A secular equation for the eigenvalues of a diagonal matrix perturbation
type: Item
instance of: scholarly article
title: A secular equation for the eigenvalues of a diagonal matrix perturbation
zbMATH Open document ID: 0861.15006
DOI: 10.1016/0024-3795(94)00314-9
author: Joel Anderson
published in: Linear Algebra and its Applications
publication date: 1997-04-29T00:00:00Z
review text: For a diagonal matrix \(D\) with complex entries \(\alpha_i\), the matrices \(P=D + x_1^*w_1+ \cdots +x^*_rw_r\), and the existence of a secular equation \(F(\lambda)\) are considered. An algorithm is presented to construct \(F\). It is shown that if \(\lambda\) is not an eigenvalue for \(D\), then \(\lambda\) is an eigenvalue for \(P\) if and only if \(F(\lambda) =0\). In the rank one self-adjoint perturbation the eigenvalues of \(P\) and \(D\) must interlace if the eigenvalues of \(D\) are distinct and the perturbation matrix is irreducible. The eigenvalues may be complex and of any multiplicity. An algorithm for determining the secular formula for a tridiagonal matrix is presented. The case of the self-adjoint perturbation of \(D\) is studied. The determination of the multiplicities of the eigenvalues of \(D\) as eigenvalues of \(P\) is discussed.
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 65F30
zbMATH DE Number: 951790
zbMATH Keywords: diagonal matrix perturbation
zbMATH Keywords: secular equation
zbMATH Keywords: eigenvalue
zbMATH Keywords: algorithm
zbMATH Keywords: tridiagonal matrix
zbMATH Keywords: self-adjoint perturbation
MaRDI profile type: Publication
full work available at URL: 0024-3795(94)00314-9
OpenAlex ID: W2045130512
cites work: A Divide and Conquer method for the symmetric tridiagonal eigenproblem
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cites work: Q3241399
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cites work: Q4039860
Wikidata QID: Q127633568
description: scientific article; zbMATH DE number 951790