# Expectation of Projection Matrix

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).

• In general, $\mathbf{P_A} = A(A^TA)^{-1}A^T$ can have negative off-diagonal entries with positive probability when $M>1$. – DiamondFrost Jun 9 '15 at 14:38