Ray patterns of matrices and nonsingularity
type: Item
instance of: scholarly article
title: Ray patterns of matrices and nonsingularity
zbMATH Open document ID: 0887.15002
DOI: 10.1016/S0024-3795(97)80057-2
published in: Linear Algebra and its Applications
publication date: 1997-12-15T00:00:00Z
review text: The authors aim at determining the nonsingularity of a complex matrix based solely on arguments of its nonzero entries. A complex \(n\times n\) matrix is called ray-nonsingular, if \(X^\circ A\) is a nonsingular matrix for all real \(n\times n\) matrices with positive entries; here the symbol \(^\circ\) denotes the entrywise product of matrices. This concept generalizes the notion of sign-nonsingularity of real matrices; cf. e.g. \textit{R. A. Brualdi} and \textit{B. L. Shader} [DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 4, 117-134 (1991; Zbl 0742.15001)]. A signed transversal product of an \(n\times n\) matrix \(A\) is just one of the \(n!\) terms that appear in the usual definition of a determinant. A sufficient condition for ray-nonsingularity is that the origin \(0\in \mathbb{C}\) is not in the relative interior of the convex hull of the signed transversal products of \(A\).
Mathematics Subject Classification ID: 15A09
zbMATH DE Number: 1089504
zbMATH Keywords: sign-nonsingularity
zbMATH Keywords: ray-nonsingularity
zbMATH Keywords: signed transversal product
author: J. J. McDonald
author: D. D. Olesky
author: M. Tsatsomeros
author: P. van den Driessche
reviewed by: Hans Havlicek
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1089504