The single ring theorem

We study the empirical measure LA of the eigenvalues of nonnormal square matrices of the form A(n) = UnTnVn, with U-n, V-n independent Haar distributed on the unitary group and T-n diagonal. We show that when the empirical measure of the eigenyalues of T-n converges, and T-n satisfies some technical conditions, L-An converges towards a rotationally invariant measure mu on the complex plane whose support is a single ring. In particular, we provide a complete proof of the Feinberg-Zee single ring theorem [6]. We also consider the case where U-n, V-n are independently Haar distributed on the orthogonal group.