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!