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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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How many eigenvalues of a random matrix are real?
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by Alan Edelman, Eric Kostlan and Michael Shub PDF
J. Amer. Math. Soc. 7 (1994), 247-267 Request permission

Abstract:

Let $A$ be an $n \times n$ matrix whose elements are independent random variables with standard normal distributions. As $n \to \infty$, the expected number of real eigenvalues is asymptotic to $\sqrt {2n/\pi }$. We obtain a closed form expression for the expected number of real eigenvalues for finite $n$, and a formula for the density of a real eigenvalue for finite $n$. Asymptotically, a real normalized eigenvalue $\lambda /\sqrt n$ of such a random matrix is uniformly distributed on the interval [-1, 1]. Analogous, but strikingly different, results are presented for the real generalized eigenvalues. We report on numerical experiments confirming these results and suggesting that the assumption of normality is not important for the asymptotic results.
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Additional Information
  • © Copyright 1994 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 7 (1994), 247-267
  • MSC: Primary 60F99; Secondary 15A18, 62H99
  • DOI: https://doi.org/10.1090/S0894-0347-1994-1231689-0
  • MathSciNet review: 1231689