Converse to the Parter--Wiener theorem: the case of non-trees
type: Item
instance of: scholarly article
title: Converse to the Parter--Wiener theorem: the case of non-trees
zbMATH Open document ID: 1114.05061
author: Charles R. Johnson
author: António Leal Duarte
published in: Discrete Mathematics
publication date: 2006-12-14T00:00:00Z
review text: Let \(A(i)\) denote the matrix obtained by deleting the \(i\)th row and column of the matrix \(A\), and let \(m_A(\lambda )\) be the multiplicity of the eigenvalue \(\lambda\) of \(A\). The Parter--Wiener theorem, as extended by the authors and \textit{C.\ M.\ Saiago} [SIAM J.\ Matrix Anal.\ Appl.\ 25, 352--361 (2003; Zbl 1067.15003)], asserts that if the graph of a Hermitian matrix \(A\) is a tree and, for some index \(j\), \(\lambda \in \sigma (A)\cap \sigma (A(j))\), then there is an index \(i\) such that \(m_{A(i)}(\lambda )=m_A(\lambda )+1\). The authors prove that this result is generally true only for trees.
Mathematics Subject Classification ID: 05C50
Mathematics Subject Classification ID: 15B57
Mathematics Subject Classification ID: 15A18
zbMATH DE Number: 5080056
zbMATH Keywords: matrix graph
zbMATH Keywords: eigenvalue multiplicity
reviewed by: Alan L. Andrew
MaRDI profile type: Publication
full work available at URL: j.disc.2005.04.025
OpenAlex ID: W2057096223
cites work: What are Schur complements, anyway?
cites work: Matrix Analysis
cites work: Q4040931
cites work: The maximum multiplicity of an eigenvalue in a matrix whose graph is a tree
cites work: The Parter--Wiener Theorem: Refinement and Generalization
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DOI: 10.1016/J.DISC.2005.04.025
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description: scientific article; zbMATH DE number 5080056