skew-symmetric bilinear form SkewSymmetricBilinearForm 2002-11-28 07:33:27 2006-02-22 23:21:05 Definition skew symmetric anti-symmetric antisymmetric % 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 A {\em skew-symmetric} (or {\em antisymmetric}) {\em bilinear form} is a special case of a bilinear form $B$, namely one which is skew-symmetric in the two coordinates; that is, $B(x,y) = -B(y,x)$ for all vectors $x$ and $y$. Note that this definition only makes sense if $B$ is defined over two identical vector spaces, so we must require this in the formal definition: a bilinear form $B: V \times V \rightarrow K$ ($V$ a vector space over a field $K$) is called {\em skew-symmetric} iff \begin{quote} $B(x,y) = -B(y,x)$ for all vectors $x, y \in V$. \end{quote} Suppose that the characteristic of $K$ is not $2$. Set $x=y$ in the above equation. Then $B(x,x)=-B(x,x)$ for all vectors $x \in V$, which means that $2B(x,x)=0$, or $B(x,x)=0$. Therefore, $B$ is an alternating form. If, however, $\operatorname{char}(K)=2$, then $B(x,y)=-B(y,x)=B(y,x)$; $B$ is a symmetric bilinear form. If $V$ is finite-dimensional, then every bilinear form on $V$ can be represented by a matrix. In this case the following theorem applies: A bilinear form is skew-symmetric iff its representing matrix is skew-symmetric. (The fact that the representing matrix is skew-symmetric is independent of the choice of representing matrix).