More generally, it turns out that when the rank of P equals the minimal number of states of an operator model that computes P, then one can prove a duality relation between operators and factorizations of P .
one can show that, for any rank factorization P = QR, the operators given byᾱ 1 = Q + p 1 ,ᾱ ∞ = p ∞ R + , andĀ a = Q + P a R + , yield an operator model for P. A key fact in proving this result is that the function P is invariant to the basis chosen to represent operator matrices.