The probability that a random real Gaussian matrix has \(k\) real eigenvalues, related distributions, and the circular law (Q1360300)

The joint eigenvalue distribution of an \(n\times n\) random matrix \(A_n\) is studied. It is assumed that the entries of \(A_n\) are independent random variables identically distributed as the standard normal one. The exact form of the distribution of complex eigenvalues is found for fixed \(n\). This distribution converges as \(n\to\infty\) to the well-known circular distribution (uniform over the unit circle) derived for the first time by \textit{V. L. Girko} [Teor. Veroyatn. Primen. 29, No. 4, 669-679 (1984; Zbl 0565.60034)]. The joint distribution formula for the real Schur decomposition of \(A_n\) is obtained. As a consequence, the probability that \(A_n\) has exactly \(k\) real eigenvalues is found. In particular, the probability that all eigenvalues of \(A_n\) are real is proved to be equal to \(2^{-n(n-1)/4}\).