Weyl-type relative perturbation bounds for eigensystems of Hermitian matrices Original Research Article

We present a Weyl-type relative bound for eigenvalues of Hermitian perturbations A+E of (not necessarily definite) Hermitian matrices A. This bound, given in function of the quantity η=short parallelA−1/2EA−1/2short parallel2, that was already known in the definite case, is shown to be valid as well in the indefinite case. We also extend to the indefinite case relative eigenvector bounds which depend on the same quantity η. As a consequence, new relative perturbation bounds for singular values and vectors are also obtained. Using matrix differential calculus techniques we obtain for eigenvalues a sharper, first-order bound involving the logarithm matrix function, which is smaller than η not only for small E, as expected, but for any perturbation.