# antisymmetric mapping

Let $U$ and $V$ be a vector spaces^{} over a field $K$. A bilinear mapping
$B:U\times U\to V$
is said to be *antisymmetric* if

$$B(u,u)=0$$ | (1) |

for all $u\in U$.

If $B$ is antisymmetric, then the polarization of the anti-symmetry
relation^{} gives the condition:

$$B(u,v)+B(v,u)=0$$ | (2) |

for all $u,v\in U$. If the characteristic of $K$ is not 2, then
the two conditions are equivalent^{}.

A multlinear mapping $M:{U}^{k}\to V$
is said to be *totally antisymmetric*, or simply antisymmetric, if
for every ${u}_{1},\mathrm{\dots},{u}_{k}\in U$ such that

$${u}_{i+1}={u}_{i}$$ |

for some $i=1,\mathrm{\dots},k-1$ we have

$$M({u}_{1},\mathrm{\dots},{u}_{k})=0.$$ |

###### Proposition 1

Let $M\mathrm{:}{U}^{k}\mathrm{\to}V$ be a totally antisymmetric, multlinear
mapping, and let $\pi $ be a permutation^{} of $\mathrm{\{}\mathrm{1}\mathrm{,}\mathrm{\dots}\mathrm{,}k\mathrm{\}}$. Then,
for every ${u}_{\mathrm{1}}\mathrm{,}\mathrm{\dots}\mathrm{,}{u}_{k}\mathrm{\in}U$ we have

$$M({u}_{{\pi}_{1}},\mathrm{\dots},{u}_{{\pi}_{k}})=\mathrm{sgn}(\pi )M({u}_{1},\mathrm{\dots},{u}_{k}),$$ |

where $\mathrm{sgn}\mathit{}\mathrm{(}\pi \mathrm{)}\mathrm{=}\mathrm{\pm}\mathrm{1}$ according to the parity of $\pi $.

Proof. Let ${u}_{1},\mathrm{\dots},{u}_{k}\in U$ be given. multlinearity and anti-symmetry imply that

$0$ | $=M({u}_{1}+{u}_{2},{u}_{1}+{u}_{2},{u}_{3},\mathrm{\dots},{u}_{k})$ | ||

$=M({u}_{1},{u}_{2},{u}_{3},\mathrm{\dots},{u}_{k})+M({u}_{2},{u}_{1},{u}_{3},\mathrm{\dots},{u}_{k})$ |

Hence, the proposition^{} is valid for $\pi =(12)$ (see cycle notation).
Similarly, one can show that the proposition holds for all
transpositions^{}

$$\pi =(i,i+1),i=1,\mathrm{\dots},k-1.$$ |

However, such transpositions generate the group of permutations, and hence the proposition holds in full generality.

## Note.

The determinant^{} is an excellent example of a totally
antisymmetric, multlinear mapping.

Title | antisymmetric mapping |

Canonical name | AntisymmetricMapping |

Date of creation | 2013-03-22 12:34:39 |

Last modified on | 2013-03-22 12:34:39 |

Owner | rmilson (146) |

Last modified by | rmilson (146) |

Numerical id | 10 |

Author | rmilson (146) |

Entry type | Definition |

Classification | msc 15A69 |

Classification | msc 15A63 |

Synonym | skew-symmetric |

Synonym | anti-symmetric |

Synonym | antisymmetric |

Synonym | skew-symmetric mapping |

Related topic | SkewSymmetricMatrix |

Related topic | SymmetricBilinearForm |

Related topic | ExteriorAlgebra |