We consider two $n \times n$ projections, $A$ and $B$. In particular, this means that $A^2 = A$ and $B^2 = B$.
Given this, I was curious on if products of projection matrices would have the same rank? For instance, if $\text{rk}(PQ) = \text{rk}(QP)$ or $\text{rk}(PQP) = \text{rk}(QPQ)$?
We have that for projection matrices, the trace is simply equal to the rank, so I was thinking that perhaps products of projection matrices will result in the same trace regardless of order.