Prove $X$ is nonsingular given $X + X^\top \succ 0 $.

Question

I am trying to find a simple way to prove $X \in \mathbb{R}^{n \times n}$ is nonsingular given $X + X^\top \succ 0 $.

$\succ 0 $ means positive definite.

Answer 1

Hint: if $Xv=0$, consider $v^T(X+X^T)v$.