# Is this Hermitian matrix positive definite?

We say an Hermitian matrix $A$ is positive if $$\bar{z}^tAz=\sum_{i,j=1}^na_{ij}\bar{z}_iz_j>0,\quad \forall z\neq 0.$$

But if we have $$z^tA\bar{z}=\sum_{i,j=1}^na_{ij}z_i\bar{z}_j>0,\quad \forall z\neq 0.$$ can we say that $A$ is positive? Prove or counterexample

Thanks!

-
It is obviously true when $A$ is real. –  akkkk Apr 16 '12 at 14:05

Replace $z$ in the second line by $w$ and then choose $w=\bar z$ to see that the two statements are equivalent.