Sign patterns that require repeated eigenvalues
type: Item
instance of: scholarly article
title: Sign patterns that require repeated eigenvalues
zbMATH Open document ID: 0795.15008
DOI: 10.1016/0024-3795(93)90225-D
author: Carolyn A. Eschenbach
author: Charles R. Johnson
published in: Linear Algebra and its Applications
publication date: 1993-10-13T00:00:00Z
review text: An sign-pattern matrix is an \(n\)-by-\(n\) matrix \(B\) whose entries are elements of the set \(\{+,-,0\}\). The sign-pattern class \(Q(B)\) associated with \(B\) is the set of all \(n\)-by-\(n\) real matrices\(A\) for which the entry of \(A\) in the intersection of row \(i\) and column \(j\) is positive (respectively, negative, zero) if the entry of \(B\) in the same location is + (respectively, \(-\), 0). For every positive integer \(k\), the authors characterize all matrices \(B\) for which \(k\) is the smallest integer with the property that every matrix \(A\) in \(Q(B)\) has an eigenvalue of multiplicity at least \(k\). The characterization is given in terms of the size of zero submatrices of \(B\), or in terms of the length of cycles in the directed graph on vertices \(\{1,\dots,n\}\) associated with \(B\) in the usual way. The graph contains an arrow starting at \(i\) and ending at \(j\) if and only if the element of \(B\) in the intersection of row \(i\) and column \(j\) is not 0. The authors discuss the relationship of the above characterization to diagonalizability (by similarity) of matrices in \(Q(B)\) and give several instructive examples.
Mathematics Subject Classification ID: 15B36
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 15A21
Mathematics Subject Classification ID: 05B20
Mathematics Subject Classification ID: 05C70
zbMATH DE Number: 428213
zbMATH Keywords: eigenvalue
zbMATH Keywords: sign-pattern matrix
zbMATH Keywords: cycles
zbMATH Keywords: directed graph
zbMATH Keywords: diagonalizability
zbMATH Keywords: similarity
reviewed by: Mark Kleiner
MaRDI profile type: Publication
cites work: Q3851094
cites work: Matrix Analysis
description: scientific article; zbMATH DE number 428213