Suppose there is some Random Matrix $A$ which exists in $\mathbb{R}^{N\times N}$. Each element in $A$ has a different $\mu$ and $\sigma$. However, all elements are independent of eachother.
$A$ is a Ginibre ensemble in that it is non-Hermitian. Furthermore, the eigenvalues of $A$ are mostly real but some are imaginary.
What are the distributions of $A^M$ where $M$ is a natural number?
So far, I have done numerical experiments by using expressions for the distributions of eigenvalues (eq 2.15) and the $U$ and $V$ matrices from the means of $A$. I approximated the distribution of $A = U \Sigma V^T$ where all randomness is contained inside $\Sigma$ and $U$ and $V$ are fixed. Then I tried to recreate the distribution of $A$ by drawing samples from the distribution for $\Sigma$ and carrying out the matrix products. However, the distributions from recreating $A$ from sampling from the eigenvalue pdfs were not similar to the original distributions of $A$. Any information or resources is appreciated.