|
|
|
|
alternating form
|
(Definition)
|
|
|
A bilinear form on a vector space (over a field ) is called an alternating form if for all , .
Since for any ,
we see that
. So an alternating form is automatically a anti-symmetric, or skew symmetric form. The converse is true if the characteristic of is not .
Let be a two dimensional vector space over with an alternating form . Let
be a basis for . The matrix associated with looks like
where
. The skew symmetric matrix has the property that its diagonal entries are all 0. is called the alternating or symplectic matrix.
is called non-singular or non-degenerate if there exist a vectors such that
. are necessarily non-zero. Note that the associated matrix is non-singular iff iff is non-singular.
In the two dimensional vector space case above, if is non-singular, we can re-scale the basis elements so that . This means that the matrix associated with 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
.
Remark. In general, it can be shown that if is an -dimensional vector space equipped with a non-singular alternating form , then can be written as an orthogonal direct sum of the alternating hyperbolic planes
. In other words, the associated matrix for has the block form
where 
Furthermore, is even. is called a symplectic vector space.
|
"alternating form" is owned by CWoo.
|
|
(view preamble)
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) |
|
|
|
|
|
|
Pending Errata and Addenda
|
|
|
|
|
|
|
|
|
|
|