Relative perturbation results for eigenvalues and eigenvectors of diagonalisable matrices
type: Item
instance of: scholarly article
title: Relative perturbation results for eigenvalues and eigenvectors of diagonalisable matrices
zbMATH Open document ID: 0913.15012
DOI: 10.1007/BF02510256
published in: BIT
publication date: 1999-04-13T00:00:00Z
review text: The authors discuss the problem of finding error bounds in a perturbed eigenpair \((\widehat\lambda,\widehat x)\) of diagonalizable matrices. An absolute perturbation bound, an extension of \textit{C. Davis} and \textit{W. M. Kahan}'s \(\sin \theta\) theorem for Hermitian matrices [Bull. Am. Math. Soc. 75, 863-868 (1969; Zbl 0175.43204)], and two relative perturbation bounds assuming that \((\widehat\lambda,\widehat x)\) is an exact eigenpair of a perturbed matrix \(D_1AD_2\) not necessarily diagonalizable, where \(D_1\) and \(D_2\) are nonsingular, are derived. A bound on the relative error in \(\widehat\lambda\) and a \(\sin\theta\) theorem based o a relative eigenvalue separation are presented. The perturbation bounds contain both the deviation of \(D_1\) and \(D_2\) from similarity and the deviation of \(D_2\) from identity.
Mathematics Subject Classification ID: 15A42
Mathematics Subject Classification ID: 65F15
zbMATH DE Number: 1228551
zbMATH Keywords: eigenvalues
zbMATH Keywords: eigenvectors
zbMATH Keywords: error bounds
zbMATH Keywords: diagonalizable matrices
zbMATH Keywords: Hermitian matrices
zbMATH Keywords: perturbation bounds
Wikidata QID: Q56335055
author: Stanley C. Eisenstat
author: Ilse C. F. Ipsen
reviewed by: Prem Narain
MaRDI profile type: Publication
cites work: Norms and exclusion theorems
cites work: Perturbation of spectral subspaces and solution of linear operator equations
cites work: Q5751854
cites work: Some new bounds on perturbation of subspaces
cites work: The Rotation of Eigenvectors by a Perturbation. III
cites work: Q4327451
cites work: Relative Perturbation Techniques for Singular Value Problems
cites work: Q3359644
cites work: Q5661059
cites work: Perturbation bounds for means of eigenvalues and invariant subspaces
cites work: Error Bounds for Approximate Invariant Subspaces of Closed Linear Operators
cites work: Error and Perturbation Bounds for Subspaces Associated with Certain Eigenvalue Problems
cites work: Computing Invariant Subspaces of a General Matrix when the Eigensystem is Poorly Conditioned
cites work: Floating-point perturbations of Hermitian matrices
cites work: Q5674306
full work available at URL: https://doi.org/10.1007/bf02510256
OpenAlex ID: W2009034807
description: scientific article; zbMATH DE number 1228551