The single ring theorem
TL;DR: In this article, the authors studied the empirical measure LAn of the eigenvalues of non-normal square matrices of the form An = UnTnVn with Un;Vn independent Haar distributed on the unitary group and Tn real diagonal.
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Abstract: We study the empirical measure LAn of the eigenvalues of non-normal 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, LAn converges towards a rotationally invariant measure µ on the complex plane whose support is a single ring. In particular, we provide a complete proof of Feinberg-Zee single ring theorem [6]. We also consider the case where Un;Vn are independent Haar distributed on the orthogonal group.
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References
•Book
An Introduction to Random Matrices
Greg W. Anderson,Alice Guionnet,Ofer Zeitouni +2 more
- 21 Dec 2009
TL;DR: The theory of random matrices plays an important role in many areas of pure mathematics and employs a variety of sophisticated mathematical tools (analytical, probabilistic and combinatorial) as mentioned in this paper.
Statistical Ensembles of Complex, Quaternion, and Real Matrices
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1.3K
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982
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