Ray patterns of matrices and nonsingularity
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 \(A\).
scientific article; zbMATH DE number 1089504