Combinatorial eigenvalues of matrices
type: Item
instance of: scholarly article
title: Combinatorial eigenvalues of matrices
zbMATH Open document ID: 0756.15015
DOI: 10.1016/0024-3795(92)90310-7
published in: Linear Algebra and its Applications
publication date: 1993-01-16T00:00:00Z
review text: Given \(A\in C^{n\times n}\), its spectrum \(\sigma(A)\) and the subset \(S\subseteq\) \{the set of diagonal entries of \(A\)\} one shows that the existence of a critical diagonal \(S\) (i.e. \(S\) and \(\sigma(A)\) have a common element) is equivalent to the matrix \(\tilde A\), obtained from \(A\) by setting its diagonal entries equal to zero, having an \(s\times t\) zero submatrix with \(s+t\geq n+1\) (equality holding when \(S\) is minimal). The study of the eigenspace of an eigenvalue associated with the critical diagonal \(S\), and observations regarding the Frobenius normal form of \(A\) and reducibility of \(A\) by a combinatorial approach are given. Several examples illustrate the theoretical treatment.
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 15A21
zbMATH DE Number: 95484
zbMATH Keywords: critical diagonal
zbMATH Keywords: eigenvalue
zbMATH Keywords: Frobenius normal form
zbMATH Keywords: reducibility
author: John S. Maybee
author: D. D. Olesky
author: M. Tsatsomeros
author: P. van den Driessche
reviewed by: Mihail Voicu
MaRDI profile type: Publication
full work available at URL: 0024-3795(92)90310-7
OpenAlex ID: W2084300474
cites work: Sign patterns that require exactly one real eigenvalue and patterns that requiren−1 nonreal eigenvalues
cites work: Combinatorial matrix theory
cites work: Systems of representatives
cites work: N-matrices
cites work: Q5342712
description: scientific article; zbMATH DE number 95484