Spectral radius and infinity norm of matrices
type: Item
instance of: scholarly article
title: Spectral radius and infinity norm of matrices
zbMATH Open document ID: 1153.15016
author: Baodong Zheng
author: Liancheng Wang
published in: Journal of Mathematical Analysis and Applications
publication date: 2008-07-21T00:00:00Z
review text: Let \(A=(a_{ij})\) be a real \(n\)-by-\(n\) matrix. Denote by \(\rho(A)\) the spectral radius and by \(\| A\| _{\infty}=\max_{1\leq i\leq n}\sum_{j=1}^{n}| a_{ij}| \) the infinity norm of \(A\). It is well known that \(\rho(A) \leq \| A\| _{\infty}\). (Actually the inequality holds for every matrix norm.) In the paper an algebraic criterion for \(\rho(A) < \| A\| _{\infty}\) is given. Namely, let \(a\) be a positive number and let \(A=(a_{ij})\) be a real \(n\)-by-\(n\) matrix. If \(\sum_{j=1}^{n}| a_{ij}| <a\) for \(i=i_1,\ldots, i_k\), then substitute all elements in these \(k\) rows and corresponding \(k\) columns by zeros. Denote this transformation by \(\varphi_a\). One of the results in the paper says that \(\rho(A)<\| A\| _\infty\) if \(\varphi_{\| A\| _{\infty}}^{n}\bigl(A\bigr)=0\). On the other hand, if \(A\) is entrywise nonnegative, the converse holds as well. There are some other related results and an application to the discrete dynamical systems.
reviewed by: Janko Bračič
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 15A60
Mathematics Subject Classification ID: 15A45
zbMATH DE Number: 5303208
zbMATH Keywords: spectral radius
zbMATH Keywords: infinity norm
zbMATH Keywords: dynamic system
zbMATH Keywords: asymptotical stability
MaRDI profile type: Publication
full work available at URL: j.jmaa.2008.05.003
OpenAlex ID: W2061293555
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DOI: 10.1016/J.JMAA.2008.05.003
description: scientific article; zbMATH DE number 5303208