Local minimal factorizations of rational matrix functions ... - Springer Link

Integr Equat Oper Th Vol. 16 (1993)

0378-620X/93/010098-3351.50+0.20/0 (c) 1993 Birkh~user Verlag, Basel

L O C A L M I N I M A L F A C T O R I Z A T I O N S OF R A T I O N A L M A T R I X F U N C T I O N S IN T E R M S OF NULL AND P O L E DATA: F O R M U L A S F O R FACTORS

M.A. Kaashoek, A.C.M. Ran* and L. Rodman** It is known that local minimal factorizations of a rational matrix function can be described in terms of local null and pole data (expressed in the form of left null-pole triples and their corestrictions) of this function. In this paper we give formulas for the factors in a local minimal factorization that corresponds to a given corestriction of the left null-pole triple. 1. I N T R O D U C T I O N Let W(A) be a given rational n x n matrix-valued function (more precisely, W(A) is a matrix whose entries are rational functions of the complex variable A with complex coefficients). It will be assumed throughout that W(A) is regular, i.e., det W(A) is not identically zero. This paper concerns factorizations

w ( ~ ) = Wl(~)w2(A),

(1.1)

where W1 (A) and W2(A) are (regular) rational matrix functions. In particular, we are interested in factorizations (1.1) that are minimal on a set a C C U { ~ } . The latter means that there is no pole-zero cancellation between the factors at the points in a (a precise definition will be given at the end of the introduction). The importance of minimal factorizations was first recognized in system theory and electrical engineering; such factorizations have been extensively studied recently (see, e.g., the monographs [BGI(], [GLR2] and [BGR2]). In the theory of factorizations of rational matrix functions and its various applications, the class of factorlzations that are minimal in the whole C U {ee} is often * The first version of this paper was written while the second author visited the College of William and Mary. ** Partially supported by the NSF grant DMS-8802836 and by the Binational United States-Israel Foundation grant.

Kaashoek,

Ran and Rodman

99

not adequate. For example, Wiener-Hopf factorizations [GKR3] or factorizations of matrix polynomials with polynomial factors axe in general not minimal on C U {oo}. To cover the latter examples, one is naturally led to a more general class of factorizations (the next best case) - minimal everywhere with the possible exception of one point and having minimal degree of non-minimality in the exceptional point (in a sense that will be made precise later on). In this paper we derive formulas for the factor W2 and its inverse for this class of factorizations, by taking as a starting point the formulas for WI obtained in [GKR3]. The formulas for 1472 and W2 --I which we derive will have the form of minimal state space realizations, and can be regarded as a far reaching generalization of the formulas given in [BGK] for the factorizations of W which are minimal everywhere. The obtained formulas are potentially applicable in many factorization problems, in particular when there is a strong connection between the factors W1 and W2 (for example, by the way of symmetry). One such application appears in the study of symmetric factorizations of symmetric rational matrix functions [RR]. As a starting point, we provide formulas for both factors W1 and W2 and their inverses in factorizations (1.1) in a more general situation where the only requirement is minimality on a given set ~. We also consider a related but somewhat different problem concerning factorizations of matrix polynomials and formulas for the factors and their inverses. In our proofs, the points of departure are the descriptions of minimal complements of rational matrix functions obtained in [GKR3] and the local minimal factorization theory of rational matrix functions developed in [BGR1]. Sections 2 and 4 are of preliminary character. \u recall there the necessary background concerning null-pole triples and minimal complements (following essentially [BGR1] and [GKR3]). The main results are stated and proved in Sections 3 and 5 assuming that the rational matrix function at hand is analytic and invertible at infinity. The polynomial problem is studied in Section 6. We conclude this introduction with some terminology and notation.

We

denote by a(S) the spectrum (= the set of eigenvalues) of a matrix or linear operator S. The scalar matrix AI

(A 6 C) will be often abbreviated to A. The algebraic direct sum

of two subspaces M and N is denoted

MSN. By Im K = {Yz I z E C p} we denote the

raxlge of a q x p matrix Y. Given a nonempty subset of the complex plane we write 7~n(a) for the set of rational n x 1 vector-valued functions with poles off c~.

i00

Kaashoek,

R a n and R o d m a n

Consider a (regular) rational m a t r i x function W(A), and its Laurent series oo

w(A) = ~

wj(~ - ~0/,

A0 E C,

j=k

where k _< 0 a n d Wj depends on A0. The Ioced pole multipIicity 8(W; A0) of W(A) at A0 is defined as follows

8(W;Ao) = 0

5(W; A0) = rank

if k = O,

wk

o

...

o

Wk+l

Wk :

...

0:

W-2

...

Wk

\W-1

ilk