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 condition numbers of eigenvalues and the pseudospectral growth rate. We
obtain that if a simple eigenvalue of a matrix polynomial is ill-conditioned in
some respects, 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.Comment: 4 figure
