Characterization of ray pattern matrix whose determinantal region has two components after deleting the origin

• Published in: Linear algebra and its applications Vol. 435; no. 12; pp. 3139 - 3150
• Main Authors: Liu, Yue, Shao, Jia-Yu, Ren, Ling-Zhi
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Inc 15.12.2011; Elsevier
• Subjects: Algebra; Determinantal region; Digraph; Exact sciences and technology; Linear and multilinear algebra, matrix theory; Mathematics; Matrix; Ray pattern; Sciences and techniques of general use; 15A48; Matrix; Digraph; 15A09; Ray pattern; Determinantal region; Permutation; Determinant; Non singular matrix; Matrix inversion; Positive definite matrix; Sign; Ray theory
• ISSN: 0024-3795; 1873-1856
• DOI: 10.1016/j.laa.2011.05.015

Ray nonsingular matrices are generalizations of sign nonsingular matrices. The problem of characterizing ray nonsingular matrices is still open. The study of the determinantal regions R A of ray pattern matrices is closely related to the study of ray nonsingular matrices. It was proved that if R A ⧹ { 0 } is disconnected, then it is a union of two opposite open sectors (or open rays). In this paper, we characterize those ray patterns whose determinantal regions become disconnected after deleting the origin. The characterization is based on three classes (F1), (F2) and (F3) of matrices, which can further be characterized in terms of the sets of the distinct signed transversal products of their ray patterns. Moreover, we show that in the fully indecomposable case, a matrix A is in the class (F1) (or (F2), respectively) if and only if A is ray permutation equivalent to a real SNS (or non-SNS, respectively) matrix.