CanonicalBasisForSymmetricBilinearForms 2005-01-08 17:34:18 2007-05-26 13:29:18 Definition bilinear form Sylvester's Law of Inertia % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here 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 $1 \le j \le n-t-p$. Furthermore, these integers are \emph{invariants} of the bilinear form. This is known as \emph{Sylvester's Law of Inertia}. $B$ is \emph{positive definite} if and only if $t = n$, $p = 0$. Such a form constitutes a \emph{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 $1 \le j \le n-r$. 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 $1 \le j \le n-r$, and $n$ is the least positive quadratic non-residue.