# 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.

## Background

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,

 and A P = P A P (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.