## 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.

Original language | English |
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Pages (from-to) | 259-292 |

Number of pages | 34 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 194 |

Issue number | 1 |

DOIs | |

State | Published - 15 Aug 1995 |