Do there exist complex square matrices that cannot be classified as either positive semi-definite, or negative semi-definite, or indefinite?
Here I am using the definitions:
Positive semi-definite: $u^* M u \ge 0\quad \forall u$.
Negative semi-definite: $u^* M u \le 0\quad \forall u$.
Indefinite: there exist $u,v$ such that $u^*Mu < 0 < v^* M v$.
Obviously these three cases cover all Hermitian matrices. But for non-Hermitian matrices it is not clear to me. For example, could there be matrices for which $u^* M u$ is always strictly complex and not real? Or perhaps sometimes not real, sometimes real and positive, but never real and negative.