A note on eigenvalues of perturbed Hermitian matrices

Abstract Let A = H 1 E ∗ E H 2 and A ∼ = H 1 O O H 2 be Hermitian matrices with eigenvalues λ 1  ⩾ ⋯ ⩾  λ k 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 H 1 and H 2 . It is shown that | λ i - λ ∼ i | ⩽ 2 ‖ E ‖ 2 η + η 2 + 4 ‖ E ‖ 2 , which improves all the existing results. Similar bounds are obtained for singular values of matrices under block perturbations.