I'm interested in the following question:
Let $\mathbf{A}\in\mathbb{R}^{N\times M}$, $M\leq N$ be an random matrix with i.i.d. entries that take strictly positive values. A continuous probability density can be assumed.
Let $\mathbf{P_A}\in \mathbb{R}^{N\times N}$ be the orthogonal projection matrix onto the column space of $\mathbf{A}$.
Question: How to show that the expectation $\mathbb{E}[\mathbf{P_A}]$ has non-negative entries? (The off-diagonals in particular; the diagonals are non-negative due to positive-definiteness).
Thanks for reading!