An algebraic-analytic approach to nonnegative bases
type: Item
instance of: scholarly article
title: An algebraic-analytic approach to nonnegative bases
zbMATH Open document ID: 0708.15003
DOI: 10.1016/0024-3795(90)90242-5
author: R. E. Hartwig
author: M. Neumann
author: Nicholas J. Rose
published in: Linear Algebra and its Applications
publication date: 1990-01-01T00:00:00Z
review text: For a nonnegative \(n\times n\) matrix A let \(W_ A\) be the algebraic eigenspace corresponding to the Perron root of A, which is equal to the spectral radius \(\rho\) (A). Using the Laurent expansion of the resolvent operator \((\lambda I-A)^{-1}\) in a deleted neighborhood of \(\lambda =\rho (A)\), the authors give new proofs for the following two results due to \textit{U. G. Rothblum} [ibid. 12, 281-292 (1975; Zbl 0321.15010)]: (i) \(W_ A\) has a basis of nonnegative vectors, (ii) \(W_ A\) has a Jordan chain of maximal length consisting entirely of nonnegative vectors. A simple and elegant algorithm for determining a nonnegative basis of \(W_ A\) (which is not necessarily a Jordan chain) is also presented.
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 15B48
zbMATH DE Number: 4163077
zbMATH Keywords: Perron-Frobenius theory
zbMATH Keywords: nonnegative matrices
zbMATH Keywords: resolvent operator
zbMATH Keywords: algebraic eigenspace
zbMATH Keywords: Perron root
zbMATH Keywords: spectral radius
zbMATH Keywords: Jordan chain of maximal length
zbMATH Keywords: nonnegative basis
reviewed by: László Mihályffy
MaRDI profile type: Publication
full work available at URL: 0024-3795(90)90242-5
OpenAlex ID: W2023934160
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description: scientific article; zbMATH DE number 4163077