Linear Transformations in n-dimensional vector space [1st reprint ed.] 0116301276188

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LINEAR TRANSFORMATIONS in n-dimensional vector space

LINEAR TRANSFORMATIONS in n-dimensional vector space AN INTRODUCTION TO THE THEORY OF HILBERT SPACE BY

H. L. HAMBURGER Professor in the University of Cologne AND

M. E. GRIMSHAW Fellow of Newnham College, Cambridge

ti f! f§ iMGPDQC %KV If

CAMBRIDGE AT THE UNIVERSITY PRESS 19 5 6

Q, K 1, and let x be any vector of $8W. If we assume the existence of an orthonormal basis gx,g2, ...,gr of TO we obtain the projection p of x on TO and the perpendicular q dropped from x on TO as p = Pmx=

£

{x,gv)gv,

q = x-

V=1

£

{x,gv)gv.

(3-31-1)

V=1

For, as before, peTO, and q is orthogonal to every vector of TO, since (q,-g/1) 0 for p = 1,2,..., r. —

3-4. We make use of the geometrical conception of projection to establish the existence of an orthonormal basis of any L.M. We then interpret (3-31-1) as giving the projection of any vector x on any L.M. TO. Every L.M. TO in has an orthonormal basis.

Theorem. TO = 0,

Proof.

with the trivial exception

We prove the theorem by induction, considering a L.M.

TO of rank r, where r > 1, and assuming that the theorem is true for any L.M. of rank r— 1. We have already verified the theorem, in 3*3, for the case r — 1. Let the vectors a1, a2, ...,ar form a basis of TO and consider the L.M. TO' = [a1, a2,..., ar_1], of rank r — 1, which does not contain ar. Then, by our assumption, TO' has an orthonormal basis g1, g2,..., gr~x and, by (3-31-1), the perpendicular qr from ar on TO' is qr = ar—

r—1 2

(ar, gv) gv.

»=i

By a remark in 3-21, qr is not zero, since ar is not contained in TO', and we may therefore write gr = || qr |j ~1qr. We now readily verify that the set of vectors g1, g2,..., gr forms an orthonormal basis of TO. Thus, the theorem is true for any L.M. of rank r. 3-41. The method described in the proof of Theorem 3-4 enables us to replace any basis a1,a2, ...,ar of a L.M. TO of rank r by an

COMPLETE ORTHONORMAL SYSTEMS

15

orthonormal basis g1, g2,..., gr by successive steps and in such a way that r i o , r , „ [a1,a2, = \gx,g2, {/i = l,2,...,r). The process is known as E. Schmidt’s orthogonalization process. It should be noted that the procedure holds, with trivial modi¬ fications, for elements a1,a2, ...,am spanning the L.M. 9ft that are not known to be linearly independent. In this case the vector q* = of- *2 (a-“,gv) = and Affect. 8*72. We prove in 8*82 that any eigen-manifold of a H.T. H reduces H completely, and in 17*51 we show that an eigen-manifold of a general L.T. A does not necessarily reduce A completely. 8*8. Theorem.

If A is any L.T. in

93^

and if W is a L.M. that

reduces A, then 93n ©9ft reduces the adjoint A* of A. Proof.

Let x be any element of 9ft and y any element of 93n ©9ft.

By^631^

(Ax,y) = (x,A*y) = 0,

since ^4a:e9ft. But this implies that H*y€93n©9ft for every y of 93n ©9ft and hence that 93 n ©9ft reduces A*. 8*81.

Corollary of Theorem

8*8. If a L.M. 9ft reduces a H.T.

H, then 9ft and 93„©9ft reduce H completely. Write ft = ©9ft. By a remark at the end of 5*02, the L.M. ft is complementary to 9ft; by Theorem 8*8, ft reduces H*\ but H = H* since H is Hermitian; and therefore, by Definition 8-7 (ii), 9ft and ft reduce H completely. Proof.

8*82. If © is any eigen-manifold of a H.T. H, then, by 8*71, © reduces H and, by Corollary 8*81, © and 93n©© reduce H com¬ pletely. This property is true, in particular, for the eigen-manifold 3E corresponding to the eigen-value zero, and for 93n©T which is the range ft.

NORMAL TRANSFORMATIONS

§9.

45

Normal transformations and UNITARY TRANSFORMATIONS

9*0. We have already seen that the coincidence of the null manifolds 36 and *3) for any H.T. has interesting implications. For example, we have proved in Theorem 8*3 that any H.T. determines a one-one correspondence of its range 0ft == ©36 with itself. We now define another class of L.T.’s, the normal transformations, having the property that 36 = ‘J) and having most of the other properties of H.T.’s, with one important exception, namely, that the eigen-values of normal transformations are not necessarily real.

Definition of a normal transformation. A L.T. Nin SS^is said to be a normal transformation if N*N = NN*. It is clear that N* is normal if N is normal. The H.T.’s are obvious examples of normal transformations since, for any H.T. H, H*H = HR* = H2. Other examples are given in 9*4 and in § 16. 9*01. Theorem. If the L.T. N is normal, then N — XI is also normal, for any value of the number A, real or not real. Proof.

We

have, by

7* 11

and

7*01,

{N — XI)* = N*-XI, and, since N*N = NN*, (N*-XI) {N-XI) = N*N — XN-XN* + XXI = {N-XI){N*-XI). 9*1.

Theorem.

If N is a normal transformation and if the null

manifolds of N and N* are 36 and 2), then 36 = 2), and hence 9ft -SS„©36. Proof,

liy e?), then N*Ny — NN*y =

0

and hence NyeSI). But

Nyetii so that Nyed1.2), and since 9i = $w©$ by Theorem 6-4, we have .?) = © so that Ny = 0. It follows that ye 36 and that c 3£. In a similar way, we show that = 0 for any £ of 36, that is, that 36^$. Hence, 36 = f). 9*11. Theorem. If A is an eigen-value of the normal transformation N, then X is an eigen-value of N*, and if (£ and (S* are the eigenmanifolds of N and N* corresponding to A and A respectively, then © = ©*. Proof.

The eigen-manifold '($ can be considered as the null

manifold of N — XI. By Theorem 9*01, N -XI is also a normal

46

UNITARY TRANSFORMATIONS

transformation, and by Theorem 9*1 its adjoint N* — AJ has the same null manifold. Thus Gs is the eigen-manifold of N* corre¬ sponding to the eigen-value A. 9*2.

Theorem.

If (j)1 and 2 are eigen-solutions of a normal

transformation N corresponding to distinct eigen-values Ax and A2, then ((j)1, (j)2) = 0. Proof.

By Theorem 9* 11, the equations N(jA = A^hiV^2 = X2f>2

imply that N*(j)x = A^1, N*f>2 — A22. Hence A 1(1,2) = (Ncf>\2) = (\N*2) = (^,A2^2) = Ajp.p), and, since Ax #= A2, we deduce that (01, (f2) = 0. 9-3.

Theorem.

If © is the eigen-manifold of a normal transforma¬

tion N corresponding to an eigen-value A, then © and 33n©S reduce both N and N* completely. Proof. By Theorem 9*11, (5 is an eigen-manifold of N* corre¬ sponding to A and therefore, by 8*71, S reduces both N and N*. Hence, by Theorem 8‘8, reduces both N* and N and so, by Definition 8*7 (ii), © and 33n©@ reduce N and N* complete^.

9'31. We have now seen that the eigen-manifolds of a normal transformation behave in the same way as those of a H.T., in that they reduce the transformation completely. We therefore go on to define the multiplicity of an eigen-value of a normal transformation as we defined, in 8‘41, the multiplicity of an eigen-value of a H.T.

Definition of the multiplicity of an eigen-value of a normal transformation. We define the multiplicity, or order, of any eigen-value of a normal transformation as the rank of the corre¬ sponding eigen-manifold. 9*4. Definition of a unitary transformation. A L.T. x' = Ux in 18 n is said to be a unitary transformation if it leaves unchanged the lengths of vectors so that (x',x') = (x,x). A unitary transformation that is real is called an orthogonal transformation. It follows immediately from Theorem 6*21 that any unitary transformation U in 58/t has rank n, since x’ = Ux — 0 if, and only if, x = 0, and therefore, by Corollary 6-41, U determines a one-one correspondence of with itself. It also follows at once from Definition 9*4 that a unitary transformation maps the unit sphere

UNITARY TRANSFORMATIONS

47

(x, x) — 1 on itself and, conversely, that any L.T. U that maps the unit sphere on itself is a unitary transformation, for we then have (Ux, Ux) = (x,x) for every x. 9*41. It follows from Definition 9-4, by (7-3-1), that, for any unitary transformation U, (x,x) = (Ux, Ux) = (U*Ux,x) = (x, U*Ux),

(9-41-1)

so that, by Theorem 6-06, U*U — I and U* = £7_1. Conversely, we see from (9-41-1) that the L.T. U is unitary if U*U = I. Let the analytical representation of the unitary transformation n

x' = Ux be x^ — 2

Then, since U*U ~1, we have by (7-42-1)

V=\

n 2

_ UpiKUflV = ^KV

fl=l

We also have (x',y‘') = (Ux, Uy) = (x, U*Uy) = (x,y), so that angles between vectors, defined by (1-42-2), as well as the lengths of vectors, are unchanged by U. Further, since

U-'U = uu-1 UU* = I,

2 V=

I,

_

n

we have

=

=

1

from which it follows that U* is a unitary transformation. Finally, from the relation U*U — UU* = I, we see by Definition 9-0 that every unitary transformation is also a normal transformation. By writing

n

g* = Z uVfluv, V= 1

_

n

g*v = 2

(9-41-2) 1

we see that the set of vectors and the set g*v each forms a complete orthonormal system. Conversely, we obtain two adjoint unitary transformations from any complete orthonormal system of vectors by taking the coordinates of the yth. vector of the system as the fith. column, or the conjugates of the coordinates as the /^th row, of the corresponding matrix. 9*5. There is a twofold geometrical interpretation of any unitary transformation. First, we may interpret it as the replacement of every vector by another of equal length in such a way that angles between vectors are unchanged. This process we may describe as a mapping of the whole space on itself, conserving the metric magnitudes, or as a generalized rotation of

about the origin.

48

UNITARY TRANSFORMATIONS

Secondly, we may interpret it as the substitution for the coordinates of every vector referred to one system of coordinate vectors in terms of new coordinates referred to a new system. This process we may describe as a transformation of the coordinate system. 9*51. Taking the first interpretation and following the procedure described in 6*02, we see that the unitary transformation x' — Ux transforms the coordinate vectors into the orthonormal system of vectors gt* of (9-41-2). This gives an illustration of the invariance of lengths and angles. 9*52. For the interpretation of a unitary transformation as a coordinate transformation it is more convenient to consider it as a substitution of the form x = Ux',

xS

(9-52-1)

p= l

This implies, by 9’41, that n

X1 = U~xx = U*x,

x'v = 2 ^~XU.

(9-52-2)

We now interpret the numbers x'v as the coordinates, in a new coordinate system {u'"}, of the vector that has the coordinates in the initial coordinate system {v/}. Thus 2 z ^ = 2 xvu'v. v=l

Substituting for x from (9-52-1) we obtain

2

=

H=\

2 fly V=1

Therefore, by (9-41-2), u'v = £ w M=

(9-52-3)

1

and these are the coordinate vectors of the new coordinate system. Their coordinates in the old system are given by the elements of the i/th column of U. 9-6. If U1 and U2 are unitary transformations then the products

i,

V = U2U W = UXU2 are also unitary transformations since, by 7-3, F* = I7f Ut, W* = Ut Ut and therefore V*V = I, W*W = I.

49

PROJECTORS

§ 10.

Projectors

10*0. We now interpret the formation of the projection of any vector x of on a L.M. 9ft of rank r, described in 3*31, as a H.T. If the vectors gr1, g2,..., gr form an orthonormal basis of 9ft and if we write, as in (3*31 * 1),

r

Pmx = S

(10-0-1)

/*= i we readily see that the transformation x' = P^x is a L.T. since, for any vectors x and y and any number a, Pm(ccx) = ctPm X, Further,

Pm(x + y) = Pmx + P^y.

(Pmx, y) = 2 (*> (P) (P> V) = (*. pmV)> M= 1

and therefore

is a H.T. We call it the projector of

on 9ft and,

as before, we call P^x the projection of x on 9ft. If we write, by 5-01, x = 2 (x,g'l)g/i+ 2 {xMjlnJ1, /t=i

/t=i

where the vectors hJ1 form an orthonormal basis of the L.M. 3S„©9ft, then it is clear that the second sum, (I-Pm)x = 2 (xM)}^, n=i

gives the projection ofxon3Sn©9ft.We have previously interpreted it as the perpendicular dropped from x on 9ft. 10*01. We obtain at once from (10-0-1) pSr*= i (x,g»)Pmg» = f (.x,gfi)gfl = pmx> /i=l l“=1 since PmgP = g*, and, further,

{PPy^

=

(pmx>yy

(10-01-1)

Conversely, as we show in Theorem 10*2, if P is a H.T. in 93n such that P2 = P, then P is a projector of the form (10-0-1). 10*1. Definition of general projection. We now introduce, in addition to the projection defined in 10*0, a more general operation, called general projection, for which the projector is denoted by Q. The projector P will be called an orthogonal projector whenever the use of the term projector alone would lead to confusion.

50

PROJECTORS

Let 2)1 and 9? be two complementary L.M.’s of 93n, that is, such that 9ft . 91 = 0 and 9ft © 9? = be written in the form

Then every vector x of 33 n may

, x = a + b,

(10-1-1)

where a e 9ft and b e 9?. We call a the projection of x ontyR parallel to 92 and b the projection of x on 9? parallel to 9ft and we write a = Qmx,

b = Q^x = (/- Qm)x = x-a.

(10-1-2)

If the L.M.’s 9ft and 92 are given, then Q^x and Q^x are uniquely defined by (10-1-1) and (10-1-2) for any x of 33n, since a relation x = a + b = a'+ b' would imply that a —a’ — b' — b, where a — a' e 9ft and 6' —6e92, and so that a — a' = b — b' = 0, since 902.92 = 0. It follows that the transformations a = Q^x and b = Q^x are linear, since we have, for vectors x and y and any number a, ocx = ocQ^x + ocQ^x,

x + y = (Qmx + Qmy) + (Q^x + Q^y),

so that Qm(ocx) = ocQ^x and Q^x + y) — Q^nx+ Q^y, with similar relations for Qgi • It is clear that 9ft is the range of Qm and that 92 is its null manifold. We call the L.T.’s

and Q^ the projectors of

93 n on 9ft parallel to 92 and on 92 parallel to 9ft. The relations (10-1-2) show that, if Qm is the projector of 93ra on 9ft parallel to 92, then I — Qgn is the projector of 93n on 92 parallel to 9ft. Reference to Definition 3-2 shows that general projection reduces to orthogonal projection when the L.M.’s 9ft and 92 are orthogonal complements; that is, when 92 = 93n©9ft. 10-11. The L.T. a = Q^x defined in 10-1 is such that Qma = a

if

ae 9)2,

Qmb = 0

if

6e92,

and hence Qlnx = Qma = a= Qmx, that is, Q^ = Qm. 10-2. Theorem. If T is a L.T. in 93n such that T2 = T, then T is a general projector, and if, in addition, T is Hermitian, there T is an orthogonal projector. Proof, (i) Let the rank of T be r, let its range be 92 and its null manifold be 1, as in Theorem 6-4. Then, by Theorem 6-4, the ranks of 92 and T are r and n — r respectively. Moreover, 92. T = 0; for, since T2 = T, we have T(Tx) — Tx = a for any vector x and Ta = a for any a of 92, and therefore, within 92, Ta = 0 only if a = 0. Thus, 92©3£ — 93n) by Theorem 4-21 and Corollary 2-61, and we

51

PROJECTORS

can write any x of 93 n in the form x = a + £, where a e 91 and 36. Hence, Tx = Ta+ TE, — a and T is Q9?, the projector of 53n on 91 parallel to 36. (ii) If, in addition, T is Hermitian, then, by Theorem 8*3, 36 = 93n©91 and therefore, by the remark at the end of 10*1, T is the orthogonal projector P,^ of 53„ on 91. Note. Reference to Theorem 9*1 shows that the condition in Theorem 10*2 that T is Hermitian may be replaced by the weaker

condition that T is normal. 10*3.

Theorem.

If Qm is the projector of 3$n on 9J1 parallel to 91,

then Qgx is the projector of 93n on 53n ©91 parallel to 53n©9)l. Proof.

The transformation Q^ is a projector, by Theorem 10*2,

since the relation Q^ the range of

implies (Q^)2 = Qe^. By Theorem 6*4,

is 93n ©91, since 91 is the null manifold of Qm, and

the null manifold of

Is

1, since 9J1 is the range of Qc^.

Therefore, by Theorem 10-2, Q^ is the projector of 53n on 53n©91 parallel to 93„©9J1. 10-4.

Theorem.

Let Q1 and Q2 be general projectors. Then Qx + Q2

is a projector if, and only if, QiQ2 = QzQi = 0. If Q1 and Q2 are both orthogonal projectors and if QxQ2 = 0, then Q2Q1 = 0 arid Qx + Q, is an orthogonal projector. Proof.

Write Q = Qx + Q2. We have Q'^Q^QiQt+QtQi+Q*

t10'4,1)

since Q\ — Qx and Q\ — Q2 by 10*11. (i) If then Q1Q2 = Q2QX = O, we have Q2 = Qx + Q2 = Q and Q is a projector by Theorem 10*2. (ii) If Q1 and Q2 are both orthogonal projectors, then Qx, Q2 and Q are all H.T.’s.

If QxQ2 = 0, then (QM* = 0.

But since

(QxQf)* = QtQ* = QiQi’this means that ^Qi = 0 and therefore that Q is a projector, by (i). Moreover, by Theorem 10-2, Q is an orthogonal projector since Q is a H.T. (iii) Conversely, if Q is a projector, we

deduce from (10-4-1)

QiQt+QtQi = o,

that

(io-4-2)

since Q2 = Q = Qx + Qz- By multiplying (10-4-2) by Qx, first from

52

PROJECTORS

the left and then from the right, and using the relation Q\ = Qx, we obtain $1Q2 + Qi Q2 Qi = 0,

Qx Q2 Qi + Q2Qi = 0,

and then, by subtracting these equations, Q1Q2-Q2Q1 =

.

0

In virtue of (104-2) we obtain QXQ2 = Q2QX = 0. 10-41. Theorem. If Qx, Q2 and Qx + Q2 are general projectors with ranges 9ftx, 9ft2 and 9ft and null manifolds ftx, ft2 and ft respectively, then 9ftx.9ft2 = 0, Proof.

9ftx©9ft2 = 9ft

and

ft1.ft2 = ft.

Write Q = Qx + Q2. By hypothesis, Q is a projector and

therefore, by Theorem 10-4, QXQ2 = Q2QX = 0. But the equation QiQ2 — 0 implies that 9ft2Sfti, so that, since 9ftx. ftx = 0, we have 9ftx. 9ft2 = 0- The two L.M.’s 9ftx and 9ft2 are therefore linearly independent and Qxx + Q2x — 0 if, and only if, ^eftx and xc^fl2, that is, if zeftx.ftg. Hence, ft = ftx. ft2. Further, if x = a1 + a2, where a1e9ft1 and a2e9ft2, we have Qxx = a1,

Q2x = a2,

Qx = a1 + a2 = x

and

9ft1©9ft2c9ft.

Also, 9ft^9ft1©9ft2 since, for any x of 93n, Qx = Qxx + Q2x, where Qxxe 9fti and Q2xe 9ft2. Hence, 9ft = 9fti©9ft2. 10*42.

Theorem.

Let Q^for p = 1, 2, ...,m, be general projectors m

with ranges 91^ and null manifolds ft/r Write Q = 2 QM and let the n=\

range and null manifold of Q be 9ft and ft respectively. Then Q is a projector if Q^ Qv = S/w Q/xfor 1 < p, v < m; the L.M.’s 9ft^ are moreover linearly independent, 9ft^. 9ft„ = {Qa, Qa) - (Qa, a) = 2 (QMa, a) = 2 (Q^a, Q^) /t=l

n= 1

^(Qva, Qva) = (a, a). The equahty sign must hold throughout in the relations above and therefore Q^a — 0 for p-f=p whenever ac 99© It follows that QflQvx = 0 for p + v and for any vector x of 2$n. Finally, since Q^ is a projector, Q*x — Q^x. This completes the proof of the theorem.

10-5.

Theorem.

Let Qx and Q2 be general projectors with ranges

9L© and 9912 and null manifolds 9© and 9© Then Qx Q2 = Q2 if, and only if, 9ft 2 c 99© and Q2 Qx = Q2 if, and only if, 9?! £ 9© Note.

The theorem implies that QXQ2 — Q2QX = Qiif, and only

if, both

and 9© = 9©

(i) If 9ft2 c 99© then, since Q2x e 9ft2 and so Q2x e 99© we have QxQ2x = Q2x for any vector x, and therefore QXQ2 = Q2. (ii) Conversely, if Qx Q2 = Q2, then Qx Q2 is a projector with range 99© But the range of Qx Q2 is contained in 99© and therefore Proof,

9ft2c9J© (iii) The ranges of the projectors I — Qx and I —Q2 are 9^ and 9£2 respectively. Hence we deduce from (i) and (ii) above that the condition 9?! c 9J2 is the necessary and sufficient condition for the equahty

{I-Qf) (I-Qi) = I-Qv

But this is equivalent to the equality

I—Q2 — Qx + Q2Qx — I — Qi> or Q2Qx = Q2. Thus, Q2QX = Q2 if, and only if, *©£9©

10-51.

Corollary of Theorem

10-5. Let Px and P2 be orthogonal

projectors with ranges 9ft t and 99© If PxP2 = P2, then P2PX — P2, W2 c 9ftx and

9?x = 9© ©9ftx £ 9i2 = 9© ©9)©

Conversely, if 9ft 2 £ 99© then PXP2 = P2PX = P2. Proof.

Since the orthogonal projectors Px and P2 are H.T.’s, the

equality Px P2 = P2 implies© = Pt = P* P* = P>© The remaining

54

PROJECTORS

assertions of the corollary now follow at once from the theorem and note of 10*5. 10-6. We now give a matrix representation for any projector. We consider first the orthogonal projection 9k of rank

Pm of

on the L.M.

r. Let g1,g2, ■■.,gr be an orthonormal basis of 9k, let a;

be any vector of $Bn and write 9K = Z

VVku\

x

= £

v=1

xvuV-

y-1

Then, by (10-0-1), P'mx= £ (x>gK)gK = £ K=1

K—\

= £ £ *„( £ 1 v=l

V *r=l

2

£ V=

1 /*=1

/

{xvVVk)V^

^= £

/i-1

where

(10-6-1)

=£(£ v=l \ *r = l

Thus the matrix representation for ■an

/

Pm is given by

P/ikPvk =(£ \K= 1

The rectangular matrix (p^) of n rows and r columns consists of the coordinates of vectors forming an orthonormal basis of 9k. Since (gl,gK) = SlK, we have n

_

2 (PvlPvk) = SlK K=1

M = 1,2, ...,r),

which means that the vectors defined by the columns of the matrix (pVK) are orthogonal to one another. Thus, this matrix may be interpreted as an incomplete unitary matrix which could be completed by the addition of n — r columns consisting of the coordinates of vectors forming an orthonormal basis of $Bn©9k. 10-7. We now consider a general projector Qm. Let 9k and 92 be L.M.’s, as in 10-1, with bases a1, a2, ...,aT and 61,62,..., 6n-r respectively, such that Qm is the projector of on 9k parallel to 92. Let T be the L.T. defined according to (6-02-1) by the relations Tvf = a* (/i = 1,2, ...,r), Tu*+r = b* {/i = 1,2, ...,n-r). (10-7-1) By 6-11, T is of rank n, the n vectors a1,a2, ...,ar, 61,62, ...,6n_r

PROJECTORS

being linearly independent since SOI.

55

= 0. Thus the inverse T~x

exists. Further, if we write T = (t^), we obtain from (6-02-3),

S

(v= 1,2,...,/•),

/i=i

(10-7-2) t>v=

2 a-i

(»> =

Writing a: = 2 x^, we deduce from (10-1-1) and (10-7-2) /i=i n

n—r

x = 2

= 2 which is the rank r of 9k, and hence this set of r elements is linearly independent, since otherwise the L.M. defined by the square bracket (10-82-1) would have rank less than the rank of 9k. It follows by Definition 4-4 that the m L.M.’s 9k/t are linearly independent and that (10-82-1) may be written in the form ra 9k 0, we write

H,+x = Then H^x = H in 3SW1, = 0 in $n©$rfl+1, and ^+1 is Hermitian in the subspace S8rfi+1 which reduces H completely. Moreover, the eigen-values of H other than 0, A1?..., A^ coincide with the eigen-values of H' +1 and the corresponding eigen-manifolds also coincide, since by Theorem 8-6 they are contained in 23r(i+1. By Theorem 12-0, the maximum value A/i+1 of the Hermitian form (H^+Xx', x') on the unit sphere || x' || = 1 in 33rfJ+1 is an eigen-value of It is therefore an eigen-value of H different from 0, A1?..., A^, and the corresponding eigen-manifold @ +1, of rank kp+1, is con¬ tained in 58r(i+1. We denote by the orthogonal projector of 93 n on @A+1 and we obtain, as before,

We also have

HP^ = A^P^ = P^+i#,

by Theorem 10-9, since @ +1 reduces H completely. In this way we obtain a succession of non-zero eigen-values A* as the maximum values of (Hx, x) on the unit spheres in the sub¬ spaces Since and A^ + A,^, we have A^neH.

.

(12*1*3)

SPECTRAL REPRESENTATIONS OF A H.T.

64

We see that there can exist no eigen-solution y of H belonging to a non-zero eigen-value w different from the XK obtained above. For such a vector would not belong either to T or to any of the &K, since by Theorem 8*6 it would be orthogonal to all vectors of these manifolds and therefore, by (12-1-2), it could only be the zero vector and no eigen-solution. Thus the eigen-values found above form a complete set of eigen-values for H. 12-2. We may summarize the results j ust obtained in the following theorem. Theorem. Every H. T. in 23n, not identically zero, has a finite set of non-zero eigen-values A1} A2,..., Am, where l^m^n. If £ is the eigen-manifold of H corresponding to zero (the null manifold of H) of rank k0, and if (&K, of rank kK, is the eigen-manifold corresponding to XK (k = 1, 2,...,m), we have m

m

33 n = T© 2 ©

n =

2

h/c’

Af=0

K=\

where the number kK is also the order of the eigen-value XK. If P0 and PK are the orthogo'nal projectors of 33 on dc and dK (1 ^ k < m) respectively, we have m P,PK = 8iKPK, ZPK = I, (12-2-1) K=0

HPk = AkPk = PkH If we take orthonormal bases in the

(0

(12-2-2)

and in 36, we obtain a complete

orthonormal system of vectors fi1,02, ..., ... ^ An (as in 14*1), and if the eigen-values of Hx in 9ft are Ax^ A^S5 ••• ^An_r, thenAs+r < A' < As/ors = 1, 2,...,n-r. Let 0'1, 0'2,...,be eigen-solutions of Hx in 9ft corresponding to the first 5—1 eigen-values Aj, A2,..., As_j of Hx. Proof.

Write

9ft' = 9ft ©&-1 = ($n©9*)©&-l = ®n©TO&-l)> and apply Theorem 14-1 to the L.M. 9ft' which has rank n - r - s + 1.

76

INEQUALITIES FOR EIGEN-VALUES

We find that Xs+r < max (//?/, y) on the unit sphere in W. But so that Pmy — y for every y of 9Ji', and therefore (Hy, y) = (HP^y^^y) = (Hxy,y) for every y of 9K'. Further, by 12-0 and 12*1, X's = ma,x(Hxy,y) on the unit sphere in TO'. Hence, As+r^ A's. We obtain the remaining inequality Ag^As by considering the Hermitian form ( — Hx,x). Its eigen-values w1 ^ w2 ^ ^ wn are given by the equations cofl = — Aw+1„/i and the eigen-values (i)l

of — Hx in yjl are given by

(O2

^ ... ^ 0)n—r

= — A^+1_r_^. Hence the inequality

^ o)’s implies - Am+1_s_r ^ - An+i-r-s’ and this gives tiie result required if we substitute s for n+ 1 — r — s. (os+r

14*21. We see the value of the previous result for the considera¬ tion of a particular H.T. if we define the H.T. by a matrix H = (a and take 9JI = [u1,u2, ...,un~r] = n

(Hx, x) =

Then n—r

_

v

alivxvxp

(Hxy, y) —

_

£

H,V=\

=

1

The form (Hxy, y) is called the reduced form of (Hx, x), and Ag is its sth eigen-value. Theorem 14*2 gives bounds for the eigen-values of a Hermitian form in terms of the eigen-values of its reduced form which are, clearly, easier to calculate. 14*3. When the H.T. H is positive definite we can deduce from Theorem 14*1 a more precise set of inequalities than those of Theorem 14*2. The new inequalities are due to Aronszajn. We need a preliminary lemma. Lemma.

such that

Let 0

for every complex number

and

Then ocy>/3fi,

(14-3-1)

/(£)ccr2-2pr + y

^0,

(14-3-3)

77

INEQUALITIES FOR EIGEN-VALUES

for all values of r and 0, with equality in the inequality on the left only for cos(

/(Co) = ccr2-2pr0 + y = 0, where | Co I = ro- Thus, (14-3-1) is verified. It follows that pr ^ r ./(ay) < ^(a + yr2).

(14-3-4)

We now deduce from (14-3-3) and (14-3-4) that

/(C) < ar2 + 2pr + y ^ (1 + r2) (a + y), and (14-3-2) is verified.

14*4.

Theorem.

(Aronszajn’s Inequalities.) Let H be a positive

definite H.T. in 33n, let 9R and 91 have the same meanings as in Theorem 14*1, and let and -^31 be the orthogonal projectors of 33 n on 9k and 91 respectively. Write = PqtjH/Cjj,

H2 = P^HPm>

and consider the Hermitian forms (I^x, x) and (H2x, x) in 9k and 91 respectively. Let the eigen-values of H in 33ra be

A1^A2^...^A,l>0 (as in 14-1), those of Hx in 9k be A^ A2> ... >K-r (as in 14‘2) and those of H2 in 9J be A'^ > A2 ^ ^ A". Then

As+ =

, A

, P.+ 2 (A-A,)«S«+1.

Av

(15-43-2)

16*3. Referring to 15*23, we see that f(H) is not necessarily Hermitian if/(if) is not a real function. We notice at once, however, that the transformation N — f(H) is a normal transformation, since, by 15*23, N* =f(H) and, by (15-3-1), NN* - N*N =f{H)f{H). Thus we are led to the normal transformations by consideration of the L.T.’s generated by the Hermitian transformations. We show in 16*5 that every normal transformation N can be written in the form N = f(H), where H is a suitable H.T. We first, however, prove the following theorem. Let N be a L.T. and {cf)K} (k = 1,2, ...,n) a complete orthonormal system in %n such that the (j)K are eigen-solutions of N Theorem.

corresponding to eigen-values lK; that is, N(pK = lK*. Then N is a normal transformation and the lK are all its eigen-values. The numbers lK are complex, not necessarily distinct, and some of them may be zero. Note.

Proof.

Since the 0* form a complete orthonormal system we can n

write any x of

as x — T] (x, (fK) (f)K, and since N(j)K = lK(pK we have /c=l

Nx= S

(Nz,y) = £ K(x>K)y) = (x,N*y), 1

«■= 1

89

NORMAL TRANSFORMATIONS

sothat

lK(y, (j>K) (j)K.

N*y = £

(16-3-1)

Hence, as in (16*1*2), NN*x — 2

— N*Nx,

(16-3-2)

K=1

and N is a normal transformation. Finally, if

is an eigen-solution

corresponding to any eigen-value o) of N, we have ^=2 (fr,K)f>K,

Nxjf= 2

*:= 1

k=

= (o S (ft,K)K,

1

/c= X

so that, for each k, either (o = lK or (fr, K) = 0. Thus a> coincides with at least one of the numbers lK. If oj = = Z2 = ... = lv, say, we see that the corresponding eigen-manifold is [i^1,02, ...,(J>V]. 16*4. The converse of Theorem 16*3 is also true and gives the spectral representation for any normal transformation. Corresponding to any normal transformation N in iSn

Theorem.

there exists a complete orthonormal system { 1 and therefore that 363+*’ = 363. Example.

for p= 1,2,

If the L.T. A is defined by the equations Auv = uv+l 1 and Aun = 0, then 36 = \un~\, 36* = [un~K+1,un~K+2,...,un]

for k = 2, 3, ...,n, and 36= 36w = for every positive integer v. In this case j = n. The matrix representing the L.T. of this example is given in (7-83-2).

17*02. Definition of the index and of the principal manifold of a L.T. The number^’ defined in Theorem 17*01 is called the index of A and the corresponding L.M. 363 is called the principal manifold of A. We denote the principal manifold by $ and its rank by k.

98

EIGEN-VALUES AND PRINCIPAL MANIFOLDS

The principal manifold may be defined as the set of all vectors that are elements either of the null manifold of A or of the null manifold of some positive power AK of A. Since we have rank XK+1 > rank XK for K^j— 1 and rank X ^ 1, we see that Jc^j. If ^ = iSn, that is, if k = n, we say that A is a nilpotent transformation, for in this case Aj — 0. The L.T. A defined in the Example of 17*01 is, clearly, a nilpotent transformation of index n. 17*1. We denote the range of AK by 9P and we write 9P = 91. We see at once that 9P+1 £= 9P, since AK(Ax) = AK+1x. In the Example of 17*01 we have 91 =

= \uK+1, uK+2,..., un]

[it2, it3,..., id1],

for k = 2, 3, ...,n— 1, and 9tn+*' = teger v. 17*11.

Theorem. $

.W

=

= 0 for every positive in¬

We have Xj. W =

0,

9*^ = 8**

(p —

1,2,...),

where the notation is that of 17*0, 17*02 and 17*1. Proof.

Let £e93 • 9ft5'; then we may write £ = Ajx, where 0 = Ai£ = A2>x.

But, by 17*02, the equation A2jx — 0 implies that

and there¬

fore that A>x = 0, since j is the index of A. Thus £ = 0 and therefore

$.W

=

0.

Now the range of Ai+1, may be interpreted as the image of 9P with respect to A. Further, 3£.9P = 0, since and ^. 9P = 0, so that, by Theorem 6*2, 9P+1 and 9P have the same rank. We have shown in 17* 1, however, that 9P+1 c 9P and therefore, by Theorem 2*6, 9P+1 = 9F. An induction argument now completes the proof of the theorem. 17*12. There is a converse of the preceding theorem. If the L.T. A has index j and if, for some positive integer v, X. 9P = 0, where SI" is the range of Av, then v^j. Theorem.

Proof.

We know that

3El’+12

Xv and, by Theorem 17*01, it will

be enough to prove that 3E*'+1 = Xv. Now, if £(*’+1) is any non-zero element of Xv+1, and if we write £ = Al'£(v+1), we find that £e 3£ and £e9P. Since, however, X. 91" = 0 we have £ = 0 and therefore £(H-i)e 36". It follows that 3£‘,+1 = Xv and v ~^j.

EIGEN-VALUES AND PRINCIPAL MANIFOLDS

99

17-13. There are relations between the 31" corresponding to the relations between the 3E". We have 31"+1