The rotation of eigenspaces of perturbed matrix pairs
type: Item
instance of: scholarly article
title: The rotation of eigenspaces of perturbed matrix pairs
zbMATH Open document ID: 1245.15016
author: Luka Grubišić
author: Ninoslav Truhar
author: Krešimir Veselić
published in: Linear Algebra and its Applications
publication date: 2012-05-11T00:00:00Z
full work available at URL: https://arxiv.org/abs/1011.4424
review text: The authors consider the generalized eigenvector problem \(Hx=\lambda Mx\) and the corresponding perturbed one \((H+ \delta H)\tilde x =\tilde \lambda(M+\delta M)\tilde x\), where \(H, M, H+ \delta H, M+\delta M\in {\mathbb C}_{n\times n}\) are positive definite matrices. They present new \(\sin \Theta\) theorems for perturbations of positive definite matrix pairs. The rotation of eigenspaces is measured in the matrix dependent scalar product. They assess the sharpness of the new estimates in terms of effectivity quotients (the quotient of the measure of the perturbation and the estimator). Known relative \(\sin \Theta\) theorems for eigenspaces of positive definite Hermitian matrices are then special cases. Numerical experiments indicate that relative \(\sin \Theta\) theorems are asymptotically sharp when the rotation is measured in the appropriate matrix dependent scalar product but not always in the ordinary Euclidean scalar product.
reviewed by: Tin-Yau Tam
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 15B57
Mathematics Subject Classification ID: 15B48
zbMATH DE Number: 6033312
zbMATH Keywords: eigenvalues
zbMATH Keywords: matrix pairs
zbMATH Keywords: rotation of eigenvectors
zbMATH Keywords: generalized eigenvector problem
zbMATH Keywords: positive definite matrices
zbMATH Keywords: perturbations
zbMATH Keywords: Hermitian matrices
zbMATH Keywords: numerical experiments
MaRDI profile type: Publication
OpenAlex ID: W1994755446
arXiv ID: 1011.4424
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DOI: 10.1016/J.LAA.2012.01.026
description: scientific article; zbMATH DE number 6033312