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July 29, 2011Annals of Mathematics133 citationsOpen Access

The single ring theorem

AGAlice GuionnetMKManjunath KrishnapurOZOfer Zeitouni

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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Cite This Study

Guionnet et al. (2011) studied this question.

synapsesocial.com/papers/6a0f8f0401be78fe815fd3a0https://doi.org/10.4007/annals.2011.174.2.10
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