A bound for the condition of a hyperbolic eigenvector matrix
type: Item
instance of: scholarly article
title: A bound for the condition of a hyperbolic eigenvector matrix
zbMATH Open document ID: 0944.15002
DOI: 10.1016/S0024-3795(98)10243-4
author: Ivan Slapničar
author: Krešimir Veselić
published in: Linear Algebra and its Applications
publication date: 2000-09-18T00:00:00Z
review text: Let \(H\) be a Hermitian positive definite matrix and \(J= \text{diag}(\pm 1)\). The starting point of the paper is the hyperbolic eigenvalue problem \(Hx=\lambda Jx\). It is known that there is always a matrix \(X\) such that \(X^*HX =\Lambda\), and \(X^*JX=J\), where \(\Lambda\) is a diagonal positive definite matrix. The matrix \(X\) is called a hyper-exchange matrix with respect to the signature matrix \(J\) and occurs in a variety of linear algebra problems. If \(\|\cdot\|\) denotes the spectral matrix norm, the spectral condition \(\kappa(A)\) of a nonsingular matrix \(A\) is given by \(\kappa (A)= \|A\|\cdot\|A^{-1}\|\). It is known that all the matrices \(X\) above have the same condition \(\kappa(X)= \|X\|^2\). A main result of the present paper is that \(\kappa(X) \leq\sqrt {\min \kappa(D^*HD)}\), where the minimum is over all nonsingular matrices \(D\) that commute with \(J\). It is also shown for which matrices \(D\) the minimum is attained and for which matrices \(H\) the bound itself is attained. Some applications are given to other related systems.
reviewed by: Stanley E. Payne
Mathematics Subject Classification ID: 15A12
Mathematics Subject Classification ID: 15A42
zbMATH DE Number: 1331377
zbMATH Keywords: perturbation of eigensolution
zbMATH Keywords: Hermitian positive definite matrix
zbMATH Keywords: hyperbolic eigenvalue problem
zbMATH Keywords: hyper-exchange matrix
zbMATH Keywords: spectral condition
MaRDI profile type: Publication
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Wikidata QID: Q127333055
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description: scientific article; zbMATH DE number 1331377