Regularization of Non-Normal Matrices by Gaussian Noise

Abstract

We consider the regularization of matrices |$M^N$| in Jordan form by additive Gaussian noise |$N^{-\gamma }G^N$|⁠, where |$G^N$| 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 |$M^N,$| we evaluate the limit of the empirical measure of eigenvalues of |$M^N+N^{-\gamma } G^N$| and show that it depends on |$\gamma$|⁠, in contrast with the case of a single Jordan block.