Local inverse spectral problems for rational matrix functions
type: Item
instance of: scholarly article
title: Local inverse spectral problems for rational matrix functions
zbMATH Open document ID: 0621.47022
DOI: 10.1007/BF01195035
published in: Integral Equations and Operator Theory
publication date: 1987-01-01T00:00:00Z
review text: Given a regular rational matrix function A(z) we describe the set of vector-valued functions g(z) of the form \(g(z)=A(z)f(z)\) for a vector- valued function f(z) which is analytic at a prescribed point \(z_ 0\). In the scalar case, the solution is quite simple, involving the order of the pole or zero of A(z) at \(z_ 0\). The matrix case is complicated when \(z_ 0\) is both a pole and a zero for \(z_ 0\). In addition to a ''zero pair'' and a ''pole pair'' for A(z) at \(z_ 0\), a new invariant called the ''coupling matrix'' is needed for a complete description. The inverse problem of deciding when a ''data set'' arises from a matrix function in this way is also solved. As an application, the authors show how to obtain state space formulas for the linear fractional map which parametrizes all solutions of a Nevanlinna-Pick interpolation problem.
Mathematics Subject Classification ID: 47A56
Mathematics Subject Classification ID: 47A10
zbMATH DE Number: 4007040
zbMATH Keywords: regular rational matrix function
zbMATH Keywords: coupling matrix
zbMATH Keywords: state space formulas for the linear fractional map
zbMATH Keywords: Nevanlinna-Pick interpolation problem
author: Joseph A. Ball
author: André C. M. Ran
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01195035
OpenAlex ID: W2042292463
description: scientific article; zbMATH DE number 4007040