  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 eigenvalue aspect
of the problem, the non-random limiting value is shown to depend explicitly on
the limiting singular value distribution of the unperturbed matrix via an
integral transforms that linearizes rectangular additive convolution in free
probability theory. The large matrix limit 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 finite $n$
fluctuations above these non-random limits.
