alternating form AlternatingForm 2006-02-23 00:22:32 2006-03-06 11:10:20 Definition alternating hyperbolic plane \usepackage{amssymb,amscd} \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} % define commands here A bilinear form $A$ on a vector space $V$ (over a field $k$) is called an \emph{alternating form} if for all $v\in V$, $A(v,v)=0$. Since for any $u,v\in V$, $$0=A(u+v,u+v)=A(u,u)+A(u,v)+A(v,u)+A(v,v)=A(u,v)+A(v,u),$$ we see that $A(u,v)=-A(v,u)$. So an alternating form is automatically a anti-symmetric, or skew symmetric form. The converse is true if the characteristic of $k$ is not $2$. Let $V$ be a two dimensional vector space over $k$ with an alternating form $A$. Let $\lbrace e_1,e_2\rbrace$ be a basis for $V$. The matrix associated with $A$ looks like \begin{center}$ \begin{pmatrix} A(e_1,e_1) & A(e_1,e_2) \\ A(e_2,e_1) & A(e_2,e_2) \end{pmatrix}=r \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}=rS, $\end{center} where $r=A(e_1,e_2)$. The skew symmetric matrix $S$ has the property that its diagonal entries are all $0$. $S$ is called the $2\times 2$ \emph{alternating} or \emph{symplectic matrix}. $A$ is called \emph{non-singular} or \emph{non-degenerate} if there exist a vectors $u,v\in V$ such that $A(u,v)\neq 0$. $u,v$ are necessarily non-zero. Note that the associated matrix $rS$ is non-singular iff $r\neq 0$ iff $A$ is non-singular. In the two dimensional vector space case above, if $A$ is non-singular, we can re-scale the basis elements so that $r=1$. This means that the matrix associated with $A$ is the alternating matrix. A two-dimensional vector space which carries a non-singular alternating form is sometimes called an \emph{alternating} or \emph{symplectic hyperbolic plane}. Some authors also call it simply a hyperbolic plane. But here on PlanetMath, we will reserve the shorter name for its cousin in the category of quadratic spaces. Let's denote an alternating hyperbolic plane by $\mathcal{A}$. \textbf{Remark.} In general, it can be shown that if $V$ is an $n$-dimensional vector space equipped with a non-singular alternating form $A$, then $V$ can be written as an orthogonal direct sum of the alternating hyperbolic planes $\mathcal{A}$. In other words, the associated matrix for $A$ has the block form \begin{center}$ \begin{pmatrix} S & \boldsymbol{0} & \cdots & \boldsymbol{0} \\ \boldsymbol{0} & S & \cdots & \boldsymbol{0} \\ \vdots & \vdots & \ddots & \vdots \\ \boldsymbol{0} & \boldsymbol{0} & \cdots & S \\ \end{pmatrix},\mbox{ where }\boldsymbol{0}= \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}. $\end{center} Furthermore, $n$ is even. $V$ is called a symplectic vector space.