If I have a positive semidefinite matrix $A$ and a negative definite matrix $B$, is it true that their Hadamard product $A\circ B$ is negative semidefinite? Ideally I am looking for a proof / a complete argument for why it is true / false that I can replicate.
Schur product theorem states that Hadamard product of two positive semidefinite matrices is positive semidefinite.
$B$ is negative definite $\implies -B$ is positive definite.
Since $$A \circ B = -(A \circ (-B)),$$ and $A \circ (-B)$ is positive semidefinite by Schur product theorem.
We conclude that $A \circ B$ is negative semidefinite.