last modified Sep 11, 1996

The Bart-Gohberg-Kaashoek-Van Dooren Theorem

Here we derive a theorem due to Bart, Gohberg, Kaashoek and van Dooren. The reader can skip the statement of this theorem if he wishes and go directly to the algebraic problem statement.


Definition A factorization
of a system [A, B, C, 1] is minimal if the statespace dimension of [A, B, C, 1] is d + d.

Theorem([BGKvD]) Minimal factorizations of a system [A, B, C, 1] correspond to projections P and P satisfying P + P = 1,

A P = P A P and (A - B C) P = P (A - BC) P.

We begin by giving the algebraic statement of the problem. Suppose that these factors exist. By the Youla-Tissi statespace isomorphism theorem, there is map

(m, m):       Statespace of the product Statespace of the original

which intertwines the original and the product system. Also minimality of the factoring is equivalent to the existence of a two-sided inverse (n, n) to (m, m). These requirements combine to imply that each of the expressions of (FAC) below is zero.

The problem

Minimal factors exist if and only if there exist m , m , n , n , a, b, c, e, f and g such that the following polynomials are zero.
A m - m a - m f c A m - m e
B - m b - m f -c + C m
(FAC) n m - 1 n m - 1
n m n m
-g + C m m n + m n - 1
Each of these expressions must equal 0. Here A, B and C are known. The problem is to solve this system of equations. That is, we want a constructive theorem which says when and how it can be solved.

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