Rational Matrix Functions With Coisometric Values on the Imaginary Line
Document Type
Article
Publication Date
1995
Abstract
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.
Recommended Citation
D. Alpay and M. Rakowski, Rational matrix functions with coisometric values on the imaginary line. Journal of mathematical analysis and its applications, vol. 194 (1995) pp 259-292
Peer Reviewed
1
Copyright
Elsevier
Comments
This article was originally published in Journal of Mathematical Analysis and Applications, volume 194, in 1995. DOI: 10.1006/jmaa.1995.1298