  We consider the eigenvalues and eigenvectors of small rank perturbations of
random $N\times N$ matrices. We allow the rank of perturbation $M$ increases
with $N$, and the only assumption is $M=o(N)$. In both additive and
multiplicative perturbation models, we prove rigidity results for the outliers
of the perturbed random matrices. Based on the rigidity results we derive the
empirical distribution of outliers of the perturbed random matrices. We also
compute the appropriate projection of eigenvectors corresponding to the
outliers of the perturbed random matrices, which are approximate eigenvectors
of the perturbing matrix. Our results can be regarded as the extension of the
finite rank perturbation case to the full generality up to $M=o(N)$.
