Bilinear form/Symmetric/Definiteness/Definition

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

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