canonical basis for symmetric bilinear forms

If $B:V\times V\rightarrow K$ is a symmetric bilinear form over a finite-dimensional vector space, where the characteristic of the field is not 2, then we may prove that there is an orthogonal basis such that $B$ is represented by

\bordermatrix{&\cr&a_{1}&0&\ldots&0\cr&0&a_{2}&\ldots&0\cr&\vdots&\vdots&% \ddots&\vdots\cr&0&0&\ldots&a_{n}\cr}

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

If $K=\mathbb{R}$ we may choose a basis such that $a_{1}=\cdots=a_{t}=1$ , $a_{t+1}=\cdots=a_{t+p}=-1$ and $a_{t+p+j}=0$ , for some integers $p$ and $t$ , where $1\leq j\leq n-t-p$ . 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 $K=\mathbb{C}$ we may go further and choose a basis such that $a_{1}=\cdots=a_{r}=1$ and $a_{r+j}=0$ , where $1\leq j\leq n-r$ .

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

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

Title	canonical basis for symmetric bilinear forms
Canonical name	CanonicalBasisForSymmetricBilinearForms
Date of creation	2013-03-22 14:56:25
Last modified on	2013-03-22 14:56:25
Owner	Mathprof (13753)
Last modified by	Mathprof (13753)
Numerical id	7
Author	Mathprof (13753)
Entry type	Definition
Classification	msc 47A07
Classification	msc 11E39
Classification	msc 15A63
Defines	Sylvester’s Law of Inertia