Two classes of symmetric sign patterns that require unique inertia.
type: Item
instance of: scholarly article
title: Two classes of symmetric sign patterns that require unique inertia.
zbMATH Open document ID: 1035.15012
author: Yanling Shao
published in: Applied Mathematics. Series B (English Edition)
publication date: 2003-11-17T00:00:00Z
review text: Sign pattern matrices (sign patterns) \(A\) of order \(n\), whose entries are from the set \(\{+, -, 0\}\) are studied. A class of real matrices, called the qualitative class of \(A\) and denoted by \(Q(A)\) is associated with each \(A\) from the set of all \(n \times n\) sign patterns. A sign pattern \(A \in Q(A)\) is a double star (respectively cycle) sign pattern if the undirected graph of \(A\) is a double star (respectively cycle) graph, possibly with loops. The inertia of the real symmetric matrix \(A\), written as \(i(A)\) is the set \(i(A)= \{i(B)=(i_+(B), i_-(B), i_0(B)) \mid B=B^T \in Q(A)\}\), where \(i_+(B)\) (respectively \(i_-(B)\), \(i_0(B)\)) denotes the number of positive (respectively negative, zero) eigenvalues of a matrix \(B\) counted with their algebraic multiplicities. In a special case, when \(i(B_1)=i(B_2)\) for all real symmetric matrices \(B_1, B_2 \in Q(A)\), it is said that the symmetric sign pattern \(A\) requires unique inertia. \textit{F. J. Hall}, \textit{Zhongshan Li } and \textit{Di Wang} [Linear Algebra Appl. 338, 153-169 (2001; Zbl 0994.15028)] have given sufficient and necessary conditions when a symmetric sign pattern requires unique inertia. These conditions are difficult to fulfill and in this work the special cases which characterize the double star and cycle sign patterns requiring unique inertia are presented and respective unique inertia are found.
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 15B48
Mathematics Subject Classification ID: 05C50
zbMATH DE Number: 2005196
zbMATH Keywords: sign pattern
zbMATH Keywords: double star sign pattern
zbMATH Keywords: cycle sign pattern
zbMATH Keywords: inertia
zbMATH Keywords: unique inertia
zbMATH Keywords: undirected graph
zbMATH Keywords: symmetric matrix
zbMATH Keywords: eigenvalues
MaRDI profile type: Publication
full work available at URL: s11766-003-0031-4
OpenAlex ID: W1983512465
cites work: Spectrally arbitrary patterns
cites work: Inertially arbitrary patterns
cites work: Symmetric sign pattern matrices that require unique inertia
cites work: Sign patterns that require real, nonreal or pure imaginary eigenvalues
cites work: Q4858123
cites work: Matrix Analysis
DOI: 10.1007/S11766-003-0031-4
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description: scientific article; zbMATH DE number 2005196