Perturbation behavior of a multiple eigenvalue in generalized Hermitian eigenvalue problems

We show that a multiple eigenvalue has different sensitivities under perturbations in a generalized Hermitian eigenvalue problem. Our result provides a solution to a question raised by Stewart and Sun. We also show how this difference of sensitivities plays a role in the eigenvalue forward error analysis after the Rayleigh-Ritz process, for which we present an approach that provides tight bounds.