Show that a matrix $A$ is positive definite if and only if $ A + A^T$ is positive definite

Question

I tried messing around with the definition of positive definite: $(Av,v) > 0$ and the fact that $A = \frac{1}{2}((A+A^T) + (A-A^T)) $, but I didn't really get anywhere.

Answer 1

Note that

$$((A+A^T)v,v)=(Av+A^Tv,v)=(Av,v)+(A^Tv,v)=(Av,v)+(v,Av)=2(Av,v).$$

Thus $(Av,v)>0$ if and only if $((A+A^T)v,v)>0$.