Rayleigh-Ritz majorization error bounds of the mixed type

TL;DR

This paper develops new majorization-based bounds for the eigenvalue changes of Hermitian matrices when using subspace approximations, extending previous vector-based bounds to multidimensional subspaces.

Contribution

It introduces novel bounds for eigenvalue perturbations using majorization, relating residual matrices' singular values and principal angles between subspaces.

Findings

Derived bounds relate eigenvalue changes to residual singular values and principal angles.

Extended vector-based bounds to multidimensional subspace settings.

Connected bounds to eigenvalue changes due to block discarding or additive perturbations.

Abstract

The absolute change in the Rayleigh quotient (RQ) for a Hermitian matrix with respect to vectors is bounded in terms of the norms of the residual vectors and the angle between vectors in [\doi{10.1137/120884468}]. We substitute multidimensional subspaces for the vectors and derive new bounds of absolute changes of eigenvalues of the matrix RQ in terms of singular values of residual matrices and principal angles between subspaces, using majorization. We show how our results relate to bounds for eigenvalues after discarding off-diagonal blocks or additive perturbations.