Matrix regularizing effects of Gaussian perturbations

The addition of noise has a regularizing effect on Hermitian matrices. This effect is studied here for H = A + V, where A is the base matrix and V is sampled from the GOE or the GUE random matrix ensembles. We bound the mean number of eigenvalues of H in an interval, and present tail bounds for the distribution of the Frobenius and operator norms of H-1 and for the distribution of the norm of H-1 applied to a fixed vector. The bounds are uniform in A and exceed the actual suprema by no more than multiplicative constants. The probability of multiple eigenvalues in an interval is also estimated.