\(J\)-unitary preserving automorphisms of rational matrix functions: State space theory, interpolation, and factorization
type: Item
instance of: scholarly article
title: \(J\)-unitary preserving automorphisms of rational matrix functions: State space theory, interpolation, and factorization
zbMATH Open document ID: 0794.15012
DOI: 10.1016/0024-3795(94)90503-7
author: Joseph A. Ball
author: Daniel Alpay
author: I. Gohberg
author: Leiba Rodman
published in: Linear Algebra and its Applications
publication date: 1994-08-31T00:00:00Z
review text: Let \(GL_ m({\mathcal R})\) be the multiplicative group of \(C^{m\times m}\) valued rational functions of one complex variable \(z\) with determinant not vanishing identically. According to state space representations, the author analyzes general inner automorphisms of \(GL_ m({\mathcal R})\) that preserve the set of matrix functions which are unitary (with respect to an indefinite scalar product) for real, or unimodular values of \(z\). As examples, the author presents a number of generalized inner automorphisms where \(GL(J,\Delta)\) with \(\Delta=R\) or \(\Delta=T\) for various choices of \(J\) is left invariant. Moreover, the author studies the homogeneous interpolation problem for rational matrix functions belonging to the associated \(J\)-unitary group with a given automorphism \({\mathcal A}\) of one of the four forms. In the end, as a development, the author characterizes the factorizations of rational matrix functions \(W\) in the associated \(J\)-unitary group such that both \(W=W_ 1W_ 2\) and \({\mathcal A}(W)={\mathcal A}(W_ 1){\mathcal A}(W_ 2)\) are minimal \(J\)-unitary factorizations.
Mathematics Subject Classification ID: 15A54
zbMATH DE Number: 553975
zbMATH Keywords: state space theory
zbMATH Keywords: interpolation
zbMATH Keywords: factorization
zbMATH Keywords: rational matrix function
zbMATH Keywords: \(J\)-unitary preserving automorphism
zbMATH Keywords: homogeneous interpolation
zbMATH Keywords: minimal factorization
reviewed by: Fang Li
MaRDI profile type: Publication
full work available at URL: 0024-3795(94)90503-7
OpenAlex ID: W1987229570
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description: scientific article; zbMATH DE number 553975