Let $A$ be a symmetric positive semidefinite matrix. Let $W$ be a diagonal matrix with the entries $w_i \in (0,1)$.
I think $$A - WAW$$ should be positive semidefinite, but I don't know how to prove it or how to find a counterexample. I think it makes sense to think about $WAW$ as a rescaling of $A$, and since all the coefficients are less than $1$, it should be less than $A$ in the appropriate sense. How would I prove this?