  We consider the eigenvalues and eigenvectors of matrices of the form M + P,
where M is an n by n Wigner random matrix and P is an arbitrary n by n
deterministic matrix with low rank. In general, we show that none of the
eigenvalues of M + P need be real, even when P has rank one. We also show that,
except for a few outlier eigenvalues, most of the eigenvalues of M + P are
within 1/n of the real line, up to small order corrections. We also prove a new
result quantifying the outlier eigenvalues for multiplicative perturbations of
the form S ( I + P ), where S is a sample covariance matrix and I is the
identity matrix. We extend our result showing all eigenvalues except the
outliers are close to the real line to this case as well. As an application, we
study the critical points of the characteristic polynomials of nearly Hermitian
random matrices.
