A Hermitian form $\langle,\rangle$ on a vector space over the complex field $\mathbb{C}$ is a function $$\langle\cdot,\cdot\rangle: V \times V \rightarrow \mathbb{C}$$
that satisfies conditions that need not be listed for the purpose of my question. A Hermitian form is positive-definite if for every $v \neq 0 \in V$, $$\langle v,v\rangle > 0$$
If a Hermitian form returns a complex number with an imaginary part, do we only look at the real part to see if the Hermitian is positive-definite? I'm under the impression that positive and negative are only defined only on the real line.
Best Answer
$\langle v,v\rangle >0$ is just a statement about real numbers. One of the two axioms for a Hermitian form is $$\langle u,v\rangle=\overline{\langle v,u\rangle}.$$ For $u=v$ this implies $$\langle v,v\rangle=\overline{\langle v,v\rangle}\in\mathbb R$$ since a complex number equal to its own conjugate must be real.