**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.

Wed Jul 3 10:27:42 PDT 1996