# 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 |