last modified Sep 11, 1996
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. |
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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.
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 |