Assuming $A$ is an $m \times n$ matrix (with $n \ge m$) of normally distributed elements with $\mu_A = 0$ and $\sigma_A = 1$, is there a mathematical formulation for the distribution of the elements of the Moore-Penrose inverse (pseudoinverse) $A^+$?
I could find empirically that in the limit for $n\to \infty$ they appear to be normally distributed, with $\mu_{A^+} = 0$ and $\sigma_{A^+} = \frac{1}{n}$ but this is definitely not the case for $n \sim m$. Is there any known description of the $A^+$ elements distribution for an arbitrary $n$ or any way to derive it?