skew-symmetric bilinear form

A skew-symmetric (or antisymmetric) bilinear form is a special case of a bilinear form $B$, namely one which is skew-symmetric in the two coordinates; 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 spaces, so we must require this in the formal definition:

a bilinear form $B:V\times V\to K$ ($V$ a vector space over a field $K$) is called skew-symmetric iff

$B(x,y)=-B(y,x)$ for all vectors $x,y\in V$.

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 $x\in V$, which means that $2B(x,x)=0$, or $B(x,x)=0$. Therefore, $B$ is an alternating form.

If, however, $\mathrm{char}(K)=2$, then $B(x,y)=-B(y,x)=B(y,x)$; $B$ is a symmetric bilinear form.

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