canonical basis for symmetric bilinear forms

If $B:V\times V\to K$ is a symmetric bilinear formover a finite-dimensional vector space, where the characteristic of the field isnot 2,then we may prove that there is an orthogonal basis such that $B$ is represented by

Recall that a bilinear form has a well-defined rank, and denote this by $r$.

If $K=ℝ$ we may choose a basis such that $a_1=\mathrm{\cdots }=a_t=1$,$a_{t+1}=\mathrm{\cdots }=a_{t+p}=-1$ and $a_{t+p+j}=0$, for some integers $p$ and $t$,where $1\le j\le n-t-p$.Furthermore, these integers are invariants of the bilinear form.This is known as Sylvester’s Law of Inertia.$B$ is positive definite if and only if$t=n$, $p=0$. Such a form constitutes a real inner product space.

If $K=ℂ$ we may go further and choose a basis such that $a_1=\mathrm{\cdots }=a_r=1$ and$a_{r+j}=0$, where $1\le j\le n-r$.

If $K=F_p$ we may choose a basis such that $a_1=\mathrm{\cdots }=a_{r-1}=1$,

$a_r=n$ or $a_r=1$;and $a_{r+j}=0$, where $1\le j\le n-r$, and$n$ is the least positive quadratic non-residue.