Let us assume we have a square matrix $A$ whose entries are sampled from a standard Gaussian distribution of mean $0$. Do we have any information about the distribution of its eigenvalues? 


Particularly, I'm aware that there are different results on symmetric gaussian matrices (or, the Gaussian orthogonal ensemble of $A$):


* The eigenvalues follow a semicircle law


Is there an equivalent result for standard, non-symmetric Gaussian matrices?



Eigenvalues and eigenvectors of Gaussian random matrices
This is the Ginibre ensemble, see [Eigenvalue statistics of the real Ginibre ensemble](https://arxiv.org/abs/0706.2020) for the eigenvalue distribution. For an $N\times N$ matrix with $N\gg 1$ there are on average $\sqrt{2N/\pi}$ eigenvalues on the real axis, uniformly in the interval $(-\sqrt N,\sqrt N$). The rest of the eigenvalues fill a disc of radius $\sqrt N$ in the complex plane, uniformly except for a depleted strip along the real axis. Here is a scatter plot of the eigenvalues for $N=100$ (taken from [arXiv:1305.2924](https://arxiv.org/abs/1305.2924)).


![](https://ilorentz.org/beenakker/MO/Ginibre_disc.png)


