A solution of a nonlinear system arising in spectral perturbation theory of nonnegative matrices
type: Item
instance of: scholarly article
title: A solution of a nonlinear system arising in spectral perturbation theory of nonnegative matrices
zbMATH Open document ID: 0882.65024
DOI: 10.1016/S0024-3795(96)00154-1
author: Uriel G. Rothblum
author: Hans Schneider
published in: Linear Algebra and its Applications
publication date: 1998-02-19T00:00:00Z
review text: This paper develops an algorithm for computing the coefficients of the fractional power series expansions of the spectral radius \(\rho(P+\varepsilon E)\) and corresponding (normalized) eigenvector of \(P+\varepsilon E\), \(P\) and \(E\) are complex \(n\times n\) and \(P+\varepsilon E\) with sufficiently small \(\varepsilon>0\) is nonnegative and irreducible, \(P\) has a single Jordan block corresponding to \(\rho(P)\) and the left and right eigenvectors \(v\) and \(w\), respectively, of \(P\) corresponding to \(\rho(P)\) satisfy \(v^TEw\neq 0\). The novel idea is the recursive solution of a system of nonlinear equations, where in each step the coefficients of the expansion of \(\rho(P+\varepsilon E)\) are considered as parameters and a related linear system is solved parametrically, and the next coefficient of the development of \(\rho(P+\varepsilon E)\) is determined from feasibility considerations for a linear system. The method is also considered under relaxed conditions. The computed coefficients also yield a preferred basis of the generalized eigenspace corresponding to \(\rho(P)\).
Mathematics Subject Classification ID: 65F15
Mathematics Subject Classification ID: 15B48
zbMATH DE Number: 1026429
zbMATH Keywords: spectral perturbation
zbMATH Keywords: nonnegative matrices
zbMATH Keywords: algorithm
zbMATH Keywords: fractional power series expansions
zbMATH Keywords: spectral radius
zbMATH Keywords: eigenvector
Wikidata QID: Q101371146
reviewed by: Erwin O. Kreyszig
MaRDI profile type: Publication
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full work available at URL: s0024-3795(96)00154-1
OpenAlex ID: W2064843668
description: scientific article; zbMATH DE number 1026429