Combined perturbation bounds: I. Eigensystems and singular value decompositions

In this paper we present some new combined perturbation bounds of eigenvalues and eigensubspaces for a Hermitian matrix H, particularly in an asymptotic sense, delta(2)(12)parallel to sin Theta((U) over tilde (1))parallel to(2)(F)+ Sigma(r)(i=1)(lambda(i) - (lambda) over tilde (i))(2) <= parallel to Delta HU1 parallel to(2)(F) + O(parallel to Delta HU1 parallel to(4)(F)), where lambda(i) denotes the eigenvalues of H and U-1 the eigensubspace corresponding to the eigenvalues lambda(i), i = 1, 2,..., r. The bound for each factor of eigensystems is optimal due to the sin Theta theorem and the Hoffman-Wielandt theorem. In addition, combined perturbation bounds for singular value decompositions and combined perturbation bounds in some, more general, measures are also obtained.