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, E r ( μ ) and E r , 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 E r , k ( μ ) , and derive an upper bound for E r ( μ ) by constructing an associated perturbation of P ( λ ) .