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alternating form (Definition)

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$,

$\displaystyle 0=A(u+v,u+v)=A(u,u)+A(u,v)+A(v,u)+A(v,v)=A(u,v)+A(v,u),$
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{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 $ 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 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... ...ts & \vdots \ \boldsymbol{0} & \boldsymbol{0} & \cdots & S \ \end{pmatrix},$ where $ \boldsymbol{0}= \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}. $

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



"alternating form" is owned by CWoo.
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See Also: symplectic vector space, every symplectic manifold has even dimension

Other names:  alternate form, alternating, symplectic hyperbolic plane
Also defines:  alternating hyperbolic plane
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Cross-references: symplectic vector space, even, block, orthogonal direct sum, quadratic spaces, category, PlanetMath, hyperbolic plane, iff, vectors, non-degenerate, non-singular, symplectic matrix, diagonal, property, matrix, basis, characteristic, converse, skew symmetric, anti-symmetric, field, vector space, bilinear form
There are 22 references to this entry.

This is version 4 of alternating form, born on 2006-02-23, modified 2006-03-06.
Object id is 7649, canonical name is AlternatingForm.
Accessed 4205 times total.

Classification:
AMS MSC: 15A63 (Linear and multilinear algebra; matrix theory :: Quadratic and bilinear forms, inner products)
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