Matrix regularizing effects of Gaussian perturbations

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