On nonnegative matrices similar to positive matrices
type: Item
instance of: scholarly article
title: On nonnegative matrices similar to positive matrices
zbMATH Open document ID: 0887.15019
DOI: 10.1016/S0024-3795(97)00362-5
author: Alberto Borobia
author: Julio Moro
published in: Linear Algebra and its Applications
publication date: 1998-04-28T00:00:00Z
review text: Let \(\widetilde{\mathcal N}^n\) denote the set of complex \(n\)-tuples \((1, \lambda_2,\cdots,\lambda_n)\) such that there exists a nonnegative matrix with Perron root \(1\) and spectrum \(\{1, \lambda_2, \cdots, \lambda_n \}\). The authors prove that \(\widetilde{\mathcal N}^n\) is star-shaped with respect to \((1,0,\cdots,0)\) and that \((1, \lambda_2, \cdots, \lambda_n) \in \widetilde{\mathcal N}^n\) is on the boundary of \(\widetilde{\mathcal N}^n\) if and only if \(\{1, \lambda_2, \cdots, \lambda_n \}\) is not the spectrum of a positive matrix. Hence, they study the problem of determining the nonnegative pattern matrices \(P\) that are similar to positive pattern matrices. They obtain some partial results which help them in giving a complete solution to the case of \(3\times 3\) matrices.
Mathematics Subject Classification ID: 15B48
Mathematics Subject Classification ID: 15B51
zbMATH DE Number: 1083226
zbMATH Keywords: pattern matrix
zbMATH Keywords: nonnegative matrix
zbMATH Keywords: positive matrix
zbMATH Keywords: Perron eigenvalue
zbMATH Keywords: stochastic matrix
zbMATH Keywords: spectrum
reviewed by: Arbind Kumar Lal
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1083226