Spectrally arbitrary patterns: Reducibility and the \(2n\) conjecture for \(n = 5\)
type: Item
instance of: scholarly article
title: Spectrally arbitrary patterns: Reducibility and the \(2n\) conjecture for \(n = 5\)
zbMATH Open document ID: 1118.15007
author: Luz Maria DeAlba
author: Irvin Roy Hentzel
author: Leslie Hogben
author: J. J. McDonald
author: Rana Mikkelson
author: Olga Pryporova
author: Kevin N. Vander Meulen
author: Bryan L. Shader
published in: Linear Algebra and its Applications
publication date: 2007-05-10T00:00:00Z
review text: A sign pattern is a square matrix whose entries are in \(\left\{ +,-,0\right\} \). A real square matrix \([a_{ij}]\) has sign pattern \([z_{ij}]\) where the \(z_{ij}:=sgn(a_{ij})\) for all \((i,j)\) [see \textit{F. J. Hall} and \textit{Z. Li}, Sign pattern matrices. Handbook of Linear Algebra (L. Hogben, ed.), CRC Press, Boca Raton (2007) for a recent survey]. A sign pattern \(Z\) is called a ``spectrally arbitrary pattern'' (SAP) if for each monic real polynomial \(q(x)\) there exists a real matrix \(A\) with sign pattern \(Z\) such that \(q(x)\) is the characteristic polynomial of \(A\) [see \textit{J. H. Drew, C. R. Johnson, D. D. Olesky} and \textit{P. van den Driessche}, Linear Algebra Appl. 308, 121--137 (2000; Zbl 0957.15012)]. In the latter paper it is conjectured that a reducible sign pattern is SAP if and only if each of the diagonal blocks is SAP, but examples given in the current paper show that neither implication is true. The authors also prove that, for \(n=5\), every \(n\times n\) SAP necessarily has at least \(2n\) nonzero entries. The cases \(n\leq4\) of this conjecture were dealt with in [\textit{T. Britz, J. J. McDonald, D. D. Olesky} and \textit{P. van den Driessche}, SIAM J. Matrix Anal. Appl. 26, 257--271 (2004; Zbl 1082.15016)]. Analogous results for nonzero patterns are also proved.
reviewed by: John D. Dixon
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 15B57
zbMATH DE Number: 5151566
zbMATH Keywords: sign pattern
zbMATH Keywords: spectrally arbitrary sign pattern
zbMATH Keywords: nonzero pattern
zbMATH Keywords: reducible sign pattern
zbMATH Keywords: irreducible sign pattern
zbMATH Keywords: potentially nilpotent
MaRDI profile type: Publication
full work available at URL: j.laa.2006.12.018
OpenAlex ID: W2118264190
Wikidata QID: Q122972300
cites work: Q4858123
cites work: Minimal Spectrally Arbitrary Sign Patterns
cites work: Spectrally and inertially arbitrary sign patterns
cites work: Spectrally arbitrary zero–nonzero patterns of order 4
cites work: Spectrally arbitrary patterns
cites work: Parallel concepts in graph theory
DOI: 10.1016/J.LAA.2006.12.018
description: scientific article; zbMATH DE number 5151566