Assume a positive semi-definite $M\times M$ matrix $A$, not with full rank, and an $M\times N$ matrix $X$, where $M>N$. The elements of $X$ are independent, zero-mean complex Gaussian with variance $1/M$.

My question is simple, what is the distribution of $X^H AX$?

From what I have seen, a matrix of form $X^H X$ is Wishart if the rows of $X$ are correlated, but in my case it is the columns.