 theorem ratio series Fact 1.11.26, 33 Fact 2.12.48, 224 Fact 12.15.5, 949 limit square root similar matrices Fact 12.18.29, 962 Fact 3.23.8, 355 Cor. 9.2.6, 684 monotonic sequence Wirtinger’s type inequality Fact 9.5.15, 692 Fact 1.12.43, 47 Fact 2.11.20, 191 spectrum norm inequality sum of squares ratio Cor. 9.2.6, 684 Fact 2.11.130, 211 inequality Fact 9.5.15, 692 positive integer Fact 2.12.49, 224 Sylvester’s identity Fact 1.11.18, 32 sum of subspaces determinant Fact 1.11.26, 33 subspace dimension Fact 3.17.1, 334 Stirling number of the theorem Sylvester’s inequality second kind Thm. 3.1.3, 279 rank of a product, 294 Fact 1.19.4, 108 sum of transfer functions Sylvester’s law of inertia subset number H2 norm definition, 554 Fact 1.19.4, 108 Prop. 16.11.6, 1287 Sylvester’s law of nullity Waring’s problem Sun defect Fact 1.11.26, 33 spectral variation of a Fact 3.13.22, 317 sum of powers of integers normal matrix Sylvester’s rank formula, Bernoulli number Fact 11.14.9, 894 294 Fact 1.12.2, 37 superdiagonal product of matrices Faulhaber polynomial definition, 280 Fact 3.13.21, 316 Fact 1.12.2, 37 superior symmetric cone inequality sequence induced by symmetric Fact 1.12.35, 46 Defn. 12.2.11, 917 Fact 1.12.36, 46 relation Prop. 12.2.10, 917 Fact 1.12.37, 47 Prop. 3.1.7, 280 supermultiplicativity sum of products symmetric difference induced lower bound Abel’s inequality definition, 1 Prop. 11.5.6, 847 Fact 2.12.15, 217 exclusive or support of a relation Fact 2.12.16, 217 Fact 1.8.3, 23 definition equality set Defn. 1.3.3, 6 Fact 2.11.1, 188 Fact 1.8.3, 23 supremum Fact 2.11.2, 188 symmetric gauge function sequence Hardy-Hilbert inequality Schur-convex function Fact 12.18.5, 958 Fact 2.12.25, 219 Fact 11.9.59, 867 Fact 12.18.6, 958 Fact 2.12.26, 219 unitarily invariant norm set Fact 2.12.27, 220 Fact 11.9.59, 867 Fact 1.8.6, 24 sum of products inequality surjective function vector norm

Toeplitz matrix Fact 11.7.20, 853 weak majorization Fact 3.25.16, 362 symmetric graph adjacency matrix Fact 4.26.2, 426 cycle Fact 1.9.7, 25 Fact 1.9.8, 26 degree matrix Fact 4.26.2, 426 forest Fact 1.9.8, 26 incidence matrix Fact 4.26.1, 426 Laplacian matrix Fact 10.18.1, 792 symmetric group group Fact 4.31.16, 435 symmetric hull definition Defn. 1.3.3, 6 relation Prop. 1.3.4, 6 symmetric matrix, see Hermitian matrix congruent matrices Fact 7.10.17, 571 definition Defn. 4.1.1, 363 eigenvalue Fact 6.10.7, 531 Fact 10.21.5, 801 factorization Cor. 7.4.10, 552 Fact 7.17.26, 608 generalized inverse Fact 8.3.3, 628 Hankel matrix Fact 4.23.2, 420 Hermitian matrix Fact 4.10.8, 378 involutory matrix Fact 7.17.39, 609 linear matrix equation Fact 4.10.2, 377 matrix exponential Prop. 15.4.8, 1188 matrix power Fact 4.10.3, 377 matrix transpose Fact 4.10.1, 377 maximum eigenvalue Fact 7.13.30, 596 minimum eigenvalue Fact 7.13.30, 596 orthogonally similar matrices Fact 7.10.16, 571 partitioned matrix Fact 4.13.12, 388 permutation matrix Fact 7.17.44, 610 quadratic form

Fact 4.10.4, 377 reverse-symmetric matrix Fact 4.10.38, 382 similar matrices Fact 7.17.42, 610 similarity transformation Fact 7.17.3, 605 Fact 7.17.4, 605 skew-Hermitian matrix Fact 4.10.8, 378 skew-symmetric matrix Fact 7.10.17, 571 Fact 7.17.42, 610 trace Fact 7.13.16, 593 symmetric relation definition Defn. 1.3.1, 5 directed graph Defn. 1.4.2, 10 intersection Prop. 1.3.2, 6 symmetric cone induced by Prop. 3.1.7, 280 symmetric set definition, 277 symmetry groups group Fact 4.31.16, 435 symplectic group determinant Fact 4.28.11, 428 quaternion group Fact 4.32.4, 438 special orthogonal group Fact 4.32.5, 438 unitary group Fact 4.31.9, 434 symplectic matrix Cayley transform Fact 4.28.12, 428 definition Defn. 4.1.6, 367 determinant Fact 4.28.10, 428 Fact 4.28.11, 428 equality Fact 4.28.1, 427 group Prop. 4.6.6, 372 Hamiltonian matrix Fact 4.28.2, 427 Fact 4.28.12, 428 Fact 4.28.13, 428 identity matrix Fact 4.28.3, 427 matrix exponential Prop. 15.6.7, 1192 matrix logarithm Fact 15.15.20, 1214 partitioned matrix Fact 4.28.9, 428 reverse permutation matrix

Fact 4.28.3, 427 skew-involutory matrix Fact 4.28.2, 427 spectrum Prop. 7.7.21, 562 unit imaginary matrix Fact 4.28.3, 427 symplectic similarity Hamiltonian matrix Fact 4.28.4, 428 Szasz’s inequality positive-semidefinite matrix Fact 10.16.39, 782

1539

Fact 2.3.11, 149 medians Fact 5.4.2, 494 volume Fact 5.4.1, 493 theorem definition, 5 thermodynamic inequality matrix exponential Fact 15.15.35, 1217 relative entropy Fact 15.15.27, 1215 third central moment inequality Fact 2.11.38, 196 Tian T idempotent matrix Fact 7.11.27, 579 T-congruence complex-symmetric matrix rank Fact 4.30.1, 430 Fact 7.10.22, 572 Fact 8.9.15, 663 T-congruence rank of a partitioned transformation matrix definition, 374 Fact 8.4.7, 634 T-congruent rank of a sum of diagonalization Kronecker products complex-symmetric matrix Fact 9.4.28, 687 Fact 7.10.22, 572 rank of partitioned matrix T-congruent matrices Fact 8.9.16, 663 definition Tikhonov inverse Defn. 4.7.4, 373 positive-definite matrix tangent Fact 10.10.45, 734 limit time-varying dynamics Fact 12.17.4, 954 commuting matrices Tao Fact 16.20.6, 1306 H¨older-induced norm determinant Fact 11.9.37, 864 Fact 16.20.6, 1306 Taussky matrix differential eigenstructure of equation intertwining matrices Fact 16.20.6, 1306 Fact 7.16.11, 603 Fact 16.20.7, 1306 Taussky-Todd state transition matrix factorization Fact 16.20.7, 1306 Fact 7.17.9, 606 trace tautology Fact 16.20.6, 1306 definition, 4 Toeplitz matrix Taylor block-Toeplitz matrix trace inequality Fact 4.23.3, 420 Fact 10.19.25, 798 definition Taylor series Defn. 4.1.3, 365 definition, 928, 1092 determinant rational function Fact 3.16.19, 332 Fact 12.14.2, 946 Fact 4.23.9, 421 tensor Fact 4.25.6, 426 Kronecker product factorization Fact 9.4.41, 689 Fact 7.17.45, 610 wedge product Hankel matrix Fact 9.4.41, 689 Fact 4.23.1, 420 tetrahedral group inverse matrix group Fact 4.23.10, 421 Fact 4.31.16, 435 lower triangular matrix tetrahedral numbers Fact 4.23.7, 421 integers Fact 15.14.1, 1209 Fact 1.12.6, 40 nilpotent matrix tetrahedron Fact 4.23.6, 421 de Gua’s theorem

1540

Tominaga

polynomial multiplication spanning path Fact 6.8.12, 521 Fact 1.9.9, 26 positive-definite matrix trace Fact 10.9.23, 730 2 × 2 matrices Fact 10.15.14, 776 Fact 3.15.10, 326 reverse-symmetric matrix 2 × 2 matrix equality Fact 4.23.5, 420 Fact 6.9.21, 528 spectrum Fact 6.9.22, 528 Fact 6.10.21, 534 3 × 3 matrix equality Fact 7.12.46, 587 Fact 6.9.27, 529 Fact 7.12.47, 587 Fact 6.9.28, 529 Fact 10.9.1, 724 adjugate Fact 10.9.23, 730 Fact 6.9.4, 525 Fact 10.10.39, 734 asymptotically stable tridiagonal matrix matrix Fact 4.23.9, 421 Fact 15.19.31, 1231 Fact 7.12.46, 587 commutator Fact 7.12.47, 587 Fact 3.23.1, 354 Fact 10.9.23, 730 Fact 4.29.2, 428 upper triangular matrix Fact 4.29.3, 429 Fact 4.23.7, 421 Fact 4.29.4, 429 Fact 15.14.1, 1209 Fact 4.29.5, 429 Tominaga Fact 7.10.19, 571 Specht’s ratio complex conjugate Fact 2.2.54, 140 transpose Tomiyama Fact 10.13.4, 761 maximum singular value Fact 10.13.15, 763 Fact 11.16.15, 902 Fact 10.13.16, 763 total least squares Fact 10.13.17, 763 linear system solution concave function Fact 11.17.1, 912 Fact 11.10.3, 869 total ordering convex function definition Prop. 10.6.17, 718 Defn. 1.3.14, 8 Fact 10.17.21, 790 dictionary ordering definition, 287 Fact 1.8.16, 25 derivative lexicographic ordering Prop. 12.10.4, 934 Fact 1.8.16, 25 Fact 15.15.4, 1212 planar example determinant Fact 1.8.16, 25 Prop. 10.4.14, 712 total response, 1251 Cor. 15.2.4, 1183 totally nonnegative matrix Cor. 15.2.5, 1183 definition Fact 3.16.26, 333 Defn. 4.1.1, 363 Fact 10.13.8, 762 Fact 15.19.23, 1228 Fact 10.16.24, 779 eigenvalues Fact 15.15.21, 1214 Fact 6.11.9, 540 diagonalizable matrix Fact 6.11.10, 540 Fact 7.15.8, 599 oscillatory matrix dimension Fact 6.11.10, 540 Fact 3.23.11, 356 totally positive matrix eigenvalue definition Prop. 10.4.13, 712 Defn. 4.1.1, 363 Fact 7.12.14, 581 rank Fact 10.21.5, 801 Fact 10.7.9, 723 Fact 10.21.10, 801 totient function Fact 10.22.25, 809 cyclic number eigenvalue bound Fact 1.20.4, 115 Fact 7.12.48, 588 Euler’s theorem Fact 7.12.49, 588 Fact 1.20.4, 115 elementary projector tournament Fact 7.9.10, 566 graph elementary reflector Fact 1.9.9, 26 Fact 7.9.10, 566 Hamiltonian cycle equalities, 287 Fact 1.9.9, 26 Frobenius norm

Fact 11.13.2, 890 Fact 11.13.3, 891 Fact 11.13.4, 891 Fact 11.14.2, 892 generalized inverse Fact 8.7.2, 650 group generalized inverse Fact 8.10.5, 670 Hamiltonian matrix Fact 4.28.7, 428 Hermitian matrix Prop. 10.4.13, 712 Cor. 10.4.10, 711 Lem. 10.4.12, 711 Fact 4.10.13, 379 Fact 4.10.22, 381 Fact 4.10.24, 381 Fact 7.13.17, 593 Fact 10.13.7, 762 Fact 10.14.1, 763 Fact 10.14.20, 766 Fact 10.14.57, 773 Fact 10.14.58, 773 Fact 15.15.26, 1215 Hermitian matrix product Fact 7.13.10, 592 Fact 7.13.11, 592 Fact 7.13.12, 592 Fact 7.13.13, 592 Fact 10.14.3, 764 Fact 10.14.16, 766 Fact 10.22.25, 809 inequality Fact 7.13.18, 593 involutory matrix Fact 7.9.2, 565 Klein’s inequality Fact 15.15.27, 1215 Kronecker permutation matrix Fact 9.4.38, 688 Kronecker product Prop. 9.1.12, 683 Fact 10.25.25, 823 Fact 15.15.42, 1217 Kronecker sum Fact 15.15.40, 1217 left-inner matrix Fact 7.13.19, 593 matrix equality Fact 3.15.38, 329 matrix exponential Cor. 15.2.4, 1183 Cor. 15.2.5, 1183 Fact 10.17.24, 791 Fact 15.12.6, 1206 Fact 15.15.4, 1212 Fact 15.15.11, 1213 Fact 15.15.30, 1216 Fact 15.15.32, 1216 Fact 15.15.33, 1216 Fact 15.15.34, 1216 Fact 15.15.35, 1217 Fact 15.15.40, 1217 Fact 15.15.42, 1217

Fact 15.16.4, 1218 Fact 15.16.5, 1218 Fact 15.17.1, 1220 Fact 15.17.4, 1221 matrix logarithm Fact 15.15.25, 1215 Fact 15.15.27, 1215 Fact 15.15.29, 1215 Fact 15.15.35, 1217 matrix power Fact 3.15.14, 326 Fact 3.15.21, 327 Fact 3.15.22, 327 Fact 6.10.28, 536 Fact 6.11.29, 544 Fact 7.12.12, 580 Fact 7.12.13, 580 Fact 10.13.15, 763 Fact 10.13.16, 763 Fact 10.13.17, 763 matrix squared Fact 7.12.12, 580 Fact 7.12.13, 580 maximum singular value Fact 7.13.15, 593 maximum singular value bound Fact 11.15.15, 897 nilpotent matrix Fact 4.22.6, 419 nonnegative matrix Fact 6.11.29, 544 normal matrix Fact 4.10.12, 379 Fact 10.13.17, 763 Fact 10.14.2, 764 normal matrix product Fact 7.13.12, 592 orthogonal matrix Fact 4.13.14, 388 Fact 4.13.15, 388 Fact 4.13.33, 391 Fact 4.14.4, 392 Fact 7.13.18, 593 Fact 7.13.21, 594 outer-product matrix Fact 7.15.5, 598 partitioned matrix Fact 10.12.11, 751 Fact 10.12.13, 751 Fact 10.12.14, 751 Fact 10.12.64, 760 Fact 10.14.54, 772 Fact 11.12.7, 888 polarized Cayley-Hamilton theorem Fact 6.9.21, 528 product of matrices Fact 3.15.17, 327 projector Fact 4.17.6, 408 Fact 4.18.6, 410 Fact 4.18.21, 415 Fact 7.9.10, 566

triangle Fact 10.14.55, 772 quadratic form Fact 10.19.25, 798 quadruple product Fact 9.4.10, 686 rank Fact 7.12.13, 580 Fact 11.13.3, 891 reflector Fact 7.9.10, 566 rotation matrix Fact 4.13.14, 388 Schatten norm Fact 11.14.1, 892 singular value Fact 7.13.14, 592 Fact 10.13.4, 761 Fact 10.21.7, 801 Fact 11.14.1, 892 Fact 11.16.2, 899 Fact 11.16.46, 908 skew-Hermitian matrix Fact 4.10.24, 381 skew-Hermitian matrix product Fact 7.13.11, 592 skew-symmetric matrix Fact 4.10.23, 381 Fact 4.10.33, 382 spectral radius Fact 6.10.28, 536 Fact 7.12.50, 588 Fact 11.15.14, 897 spectrum Fact 6.10.12, 532 square Fact 10.13.3, 761 strictly convex function Fact 10.17.19, 790 symmetric matrix Fact 7.13.16, 593 time-varying dynamics Fact 16.20.6, 1306 trace norm Prop. 11.2.4, 838 triple product Fact 3.15.12, 326 Fact 9.4.8, 686 tripotent matrix Fact 4.21.5, 418 Fact 7.9.3, 565 unitarily invariant norm Fact 11.9.46, 865 Fact 11.10.75, 881 unitarily similar matrices Fact 7.11.11, 576 unitary matrix Fact 4.13.13, 388 Fact 4.13.27, 390 Fact 4.13.28, 390 Fact 7.13.10, 592 vec Prop. 9.1.1, 681 Fact 9.4.8, 686 Fact 9.4.10, 686

trace norm

compatible norms Cor. 11.3.8, 840 definition, 838 Frobenius norm Fact 11.10.35, 873 matrix difference Fact 11.10.19, 871 maximum singular value Cor. 11.3.8, 840 positive-semidefinite matrix Fact 11.10.44, 874 Schur product Fact 11.16.47, 908 trace Prop. 11.2.4, 838 trace of a convex function Berezin Fact 10.14.48, 771 Brown Fact 10.14.48, 771 Hansen Fact 10.14.48, 771 Kosaki Fact 10.14.48, 771 Pedersen Fact 10.14.48, 771 Tracy-Singh product Kronecker product, 700 trail definition Defn. 1.4.4, 11 transfer function adjugate Fact 16.24.15, 1315 cascade interconnection Prop. 16.13.2, 1289 derivative Fact 16.24.6, 1314 feedback interconnection Fact 16.24.9, 1314 frequency response Fact 16.24.5, 1314 H2 norm Fact 16.24.18, 1316 Fact 16.24.19, 1316 Fact 16.24.20, 1316 Fact 16.24.21, 1316 Fact 16.24.22, 1316 imaginary part Fact 16.24.5, 1314 Jordan form Fact 16.24.11, 1315 parallel interconnection Prop. 16.13.2, 1289 partitioned transfer function Fact 16.24.8, 1314 real part Fact 16.24.5, 1314 realization Fact 16.24.3, 1313 Fact 16.24.4, 1314 Fact 16.24.7, 1314

1541

Fact 16.24.8, 1314 Fact 7.10.11, 571 Fact 16.24.9, 1314 similar matrices realization of inverse Prop. 7.7.13, 560 Prop. 16.13.1, 1289 Cor. 6.3.12, 505 realization of Cor. 7.4.9, 552 Cor. 7.7.22, 562 parahermitian conjugate Fact 7.10.10, 571 Prop. 16.13.1, 1289 Fact 7.10.11, 571 realization of transpose transpose of a matrix Prop. 16.13.1, 1289 definition, 286 reciprocal argument transpose of a vector Fact 16.24.4, 1314 definition, 285 right inverse transposition Fact 16.24.10, 1314 definition, 21 shifted argument transposition matrix Fact 16.24.3, 1313 definition transitive hull Defn. 4.1.1, 363 definition determinant Defn. 1.3.3, 6 Fact 4.13.18, 389 relation permutation matrix Prop. 1.3.4, 6 Fact 4.31.14, 434 transitive relation tree convex cone induced by definition Prop. 3.1.7, 280 Defn. 1.4.4, 11 definition graph Defn. 1.3.1, 5 Fact 1.9.5, 25 directed graph Trenkler Defn. 1.4.2, 10 generalized inverse intersection Fact 8.7.2, 650 Prop. 1.3.2, 6 triangle positive-semidefinite acute triangle matrix Fact 5.2.13, 480 Prop. 10.1.1, 703 altitude transitivity Fact 5.2.12, 472 rank subtractivity partial angle bisector ordering Fact 5.2.12, 472 Fact 4.30.2, 430 area star partial ordering Fact 5.2.3, 443 Fact 4.30.7, 430 Fact 5.2.4, 443 transmission zero Fact 5.2.5, 443 definition Fact 5.2.6, 443 Defn. 6.7.10, 515 areal coordinates Defn. 6.7.13, 516 Fact 5.2.11, 466 invariant zero Bandila’s inequality Thm. 16.10.8, 1282 Fact 5.2.8, 446 Thm. 16.10.9, 1282 barycenter null space Fact 5.2.11, 466 Fact 6.8.18, 523 barycentric coordinates rank Fact 5.2.11, 466 Prop. 6.7.12, 516 Blundon transpose Fact 5.2.8, 446 controllability Carnot’s theorem Fact 16.21.18, 1309 Fact 5.2.11, 466 involutory matrix center of mass Fact 7.10.8, 570 Fact 5.2.11, 466 Kronecker permutation centroid matrix Fact 5.2.11, 466 Prop. 9.1.13, 683 cosine rule Kronecker product Fact 5.2.7, 443 Prop. 9.1.3, 681 cross product Kronecker sum Fact 5.2.6, 443 Prop. 9.2.2, 683 ellipse matrix exponential Fact 5.5.3, 496 Prop. 15.2.8, 1184 Fact 5.5.4, 496 normal matrix equality Fact 7.10.10, 571

1542

triangle inequality

Fact 5.2.15, 485 Erd¨os-Mordell theorem Fact 5.2.11, 466 Euler’s inequality Fact 5.2.8, 446 fundamental triangle inequality Fact 5.2.8, 446 Gergonne point Fact 5.2.11, 466 Hadwiger-Finsler inequality Fact 5.2.8, 446 Heron’s formula Fact 5.2.7, 443 inequality Fact 5.2.14, 484 Klamkin’s inequality Fact 5.2.8, 446 Fact 5.2.10, 456 Leibniz’s inequality Fact 5.2.8, 446 median Fact 5.2.12, 472 Mircea’s inequality Fact 5.2.8, 446 Mollweide’s formula Fact 5.2.7, 443 Nagel point Fact 5.2.11, 466 Neuberg-Pedoe inequality Fact 5.2.22, 486 obtuse triangle Fact 5.2.13, 480 Ono’s inequality Fact 5.2.13, 480 Padoa’s inequality Fact 5.2.8, 446 right triangle Fact 5.2.13, 480 semiperimeter Fact 5.2.7, 443 sides Fact 5.2.23, 486 sine rule Fact 5.2.7, 443 trigonometric equalities Fact 5.2.9, 452 trigonometric inequality Fact 5.2.10, 456 Vasic’s inequality Fact 5.2.10, 456 Weitzenbock’s inequality Fact 5.2.8, 446 triangle inequality definition Defn. 11.1.1, 833 equality Fact 11.7.22, 853 Frobenius norm Fact 11.10.1, 868 linear dependence Fact 11.7.22, 853 positive-semidefinite matrix

Fact 4.23.9, 421 Fact 7.12.46, 587 Fact 7.12.47, 587 nilpotent matrix Fact 10.9.23, 730 Fact 7.19.7, 617 upper Hessenberg matrix triangular number Fact 4.24.1, 422 equalities trigonometric equalities Fact 1.12.3, 38 triangle generating function Fact 5.2.9, 452 Fact 13.4.6, 1010 trigonometric function series derivative Fact 13.4.6, 1010 Fact 2.16.22, 246 triangular numbers Fact 2.16.23, 246 integers equality Fact 1.12.6, 40 Fact 2.18.1, 254 triangularization Fact 2.18.2, 255 commutator Fact 2.18.3, 255 Fact 7.19.6, 617 infinite product commuting matrices Fact 13.10.7, 1084 Fact 7.19.5, 617 series tridiagonal matrix Fact 13.5.95, 1054 asymptotically stable trigonometric functions matrix Chebyshev polynomial of Fact 15.19.24, 1229 the first kind Fact 15.19.25, 1229 Fact 13.2.6, 985 Fact 15.19.26, 1229 Chebyshev polynomial of Fact 15.19.27, 1230 Fact 15.19.28, 1230 the second kind cyclic matrix Fact 13.2.7, 986 Fact 15.19.25, 1229 equalities definition Fact 2.16.3, 233 Defn. 4.1.3, 365 Hermite polynomial determinant Fact 13.2.10, 990 Fact 4.23.9, 421 Laguerre polynomial Fact 4.24.3, 423 Fact 13.2.9, 989 Fact 4.24.4, 423 Legendre polynomial Fact 4.24.6, 424 Fact 13.2.8, 987 Fact 4.24.7, 424 partial fractions inverse matrix expansions Fact 4.24.6, 424 Fact 13.4.10, 1014 Fact 4.24.7, 424 series Fact 4.24.8, 424 Fact 13.4.8, 1011 lower Hessenberg matrix trigonometric matrix Fact 4.24.1, 422 rank positive-definite matrix Fact 4.23.8, 421 Fact 4.24.4, 423 trigonometric sum Fact 4.24.5, 424 limit Fact 10.9.22, 730 Fact 12.18.28, 962 positive-semidefinite trinomial matrix root bound Fact 10.9.6, 726 Fact 15.21.21, 1237 Routh matrix trinomial coefficient Fact 15.19.27, 1230 polynomial Schwarz matrix Fact 2.1.10, 121 Fact 15.19.25, 1229 triple product Fact 15.19.26, 1229 equality spectrum Fact 3.15.11, 326 Fact 7.12.43, 586 Kronecker product Fact 7.12.44, 586 Prop. 9.1.5, 682 Fact 7.12.45, 586 Fact 9.4.8, 686 Fact 7.12.46, 587 trace Fact 7.12.47, 587 Fact 6.9.22, 528 Fact 10.9.23, 730 Fact 6.9.28, 529 submatrix Fact 9.4.8, 686 Fact 4.24.2, 423 vec Toeplitz matrix Prop. 9.1.9, 682 Fact 10.11.34, 738

triangular matrix

tripotent matrix

commuting matrices Fact 4.21.7, 418 definition Defn. 4.1.1, 363 diagonalizable matrix Fact 7.15.21, 600 Drazin generalized inverse Prop. 8.2.2, 626 generalized inverse Fact 8.5.4, 641 Fact 8.5.5, 642 group generalized inverse Fact 8.11.12, 676 group-invertible matrix Prop. 4.1.7, 367 Hermitian matrix Fact 4.21.3, 417 Fact 4.21.4, 418 Fact 4.21.7, 418 Fact 8.5.5, 642 idempotent matrix Fact 4.21.1, 417 Fact 4.21.7, 418 inertia Fact 7.9.3, 565 involutory matrix Fact 4.21.2, 417 Kronecker product Fact 9.4.22, 687 normal matrix Fact 4.21.4, 418 partial isometry Fact 4.21.4, 418 projector Fact 8.8.4, 652 range-Hermitian matrix Fact 8.5.4, 641 rank Fact 4.21.3, 417 Fact 4.21.5, 418 reflector Prop. 4.1.7, 367 signature Fact 7.9.3, 565 similar matrices Prop. 4.7.5, 374 Cor. 7.7.22, 562 spectrum Prop. 7.7.21, 562 trace Fact 4.21.5, 418 Fact 7.9.3, 565 unitarily similar matrices Prop. 4.7.5, 374 Trudi’s formula determinant Fact 4.25.6, 426 tuple definition, 2 Turan’s inequalities spectral radius bound Fact 6.10.28, 536 Turan’s inequality Legendre polynomial

unitarily invariant norm Fact 13.2.8, 987

coprime right polynomial fraction description four variables Prop. 6.7.15, 517 Fact 2.4.23, 181 definition twisted logarithm Defn. 6.3.1, 503 matrix logarithm determinant Fact 15.14.15, 1211 Prop. 6.3.8, 504 two-sided directional Smith form Prop. 6.3.8, 504 differential union definition, 924 boundary two-variable limit Fact 12.12.2, 937 convex function cardinality Fact 12.17.18, 957 Fact 1.8.4, 23 closed set Fact 12.12.17, 939 U Fact 12.12.19, 939 closure ULU decomposition Fact 12.12.2, 937 factorization convex cone Fact 7.17.13, 606 Fact 3.12.13, 312 Umegaki convex set relative entropy Fact 12.12.11, 938 Fact 15.15.27, 1215 definition, 1 uncontrollable eigenvalue interior controllability pencil Fact 12.12.2, 937 Prop. 16.6.13, 1263 Fact 12.12.3, 937 definition open set Defn. 16.6.11, 1263 Fact 12.12.16, 939 full-state feedback relative interior Prop. 16.6.14, 1264 Fact 12.12.13, 938 Hamiltonian span Prop. 16.17.7, 1300 Fact 3.12.11, 312 Prop. 16.17.8, 1300 unipotent matrix Lem. 16.17.4, 1299 definition Lem. 16.17.6, 1300 Defn. 4.1.1, 363 uncontrollable group multispectrum Fact 4.31.11, 434 definition Fact 15.23.1, 1243 Defn. 16.6.11, 1263 Heisenberg group uncontrollable spectrum Fact 4.31.11, 434 controllability pencil Fact 15.23.1, 1243 Prop. 16.6.16, 1264 matrix exponential definition Fact 15.14.14, 1210 Defn. 16.6.11, 1263 spectrum invariant zero Prop. 7.7.21, 562 Thm. 16.10.9, 1282 uniqueness Smith zeros inverse Prop. 16.6.16, 1264 Prop. 1.6.4, 17 uncontrollableordinary differential unobservable equation spectrum Thm. 15.8.1, 1195 invariant zero Thm. 15.8.2, 1195 Thm. 16.10.9, 1282 unit circle undamped rigid body discrete-time mass asymptotically stable Fact 7.15.31, 601 polynomial undetectable subspace Fact 15.21.1, 1234 definition unit imaginary matrix Defn. 16.5.1, 1259 congruent matrices uniform convergence Fact 4.10.37, 382 differentiable function definition, 367 Fact 12.16.16, 951 Hamiltonian matrix Weierstrass M-test Fact 4.28.3, 427 Fact 12.16.15, 951 skew-symmetric matrix unimodular matrix Turkevici’s inequality

Fact 4.10.37, 382 spectrum Fact 7.10.24, 572 symplectic matrix Fact 4.28.3, 427 unit impulse function definition, 1250 unit sphere group Fact 4.31.8, 433 unit-length quaternions Sp(1) Fact 4.32.1, 437 unital algebra generated by two matrices dimension Fact 7.11.22, 577 unitarily biequivalent matrices definition Defn. 4.7.3, 373 singular values Fact 7.11.23, 578 unitarily invariant norm anti-norm Fact 11.10.4, 869 Cartesian decomposition Fact 11.14.17, 895 commutator Fact 11.11.6, 885 Fact 11.11.7, 885 Fact 11.11.8, 885 complex conjugate transpose Fact 11.9.44, 865 contractive matrix Fact 11.10.21, 871 definition, 837 eigenvalue Fact 11.14.4, 893 Fact 11.14.5, 893 eigenvalue perturbation Fact 11.14.16, 895 fixed-rank approximation Fact 11.16.39, 906 Frobenius norm Fact 11.16.45, 907 generalized inverse Fact 11.10.74, 881 Fact 11.17.4, 909 Fact 11.17.5, 909 Heinz inequality Fact 11.10.82, 882 Hermitian matrix Fact 11.10.23, 872 Fact 11.10.30, 872 Fact 11.10.56, 876 Fact 11.10.72, 880 Fact 11.10.73, 880 Fact 11.10.76, 881 Fact 11.14.4, 893 Fact 15.17.15, 1223 H¨older’s inequality Fact 11.10.38, 873 left inverse

1543

Fact 11.17.4, 909 matrix difference Fact 11.10.20, 871 matrix exponential Fact 15.16.6, 1218 Fact 15.17.4, 1221 Fact 15.17.7, 1222 Fact 15.17.15, 1223 Fact 15.17.18, 1223 Fact 15.17.19, 1223 matrix logarithm Fact 11.10.86, 883 matrix power Fact 11.10.50, 875 maximum eigenvalue Fact 11.11.7, 885 Fact 11.11.8, 885 maximum singular value Fact 11.10.34, 873 Fact 11.11.6, 885 McIntosh’s inequality Fact 11.10.72, 880 Fact 11.10.79, 881 Minkowski inequality Fact 11.10.11, 870 normal matrix Fact 11.10.32, 872 outer-product matrix Fact 11.9.57, 866 partitioned matrix Fact 11.9.52, 865 Fact 11.12.2, 885 polar decomposition Fact 11.10.77, 881 power Fact 11.10.46, 875 product Fact 11.10.32, 872 Fact 11.10.39, 874 properties Fact 11.9.58, 867 rank Fact 11.16.39, 906 right inverse Fact 11.17.5, 909 Schatten norm Fact 11.9.11, 861 Schur product Fact 11.9.58, 867 Fact 11.10.9, 870 Fact 11.10.96, 884 Fact 11.16.50, 909 similar matrices Fact 11.9.45, 865 singular value Fact 11.9.48, 865 Fact 11.9.49, 865 Fact 11.16.39, 906 singular value perturbation Fact 11.16.40, 906 square root Fact 11.10.45, 874 Fact 11.10.51, 875 subadditive inequality

1544

unitarily left-equivalent matrices

Fact 11.10.70, 880 Fact 11.10.71, 880 submultiplicative norm Fact 11.9.58, 867 Fact 11.10.25, 872 Fact 11.10.33, 872 symmetric gauge function Fact 11.9.59, 867 trace Fact 11.9.46, 865 Fact 11.10.75, 881 unitarily left-equivalent matrices complex conjugate transpose Fact 7.11.23, 578 Fact 7.11.24, 578 definition Defn. 4.7.3, 373 positive-semidefinite matrix Fact 7.11.23, 578 Fact 7.11.24, 578 unitarily right-equivalent matrices complex conjugate transpose Fact 7.11.23, 578 definition Defn. 4.7.3, 373 positive-semidefinite matrix Fact 7.11.23, 578 unitarily similar matrices biequivalent matrices Prop. 4.7.5, 374 complex conjugate transpose Fact 7.10.21, 571 definition Defn. 4.7.4, 373 diagonal entry Fact 7.10.18, 571 Fact 7.10.20, 571 elementary matrix Prop. 7.7.23, 562 elementary projector Prop. 7.7.23, 562 elementary reflector Prop. 7.7.23, 562 group-invertible matrix Prop. 4.7.5, 374 Hermitian matrix Prop. 4.7.5, 374 Prop. 7.7.23, 562 Cor. 7.5.5, 554 idempotent matrix Prop. 4.7.5, 374 Fact 7.10.26, 572 Fact 7.10.28, 573 Fact 7.11.13, 577 Fact 7.11.16, 577 involutory matrix Prop. 4.7.5, 374 Kronecker product

Fact 9.4.13, 686 matrix classes Prop. 4.7.5, 374 matrix exponential Prop. 15.2.9, 1185 nilpotent matrix Prop. 4.7.5, 374 normal matrix Prop. 4.7.5, 374 Cor. 7.5.4, 553 Fact 7.11.8, 576 Fact 7.11.10, 576 partitioned matrix Fact 7.10.23, 572 positive-definite matrix Prop. 4.7.5, 374 Prop. 7.7.23, 562 positive-semidefinite matrix Prop. 4.7.5, 374 Prop. 7.7.23, 562 projector Fact 7.11.15, 577 range-Hermitian matrix Prop. 4.7.5, 374 Cor. 7.5.4, 553 real matrices Fact 7.11.9, 576 similar matrices Fact 7.11.10, 576 simultaneous diagonalization Fact 7.19.8, 617 simultaneous triangularization Fact 7.19.5, 617 Fact 7.19.7, 617 skew-Hermitian matrix Prop. 4.7.5, 374 Prop. 7.7.23, 562 skew-involutory matrix Prop. 4.7.5, 374 Specht’s theorem Fact 7.11.11, 576 trace Fact 7.11.11, 576 tripotent matrix Prop. 4.7.5, 374 unitary matrix Prop. 4.7.5, 374 Prop. 7.7.23, 562 upper triangular matrix Thm. 7.5.1, 553 unitary bi-equivalence transformation definition, 374 unitary group symplectic group Fact 4.31.9, 434 unitary left equivalence transformation definition, 374 unitary matrix, see orthogonal matrix additive decomposition

Fact 7.20.5, 618 block-diagonal matrix Fact 4.13.10, 388 Cayley transform Fact 4.13.25, 390 cogredient diagonalization Fact 10.20.2, 799 complex-symmetric matrix Fact 7.10.22, 572 convergent sequence Fact 10.10.38, 734 CS decomposition Fact 7.10.29, 573 definition Defn. 4.1.1, 363 determinant Fact 4.13.17, 389 Fact 4.13.20, 389 Fact 4.13.21, 389 Fact 4.13.22, 389 Fact 4.13.23, 389 Fact 11.10.83, 882 diagonal entry Fact 4.13.16, 389 Fact 10.21.14, 803 diagonal matrix Thm. 7.6.3, 557 Lem. 10.5.1, 713 discrete-time Lyapunov-stable matrix Fact 15.22.17, 1241 dissipative matrix Fact 10.10.36, 734 equalities Fact 4.13.5, 388 factorization Fact 7.17.11, 606 Fact 10.8.7, 724 Frobenius norm Fact 11.10.77, 881 geometric-mean decomposition Fact 7.10.31, 575 group Prop. 4.6.6, 372 group generalized inverse Fact 8.3.37, 632 Hermitian matrix Fact 4.13.24, 389 Fact 10.20.2, 799 Fact 15.15.38, 1217 Kronecker product Fact 9.4.22, 687 matrix exponential Prop. 15.2.8, 1184 Prop. 15.4.3, 1187 Prop. 15.6.7, 1192 Cor. 15.2.6, 1183 Fact 15.15.7, 1212 Fact 15.15.37, 1217 Fact 15.15.38, 1217 matrix limit Fact 8.3.37, 632 normal matrix Prop. 4.1.7, 367

Fact 4.13.6, 388 Fact 7.17.2, 605 orthogonal vectors Fact 4.13.9, 388 outer-product perturbation Fact 4.13.23, 389 partitioned matrix Fact 4.13.11, 388 Fact 4.13.12, 388 Fact 4.13.21, 389 Fact 4.13.22, 389 Fact 4.13.31, 391 Fact 10.8.11, 724 Fact 10.8.12, 724 Fact 10.8.13, 724 Fact 10.12.58, 759 Fact 10.12.59, 759 Fact 10.12.60, 759 Fact 11.16.13, 901 polar decomposition Fact 10.8.9, 724 positive-definite matrix Fact 11.10.83, 882 positive-semidefinite matrix Fact 10.8.2, 723 Fact 10.12.58, 759 Fact 10.12.59, 759 Fact 10.12.60, 759 Fact 11.10.3, 869 principal-angle decomposition Fact 7.10.30, 574 quaternions Fact 4.32.9, 440 Schur-Horn theorem Fact 10.21.14, 803 semicontractive matrix Fact 4.27.4, 427 Fact 10.8.11, 724 shifted-unitary matrix Fact 4.13.29, 391 simultaneous block diagonalization Fact 10.20.1, 799 simultaneous diagonalization Fact 10.20.2, 799 singular value Fact 7.12.40, 585 Fact 11.16.13, 901 Fact 11.16.26, 904 Fact 11.16.27, 904 skew-Hermitian matrix Fact 4.13.25, 390 Fact 15.15.37, 1217 spectrum Prop. 7.7.21, 562 square root Fact 10.8.2, 723 Fact 10.10.30, 733 Fact 10.10.34, 733 SU(3) Fact 4.14.3, 392

Vandermonde’s convolution subspace Fact 4.13.2, 387 Fact 4.13.3, 388 Fact 4.13.4, 388 sum Fact 4.13.20, 389 trace Fact 4.13.13, 388 Fact 4.13.27, 390 Fact 4.13.28, 390 trace of product Fact 7.13.10, 592 unitarily similar matrices Prop. 4.7.5, 374 Prop. 7.7.23, 562 upper triangular matrix Fact 7.17.11, 606 unitary right equivalence transformation definition, 374 unitary similarity singular value decomposition Fact 7.10.27, 572 Fact 8.3.23, 630 Fact 8.5.13, 643 unitary similarity transformation definition, 374 universal statement definition, 3 logical equivalents Fact 1.7.6, 22 unobservable eigenvalue definition Defn. 16.3.11, 1255 full-state feedback Prop. 16.3.14, 1256 Hamiltonian Prop. 16.17.7, 1300 Prop. 16.17.8, 1300 Lem. 16.17.4, 1299 Lem. 16.17.6, 1300 invariant zero Prop. 16.10.11, 1284 observability pencil Prop. 16.3.13, 1256 unobservable multispectrum definition Defn. 16.3.11, 1255 unobservable spectrum definition Defn. 16.3.11, 1255 invariant zero Thm. 16.10.9, 1282 observability pencil Prop. 16.3.16, 1256 Smith zeros Prop. 16.3.16, 1256 unobservable subspace block-triangular matrix Prop. 16.3.9, 1255 Prop. 16.3.10, 1255 definition

Defn. 16.3.1, 1253 equivalent expressions Lem. 16.3.2, 1253 full-state feedback Prop. 16.3.5, 1254 identity-matrix shift Lem. 16.3.7, 1255 invariant subspace Cor. 16.3.4, 1254 nonsingular matrix Prop. 16.3.10, 1255 orthogonal matrix Prop. 16.3.9, 1255 projector Lem. 16.3.6, 1254 unstable equilibrium definition Defn. 15.8.3, 1196 unstable matrix definition Defn. 15.9.1, 1198 positive matrix Fact 15.19.20, 1228 unstable polynomial definition Defn. 15.9.3, 1198 unstable subspace complementary subspaces Prop. 15.9.8, 1200 definition, 1199 idempotent matrix Prop. 15.9.8, 1200 invariant subspace Prop. 15.9.8, 1200 semistable matrix Prop. 15.9.8, 1200 up/down numbers series Fact 13.1.9, 981 upper bidiagonal matrix definition Defn. 4.1.3, 365 upper Hessenberg matrix Fact 4.24.1, 422 upper block-triangular matrix characteristic polynomial Fact 6.10.16, 533 definition Defn. 4.1.3, 365 inverse matrix Fact 3.22.8, 354 Fact 3.22.10, 354 irreducible matrix Fact 6.11.6, 540 left-invertible matrix Fact 3.18.20, 345 minimal polynomial Fact 6.10.17, 534 orthogonally similar matrices Cor. 7.5.2, 553 power Fact 3.15.30, 328 reducible matrix

Fact 6.11.6, 540

1545

matrix exponential Fact 15.12.4, 1205 Fact 15.14.1, 1209 positive-definite matrix Fact 15.14.13, 1210 Fact 10.11.50, 740 matrix power upper bound for a partial Fact 4.23.7, 421 ordering matrix product definition Fact 4.25.2, 424 Defn. 1.3.9, 7 nilpotent matrix upper Hessenberg matrix Fact 4.22.13, 420 Bell number orthogonally similar Fact 6.9.9, 526 matrices Catalan number Cor. 7.5.3, 553 Fact 6.9.9, 526 positive diagonal definition Fact 7.17.10, 606 Defn. 4.1.3, 365 positive-semidefinite determinant Fact 4.25.4, 425 matrix Fact 4.25.5, 425 Fact 10.10.42, 734 Fact 4.25.6, 426 regular pencil Fact 6.9.8, 525 Fact 7.19.4, 617 Fact 6.9.9, 526 similar matrices Fact 6.9.10, 527 Fact 7.10.3, 570 falling factorial simultaneous orthogonal Fact 6.9.9, 526 biequivalence regular pencil transformation Fact 7.19.4, 617 Fact 7.19.4, 617 simultaneous orthogonal spectrum biequivalence Fact 6.10.14, 533 transformation Toeplitz matrix Fact 7.19.4, 617 Fact 4.23.7, 421 tridiagonal matrix Fact 15.14.1, 1209 Fact 4.24.1, 422 unitarily similar matrices upper bidiagonal matrix Thm. 7.5.1, 553 Fact 4.24.1, 422 unitary matrix upper triangular matrix, Fact 7.17.11, 606 Urquhart see lower triangular generalized inverse matrix Fact 8.3.8, 628 Bruhat decomposition Fact 7.10.32, 575 characteristic polynomial V Fact 6.10.14, 533 commutator van Dam Fact 4.22.13, 420 Cauchy-Khinchin definition inequality Defn. 4.1.3, 365 Fact 3.15.39, 329 determinant Vandermonde matrix Fact 4.25.1, 424 companion matrix eigenvalue Fact 7.18.9, 613 Fact 6.10.14, 533 cycle number factorization Fact 7.18.7, 613 Fact 7.17.11, 606 determinant Fact 7.17.12, 606 Fact 7.18.5, 613 group Fact 7.18.6, 613 Fact 4.31.11, 434 Fact 7.18.7, 613 Fact 15.23.1, 1243 Fact 7.18.8, 613 Heisenberg group Fourier matrix Fact 4.31.11, 434 Fact 7.18.13, 614 Fact 15.23.1, 1243 polynomial invariant subspace Fact 7.18.12, 614 Fact 7.10.3, 570 similar matrices Kronecker product Fact 7.18.11, 614 Fact 9.4.4, 686 subset number Lie algebra Fact 7.18.7, 613 Fact 4.31.10, 434 Vandermonde’s Fact 15.23.1, 1243 convolution upper bound

1546

Vantieghem’s theorem

binomial equality Fact 1.16.13, 77 Vantieghem’s theorem prime number Fact 1.11.21, 32 variance Laguerre-Samuelson inequality Fact 2.11.37, 195 Fact 10.10.40, 734 mean Fact 2.11.37, 195 Fact 10.10.40, 734 variation of constants formula state equation Prop. 16.1.1, 1249 Vasic’s inequality triangle Fact 5.2.10, 456 vec definition, 681 Kronecker permutation matrix Fact 9.4.38, 688 Kronecker product Fact 9.4.6, 686 Fact 9.4.7, 686 Fact 9.4.9, 686 matrix product Fact 9.4.7, 686 quadruple product Fact 9.4.10, 686 trace Prop. 9.1.1, 681 Fact 9.4.8, 686 Fact 9.4.10, 686 triple product Prop. 9.1.9, 682 vector definition, 277 vector derivative quadratic form Prop. 12.10.1, 933 vector equality cosine law Fact 11.8.3, 853 generalized parallelogram law Fact 11.8.3, 853 parallelogram law Fact 11.8.3, 853 polarization identity Fact 11.8.3, 853 Pythagorean theorem Fact 11.8.3, 853 vector inequality H¨older’s inequality Prop. 11.1.6, 835 norm inequality Fact 11.7.3, 849 Fact 11.7.6, 850 vector norm Ky Fan k-norm Fact 11.7.20, 853

monotone norm Fact 11.8.1, 853 permutation matrix Fact 11.7.19, 852 Schur-convex function Fact 11.7.19, 852 symmetric gauge function Fact 11.7.20, 853 vibration equation matrix exponential Example 15.3.7, 1186 ` Viete infinite product Fact 13.10.15, 1088 Visser polynomial Fact 12.13.6, 942 volume Cayley-Menger determinant Fact 5.4.7, 495 ellipsoid Fact 5.5.14, 498 generalized hypersphere Fact 5.5.15, 498 hyperellipsoid Fact 5.5.14, 498 hypersphere Fact 5.5.13, 497 parallelepiped Fact 5.4.3, 494 Fact 5.4.4, 494 parallelotope Fact 5.4.3, 494 polytope Fact 5.4.7, 495 simplex Fact 5.4.6, 495 tetrahedron Fact 5.4.1, 493 transformed set Fact 5.4.5, 494 von Neumann symmetric gauge function Fact 11.9.59, 867 trace inequality Fact 7.13.14, 592 von Neumann–Jordan inequality norm inequality Fact 11.7.4, 849 Vosmansky’s identity binomial equality Fact 1.16.14, 86

W Wagner’s inequality

quadratic inequality Fact 2.12.41, 222 walk connected graph Fact 6.11.3, 538 definition Defn. 1.4.4, 11 graph

Fact 6.11.1, 537

nondecreasing function Fact 3.25.10, 361 partial ordering scalar inequality Prop. 3.10.2, 306 Fact 5.2.25, 486 positive-semidefinite Wallis product infinite product matrix Fact 13.10.3, 1081 Fact 10.22.12, 806 Wallis’s equality powers infinite product Fact 3.25.16, 362 Fact 13.10.6, 1084 Schur product Wallis’s formula Fact 11.16.42, 907 definite integral singular value Fact 14.2.21, 1102 Prop. 11.2.2, 837 Walsh Prop. 11.6.3, 847 polynomial root bound Fact 7.12.26, 582 Fact 15.21.8, 1235 Fact 10.13.5, 761 Wang’s inequality Fact 10.13.6, 761 scalar inequality Fact 10.21.10, 801 Fact 2.11.39, 196 Fact 10.22.14, 807 Waring’s problem Fact 10.22.28, 810 sum of powers Fact 11.15.19, 898 Fact 1.11.26, 33 Fact 11.16.23, 903 weak diagonal dominance Fact 11.16.24, 903 Fact 11.16.25, 903 theorem Fact 11.16.26, 904 nonsingular matrix Fact 11.16.27, 904 Fact 6.10.25, 535 Fact 11.16.28, 904 weak log majorization Fact 11.16.37, 905 definition Fact 11.16.38, 906 Defn. 3.10.1, 305 Fact 11.16.42, 907 eigenvalue Fact 15.17.5, 1222 Fact 10.22.13, 807 Fact 15.17.6, 1222 partial ordering strong majorization Prop. 3.10.2, 306 Fact 3.25.1, 359 positive-semidefinite Fact 3.25.2, 359 matrix symmetric gauge function Fact 10.22.13, 807 Fact 3.25.16, 362 singular value weak log majorization Prop. 11.6.2, 847 Fact 3.25.15, 362 Fact 10.22.20, 808 Weyl’s inequalities weak majorization Fact 10.21.10, 801 Fact 3.25.15, 362 Weyl’s majorant theorem weak majorization Fact 11.15.19, 898 continuous function weakly unitarily invariant Fact 3.25.17, 362 norm convex function definition, 837 Fact 3.25.8, 361 matrix power Fact 3.25.9, 361 Fact 11.9.55, 866 Fact 3.25.10, 361 numerical radius Fact 3.25.12, 361 Fact 11.9.55, 866 Fact 10.22.11, 806 Webster definition trigonometric inequality Defn. 3.10.1, 305 Fact 2.17.1, 246 diagonal entry Wedderburn Fact 7.12.26, 582 rank eigenvalue Fact 8.4.10, 634 Fact 7.12.26, 582 Fact 8.9.13, 662 Fact 10.21.10, 801 wedge product Fact 10.22.11, 806 Kronecker product Fact 10.22.12, 806 Fact 9.4.41, 689 exponential function Fact 9.4.42, 691 Fact 3.25.11, 361 tensor Karamata’s inequality Fact 9.4.41, 689 Fact 3.25.17, 362 Wei-Norman expansion matrix exponential time-varying dynamics Fact 15.17.4, 1221

Walker’s inequality

zonotope Fact 16.20.6, 1306

Fact 11.14.13, 894 Fact 10.10.47, 735 positive power of a Fact 10.11.73, 745 positive-semidefinite cogredient diagonalization primitive matrix Fact 10.20.3, 799 Fact 6.11.5, 538 matrix Weierstrass canonical Fact 10.11.38, 738 Wielandt inequality Fact 10.14.8, 764 quadratic form inequality form positive-semidefinite Fact 10.18.21, 794 pencil Prop. 7.8.3, 564 Wiles matrix inequality Fermat’s last theorem Weierstrass inequalities Fact 11.16.30, 904 Fact 1.11.39, 35 scalar inequality quadratic form Fact 2.11.123, 210 Wilker’s inequality Fact 10.18.18, 794 trigonometric inequality Weierstrass M-test scalar case Fact 2.17.1, 246 uniform convergence Fact 2.2.50, 139 Fact 12.16.15, 951 Williams Fact 2.2.52, 140 polynomial root bound weighted AM-GM scalar inequality Fact 15.21.17, 1237 Fact 2.2.53, 140 inequality Wilson’s theorem Fact 2.2.55, 141 AM-GM inequality prime number Fact 2.11.87, 204 Fact 1.11.20, 32 weighted least squares Z prime numbers Euclidean norm Z-matrix Fact 1.11.22, 32 Fact 11.17.8, 910 definition Worpitzky’s identity weighted sum Defn. 4.1.5, 367 Eulerian number inequality M-matrix Fact 1.19.5, 111 Fact 2.11.108, 208 Fact 6.11.13, 541 Wronskian Fact 2.11.109, 208 Fact 6.11.16, 542 derivative Weitzenbock’s inequality matrix exponential Fact 12.16.26, 953 triangle Fact 15.20.1, 1232 determinant Fact 5.2.8, 446 minimum eigenvalue Fact 12.16.26, 953 Weyl, 711 Fact 6.11.17, 542 singular value inequality submatrix Fact 7.12.32, 584 X Fact 6.11.14, 542 singular values Zassenhaus Xie Fact 11.15.20, 898 intertwining matrices asymptotically stable Weyl’s criterion Fact 7.16.11, 603 polynomial complex exponential Zassenhaus expansion Fact 15.18.9, 1224 Fact 12.18.27, 962 time-varying dynamics xor Weyl’s inequalities Fact 16.20.6, 1306 definition, 2 weak majorization and Zassenhaus product singular values formula Fact 10.21.10, 801 Y matrix exponential Weyl’s inequality Yamamoto Fact 15.15.19, 1214 Hermitian matrix singular value limit Zeckendorf’s theorem eigenvalue Fact 11.15.25, 899 Fibonacci number Thm. 10.4.9, 710 Young inequality Fact 1.17.1, 95 Fact 10.11.13, 736 reverse inequality zero Weyl’s majorant theorem Fact 2.2.54, 140 blocking zero singular values and weak Specht’s ratio Defn. 6.7.10, 515 majorization Fact 2.2.54, 140 full-state feedback Fact 11.15.19, 898 Young’s inequality Fact 16.24.16, 1315 Weyr form, 619 integral invariant zero Wielandt Fact 12.13.16, 944 Defn. 16.10.1, 1278 eigenvalue perturbation positive-definite matrix Thm. 16.10.8, 1282 Weierstrass

1547

observable pair Cor. 16.10.12, 1285 transmission zero Defn. 6.7.10, 515 Thm. 16.10.8, 1282 Prop. 6.7.12, 516 unobservable eigenvalue Prop. 16.10.11, 1284 zero diagonal commutator Fact 4.29.7, 429 zero eigenvalue algebraic multiplicity Fact 7.15.1, 597 defect Fact 7.15.1, 597 geometric multiplicity Fact 7.15.1, 597 Fact 7.15.2, 597 zero entry reducible matrix Fact 4.25.7, 426 Fact 4.25.8, 426 zero matrix definition, 283 maximal null space Fact 3.15.13, 326 positive-semidefinite matrix Fact 10.11.21, 737 zero of a rational function definition Defn. 6.7.1, 513 zero trace Shoda’s theorem Fact 7.10.19, 571 zeta function Apery’s constant Fact 13.5.41, 1031 cycle number Fact 13.5.40, 1031 Euler product formula Fact 13.3.1, 994 harmonic number Fact 13.5.40, 1031 integral Fact 13.3.1, 994 Stieltjes constant Fact 13.3.1, 994 zonotope convex polytope Fact 3.11.19, 310 definition, 290