The probability that a random real gaussian matrix has k real eigenvalues, related distributions, and the circular law

Let A be an n by n matrix whose elements are independent random Variables with standard normal distributions. Girko's (more general) circular law states that the distribution of appropriately normalized eigenvalues is asymptotically uniform in the unit disk in the complex plane. We derive the exact expected empirical spectral distribution of the complex eigenvalues for finite n, from which convergence in the expected distribution to the circular law for normally distributed matrices may be derived. Similar methodology allows us to derive a joint distribution formula for the real Schur decomposition of A. integration of this distribution yields the probability that A has exactly k real eigenvalues. For example, we show that the probability that A has all real eigenvalues is exactly 2(-n(n-1)/4). (C) 1997 Academic Press.