  We consider the eigenvalues and eigenvectors of finite, low rank
perturbations of random matrices. Specifically, we prove almost sure
convergence of the extreme eigenvalues and appropriate projections of the
corresponding eigenvectors of the perturbed matrix for additive and
multiplicative perturbation models. The limiting non-random value is shown to
depend explicitly on the limiting eigenvalue distribution of the unperturbed
random matrix and the assumed perturbation model via integral transforms that
correspond to very well known objects in free probability theory that linearize
non-commutative free additive and multiplicative convolution. Furthermore, we
uncover a phase transition phenomenon whereby the large matrix limit of the
extreme eigenvalues of the perturbed matrix differs from that of the original
matrix if and only if the eigenvalues of the perturbing matrix are above a
certain critical threshold. Square root decay of the eigenvalue density at the
edge is sufficient to ensure that this threshold is finite. This critical
threshold is intimately related to the same aforementioned integral transforms
and our proof techniques bring this connection and the origin of the phase
transition into focus. Consequently, our results extend the class of `spiked'
random matrix models about which such predictions (called the BBP phase
transition) can be made well beyond the Wigner, Wishart and Jacobi random
ensembles found in the literature. We examine the impact of this eigenvalue
phase transition on the associated eigenvectors and observe an analogous phase
transition in the eigenvectors. Various extensions of our results to the
problem of non-extreme eigenvalues are discussed.
