Regularization of Non-Normal Matrices by Gaussian Noise

We consider the regularization of matrices MN in Jordan form by additive Gaussian noise N{-\gamma }GN, where GN is a matrix of i.i.d. standard Gaussians and \gamma >{\tfrac {1}{2}} so that the operator norm of the additive noise tends to 0 with N. Under mild conditions on the structure of MN, we evaluate the limit of the empirical measure of eigenvalues of MN+N{-\gamma } GN and show that it depends on \gamma, in contrast with the case of a single Jordan block.