On the Perron roots of principal submatrices of co-order one of irreducible nonnegative matrices
type: Item
instance of: scholarly article
title: On the Perron roots of principal submatrices of co-order one of irreducible nonnegative matrices
zbMATH Open document ID: 1018.15019
DOI: 10.1016/S0024-3795(02)00492-5
published in: Linear Algebra and its Applications
publication date: 2003-03-11T00:00:00Z
review text: This paper is devoted to the relationships between the spectral properties of a nonnegative irreducible matrix \(A\) and those of its principal submatrices of co-order 1. Let \(\lambda(A)\) be the Perron root (spectral radius) of \(A\) and \(\lambda_{\min}(A)\) the minimum of the Perron roots of all the principal submatrices of \(A\) of co-order 1. It is well known that the interval \((\lambda_{\min} (A)\), \(\lambda(A))\) does not contain any eigenvalues of \(A\). What can one say about the spectral properties of the point \(\lambda_{ \min} (A)\) as an eigenvalue of \(A\)? When is this point the second largest real eigenvalue of \(A\)? In this paper the author completely reduces these problems to the determination of the spectral properties of the Perron root of a principal submatrix of \(A\) of co-order 1 with spectral radius \(\lambda_{\min} (A)\).
Mathematics Subject Classification ID: 15B48
Mathematics Subject Classification ID: 15A18
zbMATH DE Number: 1880059
zbMATH Keywords: Perron root spectral radius
zbMATH Keywords: nonnegative irreducible matrix
author: S. V. Savchenko
reviewed by: Rabe von Randow
MaRDI profile type: Publication
cites work: Q3254327
cites work: Q3261425
cites work: The influence of the marked reduced graph of a nonnegative matrix on the Jordan form and on related properties: a survey
cites work: On the Lidskii--Vishik--Lyusternik Perturbation Theory for Eigenvalues of Matrices with Arbitrary Jordan Structure
cites work: Ergodic properties of nonnegative matrices. I
cites work: Q5643510
cites work: Q3312749
cites work: An algebraic-analytic approach to nonnegative bases
cites work: The Growth of Powers of a Nonnegative Matrix
cites work: Perron components and algebraic connectivity for weighted graphs
cites work: Algebraic eigenspaces of nonnegative matrices
cites work: On the singular graph and the Weyl characteristic of an M-matrix
cites work: On the generalized nullspace of M-matrices and Z-matrices
cites work: Characterizations and classifications of M-matrices using generalized nullspaces
cites work: Nonnegative jordan bases
cites work: Height bases, level bases, and the equality of the height and the level characteristics of anM-matrix
cites work: On the existence of matrices with prescribed height and level characteristics
cites work: Combinatorial bases, derived jordan sets and the equality of the height and level characteristics of anM-matrix
cites work: On nonnegative jordan chains for the generalized nullspace of a singularM-Matrix*
cites work: Convexity and Concavity of the Perron Root and Vector of Leslie Matrices with Applications to a Population Model
cites work: A cone-theoretic approach to the spectral theory of positive linear operators: The finite-dimensional case
cites work: The combinatorial structure of generalized eigenspaces -- from nonnegative matrices to general matrices
cites work: Bounds on the indices of the spectral-circle eigenvalues of a nonnegative matrix
cites work: On the second real eigenvalue of nonegative and Z-matrices
cites work: Improved upper bounds for the real part of nonmaximal eigenvalues of nonnegative matrices
cites work: An Upper Bound for the Real Part of Nonmaximal Eigenvalues of Nonnegative Irreducible Matrices
cites work: Lower bounds for the first eigenvalue of certain M-matrices associated with graphs
cites work: Q3998992
full work available at URL: s0024-3795(02)00492-5
OpenAlex ID: W1994956140
description: scientific article; zbMATH DE number 1880059