The symmetric nonnegative inverse eigenvalue problem for \(5 \times 5\) matrices
type: Item
instance of: scholarly article
title: The symmetric nonnegative inverse eigenvalue problem for \(5 \times 5\) matrices
zbMATH Open document ID: 1066.15007
published in: Linear Algebra and its Applications
publication date: 2005-01-11T00:00:00Z
review text: Motivated by earlier works of \textit{L. Elsner}, \textit{R. Nabben} and \textit{M. Neumann} [Linear Algebra Appl. 271, 323-343 (1998; Zbl 0891.15019)] and \textit{J.J. McDonald} and \textit{M. Neumann} [Contemp. Math. 259, 387-407 (2000; Zbl 0965.15009)] this paper considers mainly the inverse eigenvalue problem for nonnegative symmetric matrices of order \(5\). The authors deal with the problem of finding the realizable points in the convex hull of \((1,1,1,-1,-1)\), \((1,1,0,-1,-1)\), \((1,0,0,0,-1)\), \((1,1/2,1/2,-1,-1)\), \((1,0,0,-1/2,-1/2)\), denoted by \(\mathcal{U}\). For that purpose they consider the subset \(\mathcal{U}_1\) obtained from \(\mathcal{U}\) by removing a union of two faces of \(\mathcal{U}\), one being of the Soules set and the other consisting of the trace \(0\) matrices. They show that certain matrices can not be principal submatrices of symmetric nonnegative matrices with spectrum in \(\mathcal{U}_1\). The nine \((+,0)\) patterns for which there exists an extreme matrix with spectrum in \(\mathcal{U}_1\) are analyzed.
reviewed by: C. M.da Fonseca
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 15A29
Mathematics Subject Classification ID: 15B48
zbMATH DE Number: 2126356
zbMATH Keywords: symmetric nonnegative inverse eigenvalue problem
zbMATH Keywords: Soules set
zbMATH Keywords: extreme spectrum
zbMATH Keywords: extreme matrix
author: Raphael Loewy
author: J. J. McDonald
MaRDI profile type: Publication
full work available at URL: j.laa.2003.10.018
OpenAlex ID: W2059557063
cites work: Orthogonal bases that leads to symmetric nonnegative matrices
cites work: Eigenvalues of nonnegative symmetric matrices
cites work: Row Stochastic Matrices Similar to Doubly Stochastic Matrices
cites work: The real and the symmetric nonnegative inverse eigenvalue problems are different
cites work: A note on the convexity of the realizable set of eigenvalues for nonnegative symmetric matrices
cites work: Q4352371
cites work: Extreme nonnegative matrices
cites work: A characterization of trace zero nonnegative \(5\times 5\) matrices
cites work: A note on an inverse problem for nonnegative matrices
cites work: Q4519440
cites work: Constructing symmetric nonnegative matrices
cites work: Eigenvalues of nonnegative matrices
DOI: 10.1016/J.LAA.2003.10.018
description: scientific article; zbMATH DE number 2126356