On condition numbers of polynomial eigenvalue problems with nonsingular leading coefficients

Dedicated to the memory of James H. Wilkinson (1919–1986) In this paper, we investigate condition numbers of eigenvalue problems of matrix polynomials with nonsingular leading coefficients, generalizing classical results of matrix perturbation theory. We provide a relation between the con-dition numbers of eigenvalues and the pseudospectral growth rate. We obtain that if a simple eigenvalue of a matrix polynomial is ill-conditioned in some re-spects, then it is close to be multiple, and we construct an upper bound for this distance (measured in the euclidean norm). We also derive a new expression for the condition number of a simple eigenvalue, which does not involve eigenvectors. Moreover, an Elsner-like perturbation bound for matrix polynomials is presented