Positive and negative definiteness, the inertia $(p,q,r)$, rank $p+q$, and signature $p-q$ of a real symmetric bilinear or quadratic form

Let $B$ be a symmetric bilinear form on a finite-dimensional real vector space, and write $q(v):=B(v,v)$. The form is positive definite when $q(v)>0$ for every $v\mathrm{\ne }0$, and negative definite when $q(v)<0$ for every $v\mathrm{\ne }0$. The same terminology applies to the associated quadratic form.

Suppose a basis gives a diagonal matrix with $p$ positive diagonal entries, $q$ negative diagonal entries, and $r$ zero entries. Its inertia is the triple $(p,q,r)$, its rank is $p+q$, and its signature is $p-q$. Sylvester's law of inertia proves that this triple is independent of the diagonalizing basis, and therefore justifies the notation.