The single ring theorem

Abstract

We study the empirical measure LA n of the eigenvalues of nonnormal square matrices of the form An = UnTnVn with Un, Vn independent Haar distributed on the unitary group and Tn real diagonal. We show that when the empirical measure of the eigenvalues of Tn converges, and Tn satisfies some technical conditions, LA n converges towards a rotationally invariant measure on the complex plane whose support is a single ring. In particular, we provide a complete proof of the Feinberg-Zee single ring theorem We also consider the case where Un, Vn are independently Haar distributed on the orthogonal group.