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!

  • 2
    $\begingroup$ For not-necessarily-symmetric matrices, you should look up "circular law". For rectangular matrices, you should look up "eigenvalues of Wishart matrices". $\endgroup$ – Dustin G. Mixon Oct 4 at 21:13

Your Answer

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

Browse other questions tagged or ask your own question.