NEW ESTIMATES FOR RITZ VECTORS

The following estimate for the Rayleigh-Ritz method is proved: \<(lamb da)over tilde> - lambda\\((u) over tilde, u)\ less than or equal to pa rallel to A<(u)over tilda> - <(lambda)over tilde>(u) over tilde parall el to sin angle {u; (U) over tilde}, parallel to u parallel to = 1. He re A is a bounded self-adjoint operator in a real Hilbert/euclidian sp ace, {lambda, u) one of its eigenpairs, (U) over bar a trial subspace for the Rayleigh-Ritz method, and {lambda, (u) over bar} a Ritz pair. This inequality makes it possible to analyze the fine structure of the error of the Rayleigh-Ritz method, in particular, it shows that \((u) over tilde, u)\ C epsilon(2), if an eigenvector u is close to the tri al subspace with accuracy epsilon and a Ritz vector (u) over tilde is an epsilon approximation to another eigenvector, with a different eige nvalue. Generalizations of the estimate to the cases of eigenspaces an d invariant subspaces are suggested, and estimates of approximation of eigenspaces and invariant subspaces are proved.