Perturbation theorems for the joint spectrum of commuting matrices: A conservative approach
type: Item
instance of: scholarly article
title: Perturbation theorems for the joint spectrum of commuting matrices: A conservative approach
zbMATH Open document ID: 0817.15010
DOI: 10.1016/0024-3795(94)90433-2
published in: Linear Algebra and its Applications
publication date: 1995-08-13T00:00:00Z
full work available at URL: https://pub.uni-bielefeld.de/record/1780851
review text: A joint eigenvalue of an \(m\)-tuple \(A = (A_ 1, A_ 2, \dots, A_ m)\) of complex \(n\)-by-\(n\) matrices \(A_ i\) is a vector \(\lambda \in \mathbb{C}^ m\), \(\lambda = (\lambda_ 1, \lambda_ 2, \dots, \lambda_ m)^ T\), such that there exists a nonzero vector \(x \in \mathbb{C}^ n\) with \(A_ jx = \lambda_ j x\), \(j = 1,2, \dots, m\). If \(A_ i\) are commuting, then there is at least one joint eigenvalue, \(x\) is called a joint eigenvector, and the collection of the joint eigenvalues is called the joint spectrum and is denoted by \(Sp(A)\). The effect of a perturbation of a matrix on its spectrum has been investigated for a long time and now well understood. In contrast, study of the joint spectrum \(Sp(A)\) has started only recently by using Clifford-algebra tools. After introducing the relevant definitions of the Clifford operator \(\text{Cliff} (A)\) of \(A\), the underlying spaces, and the connection between the joint spectrum of \(A\) and the spectrum of \(\text{Cliff} (A)\), the author outlines the recent perturbation theorems for the joint spectrum of commuting matrices, which have been proved using Clifford- algebra tools. After establishing bounds on \(\| \text{Cliff} (A) \|\), he proves and improves these theorems by classical means as used in the case of the standard eigenvalue problem.
Mathematics Subject Classification ID: 15A42
Mathematics Subject Classification ID: 15A66
zbMATH DE Number: 665779
zbMATH Keywords: joint spectrum of commuting matrices
zbMATH Keywords: Clifford algebras
zbMATH Keywords: joint eigenvalue
zbMATH Keywords: joint eigenvector
zbMATH Keywords: perturbation
author: Ludwig Elsner
reviewed by: Takayuki Nôno
MaRDI profile type: Publication
OpenAlex ID: W1977380800
cites work: Norms and exclusion theorems
cites work: Q3996765
cites work: A Henrici Theorem for Joint Spectra of Commuting Matrices
cites work: Bounds for iterates, inverses, spectral variation and fields of values of non-normal matrices
cites work: Q3469767
cites work: Comparison of joint spectra for certain classes of commuting operators
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description: scientific article; zbMATH DE number 665779