Componentwise pseudospectrum of a matrix
type: Item
instance of: scholarly article
title: Componentwise pseudospectrum of a matrix
zbMATH Open document ID: 1050.15009
author: A. Malyshev
author: M. Sadkane
published in: Linear Algebra and its Applications
publication date: 2004-08-06T00:00:00Z
review text: For \(\varepsilon > 0\), the \(\varepsilon\)-pseudospectrum of an \(n \times n\) matrix \(A\) is defined as \(\Lambda_\varepsilon(A) = \{ \lambda \in \mathbb{C} : \exists \Delta \in \mathbb C^{n \times n}\) such that \(\| \Delta \| _2 \leq \varepsilon\) and \(A + \Delta - \lambda I\) is singular\(\} = \{ \lambda \in \mathbb{C} : \sigma_{\min}(A - \lambda I) \leq \varepsilon \} = \{ \lambda \in \mathbb{C} : \| (A - \lambda I )^{-1} \geq \frac{1}{\varepsilon} \}\). The authors propose to define the componentwise \(\varepsilon\)-pseudospectrum by \(\Lambda_\varepsilon(A,E) = \{ \lambda \in \mathbb{C} : \exists \Delta \in \mathbb C^{n \times n}\) such that \(\| \Delta \| _2 \leq \varepsilon E\) and \(A + \Delta - \lambda I\) is singular\(\} = \{ \lambda \in \mathbb{C} : \sigma(A - \lambda I,E) \leq \varepsilon \}\). Furthermore, componentwise \(\varepsilon\)-pseudospectrum-equivalents for the third term in the definition of an \(\varepsilon\)-pseudospectrum are discussed and numerical examples are given.
Mathematics Subject Classification ID: 15A18
zbMATH DE Number: 2081759
zbMATH Keywords: matrix pseudospectrum
zbMATH Keywords: numerical examples
zbMATH Keywords: nonsingularity radius
zbMATH Keywords: componentwise structured perturbation
describes a project that uses: mctoolbox
describes a project that uses: INTLAB
describes a project that uses: Pseudospectra
MaRDI profile type: Publication
full work available at URL: j.laa.2003.10.005
OpenAlex ID: W2057902655
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DOI: 10.1016/J.LAA.2003.10.005
description: scientific article; zbMATH DE number 2081759