Let $\mathbf{X}$ be a random matrix with independent Gaussian random variable entries with different variances $v_{ij}$. Also define $\mathbf{A}=\mathbf{X}^\mathrm{H}\mathbf{X}$. Is there any relation between the distribution of the eigenvalues of $\mathbf{A}$ and moments of $\mathbf{A}$, i.e., $$\mathbf{B}_m=\mathbb{E}[\mathbf{A}^m]$$ for integers $m\geq 1$?
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2$\begingroup$ I'm not familiar with the notion of a moment being matrix-valued. Are you sure you don't mean something like ${\rm Tr}((A^m)^*A^m)$? $\endgroup$– Yemon ChoiCommented Feb 27, 2020 at 23:07
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1$\begingroup$ Moreover, can you specify what class of random matrices you are considering? Square? Symmetric/hermitian? etc $\endgroup$– Yemon ChoiCommented Feb 27, 2020 at 23:09
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$\begingroup$ If, as Yemon Choi suspects, your moments of $\mathbf{A} $ are traces of powers thereof, then you can just calculate them in the eigenbasis, where they are directly moments of the eigenvalue distribution. $\endgroup$– Michael EngelhardtCommented Feb 28, 2020 at 0:37
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$\begingroup$ What is the relation between $\mathrm{Tr}[A^m]$ and eigenvalue distribution? Could you please explain with more details? $\endgroup$– Math_YCommented Feb 28, 2020 at 7:17
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$\begingroup$ the expectation of ${\rm tr}\, A^m$ depends only on the marginal distribution of a single eigenvalue, so it contains no information on the correlations between the eigenvalues. $\endgroup$– Carlo BeenakkerCommented Feb 28, 2020 at 14:28
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