Quadratic Residual Bounds for the Hermitian Eigenvalue Problem

Let A = " M R R N # and ~ A = " M 0 0 N # be Hermitian matrices. Stronger and more general O(kRk 2) bounds relating the eigen-values of A and ~ A are proved using a Schur complement technique. These results extend to singular values and to eigenvalues of non-Hermitian matrices. (1) be Hermitian matrices. Since jjA ? ~ Ajj = jjRjj one can bound the diierence between their eigenvalues in terms of jjRjj: It is part of the folklore of numerical linear algebra that if the spectra of M and N are well separated then a residual of size jjRjj produces a perturbation of size O(jjRjj 2) in the eigenvalues. The quadratic bounds that we prove are stronger, simpler and more general than those in the literature. For an n n Hermitian matrix X let 1 (X) 2 (X) n (X) denote its ordered eigenvalues. Throughout we shall assume that A and ~ A are as in (1) and that M is m m and N is n n: Let k = k (A) and let ~ k = k (~ A): The eigenvalues of ~ If M and N have common eigenvalues there will be some freedom in the choice of indices i j : We let (X) denote the set of eigenvalues of X and use k k to denote the spectral norm (often called the 2-norm by numerical analysts).