New estimates for Ritz vectors

Abstract:

The following estimate for the Rayleigh–Ritz method is proved: \[ | \tilde \lambda - \lambda | |( \tilde u , u )| \le { \| A \tilde u - \tilde \lambda \tilde u \| } \sin \angle \{ u ; \tilde U \}, \| u \| =1. \] Here $A$ is a bounded self-adjoint operator in a real Hilbert/euclidian space, $\{ \lambda , u \}$ one of its eigenpairs, $\tilde U$ a trial subspace for the Rayleigh–Ritz method, and $\{ \tilde \lambda , \tilde u \}$ 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 $|( \tilde u , u )| \le C \epsilon ^2,$ if an eigenvector $u$ is close to the trial subspace with accuracy $\epsilon$ and a Ritz vector $\tilde u$ is an $\epsilon$ approximation to another eigenvector, with a different eigenvalue. Generalizations of the estimate to the cases of eigenspaces and invariant subspaces are suggested, and estimates of approximation of eigenspaces and invariant subspaces are proved.