Consider an asymmetric matrix $A \in {\mathbb R}^{n\times n}$ with eigenvalues $\lambda \in {\mathbb C}^+$, and $x \in {\mathbb R}^n$. I am not sure if the quadratic form $x^T A x$ is positively definite.
PS: This question is equivalent to judge the positive definiteness of $A^T + A$ given $\lambda(A) \in {\mathbb C}^+$. But I am not sure if it is correct or not.
Thanks very much!