skew-symmetric bilinear form

A skew-symmetric (or antisymmetric) 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 $B:V\times V\rightarrow K$ ($V$ 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, $\operatorname{char}(K)=2$, 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