  We prove quadratic eigenvalue perturbation bounds for generalized Hermitian
eigenvalue problems. The bounds are proportional to the square of the norm of
the perturbation matrices divided by the gap between the spectrums. Using the
results we provide a simple derivation of the first-order perturbation
expansion of a multiple eigenvalue, whose trailing term is tighter than known
results. We also present quadratic bounds for the non-Hermitian case.
