On the equality of algebraic and geometric multiplicities of matrix eigenvalues
type: Item
instance of: scholarly article
title: On the equality of algebraic and geometric multiplicities of matrix eigenvalues
zbMATH Open document ID: 1236.15023
author: N. H. Rhee
author: J. Ding
published in: Applied Mathematics Letters
publication date: 2011-12-28T00:00:00Z
review text: Let \(\mathbb{C}^{n\times n}\) denote the space of all complex \(n\times n\) matrices. For a matrix \(A\in\mathbb{C}^{n\times n}\), we denote the set of all eigenvalues of \(A\) by \(\sigma(A)\). Suppose \(\lambda\in\sigma(A)\). Then the algebraic multiplicity of \(\lambda\), denoted by \(\text{alg\,mult}_A(\lambda)\), is the number of times it is repeatedd as a zero of the characteristic polynomial. The geometric multiplicity of \(\lambda\), denoted by \(\text{geo\,mult}_A(\lambda)\), is the maximal number of linearly independent eigenvectors associated with \(\lambda\). In general we have \(\text{geo\,mult}_A(\lambda)\leq \text{alg\, mult}_A(\lambda)\). In this paper, the authors summarize seventeen equivalent conditions for the equality of algebraic and geometric multiplicities of an eigenvalue for a complex square matrix. As applications, new proofs of some important results related to mean ergodic and positive matrices are given.
reviewed by: Rodica Covaci
Mathematics Subject Classification ID: 15A18
zbMATH DE Number: 5992340
zbMATH Keywords: eigenvalue
zbMATH Keywords: algebraic multiplicity
zbMATH Keywords: geometric multiplicity
zbMATH Keywords: spectral radius
MaRDI profile type: Publication
full work available at URL: j.aml.2011.06.030
OpenAlex ID: W2013731965
cites work: Q4485695
cites work: Q3631986
cites work: Q4039989
cites work: Q4344117
cites work: Matrix Analysis
cites work: Introduction to Dynamical Systems
cites work: Characteristic polynomials of some perturbed matrices
DOI: 10.1016/J.AML.2011.06.030
description: scientific article; zbMATH DE number 5992340