# 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 CanonicalBasisForSymmetricBilinearForms 2013-03-22 14:56:25 2013-03-22 14:56:25 Mathprof (13753) Mathprof (13753) 7 Mathprof (13753) Definition msc 47A07 msc 11E39 msc 15A63 Sylvester’s Law of Inertia