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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
##### Keywords:
number of real eigenvalues; random matrix
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##### References:
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