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单词 AntisymmetricMapping
释义

antisymmetric mapping


Let U and V be a vector spacesMathworldPlanetmath over a field K. A bilinear mappingB:U×UVis said to be antisymmetric if

B(u,u)=0(1)

for all uU.

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

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

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

A multlinear mapping M:UkVis said to be totally antisymmetric, or simply antisymmetric, iffor every u1,,ukU such that

ui+1=ui

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

M(u1,,uk)=0.
Proposition 1

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

M(uπ1,,uπk)=sgn(π)M(u1,,uk),

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

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

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

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

π=(i,i+1),i=1,,k-1.

However, such transpositionsgenerate the group of permutations, and hence the proposition holds infull generality.

Note.

The determinantMathworldPlanetmath is an excellent example of a totallyantisymmetric, multlinear mapping.

Titleantisymmetric mapping
Canonical nameAntisymmetricMapping
Date of creation2013-03-22 12:34:39
Last modified on2013-03-22 12:34:39
Ownerrmilson (146)
Last modified byrmilson (146)
Numerical id10
Authorrmilson (146)
Entry typeDefinition
Classificationmsc 15A69
Classificationmsc 15A63
Synonymskew-symmetric
Synonymanti-symmetric
Synonymantisymmetric
Synonymskew-symmetric mapping
Related topicSkewSymmetricMatrix
Related topicSymmetricBilinearForm
Related topicExteriorAlgebra
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