Componentwise Perturbation Analysis of the Singular Value Decomposition of a Matrix

Abstract

A rigorous perturbation analysis is presented for the singular value decomposition (SVD) of a real matrix with full column rank. It is proved that the SVD perturbation problem is well posed only when the singular values are distinct. The analysis involves the solution of symmetric coupled systems of linear equations. It produces asymptotic (local) componentwise perturbation bounds on the entries of the orthogonal matrices participating in the decomposition of the given matrix and on its singular values. Local bounds are derived for the sensitivity of the singular subspaces measured by the angles between the unperturbed and perturbed subspaces. Determining the asymptotic bounds of the orthogonal matrices and the sensitivity of singular subspaces requires knowing only the norm of the perturbation of the given matrix. An iterative scheme is described to find global bounds on the respective perturbations, and results from numerical experiments are presented.

Keywords

• singular value decomposition (SVD)
• singular values
• singular subspaces
• perturbation analysis
• componentwise perturbation bounds