This problem is taken from Golan's linear algebra book.
Problem: Let $V$ be an inner product space over $\mathbb{R}$ and let $\alpha$ be an endomorphism of $V$. Show that $\alpha$ is positive definite if and only if $\alpha+\alpha^*$ is positive definite.
Definition: An endomorphism $\alpha$ is positive definite if and only if it is selfadjoint and satisfies the condition that $\langle \alpha(v), v\rangle$ is a positive real number for all nonzero $v\in V$.