skew-symmetric bilinear form

A skew-symmetric (or antisymmetric) bilinear formPlanetmathPlanetmath is a special case of a bilinear form B, namely one which is skew-symmetric in the two coordinatesPlanetmathPlanetmath; that is, B(x,y)=-B(y,x) for all vectors x and y. Note that this definition only makes sense if B is defined over two identical vector spacesMathworldPlanetmath, so we must require this in the formal definition:

a bilinear form B:V×VK (V a vector space over a field K) is called skew-symmetric iff

B(x,y)=-B(y,x) for all vectors x,yV.

Suppose that the characteristic of K is not 2. Set x=y in the above equation. Then B(x,x)=-B(x,x) for all vectors xV, which means that 2B(x,x)=0, or B(x,x)=0. Therefore, B is an alternating form.

If, however, char(K)=2, then B(x,y)=-B(y,x)=B(y,x); B is a symmetric bilinear formMathworldPlanetmath.

If V is finite-dimensional, then every bilinear form on V 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