On the eigenvectors belonging to the minimum eigenvalue of an essentially nonnegative symmetric matrix with bipartite graph
type: Item
instance of: scholarly article
title: On the eigenvectors belonging to the minimum eigenvalue of an essentially nonnegative symmetric matrix with bipartite graph
zbMATH Open document ID: 0668.15005
DOI: 10.1016/0024-3795(89)90569-7
author: S. H. Smith
published in: Linear Algebra and its Applications
publication date: 1989-01-01T00:00:00Z
review text: Signs of coordinates of eigenvectors belonging to the minimum eigenvalue m(B) of an essentially nonnegative symmetric matrix B whose graph G(B) is connected and bipartite are studied. As the main result the following theorem is proved: If G(B) is connected and if \(V_ 1{\dot \cup}V_ 2\) is a proper bipartition of the vertex set of G(B), then there exists an eigenvector x belonging to m(B) such that \(x_ i>0\) for \(i\in V_ 1\) and \(x_ i<0\) for \(i\in V_ 2\). The main technique is the use of the Rayleigh-quotient characterization of m(B). Using this theorem (and 2 other propositions on essentially nonnegative symmetric n-by-n matrices) the theorem by \textit{G. Constantine} [ibid. 65, 171-178 (1985; Zbl 0584.15009)] on lower bounds on the spectra of symmetric matrices with non-negative entries is proved without the restriction \(\mu\geq 0\) and without requiring the Perron-Frobenius theory. The results by \textit{M. Fiedler} [Czech. Math. J. 25(100), 607-618 (1975; Zbl 0325.15014)] on the above matrices with acyclic graph are generalized.
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 15B48
zbMATH DE Number: 4093655
zbMATH Keywords: bipartite graph
zbMATH Keywords: eigenvalues
zbMATH Keywords: Signs of coordinates of eigenvectors
zbMATH Keywords: minimum eigenvalue
zbMATH Keywords: essentially nonnegative symmetric matrix
zbMATH Keywords: Rayleigh- quotient
MaRDI profile type: Publication
full work available at URL: 0024-3795(89)90569-7
OpenAlex ID: W2007490999
cites work: Q4187840
cites work: Lower bounds on the spectra of symmetric matrices with nonnegative entries
cites work: Eigenvectors of acyclic matrices
cites work: Q3877805
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cites work: Characterization of line graphs
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description: scientific article; zbMATH DE number 4093655