  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.
