Algebraic Methods for Toeplitz-like Matrices and Operators [Reprint 2021 ed.] 9783112529003, 9783112528990

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MATHEMATICAL MATHEMATISCHE RESEARCH FORSCHUNG

Algebraic Methods for Toeplitz-like Matrices and Operators G. Heinis/ K.UosI

Band 19 Akademie-Verlag • Berlin

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Manuskripte in englischer und deutscher Sprache, die mindestens 100 Seiten und nicht mehr als 500 Seiten umfassen, können in diese Reihe aufgenommen werden. Im Interesse einer schnellen Publikation werden die Manuskripte auf fotomechanischem Weg reproduziert. Autoren, die an der Veröffentlichung entsprechender Arbeiten in dieser Reihe interessiert sind, wenden sich bitte direkt an den Akademie-Verlag. Sie erhalten dort genauere Informationen über die Gestaltung der Manuskripte und die Modalitäten der Veröffentlichung.

G. Heinig/K.Rost Algebraic Methods for Toeplitz-like Matrices and Operators

Mathematical Research • Mathematische Forschung

Wissenschaftliche Beiträge herausgegeben von der Akademie der Wissenschaften der DDR Institut für Mathematik

Band 19 Algebraic Methods forToeplitz-like Matrices and Operators by G. Heinig and K. Rost

Algebraic Methods for Toeplitz-like Matrices and Operators by Georg Heinig and Karla Rost

Akademie-Verlag • Berlin 1984

Autoren: Doz. Dr. so. nat. Georg Heinig Dr. rer. nat. Karla Rost Technische Hochschule Karl-Marx-Stadt Sektion Mathematik 9010 Karl-Marx-Stadt Reichenhainer Str. 39/41

Die Titel dieser Schriftenreihe werden vom Originalmanuskript der Autoren reproduziert.

ISSN 0138 - 3019

Erschienen im Akademie-Verlag, DDR-1086 Berlin, leipziger Str. 3-4 (C) Akademie-Verlag Berlin 1984 Lizenznummer: 202 • 100/406/84 Printed in the German Democratic Republic Gesamtherstellung: VEB Kongreß- und Werbedruck, 9273 Oberlungwitz Lektor: Dr. Reinhard Höppner Umschlag: Karl Salzbrunn LSV 1065 Bestellnummer: 763 364 9 (2182/19) 02800

INTRODUCTION The preseat volume consists of two parts. The first with (finite) matrices having a Toeplitz structure

oue is concerned (T-matrices)

or a Hankel structure (H-matrices)j the second one deals with matrices and linear operators, which are, in some sense, generalizations of T-matrices. Both parts are joined by a common algebraic principle of investigation. The study of T- and H-matrices has been an active field of research since the beginning of this century and remains it today. The first reason for this circumstance is the fact that such matrices occur in a large variety of areas in pure and applied mathematics. For example, they often appear as discretization of differential and integral equations, they arise in physical data-processing application, in the theories of orthogonal polynomials, stationary processes, moment problems and many others. The second reason is that T- and H-matrices have a lot of significant characteristic properties. In this monograph we restrict ourselves to algebraic problems concerning T - and H-matrices" 1} . We mainly make use of the following three sourcest 1) ^ast inversion algorithms. As it was shown by N . LEVINSON (in the positive definite case)for the first time, the computational complexity of T-matrix inversion can be reduced from 0(n 3 ) to 0(n 2 ) operations and the storage requirement from 0 ( n a ) to 0(n). A lot of papers is dedicated to the further development of this algorithm (see Notes and comments, Fart I). 2) Structure of T- and H-matrices. Already E. FROBENIUS proved some theorems about the rank and signature of H-matrices and connections between T- and H-matrices. This direction of investigation has been continued in the papers of I. S. IOHVIDOV and G. HEINIG. 3) Wiener-Hopf theory. The study of finite T-matrices is often motivated by some questions of the Wiener-Hopf theory, which is, in its discrete variant, the theory of infinite T-matrices. The authors' primary purpose in writing Part I is to present these three aspects under one cover. In this way some new results are obtained published here for the first time. Furthermore, some applications to root localization problems are considered. Most of the characteristic properties of T-matrices are based on the fact that AU_ - U A has at most rank two, if A is an mxn T-matrix n m q and U Q is the forward shift in C . This leads to the idea to consider matrices and operators A, for which rank (AU - VA) is small compared For analytic problems see the recent monograph in this series A. BÖTTCHER, B. SILBERMAOT [1] or U. GRBNANDER, G. SZBG'Ö [1 J. 5

with rank A, where U and V are two fixed operators. Such operators A will be referred to as UN-^oeplitz-like operators (ILO for short). In Part II it is shown that many results on finite T-matrices can be generalized to ILO, such as invertibility criteria, inversion formulae and algorithms, kernel structure properties and other. Furthermore, we shall give an abstract definition of the concept of partial indices originated in the Wiener-Hopf theory. We specify the abstract considerations for the following special classes of ILO» - matrices close to Toeplitz and Hankel (we obtain results similar to those of T. KAJLATH et al.) - matrices close to Vandermonde - matrices of the foim [ ( a ^ - ^ )

] a Q d their generalizations

- integral operators with kernel close to displacement - singular integral operators and Toeplitz operators. Furthermore, we note some applications of our approach to the problem of Wiener-Hopf factorization. In Part I we hope to draw a quite complete picture of the algebraic theoiy of T-matrices. On the other hand, the theory of ILO presented in Part II is far from being complete. Our notes offer the current state of the theory and should be viewed as a stepping stone to further development. We have occasion to thank our colleagues A. Böttcher, ü. Jungnickel and B. Silbermann for fruitful conversation. Furthermore, we should like to thank Mrs. M. Graupner for her patient excellent typing work.

Karl-Marx-Stadt, December 1983

6

The authors

CONTENTS PART I. TOBELITZ AND HANKED MATRICES 0. Preliminaries 9 0.1. Notatioas 9 0.2. Toeplitz sad Haakel matrices 10 0.3. Multiplication and difference operators 13 1. Inversion formulae 14 1.1. First inversion variant 15 1.2. Second inversion variant 19 1.3. Third inversion variant 23 1.4. Symmetric T-matrices 26 1.5. Examples 27 1.6. The transformations A and V 28 1.7. Representation using the generating function 31 1.8. Determinant representation of the fundamental solutions 33 2. Bezoutians and resultant matrices 33 2.1. Characterization of H- and T-matrices 33 2.2. Characterization of the inverse of H- and T-matrices 34 2.3. Bezoutians 35 2.4. The Bezoutian and the companion matrix 39 2.5. Resultant matrices 41 2.6. Kernel description of Bezoutians and resultant matrices 42 2.7. Root localization in the upper half-plane 43 2.8. Root localization for real polynomials 47 2.9. Bezoutian representation via Vandermonde matrices 49 2.10.Liapunov and Stein equations 51 3. Recursion formulae and inversion algorithm for 55 strongly regular T- and H-matrices 3.1. Recursion formulae - Toeplitz case 55 58 3.2. Recursion formulae - Hankel case 3.3. Inversion algorithm for strongly regular T- and 59 H-matrices 3.4. Fast solution of Toeplitz and Hankel systems of 60 equations 3.5. Inversion of perturbed T-matrices 61 3.6. Lower-upper and upper-lower factorization 62 3.7. Evaluation of the signature in the strongly regular case 67 3.8. Inversion of Bezoutians 69 4. Transformations of OD- and Hnmatrices and Bezoutians 70 4.0. Two problems 70 4.1. Möbius matrices 70 4.2. Transformations of T- and H-matrices 71 4.3. Transformations of T- and H-forms 73 4.4. Transformations of Bezoutians and related classes 74 of matrices 4.5. Root localization in the unit disc 76 4.6. Root localization of symmetric polynomial! 77 5. Kernel structure 79 5.1. U-chains 79 5.2. The kernel structure theorem 80 5.3. Characteristic polynomials and inverse problems 83 5.4. Partial indices 87 5.5. Solvability of the fundamental equations 88 5.6. iß .^-characteristic of H-matrices and 91 (q,h. ,it )-characteristic of T-matrices 5.7. Partlal~indices after one-row or one-column extensions 95 5.8. Singular extensions 99 5.9. Kernel structure of square H- and T-matrices 101 5.10.Strongly singular T- and H-matrices 102 6. T- and H-matrices with non-regular principal sections 102 6.1. Inversion algorithm - Toeplitz case 102 6.2. Inversion algorithm - Hankel case 106 7

6.3» Generalized LU-factorization 6.4. Evaluation of the signature 7. Generalized inverses of H-matrices 7.1« Two approaches for generalized inversion 7.2. Bezoutians as generalized inverses 7.3» Construction of generalized inverses with the help of characteristic polynomials 7.4-. The Moore-Penrose inverse 7*5* One-side inverses 8. Canonical representation 8.1. Definitions 8.2. The main theorem 8.3. Canonical representation of Hermitian H- and T-matrices 8.4. Canonical representation of singular extensions 8.5« Symbol description of strongly singular H- and T-matrices Notes and comments

107 111 112 112 114 117 119 120 121 121 124 127 129 130 133

PART II. TOKELITZ-L IKE OPERATORS 0. Fredholm operators 136 1. Toeplitz-like operators - first general considerations 137 1.1. Definitions 137 1.2. Criteria of invertibility 139 1.3. Solution of special equations 142 2. Inversion formulae for Toeplitz-like matrices 145 2.1. Matrices close to Toeplitz 145 2.2. Toeplitz plus Hankel matrices 152 2.3. Matrices close to Vandermonde 155 2.4. Matrices with a small rank diagonal reduction 159 3. Inversion algorithms for Toeplitz-like matrices 161 3.1. General recursion foimulae for the fundamental equations 161 3«2. Inversion of matrices close to Toeplitz 163 3.3« Inversion of matrices close to Vandermonde 165 3.4. Inversion of matrices with a small rank diagonal 166 reduction 3«5. A modification of the recursion 168 4. Inversion of integral operators with displacement kernel 168 and their generalizations 4.1. Preliminaries 168 4.2. The class of JJ-Toeplitz-like operators 169 4.3. Sahnovich operators 172 4.4. further classes of JJ-Toeplitz-like operators 175 4.5. Generating functions 176 4.6. DD-Toeplitz-1ike operators 177 4.7. Non-Fredholm operators 178 4.8. Integral equations on the half-line 180 5. Singular integral and Toeplitz operators 181 5.1. The algebra of singular integral operators 181 5.2. Inversion of singular integral operators 183 5*3» Inversion of Toeplitz operators 184 6. Kernel structure and partial indices of TLO 186 6.1. Kernel structure theorem 186 6.2. Duality 191 6.3. Partial indices 192 6.4. Structure of the reduction 193 6.5. Half-stability of partial indices 197 6.6. Application to the factorization problem 198 6.7. Examples 200 Uotes and comments 201 REFERENCES 204 SUBJECT INDEX 211 NOTATION INDEX 212 8

P All

I

TOEELITZ M B HANKEL MATRICES 0. PRELIMINARIES 0.1«

Notations»

First of all let us introduce and discuss some nota-

tions. As usual denote IN = {0,1,...} the natural numbers Z the integers IR the real numbers C the complex numbers, U = C U {oo} ® = {X € C I |\|= 1} the unit circle. is a linear space then E n = B x...x B. denotes the cartesian * >— n mxn product space, B denotes the space of mxn matrices with entries If E

1

from E. The space

Ea

will be identified with the space of column

ax

vectors E ^. In this sense

(x^p =

always means a 1 11

column vector. The space of row vectors E * Eq

and

will also be denoted by

[x^lp = [Xp ••• x ] always means a row vector. For matrices

we use the designation ill PP

[«"iklpr ^ . K l c ^ * [ a i J p- ,= . p- k=r,...,s The transposed of a matrix A will be denoted by

m A . For sake of

simplicity we shall often omit the boundaries of numeration by representing a vector or a matrix iu the component form. For most of our purposes it is convenient to numerate the components of a vector beginning with zero. That means for x 6 C n numerate ration By

x = (x^)^ [ a , ^

e

1

. Analogously! fox

mat Pic© S

W© US©

"fch© QUID.©—

f1. we shall denote the canonical basis of Cn

a n-1 k : = ^6ik^i=o '

6

ik : = U

For short we shall often write In this part

HXQ

we shall

0 :i ± k :i = k* e^

instead of e£.

we only deal with the complex scalar case

In this case any linear operator A t C

Q

1

d"

E = C.

corresponds with the

matrix ia the canonical basis 1L 1= [©iAe^]^-1 ° 1 e ® m x a

and vice

versa. In all what follows we shall always identify the operator

A 9

with the corresponding matrix X , The Kernel and image of an operator ker A «= { x : Ax = 0}

and

A w i l l be denoted by

im A «= { y = Ax}.

In the following we shall often uae the f a c t that the space

Ca

is

isomorphic to the space of polynomials with complex c o e f f i c i e n t s and degree

< n-1. This space w i l l be denoted by

x = (x k )£~ 1 6 ®a Here the variable

denote

x ( \ ) «= xQ +

® a ( v ) . Given

+ . . . + x ^ X 1 1 " 1 € C D (\).

\

w i l l be regarded as a complex variable. Moreover,

B(\)

w i l l denote the corresponding subset of C a ( \ ) .

we define

If

B c Ca

then

Introducing the vectors

l a ( x ) :=

one can write

x(\) = l Q ( x ) T x .

Let us introduce an analogous notation f o r matrices. Supposing A

= [ a i k ] o " 1 o" 1

we

defiQe

m-1 n-1 . v A(X,n) «= I £ a^vV . i=o k=o Obviously, A(A,|i)

A(\,|i) = l m ( x ) T A l a ( | i ) .

The polynomial in two variables

w i l l be called generating function of the matrix

A. Some pro-

perties of the generating function w i l l be established in Subsection 1.7. 0.2.

Toeplitz and Hankel matrices.

bers

(p»Q. £ i | p < a) al

^n(a)

To any sequence of complex num-

we associate the matrices

®1-1

®l-n+1

a l+1 a l . . ' ' ' al+m-1

(0.1)

C mxa

'

al+m-n

and

. ( a ) «=

°1 ®1+1 a l+2 a l+1 a l + 2 ®l+2

a l+m-1

10

®l+n-1 CmxQ a

l+m+Q_2

(0.2)

Here one h&s to put

a^ t= 0

for

i < p

and

1. > q. M&tricos of tlis

form (0.1) are called Toeplitz matrices ("T-matrices" f o r short), matrices of the form Co.2) Hankel matrices ("H-matrices" f o r short). These two classes are the main subject of this part.

Furthermore,

vie shall consider classes of matrices, which are related to the inverses of T- and H-matrices. For sake of simplicity we set \n(a)

,=

'

Tn

'=

TnnW»

The class of mxn T-matrices w i l l be denoted by ¿T(m,n), the class of mxn H-matrices by m,n). In particular, we set ^n)

¿H^n)

t=^Ja,a),

i=9^n,n).

Let us introduce some special matrices, which are important f o r further investigation. Denote i a t = t 5 i,k+l^i=d

m-1

6

T

>'

a

)'

k=o,...,n-1 We discuss some special cases of the matrices I Q 1= I ° Q

i s the nxn identity matrix ; 0 1. u „ « = 1*1 n n n nn 1 0

are the matrices of the shift operators in u

n^o"

1

=

Furthermore,' in case from tions By

®a Ba JQ

into onto

x q-2>;

m > n and in case

. Obviously,

0 1. nn © :

USxk)o"1 =

0 < 1 < m-n

^-I'0)' are imbeddings

m < n and m-n < 1 < 0

C*. Finally, l e t us remark that

we denote the matrix of the counteridentity

are r e s t r i c = ^

.

n-1 Jn(xjj)

Jn

1)

Here and in a l l what follows empty places in a matrix have to be replaced by zeros.

11

Obviously,

J* = I Q . The counteridentity realizes a simple connection

between T- and H-matrices. PROPOSITION 0.1« JmA € 3?(m,n)

If

A 6