For projection matrices $I-P$ and $P$, $(I-P)A=0$ implies $A=PC$?

Question

For $p\times p$ projection matrices, $I-P$ and $P$, $(I-P)A=0$ holds.

If so, $A$ should be $PC$ for non-trivial $P\neq I$ and a non-zero matrix $A$, where $C$ is any $p\times p$ constant matrix.

Any comments would be appreciated.

Answer 1

Yes, it does. Assume $A \in M_{p \times n}(\mathbb{F})$ and write $U = \ker(P)$ and $V = \operatorname{im}(P)$. Then $U \oplus V = \mathbb{F}^p$ and $I - P$ is the projection onto $U$ with image $U$ and kernel $V$.

The equation $(I - P)A = 0$ means that the image of $A$ is contained in the kernel of $I - P$ which is $V$. Hence, every column of $A$ can be written as a linear combination of the columns of $P$ (as $\operatorname{im}(P) = V$). Hence, we can write $A = PC$ where $C \in M_{p \times n}(\mathbb{F})$ and the $i$-th column of $C$ consists of the coefficients of a linear combination of the columns of $P$ which gives us the $i$-th column of $A$.