On Stewart's Perturbation Theorem for SVD

TL;DR

This paper presents a new variant of Stewart's perturbation theorem for the singular value decomposition, providing sharper bounds using spectral and unitarily invariant norms, with applications to Sylvester equations.

Contribution

A novel version of Stewart's theorem for SVD perturbations using spectral and unitarily invariant norms, improving bound sharpness and applicability.

Findings

Derived sharper perturbation bounds for SVD subspaces.

Extended bounds to solutions of Sylvester equations.

Compared new bounds with existing Stewart's theorem results.

Abstract

This paper establishes a variant of Stewart's theorem (Theorem~6.4 of Stewart, {\em SIAM Rev.}, 15:727--764, 1973) for the singular subspaces associated with the SVD of a matrix subject to perturbations. Stewart's original version uses both the Frobenius and spectral norms, whereas the new variant uses the spectral norm and any unitarily invariant norm that offer choices per convenience of particular applications and lead to sharper bounds than that straightforwardly derived from Stewart's original theorem with the help of the well-known equivalence inequalities between matrix norms. Of interest in their own right, bounds on the solution to two couple Sylvester equations are established for a few different circumstances.