Definition A factorization
of a system [A,B,C,1] is minimal if the statespace dimension of [A,B,C,1] is .
Theorem([BGKvD]) Minimal factorizations of a system [A,B,C,1] correspond to projections and satisfying ,
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 to . These requirements combine to imply that each of the expressions of (FAC) below is zero.