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 .
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|
|Date of creation||2013-03-22 13:10:47|
|Last modified on||2013-03-22 13:10:47|
|Last modified by||sleske (997)|
|Synonym||antisymmetric bilinear form|
|Synonym||anti-symmetric bilinear form|