I do research in statistics and am not sure whether the following is considered research level or not in mathematics. If it isn't, I'm happy because that means the answer is probably known and I can use it. If it is, any ideas for how to solve the problem are appreciated.

Suppose $Y$ is a real, $n \times k$ random matrix from a continuous distribution and that $Q$ is an $n\times n$ projection matrix of rank $q \leq n$. Let $C$ be a real, $n \times k$ matrix with strictly positive entries. What can be said about the rank of $(QY)\circ C$? Here, $\circ$ denotes the elementwise, or Hadamard, product.

I'm mostly interested in the nontrivial cases where $Q\neq I_n$ and the $C_{ij}$ are not all unity. My intuition tells me the rank is unchanged by the Hadamard product but I can't prove it, or find a proof in the literature.