Absolute variation of Ritz values, principal angles and spectral spread

TL;DR

This paper derives sharp bounds on the variation of Ritz values and spectral spread of self-adjoint matrices, using principal angles between subspaces, partially confirming existing conjectures.

Contribution

It introduces new bounds for Ritz value variations based on principal angles and spectral spread, advancing understanding of subspace perturbations.

Findings

Established sharp upper bounds for singular value differences of Ritz values.

Connected Ritz value variation bounds to principal angles and spectral spread.

Partially confirmed conjectures by Knyazev and Argentati.

Abstract

Let $A$ be a $d\times d$ complex self-adjoint matrix, $\mathcal{X},\mathcal{Y}\subset \mathbb{C}^d$ be $k$-dimensional subspaces and let $X$ be a $d\times k$ complex matrix whose columns form an orthonormal basis of $\mathcal{X}$. We construct a $d\times k$ complex matrix $Y_r$ whose columns form an orthonormal basis of $\mathcal{Y}$ and obtain sharp upper bounds for the singular values $s(X^*AX-Y_r^*\,A\,Y_r)$ in terms of submajorization relations involving the principal angles between $\mathcal{X}$ and $\mathcal{Y}$ 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 $\mathcal{X}$ and $\mathcal{Y}$, that partially confirm conjectures by Knyazev and Argentati.