Spectrally arbitrary patterns: Reducibility and the \(2n\) conjecture for \(n = 5\)
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.
scientific article; zbMATH DE number 5151566