Bilinear form/Symmetric/Definiteness/Definition

Definiteness of symmetric bilinear form

Let ${}V$ be a real vector space, endowed with a symmetric bilinear form ${}\left\langle -,-\right\rangle$. This bilinear form is called

1. positive definite, if ${}\left\langle v,v\right\rangle >0$ holds for all ${}v\in V$, ${}v\neq 0$.
2. negative definite, if ${}\left\langle v,v\right\rangle <0$ holds for all ${}v\in V$, ${}v\neq 0$.
3. positive semidefinite, if ${}\left\langle v,v\right\rangle \geq 0$ holds for all ${}v\in V$.
4. negative semidefinite, if ${}\left\langle v,v\right\rangle \leq 0$ holds for all ${}v\in V$.
5. indefinite, if ${}\left\langle -,-\right\rangle$ is neither positive semidefinite nor negative semidefinite.