Is symmetry a necessary condition for positive (or negative) definiteness?
If not:
It can be proved that if $\mathbf{A} \in \mathbb{R}^{m\times m}$ is a square (non-symmetric) matrix, then $$ \mathbf{z'Az=z'Bz},~~\mathbf{B=B'= \frac{A+A'}{2}} $$
On the other hand, a positive definite matrix is a symmetric matrix for which:
$$\mathbf{z'Bz}>0,~~ \mathbf{z\ne 0}$$
Can we imply that $\mathbf{A}$ which is a non-symmetric matrix, is positive definite?