Spectra with positive elementary symmetric functions
type: Item
instance of: scholarly article
title: Spectra with positive elementary symmetric functions
zbMATH Open document ID: 0778.15006
DOI: 10.1016/0024-3795(93)90535-V
author: Charles R. Johnson
author: D. D. Olesky
author: M. Tsatsomeros
author: P. van den Driessche
published in: Linear Algebra and its Applications
publication date: 1993-08-08T00:00:00Z
review text: Let \(A\) denote an \(n\times n\) matrix with all diagonal entries nonzero and for which the length of the longest cycle in its directed graph is \(k\), \(2\leq k\leq n\). If, in addition, all the cycles in the directed graph of \(-A\) are signed negatively, then the elementary symmetric functions of the eigenvalues of \(A\) are positive and the authors ask whether its eigenvalues lie in the region \(\{z:-\pi+\pi/k<\arg z<\pi-\pi/k\}\). Their approach, in order to investigate the questions that they posed, is to obtain conditions on the location of roots of polynomials in Section 2 and then apply them to the characteristic polynomials of matrices. They prove that it is true for \(k=n-1\) in Section 3. In Section 4, the authors provide a counterexample for the case \(k=n-3\) and discuss related questions for more general classes of matrices with restricted length of the longest cycle.
reviewed by: Yueh-er Kuo
Mathematics Subject Classification ID: 15A18
zbMATH DE Number: 179143
zbMATH Keywords: positive sign-stable matrix
zbMATH Keywords: qualitative \(P\)-matrices
zbMATH Keywords: cycle
zbMATH Keywords: directed graph
zbMATH Keywords: elementary symmetric functions
zbMATH Keywords: eigenvalues
zbMATH Keywords: roots of polynomials
zbMATH Keywords: characteristic polynomials
zbMATH Keywords: counterexample
MaRDI profile type: Publication
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full work available at URL: 0024-3795(93)90535-v
OpenAlex ID: W2035095781
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description: scientific article; zbMATH DE number 179143