skew-symmetric bilinear form
A skew-symmetric (or antisymmetric) bilinear form is a special case of a bilinear form , namely one which is skew-symmetric in the two coordinates; that is, for all vectors and . Note that this definition only makes sense if is defined over two identical vector spaces, so we must require this in the formal definition:
a bilinear form ( a vector space over a field ) is called skew-symmetric iff
for all vectors .
Suppose that the characteristic of is not . Set in the above equation. Then for all vectors , which means that , or . Therefore, is an alternating form.
If, however, , then ; is a symmetric bilinear form.
If is finite-dimensional, then every bilinear form on 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).
Title | skew-symmetric bilinear form |
Canonical name | SkewsymmetricBilinearForm |
Date of creation | 2013-03-22 13:10:47 |
Last modified on | 2013-03-22 13:10:47 |
Owner | sleske (997) |
Last modified by | sleske (997) |
Numerical id | 9 |
Author | sleske (997) |
Entry type | Definition |
Classification | msc 15A63 |
Synonym | antisymmetric bilinear form |
Synonym | anti-symmetric bilinear form |
Related topic | AntiSymmetric |
Related topic | SymmetricBilinearForm |
Related topic | BilinearForm |
Defines | skew symmetric |
Defines | anti-symmetric |
Defines | antisymmetric |