Let $A\in M_n(\Bbb R)$. Then we say for all $x\in\Bbb R^n$, $x^TAx$ is quadratic form. (wiki said so as well.)
However, do we concern about the complex situation? Namely, do we define
Let $A\in M_n(\Bbb C)$. Then we say for all $x\in\Bbb C^n$, $x^*Ax$ is quadratic form.
I'm not sure I've seen this definition. Why this definition is rarely seen?
PS: But why we sometimes consider things like $x^*Ax$, but not give the name "quadratic form" for it?