I recently came across this following claim about projection matrices, which I am unsure it is true. However, I cannot seem to find a valid counterexample.
Let $U \in \mathbb{R}^{n \times k}$ be a full column rank rectangular matrix where $k<n$. Define the projection matrix $P_U=U(U^TU)^{-1}U^T$. Now let $M \in \mathbb{R}^{n \times n}$ be a positive semidefinite matrix of rank $k<n$. The claim is that if both $MP_U$ and $(I-P_U)M$ are nonzero, then
$$
(I-P_U)MP_U \neq \mathbf{0}.
$$
The claim is clearly false if we allow $k=n$ as $M=I$ would be a valid counterexample. However, I can't seem to come up with a valid counterexample for $k<n$.
Appreciate any ideas/thoughts on this matter.
Best Answer
This isn't true. Counterexample: let $n=4,\,k=2$ and $$ U=\pmatrix{1&0\\ 0&1\\ 0&0\\ 0&0}, \ P_U=\pmatrix{1\\ &1\\ &&0\\ &&&0}, \ M=\pmatrix{0\\ &1\\ &&1\\ &&&0}. $$ Then $(I-P_U)M=\operatorname{diag}(0,0,1,0)\ne0$ and $MP_U=\operatorname{diag}(0,1,0,0)\ne0$ but $(I-P_U)MP_U=0$.