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I am interested in the distribution of the eigenvalues of matrices that are sampled from the matrix normal distribution.

I am sampling from $p(X \mid M,U,V)$ and let's assume that I know the eigenstructure of $M$. The way I sample from the distribution is $M + \sqrt{U}*randn(d,d)*\sqrt{V}$, where $randn(d,d)$ is the MATLAB notation for a $d \times d$ matrix with each element being i.i.d. $N(0,1)$.

When I check the eigenvalues of the samples, I get the following: enter image description here

The red dots correspond to the eigenvalues of the samples and the big green dots correspond to the eigenvalues of the mean matrix $M$.

It looks like there may be some related theory for this problem since each eigencluster looks like proper 2-d Gaussians. If you can point to the related references/theory that would be really nice. As a side note, I am an electrical engineer, so it would be better for me if the references are more accessible (i.e. less rigorous).

Thanks a lot in advance.

edit: For the figure below, M, U and V are as follows. enter image description here

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  • $\begingroup$ what are U and V? and what is M in the figure? $\endgroup$
    – ofer zeitouni
    Commented Dec 19, 2013 at 15:30
  • $\begingroup$ @oferzeitouni , If you can check the wikipedia link for the matrix normal distribution, M is the mean matrix, U is the covariance amongst the rows and V is the covariance amongst the columns. $\endgroup$
    – jkt
    Commented Dec 20, 2013 at 5:44
  • $\begingroup$ What are U and V in the plot you give? the identity matrix? $\endgroup$
    – ofer zeitouni
    Commented Dec 20, 2013 at 7:37
  • $\begingroup$ @oferzeitouni, please see the edit. U is a scaled identity matrix, V is arbitrary. $\endgroup$
    – jkt
    Commented Dec 20, 2013 at 8:21
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    $\begingroup$ So you are dealing with square matrices. This slightly simplifies matters. Not worrying about issues of regularity, the answer should be the given by the Brown measure of your matrix, which can be computed. See the [paper ](arxiv.org/pdf/math/9912242.pdf) for details, and [arxiv.org/pdf/0909.2214v2.pdf] and [arxiv.org/pdf/1208.5100.pdf] for examples where the convergence is proved. $\endgroup$
    – ofer zeitouni
    Commented Dec 20, 2013 at 9:18

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