How many eigenvalues of a random matrix are real?

Authors:
Alan Edelman, Eric Kostlan and Michael Shub

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

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be an matrix whose elements are independent random variables with standard normal distributions. As , the expected number of real eigenvalues is asymptotic to . We obtain a closed form expression for the expected number of real eigenvalues for finite , and a formula for the density of a real eigenvalue for finite . Asymptotically, a real normalized eigenvalue 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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DOI:
https://doi.org/10.1090/S0894-0347-1994-1231689-0

Article copyright:
© Copyright 1994
American Mathematical Society