The singular values and vectors of low rank perturbations of large rectangular random matrices

Abstract

In this paper, we consider the singular values and singular vectors of finite, low rank perturbations of large rectangular random matrices. Specifically, we prove almost sure convergence of the extreme singular values and appropriate projections of the corresponding singular vectors of the perturbed matrix.As in the prequel, where we considered the eigenvalues of Hermitian matrices, the non-random limiting value is shown to depend explicitly on the limiting singular value distribution of the unperturbed matrix via an integral transform that linearizes rectangular additive convolution in free probability theory. The asymptotic position of the extreme singular values of the perturbed matrix differs from that of the original matrix if and only if the singular values of the perturbing matrix are above a certain critical threshold which depends on this same aforementioned integral transform.We examine the consequence of this singular value phase transition on the associated left and right singular eigenvectors and discuss the fluctuations of the singular values around these non-random limits.