Minimum number of distinct eigenvalues allowed by a sign pattern
type: Item
instance of: scholarly article
title: Minimum number of distinct eigenvalues allowed by a sign pattern
zbMATH Open document ID: 1503.15046
author: Yanyan Li
published in: Linear Algebra and its Applications
publication date: 2022-10-07T00:00:00Z
review text: This manuscript deals with the study of the minimum number of distinct eigenvalues allowed by a square sign pattern. A sign pattern is a symbolic matrix \(\mathcal{A}\) with entries \(\{+, -, 0\}\), which represents the combinatorial structure of a real matrix, taking into account the signs of the entries. The qualitative class of a sign pattern \(\mathcal{A}\), denoted by \(Q(\mathcal{A})\), is the set of all real matrices \(A\) such that the signs of their entries coincide with the sign pattern \(\mathcal{A}\). For a matrix \(A \in \mathbb{R}^{n \times n}\), let \(q(A)\) be the number of distinct eigenvalues of \(A\). For a sign pattern \(\mathcal{A}\), \[ q(\mathcal{A})=\min \{ q(A) \ : \ A \in Q(\mathcal{A}) \}. \] The authors study \(q(\mathcal{A})\), give some definitions and determine how the minimum number of distinct eigenvalues changes under certain manipulations of the sign pattern. They also develop some techniques to help determining the number \(q(\mathcal{A})\) and present properties of the directed graphs of \(n \times n\) sign patterns \(\mathcal{A}\) with \(q(\mathcal{A})=1\). The number \(q(\mathcal{A})\) for \(2 \times 2\) and irreducible \(3 \times 3\) sign patterns \(\mathcal{A}\) is determined. The paper ends with an appendix providing a list of all irreducible \(3 \times 3\) sign patterns \(\mathcal{A}\) according to the value of \(q(\mathcal{A})\).
Mathematics Subject Classification ID: 15B35
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 15A29
Mathematics Subject Classification ID: 05C50
zbMATH DE Number: 7597866
zbMATH Keywords: sign pattern
zbMATH Keywords: eigenvalue
zbMATH Keywords: digraph
zbMATH Keywords: inverse eigenvalue problem
reviewed by: Juan Ramón Torregrosa Sánchez
MaRDI profile type: Publication
full work available at URL: j.laa.2022.09.002
OpenAlex ID: W4294958041
Wikidata QID: Q114151190
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cites work: Q4802219
DOI: 10.1016/J.LAA.2022.09.002
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description: scientific article; zbMATH DE number 7597866