Is there a necessary condition for the projection of two matrices to be the same?

Question

Take $\textbf{A},\textbf{B} \in \mathbb{R}^{d \times d}$ with $\textbf{A} \neq \textbf{B}$ and $d > 1$. Let $\textbf{P}_M$ be some $d\times d$ projection matrix. Is there a necessary condition for $\textbf{P}_M \boldsymbol{A} = \textbf{P}_M \textbf{B}$?

Answer 1

Let $M_i$ denote the $i$-th column of a matrix $M$.

A necessary (and sufficient) condition is that for all $i$, $$A_i-B_i\in\operatorname{Ker}P_M$$ A projection is entirely determined by its image and kernel. If you know them, then this is a handy criterion.