First Order Spectral Perturbation Theory of Square Singular Matrix Polynomials Articles
Overview
published in
publication date
 February 2010
start page
 892
end page
 910
issue
 4
volume
 432
Digital Object Identifier (DOI)
International Standard Serial Number (ISSN)
 00243795
Electronic International Standard Serial Number (EISSN)
 18731856
abstract

We develop first order eigenvalue expansions of oneparametric perturbations of square singular matrix polynomials. Although the eigenvalues of a singular matrix polynomial P(lambda)
are not continuous functions of the entries of the coefficients of the
polynomial, we show that for most perturbations they are indeed
continuous. Given an eigenvalue lambda0 of P(lambda) we prove that, for generic perturbations M(lambda) of degree at most the degree of P(lambda), the eigenvalues of P(lambda)+ϵM(lambda) admit covergent series expansions near lambda0 and we describe the first order term of these expansions in terms of M(lambda0) and certain particular bases of the left and right null spaces of P(lambda0). In the important case of lambda0 being a semisimple eigenvalue of P(lambda) any bases of the left and right null spaces of P(lambda0)
can be used, and the first order term of the eigenvalue expansions
takes a simple form. In this situation we also obtain the limit vector
of the associated eigenvector expansions.