Rational Matrix Functions with Coisometric Values on the Imaginary Line

Abstract

Given signature matrices J 1 and J 2, we obtain a necessary and sufficient condition for a rational matrix function W analytic at infinity to satisfy equation J 1 = W( z) J 2 W( 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 J 1 = J 2. 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 = − DJ 2 B*. We call G the hermitian matrix associated with the realization. The minimal factorizations W = W 1 W 2, where W 1 and W 2 satisfy equations J 1 = W 1( z) J 1 W 1( z)* and J 1 = W 2( z) J 2 W 2( 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 = W 1 W 2 with W 1 the multiplicative inverse of a Blaschke-Potapov factor.