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.