alternating form

A bilinear form $A$ on a vector space $V$ (over a field $k$ ) is called an 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 $\{e_{1},e_{2}\}$ be a basis for $V$ . The matrix associated with $A$ looks like

$\begin{pmatrix}A(e_{1},e_{1})&A(e_{1},e_{2})\\ A(e_{2},e_{1})&A(e_{2},e_{2})\end{pmatrix}=r\begin{pmatrix}0&1\\ -1&0\end{pmatrix}=rS,$

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 $2\times 2$ alternating or symplectic matrix.

$A$ is called non-singular or 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 $r S$ 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 alternating or 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 $\mathcal{A}$ .

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

$\begin{pmatrix}S&\boldsymbol{0}&\cdots&\boldsymbol{0}\\ \boldsymbol{0}&S&\cdots&\boldsymbol{0}\\ \vdots&\vdots&\ddots&\vdots\\ \boldsymbol{0}&\boldsymbol{0}&\cdots&S\\ \end{pmatrix},\mbox{ where }\boldsymbol{0}=\begin{pmatrix}0&0\\ 0&0\end{pmatrix}.$

Furthermore, $n$ is even. $V$ is called a symplectic vector space.

Title	alternating form
Canonical name	AlternatingForm
Date of creation	2013-03-22 15:42:17
Last modified on	2013-03-22 15:42:17
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	7
Author	CWoo (3771)
Entry type	Definition
Classification	msc 15A63
Synonym	alternate form
Synonym	alternating
Synonym	symplectic hyperbolic plane
Related topic	SymplecticVectorSpace
Related topic	EverySymplecticManifoldHasEvenDimension
Defines	alternating hyperbolic plane