The change in multiplicity of an eigenvalue of a real symmetric matrix resulting from the changes in edge values around a classified vertex in a tree
type: Item
instance of: scholarly article
title: The change in multiplicity of an eigenvalue of a real symmetric matrix resulting from the changes in edge values around a classified vertex in a tree
zbMATH Open document ID: 1402.15007
published in: Acta Mathematica Vietnamica
publication date: 2018-11-01T00:00:00Z
review text: Let \(S(T)\) be a set of real symmetric matrices whose graph of off-diagonal entries is a given tree \(T\), and for \(A\in S(T)\) let \(m_A(\lambda)\) denote the multiplicity of \(\lambda\) as an eigenvalue of \(A\). The results show that when values of the edges incident to a particular vertex are simultaneously changed, then the multiplicity of \(\lambda\) changes by at most one. In addition, when an edge \(e\) is removed from \(T\) then \(m_A(\lambda)-1\leq m_{A_e}(\lambda)\leq m_A(\lambda)+2\) by interlacing, where \(A_e\in S(T-e)\). The edge \(e\) is called 2-Parter, Parter, neutral or downer if \(m_{A_e}(\lambda)-m_A(\lambda)\) is equal to \(2\), \(1\), \(0\) or \(-1\), respectively. It turns out that 2-Parter, Parter and downer edges cannot be incident to each other and that, moreover, a downer edge must be at distance at least two from any Parter or 2-Parter edge.
reviewed by: Dragan Stevanović
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 05C50
Mathematics Subject Classification ID: 05C05
zbMATH DE Number: 6970045
zbMATH Keywords: tree
zbMATH Keywords: real symmetric matrix
zbMATH Keywords: eigenvalues
zbMATH Keywords: multiplicity
zbMATH Keywords: edges
author: Kenji Toyonaga
MaRDI profile type: Publication
full work available at URL: s40306-018-0276-x
OpenAlex ID: W2807915313
cites work: Q4902474
cites work: The Parter--Wiener Theorem: Refinement and Generalization
cites work: The change in eigenvalue multiplicity associated with perturbation of a diagonal entry
cites work: The change in multiplicity of an eigenvalue of a Hermitian matrix associated with the removal of an edge from its graph
cites work: The classification of edges and the change in multiplicity of an eigenvalue of a real symmetric matrix resulting from the change in an edge value
DOI: 10.1007/S40306-018-0276-X
description: scientific article; zbMATH DE number 6970045