First Order Spectral Perturbation Theory of Square Singular Matrix Polynomials Articles uri icon

publication date

  • February 2010

start page

  • 892

end page

  • 910


  • 4


  • 432

International Standard Serial Number (ISSN)

  • 0024-3795

Electronic International Standard Serial Number (EISSN)

  • 1873-1856


  • We develop first order eigenvalue expansions of one-parametric 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.