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We consider the eigenvalues and eigenvectors of finite, low rankperturbations of random matrices. Specifically, we prove almost sureconvergence of the extreme eigenvalues and appropriate projections of thecorresponding eigenvectors of the perturbed matrix for additive andmultiplicative perturbation models. The limiting non-random value is shown todepend explicitly on the limiting eigenvalue distribution of the unperturbedrandom matrix and the assumed perturbation model via integral transforms thatcorrespond to very well known objects in free probability theory that linearizenon-commutative free additive and multiplicative convolution. Furthermore, weuncover a phase transition phenomenon whereby the large matrix limit of theextreme eigenvalues of the perturbed matrix differs from that of the originalmatrix if and only if the eigenvalues of the perturbing matrix are above acertain critical threshold. Square root decay of the eigenvalue density at theedge is sufficient to ensure that this threshold is finite. This criticalthreshold is intimately related to the same aforementioned integral transformsand our proof techniques bring this connection and the origin of the phasetransition into focus. Consequently, our results extend the class of `spiked'random matrix models about which such predictions (called the BBP phasetransition) can be made well beyond the Wigner, Wishart and Jacobi randomensembles found in the literature. We examine the impact of this eigenvaluephase transition on the associated eigenvectors and observe an analogous phasetransition in the eigenvectors. Various extensions of our results to theproblem of non-extreme eigenvalues are discussed.

The eigenvalues and eigenvectors of finite, low rank perturbations of large random matrices