An upper bound for \(\| A^{-1}\|_{\infty}\) of strictly diagonally dominant \(M\)-matrices
type: Item
instance of: scholarly article
title: An upper bound for \(\| A^{-1}\|_{\infty}\) of strictly diagonally dominant \(M\)-matrices
zbMATH Open document ID: 1126.15022
author: Guanghui Cheng
author: Ting-Zhu Huang
published in: Linear Algebra and its Applications
publication date: 2007-09-14T00:00:00Z
review text: A square \(n\times n\) matrix \(A\) is called a nonsingular \(M\)-matrix if there exists an \(n\times n\) nonnegative matrix \(P\) such that \(A=sI-P,\) where \(I\) is the identity matrix and \(s>\rho(P)\), \(\rho(P)\) is the spectral radius of the matrix \(P\). It is clear that \(A^{-1}\) is a nonnegative matrix. In this paper, the upper bound of \(\| A^{-1}\|_\infty\) is improved and a new lower bound of \(\rho(A^{-1})^{-1}\) is obtained where the Perron eigenvalue of \(A^{-1}\) is denoted by \(\rho(A^{-1})\). Here it is necessary to point out that the authors in the abstract of this paper say that they give a sharp upper bound for \(\| A^{-1}\|_\infty\), this is not correct (see Example 3.1 of the paper).
Mathematics Subject Classification ID: 15A42
Mathematics Subject Classification ID: 15B48
Mathematics Subject Classification ID: 15A60
zbMATH DE Number: 5190969
zbMATH Keywords: diagonal dominance
zbMATH Keywords: \(M\)-matrix
zbMATH Keywords: inverse \(M\)-matrix
zbMATH Keywords: norm
zbMATH Keywords: nonnegative matrix
zbMATH Keywords: spectral radius
zbMATH Keywords: Perron eigenvalue
MaRDI profile type: Publication
full work available at URL: j.laa.2007.06.001
OpenAlex ID: W2072451235
cites work: Q4040931
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DOI: 10.1016/J.LAA.2007.06.001
description: scientific article; zbMATH DE number 5190969