# Eigenvalue distribution of random matrices

Let $$M_{m \times n}$$ be a matrix with real entries $$m_{ij}$$ which are Gaussian independently distributed. If we further assume that $$m = n$$, $$m_{ii} \sim N(0, 2)$$, $$m_{ij} \sim N(0, 1), i\neq j$$, and the matrix $$M$$ is symmetric, then there is a nice formula for the joint probability distribution of the eigenvalues $$\lambda_j$$ of $$M$$ (see Theorem 2.5.2 of this book). My questions are:

(a) In the above result, if we just drop the assumption that $$M$$ is symmetric, but still diagonalizable, is something still known?

(b) Ultimately, I am trying to get some insight into the problem where we further drop the assumption that $$m = n$$, and whether in that case something is known for the singular values of the matrix $$M$$.

Possibly it is too much to hope for a clean looking result in the latter cases, but may be something could be said for large matrices? Any reference/insight will be deeply appreciated. Thanks in advance!

• For not-necessarily-symmetric matrices, you should look up "circular law". For rectangular matrices, you should look up "eigenvalues of Wishart matrices". – Dustin G. Mixon Oct 4 at 21:13