Relative Perturbation Techniques for Singular Value Problems

A technique is presented for deriving bounds on the relative change in the singular values of a real matrix (or the eigenvalues of a real symmetric matrix) due to a perturbation, as well as bounds on the angles between the unperturbed and perturbed singular vectors (or eigenvectors). The class of perturbations considered consists of all δ B for which B + δ B = DL BDR for some nonsingular matrices DL and DR. This class includes componentwise relative perturbations of a bidiagonal or biacyclic matrix and perturbations that annihilate the off-diagonal block in a block triangular matrix. Many existing relative perturbation and deflation bounds are derived from results for this general class of perturbations. Also some new relative perturbation and deflation results for the singular values and vectors of biacyclic, triangular, and shifted triangular matrices are presented.