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Let $M^+$ represents the pseudo-inverse of matrix $M$; $M^+=(M^hM)^{-1}M^h$, where $h$ denotes the conjugate transpose. We assume that the elements of $M$ are complex Gaussian with zero mean and unit variance. We can show that $M^+ (M^+)^h= (M^hM)^{-1}$, and $M^+ (M^+)^h$ has an inverse Wishart distribution. We define $|| M^+||$ as the norm of $M^+$, for which I think we have $|| M^+||^2= \text{tr}(M^+ (M^+)^h)$. Let $[M^+ (M^+)^h]_{i}$ denotes the diagonal elements of $M^+ (M^+)^h$.

Question: How to derive the distribution of $\frac{[M^+ (M^+)^h]_{i}}{ || M^+||^2}$ ? what is the distribution of $|| M^+||^2$ ?

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