$D_1 \Sigma D_1 \succeq D_2 \Sigma D_2$ with $\Sigma$ being positive semi-definite and $D_1 \succeq D_2$ diagonal matrices

Question

Let $\Sigma \in \mathbb{R}^{n \times n}$ be a positive semi-definite matrix, and $D_1, D_2 \in \mathbb{R}^{n \times n}$ be two diagonal matrices with $(D_1)_{ii} \geq (D_2)_{ii} \geq 0$ for any diagonal entry $i \leq n$.

I wonder if it is true that $D_1 \Sigma D_1 \succeq D_2 \Sigma D_2$, i.e., $D_1 \Sigma D_1 - D_2 \Sigma D_2$ is a positive semi-definite matrix?

I feel it is true because $D_1 y$ just scales a vector $y$ more in some directions.

Answer 1

The answer is no. As a counterexample, consider $$ \Sigma = \pmatrix{1&1\\1&1}, \quad D_1 = \pmatrix{1&0\\0&1}, \quad D_2 = \pmatrix{1&0\\0&0}. $$