Andrew Knyazev
Department of Mathematical Sciences
University of Colorado Denver and Health Sciences Center

Majorization for Changes in Ritz Values and Canonical Angles Between Subspaces

We prove that the absolute value of the difference of the squares of the cosines/sines of angles between subspaces is majorized by the sines of the angles between the perturbed subspaces, with a constant of one. We show that this result can be interpreted as a bound on the change of the Ritz values in the Rayleigh-Ritz method with the change of the trial subspace, in a particular case where the Rayleigh-Ritz method is applied to an orthogonal projector. We then prove the general result for an arbitrary Hermitian operator, not necessarily a projector, where the constant becomes the difference between the largest and the smallest eigenvalues of the operator. Our proof is based on an extension of an arbitrary Hermitian operator to an orthogonal projector.

[1] A. V. Knyazev and M. E. Argentati, Majorization for Changes in Angles Between Subspaces, Ritz values, and graph Laplacian spectra, SIMAX, 29 (2006) pp. 15-32. http://dx.doi.org/10.1137/060649070