2
$\begingroup$

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?

$\endgroup$
2
  • $\begingroup$ I have answered this for the ensemble of nonsymmetric matrices, but do note that your statements on the symmetric case (GOE) are not correct: the eigenvalues follow a semicircle law, not a Gaussian distribution. $\endgroup$ Mar 28 '20 at 11:42
  • $\begingroup$ You are right. Thanks for the answer, I'll edit for future readers! $\endgroup$
    – Alfred
    Mar 30 '20 at 17:22
4
$\begingroup$

This is the Ginibre ensemble, see Eigenvalue statistics of the real Ginibre ensemble 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).

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.