The single ring theorem

Abstract

We study the empirical measure $L_{A_n}$ of the eigenvalues of nonnormal square matrices of the form $A_n=U_nT_nV_n$ with $U_n,V_n$ independent Haar distributed on the unitary group and $T_n$ real diagonal. We show that when the empirical measure of the eigenvalues of $T_n$ converges, and $T_n$ satisfies some technical conditions, $L_{A_n}$ 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 [FZ]. We also consider the case where $U_n,V_n$ are independently Haar distributed on the orthogonal group.