Distance bounds for prescribed multiple eigenvalues of matrix polynomials

In this paper, motivated by a problem posed by Wilkinson, we study the coefficient perturbations of a (square) matrix polynomial to a matrix polynomial that has a prescribed eigenvalue of specified algebraic multiplicity and index of annihilation. For an n×n matrix polynomial P(λ) and a given scalar μ∈C, we introduce two weighted spectral norm distances, Er(μ) and Er,k(μ), from P(λ) to the n×n matrix polynomials that have μ as an eigenvalue of algebraic multiplicity at least r and to those that have μ as an eigenvalue of algebraic multiplicity at least r and maximum Jordan chain length (exactly) k, respectively. Then we obtain a lower bound for Er,k(μ), and derive an upper bound for Er(μ) by constructing an associated perturbation of P(λ).