Let $A$ be a non-symmetric matrix of real numbers, $A\in\mathbb{R}^{n\times n}$.
If the quadratic form $x^\intercal Ax$ is positive definite in the sense that $x^\intercal Ax>0$ $\forall x\neq 0\in\mathbb{R}^n$, then the real part of all eigenvalues of $A$ is positive, $\Re(\lambda_i)>0$ for $i=1,\dots,n$.
However, if $\Re(\lambda_i)>0$ for $i=1,\dots,n$, it is not necessarily true that $x^\intercal Ax>0$ $\forall x\neq 0\in\mathbb{R}^n$.
Now, consider the symmetric part of $A$, $A_s=\frac{1}{2}(A+A^\intercal)$. We know by definition that $x^\intercal Ax=x^\intercal A_sx$.
What can we say about $x^\intercal Ax$ if:
- all eigenvalues of $A_s$ are positive
- at least one eigenvalue of $A_s$ is negative
Can we say that the $x^\intercal Ax$ is positive definite in the first case, and that it is certainly not positive definite in the second case?