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Let X be a complex random matrix, which has the probability function (drawn from the ensemble) V($XX^\dagger$), where V(x) is some function which guaranties good behavior at infinity. Note the unitary invariance of the ensemble.

I already know the (ensemble) averaged eigenvalue density function $\rho(z)$ of X.

Now I want to calculate the averaged eigenvalue density of $XX^\dagger$, where X is from the same ensemble V($XX^\dagger$). That is, how to represent the eigenvalue density function of $XX^\dagger$ in terms of $\rho(z)$?

$XX^\dagger$ is Hermite so it has real eigenvalues only. X has complex eigenvalues but due to invariance, its density function has rotation symmetry.

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  • $\begingroup$ this is asking for a relation between the singular values and the eigenvalues of $X$, which does not seem hopeful. $\endgroup$
    – Carlo Beenakker
    Commented Sep 21, 2013 at 10:30

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The relation between singular values and eigenvalues in the invariant case is evaluated in this paper, under some technical conditions: http://arxiv.org/abs/0909.2214 (see http://annals.math.princeton.edu/2011/174-2/p10 for the definitive version).

I am confused by the way you phrase your question though: it is easy to describe the density of $XX^*$ in terms of $V$. The hard part is actually to compute the eigenvalues of X (what you call $\rho$). So you seem to know the hard task and want to perform the easy task...

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