The extension of Weyl-type relative perturbation bounds

We extend several relative perturbation bounds to Hermitian matrices that are possibly singular, and also develop a general class of relative bounds for Hermitian matrices. As a result, corresponding relative bounds for singular values of rank-deficient $m\times n$ matrices are also obtained using the Jordan-Wielandt matrices. We also present that the main relative bound derived would be invariant with respect to congruence transformation under certain conditions, and compare its sharpness with the Weyl's absolute perturbation bound.