Problem
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Given $X \in \mathbb{R}^{n \times n}$ where $X\_{ij} \sim \mathcal{N}(\mu\_{ij}, \sigma\_{ij}^2 I)$


Find the marginal distribution of each eigenvalue, using whatever you can.


Background
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In my field, I have a Bayesian inference framework that will obtain the $X$ distribution, but what we really need is the eigenvalue distribution of the matrix $X$.


Question
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1. Is this problem well defined?
2. Is the only way to infer it through sampling in such **high dimensional** space and then for each realization of the matrix, do the eigenvalue decomposition?



How to infer the eigenvalue distribution from matrix where each entry has a known Gaussian distribution?
Here is a $2 \times 2$ example to consider. Since each entry has its own distribution, let three of them have standard deviation $0$ and the matrix be $\left[ \begin {array}{cc} 0&X\_{12}\\ 1&0\end {array} \right]$ with the mean and standard deviation of $X\_{12}=\mathcal{N}(\mu, \sigma^2)$ to be determined. The eigenvalues are either 


* $\pm t$ if $X\_{12}=-t^2 \leq 0$ or
* $\pm ti$ if $X\_{12}=t^2 \geq 0$


So the mean is distributed as the trace (which is always the case). 


For $\mu=0$ and $\sigma=1$ one will have the eigenvalues real or pure imaginary with equal probability and inside the unit circle with probability about $\frac23.$ Is that the kind of result you seek?


There is a very large literature on random matrices although it usually concentrates on the matrix being symmetric or Hermetian and drawn from $\mathbb{R}^{n \times n}$ with respect to some Gaussian distribution.


