Remarks on the perturbation of analytic matrix functions
type: Item
instance of: scholarly article
title: Remarks on the perturbation of analytic matrix functions
zbMATH Open document ID: 0605.47013
DOI: 10.1007/BF01204631
author: B. Najman
published in: Integral Equations and Operator Theory
publication date: 1986-01-01T00:00:00Z
review text: Let \(A(\lambda)\) be a hermitian matrix function, analytic in a neighbourhood of \(\lambda_ 0=0\) and let \(B(\lambda,\epsilon) = A(\lambda) + \epsilon H(\lambda)\) be its perturbation. We say that \({\tilde\lambda}\) is an eigenvalue of \(B(\cdot,\epsilon)\) if det \(B({\tilde\lambda},\epsilon)=0.\) Assume that \(\lambda_ 0\) is an eigenvalue of \(B(\cdot,0)=A(\cdot)\) of partial multiplicities \(m_ 1\leq...\leq m_ r\) and let \(m=m_ 1+...+m_ r\). The main result states that there are exactly m eigenvalues \(\lambda_ j(\epsilon)\) of \(B(\lambda,\epsilon)\) for small \(\epsilon\) and that these m eigenvalues split into r groups of \(m_ j\) \((j=1,...,r)\) eigenvalues which have Puiseux expansions provided an algebraic condition on \(H(0)\) holds. No symmetry assumption on \(H(\lambda)\) is needed. The method of the proof is rather simple, particularly in comparison with an earlier result of I. Gohberg, P. Lancaster and L. Rodman where the same result is proved for hermitian \(H(\lambda)\). However in the paper is an error; the proof is correct only if \(m_ 1<...<m_ r\). If this is not the case, the result is still generically true, i.e. there is a dense open set of matrices \(H(0)\) for which the result holds as stated. Nevertheless it is possible to find examples where \(m_ 1=m_ 2\) and \(\lambda_ j(\epsilon)\) form one group of \(m_ 1+m_ 2\) eigenvalues.
Mathematics Subject Classification ID: 47A56
Mathematics Subject Classification ID: 47A10
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 47A70
zbMATH DE Number: 3977735
zbMATH Keywords: hermitian matrix function
zbMATH Keywords: Puiseux expansions
MaRDI profile type: Publication
cites work: Q4059471
cites work: Matrices and indefinite scalar products
cites work: Q5526485
cites work: Q5342712
cites work: Q5683231
full work available at URL: https://doi.org/10.1007/bf01204631
OpenAlex ID: W4249255402
description: scientific article; zbMATH DE number 3977735