Quadratic matrix polynomials with a parameter
type: Item
instance of: scholarly article
title: Quadratic matrix polynomials with a parameter
zbMATH Open document ID: 0623.15003
DOI: 10.1016/0196-8858(86)90036-9
author: I. Gohberg
author: P. Lancaster
author: Leiba Rodman
published in: Advances in Applied Mathematics
publication date: 1986-01-01T00:00:00Z
review text: Consider a matrix polynomial of the form \(L(\lambda)=A\lambda^ 2+\epsilon B\lambda +C\) where A, B and C are positive definite \(n\times n\) matrices and \(\epsilon\) is a non-negative parameter. Such polynomials arise in a natural way in the study of damped vibrating systems. The present paper examines the behaviour of the (not necessarily real) zeroes of \(d(\lambda,\epsilon)=\det L(\lambda)\) as the parameter \(\epsilon\) varies. First it is shown that for almost all L we have the ''generic case'': d(\(\lambda\),\(\epsilon)\) has at least 2n-1 distinct zeroes in the complex plane for each positive real \(\epsilon\), and it has 2n zeroes except for finitely many \(\epsilon\). It is then shown that the number of exceptional values of \(\epsilon\) always lies between n and \(2n^ 2\). Finally (Theorem 8.2), it is shown that if \(A=I\), and C has n different eigenvalues, then \(L(\lambda)=(I\lambda -Z_ 2(\epsilon))(I\lambda -Z_ 1(\epsilon))\) where \(I\lambda -Z_ 1(\epsilon)\) is the unique analytic continuation of \(I\lambda -iC^{1/2}\). If we suppose that C has been diagonalized, then the entries of \(Z_ 1(\epsilon)\) are analytic functions of the real variable \(\epsilon\) with no poles or branch points, except for possible branch points of multiplicity 2 in the diagonal entries.
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 15A22
Mathematics Subject Classification ID: 15A60
Mathematics Subject Classification ID: 70J25
Mathematics Subject Classification ID: 30C15
Mathematics Subject Classification ID: 34A30
zbMATH DE Number: 4010685
zbMATH Keywords: generalized eigenvalues
zbMATH Keywords: matrix polynomial
zbMATH Keywords: damped vibrating systems
MaRDI profile type: Publication
full work available at URL: 0196-8858(86)90036-9
OpenAlex ID: W2000031927
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description: scientific article; zbMATH DE number 4010685