I am considering real $n$-by-$m$ matrices of the following type:

$$ M=SM^\prime,\\ M^\prime_{ij}\sim^{iid}N(0,1). $$

Here, $S$ is a fixed $n$-by-$n$ matrix and the entries of $M^\prime$ (same size as $M$) are just i.i.d Gaussian. It is important that the considered matrices are rectangular and not simply square. $S$ can be identity but, ideally, should be an arbitrary full-rank matrix.

As far as I know, in the special case of $S=I, n=m$ this construction is called the real Ginibre ensemble. Can anyone suggest some literature/references for the more general case? I'm particularly interested in spectral properties of these matrices such as singular value/vector distributions.

  • 2
    $\begingroup$ the Ginibre ensemble refers to eigenvalues; for singular values you want the Wishart ensemble (the distribution of eigenvalues of the symmetric random matrix $MM^t$) $\endgroup$ Commented Apr 12, 2016 at 15:19
  • $\begingroup$ @CarloBeenakker Wow. That is great, thanks a lot. I knew about the Wishart ensemble but its definition somehow didn't relate to the definition of the singular values in my head. So it seems that the Wishart ensemble covers the described model in full generality, right? $\endgroup$
    – Vossler
    Commented Apr 12, 2016 at 16:10
  • 1
    $\begingroup$ yes, you do want $SS^t$ to be a positive definite matrix and you do want $n\leq m$ (otherwise there are singular values that are identically zero) $\endgroup$ Commented Apr 12, 2016 at 16:27


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