First order spectral perturbation theory of square singular matrix polynomials

We develop first order eigenvalue expansions of one-parametric perturbations of square singular matrix polynomials. Although the eigenvalues of a singular matrix polynomial P(λ) 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 λ0 of P(λ) we prove that, for generic perturbations M(λ) of degree at most the degree of P(λ), the eigenvalues of P(λ)+ϵM(λ) admit covergent series expansions near λ0 and we describe the first order term of these expansions in terms of M(λ0) and certain particular bases of the left and right null spaces of P(λ0). In the important case of λ0 being a semisimple eigenvalue of P(λ) any bases of the left and right null spaces of P(λ0) 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.