Q:

If $A\succeq{B}\succeq0$ and $C\succeq{D}\succeq0$ then $A \circ C\succeq B \circ D \succeq0$

Let $A, B,C,D \in M_n$ are Hermitian.
We define

$A \circ B = [{a_{ij}}{b_{ij}}]$.
$A\succeq{B}$ Iff $A-B$ positive semidefinite.

suppose $A\succeq{B}\succeq0$ and $C\succeq{D}\succeq0$.
Why does $A \circ C\succeq B \circ D \succeq0$

A:

Just apply Schur product theorem to $A\circ C=(A-B)\circ C+B\circ (C-D)+B\circ D$.

