Bilinear form/Symmetric/Type/Definition

Type of a symmetric bilinear form

Let ${}V$ be a finite-dimensional real vector space, endowed with a symmetric bilinear form ${}\left\langle -,-\right\rangle$. We say that this bilinear form has the type

$(p,q),$

where

${}p:={\max {\left(\dim _{\mathbb {R} }{\left(U\right)},U\subseteq V,\,\left\langle -,-\right\rangle {|}_{U}{\text{ positive definite}}\right)}}\,,$

and

${}q:={\max {\left(\dim _{\mathbb {R} }{\left(U\right)},U\subseteq V,\,\left\langle -,-\right\rangle {|}_{U}{\text{ negative definite}}\right)}}\,.$