Physical interpretation of Projection matrix operations $PAP$ and $(I-P)A(I-P)$

Question

I know that a Projection matrix ($P$) projects a vector onto the column space of another matrix, say $B$ ( $P$ being formed from $B$, $P=B(B^TB)^{-1}B^T$). And $(I-P)$ projects a vector onto the null space of $B^T$. So, if we apply a projection matrix $P$ to another matrix (say, $A$), then the operation $PA$ should project all the vectors in the column space of $A$ to the column space of $B$ (I hope this interpretation is correct). Similarly, the projection should be onto the null space of $B^T$, if the operation is $(I-P)A$. But what happens if we apply the same projection matrix from the right-hand side to the obtained projections (meaning the operations $PAP$ and $(I-P)A(I-P)$)? These kind of operations have been used in quantum chemistry. Thank you.