On matrices with Perron--Frobenius properties and some negative entries
type: Item
instance of: scholarly article
title: On matrices with Perron--Frobenius properties and some negative entries
zbMATH Open document ID: 1078.15018
author: Charles R. Johnson
author: Pablo Tarazaga
published in: Positivity
publication date: 2005-06-13T00:00:00Z
review text: The authors study the set \(PF_n\) of \(n \times n\) real matrices for which there is a positive eigenvalue, of multiplicity 1, that strictly dominates all other eigenvalues and for which there is both an entry-wise positive left and right eigenvector. Such matrices may have negative entries and generalize the primitive matrices in important ways. They give a characterization of \(PF_n\) and prove that it is connected. Given a matrix \(A \in PF_n\), the authors consider two circular cones centered at the ray from the origin through \(A\) in matrix space. The first cone, \(C_{\alpha}(A)\), is determined by that matrix angularly nearest to \(A\) that does not have a strictly dominant positive eigenvalue and \(\alpha\) is the cosine of that angle. The second, \(C_{\beta}(A)\), is determined by that matrix angularly nearest to \(A\) that does not have positive eigenvectors associated with any dominant eigenvalue, and \(\beta\) is the corresponding cosine. In the paper, \(\alpha\) and \(\beta\) are calculated in some cases. Finally, the authors describe several ways of constructing this type of matrices, including a very geometric one.
Mathematics Subject Classification ID: 15B48
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 15A60
zbMATH DE Number: 2175431
zbMATH Keywords: positive matrices
zbMATH Keywords: Perron-Frobenius property
zbMATH Keywords: cones of matrices
zbMATH Keywords: positive eigenvalue
zbMATH Keywords: dominant eigenvalue
reviewed by: Juan Ramón Torregrosa Sánchez
MaRDI profile type: Publication
full work available at URL: s11117-003-3881-3
OpenAlex ID: W2041837712
cites work: Q4326384
cites work: Perron-Frobenius theorem for matrices with some negative entries
DOI: 10.1007/S11117-003-3881-3
description: scientific article; zbMATH DE number 2175431