ar X iv : 0 90 9 . 22 14 v 1 [ m at h . PR ] 1 1 Se p 20 09 The single ring theorem

We study the empirical measure L A n of the eigenvalues of non-normal square matrices of the form A n = U n D n V n with U n ,V n independent Haar distributed on the unitary group and D n real diagonal. We show that when the empirical measure of the eigenvalues of D n converges, and D n satisfies some technical conditions, L A n converges towards a rotationally invariant measure on the complex plan whose support is a single ring. In particular, we provide a complete proof of Feinberg-Zee single ring theorem [5]. We also consider the case where U n ,V n are independent Haar distributed on the orthogonal group.