Absolute Variation of Ritz Values, Principal Angles, and Spectral Spread

Abstract

Let A be a d×d complex self-adjoint matrix, X,Y⊂Cd be k-dimensional subspaces and let X be a d×k complex matrix whose columns form an orthonormal basis of X. We construct a d×k complex matrix Yr whose columns form an orthonormal basis of Y and obtain sharp upper bounds for the singular values s(X∗AX−Y∗rAYr) in terms of submajorization relations involving the principal angles between X and Y and the spectral spread of A. We apply these results to obtain sharp upper bounds for the absolute variation of the Ritz values of A associated with the subspaces X and Y, that partially confirm conjectures by Knyazev and Argentati.