I wanted to ask why is positive definite semi-ordering is well defined only for Hermitian matrices (or symmetric matrices if restricted to the reals)?
I saw an extension to the definition of positive definite matrices to the asymmetrical case. Namely, we say that a real (unnecessary symmetric) matrix $M$ is positive definite in the wide sense if and only if $z^TMz>0$ for all non-zero vectors $z$. This is equivalent to that the symmetrical part of $M$, which is $\frac{M+M^T}{2}$, is positive definite in the narrow sense.
Why is not possible to define a semi-ordering using this extended definition without considering self-adjoint matrices?
Thank yo uall in advance, Ziv Goldfeld