Improvement of Frobenius bounds for the Perron root of a non-negative matrix
type: Item
instance of: scholarly article
title: Improvement of Frobenius bounds for the Perron root of a non-negative matrix
zbMATH Open document ID: 0944.15006
published in: Computational Mathematics and Mathematical Physics
publication date: 2000-06-19T00:00:00Z
review text: According to the Perron-Frobenius theorem, the spectral radius \(r(A)\) of a nonnegative matrix \(A\) is its eigenvalue. The number \(r(A)\) is referred to as the Perron root of \(A\). Frobenius established that \(s\leq r(A)\leq S\), where \(s\) and \(S\) are the least and the greatest row sums of the elements of \(A\). This paper presents a method for constructing second-order matrices whose Perron roots are the upper and lower bounds for the Perron root of a given nonnegative matrix of arbitrary order. The bounds for the Perron root that are obtained by this method are shown to be better than both Frobenius bounds and those obtained by subsequent investigators.
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 15A21
Mathematics Subject Classification ID: 15B48
zbMATH DE Number: 1440448
zbMATH Keywords: nonnegative matrix
zbMATH Keywords: Perron root
zbMATH Keywords: Frobenius bounds
zbMATH Keywords: spectral radius
zbMATH Keywords: eigenvalue
author: Yury A. Al'pin
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1440448