The extension of Weyl-type relative perturbation bounds

TL;DR

This paper extends relative perturbation bounds to possibly singular Hermitian matrices, introduces a general class of such bounds, and compares their sharpness with classical Weyl bounds, enhancing understanding of matrix perturbations.

Contribution

It develops new relative bounds for Hermitian matrices, including singular and rank-deficient cases, and analyzes invariance properties under congruence transformations.

Findings

Derived bounds are invariant under certain transformations

Extended bounds apply to singular and rank-deficient matrices

Compared sharpness with Weyl's absolute perturbation bounds

Abstract

We extend several relative perturbation bounds to Hermitian matrices that are possibly singular, and also develop a general class of relative bounds for Hermitian matrices. As a result, corresponding relative bounds for singular values of rank-deficient $m\times n$ matrices are also obtained using the Jordan-Wielandt matrices. We also present that the main relative bound derived would be invariant with respect to congruence transformation under certain conditions, and compare its sharpness with the Weyl's absolute perturbation bound.