Correction of Errors in the First-Order Perturbation Expansions of Singular Vectors

TL;DR

This paper identifies and corrects significant errors in the first-order perturbation analysis of singular vectors as presented in Stewart's influential book, improving the accuracy of theoretical results in matrix perturbation theory.

Contribution

It provides corrected formulas for first-order perturbation expansions of singular vectors, addressing errors in Stewart's 2001 work.

Findings

Corrected first-order perturbation formulas for singular vectors

Improved theoretical accuracy in matrix perturbation analysis

Clarified previous misconceptions in Stewart's analysis

Abstract

This note corrects some serious errors in the first-order perturbation analysis of singular vectors as published by G.W. Stewart in his book "Matrix Algorithms Volume II: Eigensystems", SIAM, 2001.