Rational Matrix Functions with Coisometric Values on the Imaginary Line

Given signature matrices J1 and J2, we obtain a necessary and sufficient condition for a rational matrix function W analytic at infinity to satisfy equation J1 = W(z)J2W(z)* on the imaginary axis. The condition is based on a Lyapunov equation involving matrices in an observable realization of W and generalizes the fact well known in the case where J1 = J2. If the condition is satisfied, for every observable realization (A, B, C, D) of W there exists a unique possibly singular hermitian matrix G such that G satisfies the Lyapunov equation and CG = −DJ2B*. We call G the hermitian matrix associated with the realization. The minimal factorizations W = W1W2, where W1 and W2 satisfy equations J1 = W1(z)J1W1(z)* and J1 = W2(z)J2W2(z)* for z on the imaginary line, can be characterized in terms of decompositions of the state-space into subspaces determined by a possibly indefinite inner product induced by G. As a corollary, we obtain a sufficient condition for existence of a minimal factorization W = W1W2 with W1 the multiplicative inverse of a Blaschke-Potapov factor.