Consider an arbitrary finite set of orthogonal-projection matrices (symmetric, idempotent, etc.) in $\mathbb{R}^{n\times n}$.  

We draw two matrices $Q,P$ uniformly and i.i.d. from this set.


**Question:** Is the following expected matrix $\mathbb{E}\_{P,Q} \left[I-2P+QPQ\right]$ necessarily positive semi-definite?


Clearly, one such matrix is not necessarily PSD. For instance,
$P=\begin{bmatrix}1 & 0 \\ 0 & 0\end{bmatrix},Q=\begin{bmatrix}0 & 0 \\ 0 & 1\end{bmatrix}$ do not yield a PSD matrix together, but I'm asking about the expectation rather than about an arbitrary instantiation.



Expected matrix created from two random orthogonal-projection matrices
Found a counter example.


If we take the set to be $\left\{
\begin{bmatrix}1 & 0 \\ 0 & 0\end{bmatrix},
\frac{1}{2}\begin{bmatrix}1 & 1 \\ 1 & 1\end{bmatrix}
\right\}$, then we have
$\mathbb{E}\_{P,Q} \left[I-2P+QPQ\right]=
\frac{1}{16}\begin{bmatrix}1 & -5 \\ -5 & 11\end{bmatrix}$, whose eigenvalues are $-0.06694174, 0.81694174$.


