antisymmetric mapping

Let U and V be a vector spacesMathworldPlanetmath over a field K. A bilinear mapping B:U×UV is said to be antisymmetric if

B(u,u)=0 (1)

for all uU.

If B is antisymmetric, then the polarization of the anti-symmetry relationMathworldPlanetmathPlanetmath gives the condition:

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

for all u,vU. If the characteristic of K is not 2, then the two conditions are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath.

A multlinear mapping M:UkV is said to be totally antisymmetric, or simply antisymmetric, if for every u1,,ukU such that


for some i=1,,k-1 we have

Proposition 1

Let M:UkV be a totally antisymmetric, multlinear mapping, and let π be a permutationMathworldPlanetmath of {1,,k}. Then, for every u1,,ukU we have


where sgn(π)=±1 according to the parity of π.

Proof. Let u1,,ukU be given. multlinearity and anti-symmetry imply that

0 =M(u1+u2,u1+u2,u3,,uk)

Hence, the propositionPlanetmathPlanetmathPlanetmath is valid for π=(12) (see cycle notation). Similarly, one can show that the proposition holds for all transpositionsMathworldPlanetmath


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


The determinantMathworldPlanetmath 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