# Majorization bounds for Ritz values of self-adjoint matrices

## TL;DR

This paper develops new bounds for Ritz value changes in self-adjoint matrices using submajorization, confirming recent conjectures and improving existing theorems, with implications for eigenvalue perturbation analysis.

## Contribution

It introduces novel submajorization-based bounds for Ritz value variations and confirms recent conjectures, extending known results to arbitrary subspaces.

## Key findings

- Proved conjectures by Knyazev, Argentati, and Zhu.
- Extended bounds from one-dimensional to arbitrary subspaces.
- Improved the $ an heta$ theorem of Davis and Kahan.

## Abstract

A priori, a posteriori, and mixed type upper bounds for the absolute change in Ritz values of self-adjoint matrices in terms of submajorization relations are obtained. Some of our results prove recent conjectures by Knyazev, Argentati, and Zhu, which extend several known results for one dimensional subspaces to arbitrary subspaces. In addition, we improve Nakatsukasa's version of the $\tan \Theta$ theorem of Davis and Kahan. As a consequence, we obtain new quadratic a posteriori bounds for the absolute change in Ritz values.