How Many Eigenvalues of a Random Matrix are Real?

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.