Schur product theorem

Question

The theorem states that the Hadamard product of two positive definite matrices $ A \circ B$ is also positive definite. Can I make any statement about a the Hadamard product of a positive definite matrix $A$ and a non-positive definite matrix $C$? Specifically, does it follow that the result is not positive definite?

Answer 1

$$A=\begin{pmatrix}1&0\\0&1\end{pmatrix},\qquad B=\begin{pmatrix}1&3\\3&1\end{pmatrix}.$$

Here $A$ and $A\circ B$ are positive definite, $B$ is not. Any positive definite diagonal matrix will do in place of $A$, and it even still works if the off-diagonal entries of $A$ are nonzero but small.