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 $⟨-,-⟩$. We say that this bilinear form has the type

$(p,q),$

where

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

and

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