Let $A$ be a random matrix following multivariate normal distribution $N(\mu, \Sigma)$.



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> What is the distribution of the eigenvalues of $A^TA$?
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A reference to the literature would be most welcome.



What is the distribution of eigenvalues of $A^TA$, where $A \sim N(\mu, \Sigma)$?
Unlike in the case $\mu=0$, $\Sigma=I$, there is no simple closed-form expression for the eigenvalue distribution. In the limit of large matrices the eigenvalue density follows from the Pastur equation, see [Spectral density of the non-central correlated Wishart ensembles](https://arxiv.org/abs/1406.4184) (section VI and figure 2).


The nonzero $\mu$ is a rank-one perturbation, which leaves the main spectrum unaffected but causes a single eigenvalue to separate from the spectrum, see [Eigenvalue separation in some random matrix models.](https://arxiv.org/abs/0810.1554)


