Integral Equations and Operator Theory Vol. 14 (1991)

0378-620X/91/040564-2251.50+0.20/0 (c) 1991Birkh~user Verlag, Basel

GENERALIZED PSEUDOINVERSES OF MATRIX VALUED FUNCTIONS

Marek Rakowski We define the pseudoinverse (resp. a generalized pseudoinverse) of a matrix-valued function F to be the function F • such that, for each )~ in the domain of F , F• is the inverse (resp. a generalized inverse) of the matrix F(2). We derive a state space formula for a generalized pseudoinverse of a rational matrix function without a pole or zero at infinity. This derivation makes use of the theorem characterizing the factorization of a nonregular rational matrix function W in terms of the decomposition of the state space of a realization of W. We also give a formula for a generalized pseudoinverse of an arbitrary rational matrix function in the form of a centered realization. We indicate some applications of generalized pseudoinverses of matrix valued functions.

1 INTRODUCTION Generalized inverses of matrices and, more generally, of linear operators acting between Banach spaces, are now a well accepted concept in various branches of mathematics, from linear algebra to operator theory (see e.g. [BG], [TL]). They have numerous applications, especially in statistics and numerical analysis. A matrix B E C "•

is a generalized inverse of a matrix A E C '~• ABA

Plainly, if A has a

right (or left) inverse,

(1.1)

= A.

then B is is a

if

right (or left)

inverse of A. In

particular B = A -1 whenever A is invertible. More generally, let X, Y be vector spaces over C and let A: X -* Y be a linear transformation.

A linear transformation B: Y -* X which satisfies (1.1) is called a gen-

eralized inverse of A. It is easy to see that A always has a generalized inverse. Indeed, choose subspaces W and Z of X and Y such that X = W~-Ker A and Y = Z~-Ran A. Let Aol: Ran A ~ W be the inverse of A I W , and let B: Y ~ X be any linear transformation which agrees on Ran A with A~ 1. Then B satisfies equality (1.1). The discussion in the preceding paragraph shows that a generalized inverse of a linear transformation A is unique if and only if A is bijective, that is, if and only if A is

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565

invertible. Suppose A fails to be bijective, and let B be a generalized inverse of A. Then Ker B f3 Ran A = (0). Also, Ran (BIRan A) n Ker A = (0) and if 9 E X , A ( x - B A z ) = O. Hence X = Ran (BIRan A)~Ker A. Suppose now that, in addition, A is a generalized inverse of B, that is B A B = B.

(1.2)

Then Y = Ran (AlRan B ) + K e r B and, since Set B f3 Ran A = (0), Y = Ran A-]-Ker B. By symmetry, X = Ker A-i-Ran B. Let A0: Ran B ~ Ran A be the restriction of A. Then A0 is bijective and, by (1.1), BIRan A is a generalized inverse of A0. So B[Ran A = Ao 1 and B = A o l Q where Q is the projection of Y along Ker B onto Ran A. Conversely, let W and Z be such that X = W + K e r A and Y = Z4-Ran A. Let Aoi:Ran A --+ W be the inverse of A I W , let Q be the projection of Y along Z onto Ran A, and let B = A o l Q . Then equalities (1.1) and (1.2) hold. Thus, B is a generalized inverse of A such that (1.2) holds if and only if B = A o l Q , where Q is a projection of Y onto Ran A and Ao z is the inverse of the restriction of A to some subspace of X complementary to Ker A. Let A E C " •

be a matrix, and let B be a generalized inverse of A. The preceding

discussion shows that A is a generalized inverse of B if and only if rank A = rank B (cf. Theorem 2, Chapter 1, in [BG]). A generalized inverse B of a matrix A such that (1.2) holds is called a reflexive generalized inverse of A in the literature (see [R]). We will often denote a reflexive generalized inverse of a matrix A by A*. It follows from the discussion in the preceding paragraph that if Ai and A~ are reflexive generalized inverses of a matrix A and Ran A i = Ran A~ and Ker A i = Ker A~, then A i = A~. We will use this fact later. Let A E C "xn be a matrix. By Theorem 1 in [P], there exists a'unique reflexive generalized inverse B of A such that (AB)* = AB,

(1.3)

(BA)* = B A .

(1.4)

This matrix B is called the Moore-Penrose (generalized) inverse of A. More generally, let X, Y be Hilbert spaces and let T be a continuous finear mapping of X into Y with closed range. Let Q be the orthogonal projection of Y onto Ran T, and let Tol: Ran T ~ (Ker T) • be the inverse of T[(Ker T) •

The operator T0--1@is called the Moore-Penrose inverse of T

(see [TL], [DW]). Below, we define the pseudoinverse (resp. a generalized pseudoinverse) of a matrix valued function F to be a function G such that the value of G at each point A in the domain of F is the inverse (resp. a generalized inverse) of the matrix F(A). This terminology is not standard. The function which we call the pseudoinverse of a function F is sometimes

566

Rakowski

called the inverse of F.

Since below the dependence of the matrix valued function

F(z)

on the variable z is important, and we will be interested at the same time in matrices and matrix valued functions, the use of the term "pseudoinverse" rather than "inverse" should help avoid confusion. The language of pseudoinverses and generalized pseudoinverses fits naturally into the theory of analytic and meromorphic matrix valued functions. We will pay special at~ tention to meromorphic matrix valued functions on the extended complex plane Coo, i.e. the rational matrix functions. Such functions have simple representations in the form of realizations. In the paper we derive a state space formula for a generalized pseudoinverse of a rational matrix function W without a pole at infinity (Theorem 3.10). This formula can be used whenever the function W is analytic at infinity and has a generalized pseudoinverse which is also analytic at infinity (cf. Theorem 5.1). We also give a formula in the form of a centered realization for a generalized pseudoinverse of an arbitrary rational matrix function (Theorem 4.2). We give two independent proofs of the formulas for a generalized pseudoinverse. One proof (Theorem 4.2) is based on a computation and the properties of generalized inverses. The other proof (Theorem 3.10) depends on factorization of rational matrix functions. The advantage of the proof of Theorem 3.10 is that it gives an insight into the structure of a reflexive generalized pseudoinverse of a rational matrix function. Also, the results on factorization of rational matrix functions in Section 3 are of interest in their own right. In particular, Theorem 3.1 generalizes to the nonregular case the Bart-Gohberg-Kaashoek theorem on factorization of rational matrix functions (Theorem 1.1 in [BGK]). It characterizes factorizations of a nonregular rational matrix function W in terms of the decomposition of the state space of a realization of W into subspaces which are invariant under the state space operator A and the "associate" state space operator A •

The operator A • for a nonregu-

lar rational matrix function generalizes a concept introduced in [BGK] for regular rational matrix functions. The paper is organized as follows. Section 2 contains the basic definitions. In Section 3 we derive the state-space formula for a generalized pseudoinverse of a rational matrix function without a pole or zero at infinity. In Section 4 we give a formula for a generalized pseudoinverse of an arbitrary rational matrix function. In Section 5 we point out some applications of generalized pseudoinverses of matrix valued functions.

2 BASIC DEFINITIONS Let X, Y be vector spaces over C and let

L(X, Y)

be the space of linear trans-

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567

formations from X into Y. Let a be a subset of the extended complex plane Co., and let

F: o" ~ L(X, Y) be a function. If for every z 9 tr the linear transformation F(z) is invertible, we will say that F is ~ and the function G: ~r ~ L(Y, X ) defined by o ( z ) = F ( z ) -~ win be called the pseudoinverse of F. A function G: a ---+L(Y, X ) such that F(z)G(z)F(z) = F(z) for every z 9 a will be called a ~

(2.1)

lzi.cal.dgJa_~e,l~ of F. We will usually denote a

pseudoinverse or a generalized pseudoinverse of a matrix valued function F by F x. We illustrate the above definitions with the following example. E x a m p l e 2.1 0, 0 < r

Let X = Y = C, and choose a standard basis for C. Let ~r = {reir r _>

7r}. For e v e r y z E o',let F(z) map each point x E C t o y =

z2z, s o t h a t the

matrix of the linear transformation F(z) is [ z 2 ]. Since each linear transformation from C into C can be described by the matrix [ A2 ] for some A 9 a, and since matrices [ Al ] and [AI] are different if A1, A, 9 ~r and A1 # A,, the function F: a --* L(X, Y) is bijective, and hence invertible. Since the linear transformation F(0) is not invertible, the function F is not pseudoinvertible. Now the function F• a ~ L(Y, X) such that F•

Y ~ X can be

described by the matrix [z-2],

if

z#0

[1],

if

z = 0

is a generalized pseudoinverse of F. We note that if F• ~ --+ L(Y, X ) is such that for every z 9 ~r the transformation .F•

sends y to

z-2y,

if

z#0

0,

if

z = 0,

then ~-,x is another generalized pseudoinverse of F. Also, if a = {reir r > 0, 0 < r < ~r}, then the function F is pseudoinvertible. [] Suppose now X , Y are Banach spaces and denote by B L ( X , Y ) the space of bounded linear operators from X into Y. Let a be a subset of Coo. If F: ,, --* BL(X, Y) is pseudoinvertible, then, by the Open Mapping Theorem, the pseudoinverse of F takes values in BL(Y, X). The following proposition characterizes functions from cr into BL(X, Y) which have a generalized pseudoinverse with values in BL(Y, X).

Proposition

2.2 Let X, Y be Banach spaces and let F be a function from a subset ~ of Coo into BL(X, Y). Then F has a generalized pseudoinverse G: a ---, BL(Y, X ) if and only if for every z E ~ the subspaces Ker F(z) and Ran F(z) are topologically complemented.

568

Rakowski

Proof

The proposition follows from Theorem 12.9, Chapter IV, in [TL] after considering

the values of F at each point of ~. [] In the sequel we specialize X and Y to be finite dimensional Euclidean spaces over C. Then

L(X, Y) shrinks to BL(X, Y). We will also assume that we have fixed bases for X

and Y, and identify every linear transformation F(A): C "• Then functions with values in

~ C " x l with a matrix in C ' •

L(X, Y) can be identified with matrix valued functions.

Let ~r C Coo and let F: a --* C "*x" be a function. Since F(A) has a generalized inverse for every ~ E a, F has a generalized pseudoinverse. If a matrix F(A) has a left (resp. right) inverse for each A E er, we will call a generalized pseudoinverse of F a left (resp. right) psendoinverse of F and say that F is left (resp. right) pseudoinvertible. We define a Moore-Penrose pseudoinverse of F to be a generalized pseudoinverse G of F such that G(A) is a More-Penrose inverse of F()t) for each A E ~. After considering the value of F at each point of *,, one can see the following. P r o p o s i t i o n 2.3

Each matrix valuedfunction has a unique Moove-Penrosepseudoinvevse.

We note that Propositions 2.2 and 2.3 are not concerned with continuity or analyticity of a function F or its pseudoinverse. In fact, simple examples show that the MoorePenrose pseudoinverse of an analytic matrix valued function may fail to be analytic. Let A be a matrix. Among all generalized inverses of A one can distinguish those which have the same rank as A. As mentioned in the introduction, these are precisely the reflexive generalized inverses of A, that is g e n e r a ~ e d inverses of A for which A is a generalized inverse. If a function G is a generalized pseudoinverse of a function F: ~ --* C m•

such that

Rank G(A) = Rank F(A) for each A E ~r, we will call G a reflexive generalized useudoinverse of F.

It follows from Proposition 2.3 that every m~trix valued funcs

has a reflexive

generalized pseudoinverse. Clearly, a function G is a reflexive generalized pseudoinverse of a function F if and only if G is a generalized pseudoinverse of F and F is a generalized pseudoinverse of G.

3 PSEUDOINVERSES

OF RATIONAL

MATRIX

FUNCTIONS

In this section we will assume that ~r is a subset of C ~ whose complement consists of a finite number of points, and consider functions W: a ~

L(C '~• C '~xl) such that W is

analytic in a and, if ~ C C ~ \ a , W either has a pole at ~ or can be analytically continued to A. Such functions, called rational matrix functions, can be represented as matrices whose entries are scalar rational functions. We will denote by 7Z the field of scalar rational functions

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569

and by 7s '~xn the space of m • n rational matrix functions. Plainly, a function W G 7s215 has a pole at some point A G Coo if and only if some entry in the matrix representation of W has a pole at A. The function W has a zero at A if there exists a ~ E 7s215 analytic and nonvanishing at A, such that (Wqt)(A) = 0 and

w e # 0 whenever r e 7s

is analytic at ~ and r

= r

If W is ~nalytic at ~, such r

exists if and only if Rank W(A) < Rank W(zo) for some z0 in a neighborhood of A. It can be easily shown that each rational matrix function has a finite number of zeros in Coo. It is well known from systems theory that if a function W E 7sm• is analytic at infinity then there exist matrices A, B, C, D such that

W ( z ) = D + C(z - A)-XB.

(3.1)

A representation of W of the form (3.1) is called a realization of W and is usually written down as (A, B, C, D).

A realization (A, B, C, D) of W is said to be minimal if W has

no realization (A, B, 0 , D) with the size of A smaller than the size of A. The matrix A in a realization (A, B, C, D) of A represents a linear transformation (called a state space operator) acting on a finite dimensional Euclidean space (called the state space). Thus, a minimal realization of W is one having the smallest possible state space. Suppose a rational matrix function W is pseudoinvertible. Then the pseudoinverse of W is a rational matrix function. In fact, it can be found by computing the inverse of a matrix over 7s If W is analytic and does not have a zero at infinity, and (A, B, C, D) is any realization of W, the pseudoinverse W • of W can be more efficiently computed using the formula

W•

where

A •

=

A

-

= D -~ - D-~C(z - A•

(3.2)

BD-1C (see [BGK]; cf. [S]).

Below, we derive a formula for a rational matrix function which is a generalized pseudoinverse of a given function W E 7s215

We begin with the following factorization

theorem for rational matrix functions. This theorem extends Theorem 1.1 in [BGK]. We note that the hypotheses in Theorem 3.1 below are stronger then those in Theorem 1 in [C1] and Theorem 4.2 in [V] in that we require that the function W have no zero at infinity. Similarly as in [V], we factor W as W, W2 where W2(A) maps the input space of W(A) into C ~• with k equal to the normal rank of W, i.e. the rank of W(zo) at any point z0 which is not a pole or a zero of W. If O1 = (A~,B1, C1,D1) and O2 = (A2, B2, C2,D2) are realizations of rational matrix functions W1 and Wz such that the product W1W2 is defined, we will denote by O102 the realization

570

Rakowski

of the rational matrix function W = WzW2 (see [BGK] or Lemma 4.1 in [V]).

Let 0 = ( A , B , C , D ) be a realization of a function W E 7E"~• with the normal rank k, and suppose D = DID2 where D1 E C "xk has full column rank and D~ C k• has full row rank. Let XIA-X2 be a decomposition of the state space of the realization, and let

T h e o r e m 3.1

A=

[A21

A~2

'

B=

B2

'

be the matrix representations of A, B, C with respect to this decomposition. Choose a left inverse D~ L of Dx and a right inverse D~ R of D2. Let 01 = ( AI,, B1D~ R, 01, Dr),

(3.5)

e , = (A~,, B,, D?LC~, D,),

(3.6)

let

and let A • = A - BD~RD~LC. Then O = 6)10~ if and only if the following three conditions hold: (i) the row span orB1 is contained in the row span olD; (it) the column span of C2 is contained in the column span of D; (iii) A(X1) C Xl and A• R e m a r k 3.2

C X~.

The choice of a left inverse D~-L of D1 and a right inverse D~ R of D2 in

Theorem 3.1 has no significance. Indeed, the rows of D~ form a basis for the row span of D and the columns of D1 form a basis for the column span of D. Hence, if conditions (i) "and (it) are satisfied, there exist unique matrices B1 and C2 such that B1 = 131D2 and C~ = DIC2. Consequently, the matrices BID~ R and D~LC2 are uniquely determined by B1, C2, D1, D2. Hence realizations (3.5) and (3.6), and condition (iii) (cf. formula (3.9) below), are uniquely determined by B1, C2, D1, D~. R e m a r k 3.3

If in Theorem 3.1 D = /91D2 with

bl E C "•

b~ E C k•

and ~

then

.

D1 = DxS and D2 = S-1D2 for some nonsingular matrix S. So 0 = 0 1 0 : where

O, = (A~,, B,b~ -R, C,, b,) and

6~ = with D; L = S-1D~ L and ~ R

(A~2, B~,

b;Lc~,b~)

= D~RS. It follows that we can fix Dr, D2, D~"L, D~ R and find factorizations of W related to a given realization by considering different decompositions of the state space of the realization.

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571

P r o o f of T h e o r e m 3.1

Suppose $ = 0 1 0 , . Then

[

OLD,)

BID-aD2

and conditions (i) and (ii) hold. Also, since A,, = 0, A(X~) C

=

So A•

C

Xl. Now

~

-B,D~RD~LCI

A,, ~ B , D ~ a D ~ L C ,

]"

X,.

Conversely, suppose conditions (i)-(iii) hold and

let B1, C, be as in Remark 3.2.

Then B1 = i~lD2

= B1D2D~RD2 = B1D~RD2 and c , = D10,

= D1D~LD10, = D1D~LC,. Since

A(X~) C Xl,

A,~ = 0. Since A•

C X, and

[ A n - B1D;RD~C1

A•

= [

-B,D~RD~LC1

AI, - B,D;RD~LC,] An B,D~"D~LC, J '

(3.9)

A1, = B1D[RD~LC,. Thus, the equality (3.7) holds and O = O15,. [] We note that if the matrix D is square and nonsingnlar, then conditions (i) and (ii) in Theorem 3.1 become trivial. In this case Theorem 3.1 reduces to Theorem 1.1 in [BGK]. If W 6 T~"*•

is a function without a pole or zero at infinity, a realization

(A, B, C, D) of W whose state space admits a decomposition XI+X, such that conditions (i)-(iii) in Theorem 3.1 are satisfied for some matrices D1, D,., D~L, D~n will be said to be factorable. We emphasize that Theorem 3.1 does not describe all factorizations of a function W 6 7s "~•

but only factorizations which can be obtained from a given realization of W.

In particular, if a realization (A, B, C, D) of W is minimal, Theorem 3.1 characterizes all minimal factorizations of W, that is factorizations in which no zero-pole cancellation occurs (see Theorem 4.8 in [BGK] for the discussion of the regular case). We illustrate this remark with an example (cf. Example 3.1 in [BGKV]).

572

Rakowski

E x a m p l e 3.4

Let W(z) =

Then

[l+ljz 0 0

1 + 1/z 1

i]

9

is a minimal reali~.ation of W. Suppose W = W1W2 where W1 E 7~.3x2 and the dimensions of the state spaces in some realizations 01 and 07 of W1 and W2, respectively, add up to 2. If the state space of e l has dimension 2 (that is, W2 E C3•

then, by Theorem 3.1,

the row span of B is contained in the row span of D, a contradiction. If the state space of 01 has dimension 0, the column span of C is contained in the column span of D, a contradiction. Suppose the dimension of the state space of O1 is 1. The only subspace of C 2•

of dimension 1 which is invariant under A is X1 = span {[ 1

subspace of C 2•

0IT}. If X2 is any

complementary to )(1, there exists a constant a such that B partitions

with respect to the decomposition Xld-X~ of the state space as

[~ a ~

]

Since span B1 is not contained in the row span of D, this contradicts Theorem 3.1. Now we may take a non-minimal realization of W

([i

0 0

and put X1 = span Then AX1 C Xx. If

[i0 [i0i] [i 0 i]) {[i] [i]} {[i]}

0 -1

,

1 0

0 -1

,

,

,

0

,

Da =

,

,

(3.10)

1

X~ = span

.

D2 =

and D~-z' =

~

then A• =

,

D~-R =

[_11!]

,

0 -1 1 0 and A x x 2 C X2. Thus, the realization (3.10) of W is factorable. By Theorem 3.1, W =

W1W2 where

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573

and

[]

The realization (3.10) of the function W in Example 3.4 is factorable. This represents, in a sense, a typical situation.

Proposition 3.5

E v e r y nonzero rational m a t r i x f u n c t i o n without a pole or zero at infinity

has a factorable realization.

Proof

Let W E ~,,tx. be a nonzero function without a pole or zero at infinity, and let

k be the normal rank of W. By Theorem 3.1 in [CR], we can factor W as A _ E A + where A_ E TC"xa is analytic in a neighborhood of infinity, and has a left pseudoinverse in R kxm which is analytic in a neighborhood of infinity. Let W1 = A_ and let W2 = E A + . Then W

= W 1 W 2 and W1, W~ have neither poles nor zeros at infinity.

Choose a realization

(Ai, Bi, Ci, Di) for Wi (i = 1, 2), and let | be as in (3.3). Then O is a factorable realization of W.

[] We show now that if a function W E 7~~•

is analytic at infinity and W(oo) is

left invertible, a left pseudoinverse of W can be easily obtained from a left inverse of Proposition

3.6

Let (A, B , C, D ) be a realization of a f u n c t i o n W E 7Zm•

D - L is a left inverse o f the m a t r i x D . A • = A - BD-ZC

Proof

and suppose

= D -L - D-ZC(z - A•

- z with

is a left pseudoinverse of W .

Let~=C\(*r(A) W•

Then W •

W(oo).

Ua(ax)).

Then, f o r e a c h z G ~ r ,

= D-LD + D-LC(z - A)-IB _ D-r,C(z _ Ax)-tBD-Z _ n-LV(z

- Ax)-~BD-ZC(z

D

_ A)-IB

= I+ D-LC(z - A)-IB - D-~C(z - A• - D-LC(z - A• = I+ D-LC(z-

_ A)-IB

AX)-l(z- A • - z + A)(z-

- D-LC(z - A•

A)-XB

_ A)-IB

=I.

[]

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Rakowski

If W and W x are as in Proposition 3.6 then, clearly, W is a fight pseudoinverse of W • Thus, W • is a reflexive generalized pseudoinverse of W. A computation similar to that above proves the following analogue of Proposition 3.6.

Proposition 3.7 Let (A, B, C, D) be a realization of a function W E 7~"~• and suppose D - s is a right inverse of the matriz D. Then W•

= D-s-

D-SC(z- A •

with

A • = A - B D - R C is a right pseudoinverse of W . We shall need below the following characterization of reflexive generalized inverses of a matrix.

Lemma 3.8 Let D 6 C "x'~ be a matriz of rank k, and suppose D = DxDr with D1 e C 'nxk and Dr 6 C k•

Then D t is a reflezive generalized inverse o l D if and only if D ~ = D~SD -L

for some left inverse D~ ~ of D1 and some right inverse D~ s of D~. Proof

Choose a left inverse D~-~ of D1 and a fight inverse D~ s of Dr. Then

D I D ~ D [ a D ~ L DID2 -- D~D2 and D~nD-J'D1D2D~RD? L = D [ R D J ", so D* = D ~ a D J " is a reflexive generalized inverse of D. Conversely, suppose D* is a reflexive generalized inverse of D. Then D, D2D*D1Dr = D1D~ and D~D* D1 = Ik where Ik is the k • k identity matrix. Since Rank D* = k, D* = / ) 2 b l

for some/)2 E C "•

and D1 6 C kxl. Then S = D2/)2 is anonsingular k • k matrix and D~-R = D2S -1 is a right inverse of D2. Also, /)1D1 = S -1 and D1 n = S/)I is a left inverse of Dr. Plainly, D* = D~RD~ r. [] We give now a formula for a generalized pseudoinverse of a rationM matrix function W in terms of a factorable realization of W.

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575

L e m m a 3.9 Let W 6 7~" •

be a function analytic and without a zero at infinity and let

(A, B, C, D) be a factorable realization of W . Then

WX(z) = D ~ _ D t C ( z -

AX)-aBD ~

with A • = A - B D $ C is a reflexive generalized pseudoinverse of the function W whenever D ~ is a reflexive generalized inverse of the matrix D.

Proof

Let k be the normal rank of W .

Choose D I E

C '~xk and D2 G C h• such that

D = DID2 and let D: be a reflexive generalized inverse of D. By L e m m a 3.8, there exist a right inverse D ~ R of D2 and a left inverse D~ -L of DI such that D ~ = D~IZDI r'. Let Xa-]-X2 be a decomposition of the state space of the realization (A, B, C, D) for which conditions (i)(iii) in Theorem 3.1 cma be satisfied, and partition A, B, C with respect to the decomposition X 1 4 X 2 as in (3.4). In view of Remark 3.3, W = W 1 W , where W1 and W, have realizations

as in (3.5) and (3.6). By Propositions 3.6 and 3.7,

W?(z) =

-

(z - ( A , 1 -

D;"D;

is a left pseudoinverse of W1 and

W:(z) =

(.4,,-

D; "

is a fight pseudoinverse of W2. A straightforward computation shows that -1

0

In view of (3.8), it follows that W x = W ~ W ~ . Now it can be easily verified that W • is a generalized pseudoinverse of W and W is a generalized pseudoinverse of W x. [] Lemma 3.9 yields the following result. T h e o r e m 3.10 Let W 6 7T "• be a function analytic and without a zero at infinity and let (A, B, G, D) be a realization of W . Then W•

= o~ - D~C(z - A•

with A • = A - B D t C is a reflexive generalized pseudoinverse of the function W whenever D t is a reflexive generalized inverse of the matriz D.

576

Proof

Rakowski

We may assume W ~ 0. Since Theorem 3.2 in [BGK] holds in the nonregular case,

there exists a nonsingular matrix S such that W ( z ) = (SAS -~, SB, C S - a , D)

=D+[O

Co C2] z -

Ao

Ass|]

o

A2 J /

= D + 6'o(z - A o ) - l B o

with the realization (Ao,/3o, Co, D) minimal By Proposition 3.5, there exists a factorable realization (A, B, C, D) of W. In view of Theorem 3.2 in [BGK], we may assume

=

;~o o

4-/, A2 J

,[o

~'0

0,],D

(3.11)

with (Ao,/3o, Co, D) a minimal reaUzation of W. Let O ~ be a reflexive generalized inverse of the matrix D. By Lemma 3.9, H(z) = D ~ - D~0(z - ; ~ • with .~l• = .4 - B D t C ' is a reflexive generalized pseudoinverse of W. By (3.11),

H(z) = D~ - m 6 o ( z - ~g)-~boD~ where .4~ = Ji0 - ~toDt0o. Since any two minimal realizations of a rational matrix function are similar, there exists a nonsingular matrix T such that ( ~4o, Bo, 0o, D) = ( T A o T - ' , T Bo, C O T - ' , D).

Hence H(z) = D ~ - D)CoT-I(z - T(Ao - BoDtCo)T-')-ITBoD t = D ~ - D)Co(z - (Ao - B o D ) C o ) ) - I B o D ~ = D ~ _ DtC(z - A• = W•

and the proof is complete. ra

4 F U N C T I O N S W I T H P O L E S O R Z E R O S AT I N F I N I T Y In this section we derive a formula for a generalized pseudoinverse of an arbitrary rational matrix function, possibly with a pole or zero at infinity. A function W with a pole at infinity can be represented as W ( z ) = D + C c ( z - A c ) - ~ B c + C ~ ( z -x - A ~ , ) - ' B ~ ,

(4.1)

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577

where A~. is a nilpotent matrix (see Theorem 2.1 in [BGK] for the regular case and Theorem 1.7 in [G2] for a general case). A representation (4.1) of W generalizes that in (3.1) and is also called a realization of W. If a 6 C is not a pole or a zero of a function W 6 TC'~• W can be also represented (see [GK]) as W(z) = D~, + (z - a)C(zG - A)-IB. (4.2) A representation (4.2) of W is called a realization centered at c~, or a centered realization. We will require that matrices A and G in a realization (4.2) be such that the matrix aG - A is invertible. One can obtain the realization (4.2) of W from the realization (4.1) by choosing an appropriate a and rewriting (4.1) (see [K]) as W(z) =D + Cc(a - A o ) - I B c + aC,o(I - aAr

C'](Z[l A-]-[Ac '])

-(z -

(4.3)

On the other hand if W has a realization (4.2) then, by Theorem 3 (Chapter XlI) in [G] (see also Theorem 5.1 in [GKL]), there exist nonsingular matrices T and S such that T ( z G - A)S = [ z - Ac

zAoo - I]

with Aoo a nilpotent matrix. For such T and S, -1

W(z) = Dr + (z - a)CS [ z - Ac

zAoo - I

TB.

(4.4)

If matrices A and G are such that the matrix aG - A is invertible then, comparing (4.3) and (4.4), one can find matrices Gc, Ca., Bo, B~, D such that (4.1) holds. Thus, representations (4.1) and (4.2) of W are equivalent. For convenience we will work in this section with centered realizations. Also, we will denote the realization D + (z - a)C(zG - A ) - I B of a rational matrix function W by (G, A, B, C, D, a). The following proposition shows the multiplication rule for rational matrix functions represented by centered realizations. Proposition 4.1 If 01 = (G1,At, B1,C1, D~,a) and 02 = (G,., A,.,B,, C2, D,, a), are realizations of rational matrix functions W 1 and W2 such that the product WI W2 is defined, then 010,=([G1-BIC,.] 0 G,

J'

[A1-aB1C,] [BID,] 0 A, J' B, j , [ C 1

D1C,],D1D,,a

)

(4.5)

578

Rakowski

is a realization o f the function W = Wx W~.

Proof

We have

Wl(z)W,(z) =DID,

+ (z - a ) ( D 1 C , ( z G ,

- A,)-'B,

+ CI(zG1 - A1) - 1 B 1 D , )

+ (z - a)'e,(zg~ - A,)-*B,C,(zG,

- A,)-*B,

= D,D,+(z-a)[C, D,C,] [(za~- A~)-~ ( z - a ) ( z a ~ - &)-~B~C,(za,- A,) -~1 o

(za,-A,)

-1

[ B,D,] t B,

]

--DID,+(z-a)[C, D~C2][ZG*oAI-(:~a)B,C,_A,]J'-' [BID,] B, Hence the right-hand side of (4.5) is a realization of W. El We give now a formula for a reflexive generalized pseudoinverse of an arbitrary rational matrix function. T h e o r e m 4.2

Let (G, A, B , C, D, a) be a realization of a function W 9 T~"•

Choose a

reflexive generalized inverse D $ o f D, and let G x = G + BD$C,

A • = A + aBD~C.

Then

WX(z) = D ~ - (z - a ) D * C ( z G x - A x ) - I B D ~ is a reflexive generalized pseudoinverse o f the function W .

Proof

We have

W X ( z ) W ( z ) D ~ = D$[DD: + (z - a ) C ( z G - A ) - ~ B D ~] - (z - a ) D $ C ( z G x - A x ) - I B D * C [ D D * + (z - a ) C ( z G - A ) - I B D ~] = D* + (z - a)D$[(zG - A ) -~ - ( z G x - AX) -1 - (z - a ) ( z G • - A x ) - i B D ~ C ( z G = D* + ( z -

- A)-a]BD ~

a ) D ' e ( z G x - AX) -1

[zG x - A x - z G + A - (z - a ) B D * C ] ( z G - A ) - ~ B D ~ -_ D ~.

Hence W x w w

x = W x. Since the normal ranks of W and W x coincide, W is a reflexive

generalized pseudoinverse of W • So W • is a reflexive generalized pseudoinverse of W.

[]

579

Rakowski

The generalized pseudoinverses W • of a rational matrix function W which have been indicated in Theorems 3.10 and 4.2 have a special property. Namely, their defect is equal to 0, that is, their left and fight annihilators are spanned (over T~) by constant matrix polynomials. This follows from the fact that the rank of Dt is equal to the normal rank of W • and L W •

- 0 and W X ( z ) R = 0 whenever L D ~ = 0 and D t R = O. In fact, a more

general statement is true. We prove it for centered realizations. The same proof is valid for functions without a pole or zero at infinity and state space realizations. T h e o r e m 4.3

Let (G, A, B , C, D, a) be a realization of a function W E ~"~•

function W • E 7~n•

Then the

with no defect and without a pole or zero at a is a refleeive generalized

pseudoinverse of the function W i f and only i f

W•

= D ~ - (z - a ) D $ C ( z G • - A •

(4.6)

*,

where D ~ is a generalized inverse o l D and G • = G + B D $ C , A • = A + a B D * C .

Proof

Suppose first that W • is as in (4.6). By the remark preceding the theorem, W •

has no defect. Since the matrix a G - A is invertible, the function ( z G • - A •

is analytic

at (x. Hence W x is analytic at ~ and Rank W •

= max (Rank W•

z E Coo, W • is analytic at z}.

So W • has neither a pole nor a zero at a. Conversely, suppose that W x E ~'*•

is a reflexive generalized inverse of W with

no defect and without a pole or zero at a. Then D* = W x (ct) is a reflexive generalized inverse of D. Let H x be the function represented by the fight hand side of (4.6). Since W • and H • have no defect, it follows that Ran WX(~) = Ran H• Ker W•

= Ker H•

= Ran D: and

= Ker D* for each ;~ E Co, which is not a pole or zero of W x or

H • . So, by the properties of generalized inverses (see introduction), W • (,~) = H • (,~)for all but a finite number of points )t. Hence W x = H x .

[] Theorem 4.3 has the following corollary. If W E Ts"*•

we denote by g(W) the

McMillan degree of W, i.e. the smallest possible dimension of the domain of the operator A in a realization (G, A, B, C, D, a) of W. C o r o l l a r y 4.4 Every rational matriz function W has a generalized pseudoinverse with no defect. Moreover, i f W x is a reflexive generalized pseudoinverse o f W and W • has no defect, then 6(W • < 6(W).

580

Rakowski

5 SOME APPLICATIONS We discuss briefly some applications of generalized pseudoinverses of rational matrix functions. One is connected with locating zeros of a rational matrix function and bounding their multiplicities. Let W 6 7s mx" and let q be a scalar polynomial such that P = q W is a matrix polynomial. Let b be the Smith's normal form of P (see [M]). Then there exist unimodular matrix polynomials E and F such that P = E D F . So W = E D F , where D = (l/q)/). The factorization E D F is called a Smith-McMillan fnctorizktion of W. In this factorization, the poles and zeros of the nonzero diagonal entries of D are poles and zeros in the finite plane of the function W. The matrix polynomial D obtained in this construction is unique. It is called the Smith-McMillan form of W. The orders to which nonzero diagonal entries of D vanish at A ~re called partial multiplicities of the zero of W at A. The sum of partial multiplicities of the zero of W at A is called the (total) multiplicity of the zero of W at A. Similarly, the orders of poles at A of nonzero diagonal entries of the Smith-McMillan form of W are called the partial multiplicities of the pole of W at A. The sum of partial multiplicities of the pole of W at A is called the (total) multiplicity of the pole of W at A. Partial and total multiplicities of the pole and zero of W at infinity are defined to be the partial and total multiplicities of the pole a n d zero at 0 of a rational matrix function H ( z ) =_ W ( z - 1 ) . A more detailed discussion of the zero and pole structure of a rational matrix function can be found in [BR1] and [BR2].

Theorem 5.1 Let the function W x 6 T~"• W 6 7Z"~•

be a generalized pseudoinverse of" a function

Then each zero of the function W is a pole of the function W • . Moreover, the

multiplicity of the zero of W at a point )~ 6 Co~ is at most equal to the multiplicity of the pole o f W • at A. Proof

Let A be a zero of the function W. We assume without loss of generality that A 6 C.

Let E D F and E • 2 1 5

• be Smith-McMillan factorizations of W and W x, respectively.

Then, for each z which is not a pole of W or W • , D(z)F(z)EX(z)D•215 Let E = F E x and let ~' = F X E .

=

D(z).

(5.1)

Let k,k be the normal ranks of W and W • and let

D1 6 T~kx~, E1 6 T~kxr*, D~ 6 T~~xs F1 6 T~~xk be functions formed from the upper left corners of D, E, D x , F. Then (5.1) implies E,I(z)D~ (z)~'l(z)Dl(z) = I

(5.2)

Rakowski

581

for each z which is not a pole of D or D x . By analytic continuation, equality (5.2) holds for all z E Coo. Suppose the last l diagonal entries of D1 vanish at A to the orders ~1, ~ , ..., ~t. Then, for i = 1, 2, ..., l, el(Z) = (z - A)-'i Da(z)ek-t+i is 'a functionanalytic and nonzero at A, and

has a pole at A of multiplicity ai. By Corollary 2.4 in [BGR], the function E1D~(~'I has a pole at A of multiplicity at least ax + a~ + . . . + at. Hence D~( has a pole at ~ of multiplicity at least

~1 + ~

+ ... + ~l.

[] In view of Theorems 3.10 and 4.2, Theorem 5.1 has the following two corollaries. C o r o l l a r y 5.2

Let (A, B, C, D) be a realization of a function W E T~" x " without a pole

or zero at infinity, and let D ~ be a reflexive generalized inverse of the matrix D. Then each zero of W is an eigenvalue of the matrix A • = A - B D $ C .

Also, the multiplicity of the zero

of W at A is bounded by the number of eigenvalues of A ~ equal to A i counted according to multiplicities. C o r o l l a r y 5.3

Let W E ~,,,•

be a function without a pole or zero at a point a E C.

Let ( G, A, B, C, D, a) be a realization of W , let D ~ be a reflexive generalized inverse of the matrix D, and let G • = G + BD$C,

A • = A + ~BD$C.

I f W has a zero at a point A E Coo, then the matrix

{

AG • - A •

if A E C

Gx,

if A = r162

is singular. Another application of generalized pseudoinverses of matrix valued functions is connected with factorization theory. P r o p o s i t i o n 5.4 Let W, Q, H be matrix valued functions defined on a subset tr of Coo, and let Q• and H x be generalized pseudoinverses of Q and H. Then the equation QXH = W

(5.3)

582

Rakowski

has a solution if and only if

(5.4)

QQXWH• H = W. Moreover, if equality (5.4.) holds, X is a solution of (5.3) if and only if X = Q•

• + Y-

Q•

•

for some matrix valued function Y. Proposition 5.4 remains valid when all functions are required to be rational. Theorem

5.5 Let W, Q, H be rational matrix functions. Let rational matrix functions Q•

and H • be generalized pseudoinverses of the functions Q and H. Then equation QXH = W

(5.5)

can be solved by a rational matrix function if and only if QQ•215

= W.

(5.6)

Moreover, if equality (5.6) holds, a rational matrix function R is a solution of equation (5.5) if and only if R = Q•

• + Y-

Q•

•

for some rational matrix function Y. Proposition 5.4 and Theorem 5.5 can be proved in the same way as Theorem 2 in [P]; we omit the details of the proofs. Theorem 5.5 can be used together with Theorems 3.10 and 4.2 to study factorization problems for rational matrix functions. For instance, one way to approach the causal factorization problem in [CP] of determining whether the equation

W(z) = Z ( z ) H ( z ) ,

(5.7)

W and H rational matrix functions, can be solved by a rational matrix function analytic at infinity is as follows. Using Theorem 3.10 (or 4.2), find a generalized psendoinverse H • of H. By Theorem 5.5, equation (5.7) has a solution if and only if

WH•

= W.

(5.8)

If equality (5.8) is satisfied, the problem of finding a causal solution is equivalent to finding a rational matrix function Y such that the function

WH • + Y(I-

HH •

Rakowski

583

is analytic at infinity. A generalized pseudoinverse of a rational matrix function H can be used to find a left annihilator of a function H E 7"C~• that is the set H ~ = { r e 7~1•

4 , / / = 0}.

Indeed, ~b E H ~ if and only if ~bsolves equation (5.5) with Q = 1 and W an 1 • n zero matrix function. By Theorem 5.5, this happens if and only if = ~b(I - H H • ),

where H x is a generalized pseudoinverse of the function H and ~b E 7~lxm is arbitrary. Thus, H ~ coincides with the row span (over 7"s of a rational matrix function I - H(z)H• The right annihilator of H can be found similarly. Another application of generalized pseudoinverses is connected with Wiener-Hopf factorization of rectangular matrix functions (see [CR] and, for a comprehensive treatment of the regular case, [CG]). A sufficient condition for existence of a canonical Wiener-Hopf factorization of a function W G 7T~• in terms of a realization of W can be easily obtained from Theorem 3.1. ACKNOWLEDGEMENTS I am grateful to Professor Kevin Glancey for discussions which initiated the research and shaped the results summarized in this paper. In fact, many of the results here are an offshoot of factorization theorems in [CR]. I wish to thank Professor Joseph A. Ball for his observations and comments. I wish to thank Professor Kenneth B. Hannsgen for his suggestions on improvement of the exposition. REFERENCES

[BG] A. Ben-Israel and T. N. E. Greville, Generalized Inverses: Theory and Applications, John Wiley & Sons, New York London Sydney Toronto, 1974. [BGK] H. Bart, I. Gohber$ and M. A. Kaashoek, Minimal Factorization of Matri~ and Operator Functions, Operator Theory: Advances and Applications, vol. 1, Birkh~user Verlag, Basel-Boston-Stuttgart, 1979. [BGKV] H. Bart, I. Gohberg, M. A. Kaashoek and P. Van Dooren, Factorization of Transfer Functions, S I A M J. Control and Optimization, vol. 18, no. 6, pp. 675-696, November 1980. [BGR] J. A. Ball, I. Gohberg and L. Rodman, Realization and Interpolation of Rational Matrix Functions, in Topics in Interpolation Theory of Rational Matrix Functions (ed. I. Gohberg), Operator Theory: Advances and Applications, vol. 33, Birkhs Verlag, Basel Boston Berfin, pp. 1-72, 1988.

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[BR1] J. A. Ball and M. Rakowski, Zero-Pole Structure of Nonregular Rational Matrix Functions, to appear.

[BR2] J. A. Ball and M. Rakowski, Null-Pole Subspaces of Nonregular Rational Matrix Functions, submitted for publication. [C1] N. Cohen, On Minimal Factorization of Rational Matrix Functions, Integral Equations and Operator Theory, vol. 6, pp. 647-671, 1983. [C2] N. Cohen, On Spectral Analysis and Factorization of Rational Matrix Functions, PhD Dissertation, Weizmann Institute of Science, Rehovot, Israel, August 1984.

[CG]

K. Clancey and I. Gohberg, Factorization of Matrix Functions and Singular Integral Operators, Operator Theory: Advances and Applications, vol. 3, Birkhs Verlag, Basel-Boston-Stuttgart, 1981.

[CP] G. Conte and A. M. Perdon, On the Causal Factorization Problem, IEEE Transactions on Automatic Control, vol. AC-30, no. 8, pp. 811-813, August 1985. [CR] K. Clancey and M. Rakowski, Factorization of Rectangular Matrix Functions Relative to a Contour, unpublished note.

[DW]

C. A. Desoer and B. H. Whalen, A Note on Pseudoinverses, 3". Soc. Indust. Appi. Math., vol. 11, no. 2, pp. 442-447, June 1963.

[G] F. R. Gantmacher, The Theory of Matrices, vol. 2, Chelsea Publishing Company, New York, 1971.

[GK] I.

Gohberg and M. A. Kaashoek, Regular Rational Matrix Functions with Prescribed Pole and Zero Structure, in Topics in Interpolation Theory of Rational Matrix Functions (ed. I. Gohberg), Operator Theory: Advances and Applications, vol. 33, Birkhs Verlag, Basel Boston Berlin, pp. 109-122, 1988.

[GKL] I. Gohberg, M. A. Kaashoek and P. Lancaster, Theory of Regular Matrix Polyno-

mials and Band Toeplltz Operators, Integral Equations and Operator Theory, vol. 11, no. 6, pp. 776-882, 1988.

[K] J.

K. Kang, Interpolation by Rational Matrix Functions with Minimal McMillan degree, PhD Dissertation, Blacksburg, Virginia, April 1990.

[M] C. C. MacDuffee, The Theory of Matrices, reprint by Chelsea Publishing Company, New York 1946. [P] R. Penrose, A Generalized Inverse for Matrices, Proc. Cambridge Philos. Soc., vol. 51, pp. 406-413, 1955.

[R]

C. A. Rohde, Some Results on Generalized Inverses, SIAM Review, vol. 8, no. 2, pp. 201-205, April 1966.

IS] L. M. Silverman, Inversion of Multivariable Linear Systems, IEEE Transactions

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on Automatic Control, vol. AC-14, no. 3, pp. 270-276, June 1969. [TL] A. E. Taylor and D. C. Lay, Introduction to Functional Analysis, Second Edition, John Wiley & Sons, New York Chichester Brisbane Toronto, 1980. [V] P. Van Dooren, Factorization of a Rational Matrix: the Singular Case, Integral Equations and Operator Theory, vol. 7, pp. 704-741, 1984. Department of Mathematics Virginia Tech Blacksburg, VA 24061 USA

Submitted:

July I0, 1990