The determinantal regions of complex sign pattern matrices and ray pattern matrices
type: Item
instance of: scholarly article
title: The determinantal regions of complex sign pattern matrices and ray pattern matrices
zbMATH Open document ID: 1066.15004
author: Hai-Ying Shan
author: Jia-Yu Shao
published in: Linear Algebra and its Applications
publication date: 2005-02-23T00:00:00Z
review text: Let \(A\) be a complex square matrix and let \(S_A\) (resp. \(R_A\)) be the set of determinants of the matrices in the complex sign (resp. ray) pattern class of \(A\). \textit{C. A. Eschenbach}, \textit{F. J. Hall} and \textit{F. Z. Li} [Bull. Aust. Math. Soc. 57, No. 1, 159--172 (1998; Zbl 0951.15021)] mentioned several open problems concerning the determinantal region \(S_A\). The authors show that the boundary points of the region \(S_A\) lie always on some axis of the complex plane. Moreover, if \(S_A\) (resp. \(R_A\)) is not contained in any line through the origin, then all nonzero boundary points of \(S_A\) (resp. \(R_A\)) are not in \(S_A\) (resp. \(R_A\)). Some consequences are discussed. The authors also find bounds for the number of determinantal regions. In the case of a nonsingular matrix \(A\), all the \(21\) possible determinantal regions \(S_A\) are determined and listed. Some research problems concerning the regions \(R_A\) are proposed.
Mathematics Subject Classification ID: 15A15
Mathematics Subject Classification ID: 15A09
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 15B57
zbMATH DE Number: 2137797
zbMATH Keywords: complex sign pattern
zbMATH Keywords: complex ray pattern
zbMATH Keywords: determinants
zbMATH Keywords: determinantal regions
reviewed by: C. M.da Fonseca
MaRDI profile type: Publication
full work available at URL: j.laa.2004.08.023
OpenAlex ID: W2086785547
Wikidata QID: Q112882145
cites work: Qualitative Economics and the Scope of the Correspondence Principle
cites work: Combinatorial matrix theory
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cites work: Q5638112
DOI: 10.1016/J.LAA.2004.08.023
description: scientific article; zbMATH DE number 2137797