I have a known $n \times m$ matrix $A$ that can be decomposed according to $$ A = X B + Y C$$ where $X$ and $Y$ are unknown $n \times m$ matrices and $B$ and $C$ are known $m \times m$ matrices. $C$ is invertible but $B$ is potentially singular.
Now for the interesting part: I also know that any column of $X$ that is nonzero implies that the corresponding column of $Y$ is zero and vice-versa.
I feel like I ought to be able to solve for $X$ and $Y$ given these restrictions (probably through some pseudo-inverse or rank-nullity property, but I'm guessing about that) but haven't been able to do so. Can we solve for $X$ and $Y$? If so, how? If not, can you show that $X$ and $Y$ aren't unique?
Edit: This generalizes when you notice that we can rewrite it as $A = Z D$ where $Z = \begin{bmatrix} X & Y \end{bmatrix}$ and $D = \begin{bmatrix} B \\ C \end{bmatrix}$. The zero restrictions become tricky with this notation though. And its not obvious to me how to solve this given that $D$ won't be invertible.