The approximation of an eigenvector by ritzvectors

Abstract

Eigenvalue algorithms belonging to the class of the Rayleigh-Ritz methods (Krylov-space methods for example) use `projections' on subspaces to produce approximations to eigenvalues and eigenvectors of a matrix. This paper focuses on the eigenvectors. Two angles are important when considering an eigenvector: the angle between the eigenvector and the best approximating Ritzvector and the angle between the eigenvector and the subspace involved. It is studied how an upperbound for the first angle can be expressed in terms of the second one. This results in a theoretical expression for the case of two-dimensional subspaces and a conjecture for higher dimensional subspaces supported by numerical experiments.Matrices;Eigenvalues;Vectorization;Ritz Values;mathematics