Let $\mathbf{A}$ be a random matrix, is there any relation between the distribution of the eigenvalues of $\mathbf{A}$ and moments of $\mathbf{A}$, i.e., $$\mathbb{E}[\mathbf{A}^m]$$ for integers $m\geq 1$?

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 $m\geq 1$?

Let $\mathbf{A}$ be a random matrix, is there any relation between the distribution of the eigenvalues of $\mathbf{A}$ and moments of $\mathbf{A}$, i.e., $$\mathbb{E}[\mathbf{A}^m]$$ for integer $m\geq 1$?

Let $\mathbf{A}$ be a random matrix, is there any relation between the distribution of the eigenvalues of $\mathbf{A}$ and moments of $\mathbf{A}$, i.e., $$\mathbb{E}[\mathbf{A}^m]$$ for integers $m\geq 1$?