Let $Q$ be a projection matrix, thus symmetric and idempotent, and $W$ symmetric semidefinite positive matrix.
Prove:
$Q^{T}WQ = WQ$
If the product $WQ$ commutes, the proof is straightforward, because $Q$ is idempotent, but I do not know how to prove it.
Best Answer
Even if $Q=Q^T$, which may not hold (see my comment above), this does not hold in general. Take for instance $$Q=\begin{pmatrix} 1 & 1\\ 1 & 1 \end{pmatrix}, \; W=\begin{pmatrix} 1 & 0\\ 0 & 2 \end{pmatrix}.$$ Then $Q^T W Q = 3 Q$, while $WQ = \begin{pmatrix} 1 & 1\\ 2 & 2 \end{pmatrix}$.
On the other hand, as noted, it holds if we also assume that $W$ and $Q$ commute.