Cluster-robust accuracy bounds for Ritz subspaces

Given an approximating subspace for a Hermitian matrix A, the Rayleigh-Ritz procedure is commonly used to compute a few approximate eigenvalues (called Ritz values) and corresponding approximate eigenvectors (called Ritz vectors). In this paper, new bounds on the canonical angles between the invariant subspace of A associated with its few extreme (smallest or largest) eigenvalues and its approximating Ritz subspace in terms of the differences between Ritz values and the targeted eigenvalues are obtained. From this result, various bounds are readily available to estimate how accurate the Ritz vectors computed from the approximating subspace may be, based on information on approximation accuracies in the Ritz values. The result is helpful in understanding how Ritz vectors move towards eigenvectors while Ritz values are made to move towards eigenvalues.