How do I prove positive definiteness for a matrix difference? 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?

Choose $A = \begin{bmatrix} 200 & 100 \\\ 100 & 100 \end{bmatrix}$, $W = \frac{1}{10}\begin{bmatrix} 1 & 0 \\\ 0 & 9 \end{bmatrix}$, then $$A-WAW = \begin{bmatrix} 198 & 91 \\\ 91 & 19 \end{bmatrix},$$ which is not $\geq 0$ (since $198\cdot19-91^2 < 0$).