What question did this study set out to answer?

This work aims to analyze the convergence of empirical measures of eigenvalues for specific square matrices.

July 29, 2011Open Access

The single ring theorem

AGAlice GuionnetCentre National de la Recherche Scientifique MKManjunath KrishnapurIndian Institute of Science Bangalore OZOfer ZeitouniCourant Institute of Mathematical Sciences

Key Points

This work aims to analyze the convergence of empirical measures of eigenvalues for specific square matrices.
Examined eigenvalues of nonnormal square matrices of the form An = UnTnVn.
Un and Vn are Haar distributed on the unitary group, while Tn is a real diagonal.
Provided proofs and considered cases where distributions are on the orthogonal group.
Demonstrated that LA n converges to a rotationally invariant measure on the complex plane.
Resulting measure's support forms a single ring when Tn's eigenvalue measures converge.
Presented a complete proof of the Feinberg-Zee 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.

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