next up previous contents
Next: The problem Up: Example: The Bart-Gohberg-Kaashoek-Van Dooren Previous: Example: The Bart-Gohberg-Kaashoek-Van Dooren



Definition A factorization


of a system [A,B,C,1] is minimal if the statespace dimension of [A,B,C,1] is tex2html_wrap_inline4798 .

Theorem([BGKvD]) Minimal factorizations of a system [A,B,C,1] correspond to projections tex2html_wrap_inline4032 and tex2html_wrap_inline4034 satisfying tex2html_wrap_inline4044 ,


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


which intertwines the original and the product system. Also minimality of the factoring is equivalent to the existence of a two-sided inverse tex2html_wrap_inline4808 to tex2html_wrap_inline4810 . These requirements combine to imply that each of the expressions of (FAC) below is zero.

Wed Jul 3 10:27:42 PDT 1996