I have a postulate that states
$$ z^{*T} = -z $$ $$\begin{pmatrix} z_{11}^* &z_{12}^* \\ z_{21}^* & z_{22}^* \end{pmatrix} = \begin{pmatrix} -z_{11} & -z_{12} \\ -z_{21} & -z_{22} \end{pmatrix}$$
If this is true, the conlcusion was, that $z_{11}$ and $z_{22}$ would bouth be imaginary and if $z_{12} = z_{21}$ every matrix element would be imaginary. But there is no proof, so I wanted to ask why is that true?
If $z_{11}=x_{11}+iy_{11}$
$$z_{11}^*=-z_{11} \iff x_{11}-iy_{11}=-x_{11}-iy_{11} \iff x_{11}=-x_{11} \iff x_{11}=0 $$
do the same for $z_{22}$