Suppose there is a (non-symmetric) real square matrix $A$ with symmetric part $A+A^T$.
What are some conditions on $A$ that are sufficient for $A+A^T$ to be positive definite?
For example, if the eigenvalues of $A$ are strictly positive is $A+A^T$ positive definite? (EDIT: This part of the question is answered in the negative in the comments).
This would then give the result I actually want which is that given two positive definite matrices $C$ and $D$ it follows that the symmetric part of $CD$ is also positive definite. (EDIT: But I think it is still not clear if $CD+DC>0$ - this is (perhaps) a slightly more special case than $A+A^T$ with $A$ having positive eigenvalues.)