A note on eigenvalues of perturbed Hermitian matrices

LetA=H1E∗EH2andA∼=H1OOH2 be Hermitian matrices with eigenvalues λ1 ⩾ ⋯ ⩾ λk and λ∼1⩾⋯⩾λ∼k, respectively. Denote by ∥E∥ the spectral norm of the matrix E, and η the spectral gap between the spectra of H1 and H2. It is shown that|λi-λ∼i|⩽2‖E‖2η+η2+4‖E‖2, which improves all the existing results. Similar bounds are obtained for singular values of matrices under block perturbations.