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:

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 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.