On a perturbation bound for invariant subspaces of matrices
type: Item
instance of: scholarly article
zbMATH Open document ID: 1306.15011
DOI: 10.1137/130912372
author: Michael Karow
author: Daniel Kressner
published in: SIAM Journal on Matrix Analysis and Applications
publication date: 2014-10-15T00:00:00Z
full work available at URL: https://depositonce.tu-berlin.de//handle/11303/7278
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 15A24
zbMATH DE Number: 6356188
zbMATH Keywords: invariant subspace
zbMATH Keywords: perturbation theory
zbMATH Keywords: pseudospectra
zbMATH Keywords: quadratic matrix equation
MaRDI profile type: Publication
OpenAlex ID: W2091382115
title: On a perturbation bound for invariant subspaces of matrices
review text: A subspace \(\mathcal X \subset \mathbb{C}^n\) is invariant under a matrix \(A\in\mathbb{C}^{n\times n}\) if NEWLINE\[NEWLINE A\mathcal X \subset \mathcal X\,. NEWLINE\]NEWLINE If the columns \(X\in\mathbb{C}^{n \times k}\) form an orthonormal basis of \(\mathcal X\), then we obtain the existence of \(A_{11}\in\mathbb{C}^{k \times k}\) such that \(AX=XA_{11}\). Using the block Schur decomposition, the matrix \(X\) is extended to a unitary matrix \([X,X_{\perp}]\) such that NEWLINE\[NEWLINE A[X,X_{\perp}]=[X,X_{\perp}]\left[\begin{matrix} A_{11} & A_{12}\\ 0 & A_{22}\end{matrix}\right]. NEWLINE\]NEWLINE This implies \(\sigma(A)=\sigma(A_{11}) \cup \sigma(A_{22})\), where \(\sigma(\cdot)\) denotes the spectrum of a matrix.NEWLINENEWLINEThroughout this paper, it is assumed that NEWLINE\[NEWLINE \sigma(A_{11}) \cap \sigma(A_{22})=\emptyset . NEWLINE\]NEWLINE This is a necessary and sufficient condition for the Lipschitz continuity of \(\mathcal X\) with respect to perturbations in \(A\).NEWLINENEWLINEThe authors consider the block triangular matrix \(A\) as follows NEWLINE\[NEWLINE A=\left[\begin{matrix} A_{11} & A_{12}\\ 0 & A_{22}\end{matrix}\right],\quad A_{11}\in\mathbb C^{k \times k},\quad A_{22}\in\mathbb C^{(n-k)\times(n-k)}, NEWLINE\]NEWLINE such that \(\sigma(A_{11}) \cap \sigma(A_{22})=\emptyset\), then, under suitable conditions, for all \(\epsilon \geq 0\), NEWLINE\[NEWLINE \sigma_\epsilon(A) \subset \sigma_{g(\epsilon)}(A_{11}) \cup \sigma_{g(\epsilon)}(A_{22})\,, NEWLINE\]NEWLINE where \(g(\epsilon)=\sqrt{\epsilon(\epsilon+\|A_{12}\|_2)}\). Also, the authors investigate the effect of perturbations on invariant subspaces of a nonsymmetric matrix \(A\).
reviewed by: Ali Morassaei
description: scientific article; zbMATH DE number 6356188