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How many eigenvalues of a random matrix are real? (English) Zbl 0790.15017
The expected number of real eigenvalues of an \(n \times n\) random matrix is \(E_ n=(1-(-1)^ n)/2+\sqrt 2 P^{(1-n,3/2)}_{n-2}(3)\), where \(P\) denotes the Jacobi polynomial, and \(\lim_{n \to \infty}E_ n=\sqrt {2/\pi}\). Asymptotically, a real normalized eigenvalue \(\lambda/ \sqrt n\) is uniformly distributed on \([-1,1]\). Analogous results are obtained for a pencil of two \(n \times n\) random matrices. Some numerical experiments confirm the obtained results.
Reviewer: M.Voicu (Iaşi)

MSC:
15B52 Random matrices (algebraic aspects)
15A18 Eigenvalues, singular values, and eigenvectors
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