Majorization bounds for SVD

Given an approximating singular subspace of a matrix, in this paper, two kind of majorization type bounds on the singular value errors by the canonical angles between the singular subspaces and its approximations are obtained. From these results, based on the information about approximation accuracies of a pair of approximate singular subspaces, several bounds can be directly obtained to estimate how accurate the approximate singular values are. These results are helpful to understand how approximate singular values converge to the corresponding exact singular values in the projection subspace type algorithms.