Bounds for the Perron root of a nonnegative matrix involving the properties of its graph
type: Item
instance of: scholarly article
title: Bounds for the Perron root of a nonnegative matrix involving the properties of its graph
zbMATH Open document ID: 0853.15012
DOI: 10.1007/BF02305101
published in: Mathematical Notes
publication date: 1996-12-03T00:00:00Z
review text: New bounds for the Perron root of a nonnegative matrix are given. By the Perron-Frobenius theorem, the spectral radius of a nonnegative matrix is one of its eigenvalues. This eigenvalue is called the Perron root of the matrix. Frobenius gave bounds for the Perron root in terms of the minimal sum and the maximal sum of the elements of the matrix over rows. The bounds given here involve properties of the digraph associated with the matrix. Let \(s_i\) be the sum of elements of the \(i\)th row. The weight of a path \(i_1, \dots, i_k\), \(i_{k + 1}\) in the graph of the matrix is the product \(s_{i_1} \dots s_{i_k}\) and the mean weight of the path is the geometric mean \((s_{i_1} \dots s_{i_k})^{1/k}\). The main result is: if \(A\) is a nonnegative matrix without zero rows, \(\nu\) the minimal and \(w\) the maximal mean weight of simple contours of its graph, then the Perron root \(r\) satisfies the inequalities \(\nu \leq r \leq w\). If \(A\) is indecomposable, then \(\nu < r < w\).
Mathematics Subject Classification ID: 15A42
Mathematics Subject Classification ID: 15B48
Mathematics Subject Classification ID: 05C50
zbMATH DE Number: 898968
zbMATH Keywords: geometric mean
zbMATH Keywords: cycle
zbMATH Keywords: Perron root
zbMATH Keywords: nonnegative matrix
zbMATH Keywords: Perron-Frobenius theorem
zbMATH Keywords: spectral radius
zbMATH Keywords: eigenvalues
zbMATH Keywords: digraph
author: Yury A. Al'pin
reviewed by: Giacomo Bonanno
MaRDI profile type: Publication
cites work: Q5331504
cites work: Matrix Analysis
description: scientific article; zbMATH DE number 898968