# canonical basis for symmetric bilinear forms

If $B:V\times V\to K$ 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

$$\begin{array}{cc}\hfill \hfill & \hfill \hfill \\ \hfill \hfill & \hfill {a}_{1}\hfill & \hfill 0\hfill & \hfill \mathrm{\dots}\hfill & \hfill 0\hfill \\ \hfill \hfill & \hfill 0\hfill & \hfill {a}_{2}\hfill & \hfill \mathrm{\dots}\hfill & \hfill 0\hfill \\ \hfill \hfill & \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\ddots}\hfill & \hfill \mathrm{\vdots}\hfill \\ \hfill \hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \mathrm{\dots}\hfill & \hfill {a}_{n}\hfill \end{array}$$ |

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 ${a}_{1}=\mathrm{\cdots}={a}_{t}=1$,
${a}_{t+1}=\mathrm{\cdots}={a}_{t+p}=-1$ 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 $K=\u2102$ 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.

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 |