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Is there any closed form distribution formula for the distribution of the eigenvalues of $\mathbf{X}^\mathrm{H}\mathbf{X}$ where the entries of $\mathbf{X}$ are independent Gaussian random variables with different variances $\sigma_{ij}^2$. Moreover, in the diagonal factorization $\mathbf{X}^\mathrm{H}\mathbf{X}=\mathbf{U}\mathbf{\Lambda}\mathbf{U}^{\mathrm{H}}$, the unitary matrix $\mathbf{U}$ has Haar distribution? Is it independent of $\mathbf{\Lambda}$?

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    $\begingroup$ no, $U$ is not uniformly distributed in the unitary group (no Haar distribution) and eigenvectors and eigenvalues are not independent. There is not much more to say in this completely general case. $\endgroup$ Feb 28, 2020 at 9:19
  • $\begingroup$ Is there any result for non-identical cases? $\endgroup$
    – Math_Y
    Feb 28, 2020 at 10:42

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More of a comment to indicate how hopeless this is: Consider the edge case of $\sigma_{ij} = \lambda_i \delta_{ij}.$ The unitary matrix then is the identity matrix, so is quite far from Haar distributed. The eigenvalues are distributed normally with the covariance matrix $\mathbf{X},$ also quite far from the "standard" case. An interesting question is whether there is a less trivial case when there is a discrete set of $\mathbf{U}.$

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Q: "Is there any result for non-identical cases?"

The eigenvalue distribution function is known if $\sigma^2_{ij}=w_i$ depends only on the row index, see for example arXiv:1310.2467

$$P(x_1,x_2,\ldots x_n)\propto\prod_{i<j}|x_i-x_j|^\beta\prod_k x_k^{1-2/\beta}\int dU \exp\left(-\tfrac{1}{2}\beta\,{\rm tr}\,UXX^H U^{\rm H}W^{-1}\right),$$ where $W={\rm diag}\,(w_1,w_2,\ldots w_n)$ and $\beta=1$ for real matrix elements, $\beta=2$ for complex matrix elements.

The measure $dU$ is the Haar measure in ${\rm O}(n)$ for $\beta=1$ and in ${\rm U}(n)$ for $\beta=2$. This is the socalled Harish-Chandra-Itzykson-Zuber integral, which has a closed-form expression for $\beta=2$ (it’s a ratio of determinants). Note that the integral depends only on the eigenvalues $x_i$ of $XX^H$. For $W$ proportional to the identity matrix we recover the usual Wishart distribution.

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