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单词 SkewsymmetricBilinearForm
释义

skew-symmetric bilinear form


A skew-symmetric (or antisymmetric) bilinear formPlanetmathPlanetmath is a special case of a bilinear form B, namely one which is skew-symmetric in the two coordinatesPlanetmathPlanetmath; 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 spacesMathworldPlanetmath, so we must require this in the formal definition:

a bilinear form B:V×VK (V a vectorspace over a field K) is called skew-symmetric iff

B(x,y)=-B(y,x) for all vectors x,yV.

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 xV, which means that 2B(x,x)=0, or B(x,x)=0. Therefore, B is an alternating form.

If, however, char(K)=2, then B(x,y)=-B(y,x)=B(y,x); B is a symmetric bilinear formMathworldPlanetmath.

If V is finite-dimensional, then every bilinear form on V can be representedby 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 thechoice of representing matrix).

Titleskew-symmetric bilinear form
Canonical nameSkewsymmetricBilinearForm
Date of creation2013-03-22 13:10:47
Last modified on2013-03-22 13:10:47
Ownersleske (997)
Last modified bysleske (997)
Numerical id9
Authorsleske (997)
Entry typeDefinition
Classificationmsc 15A63
Synonymantisymmetric bilinear form
Synonymanti-symmetric bilinear form
Related topicAntiSymmetric
Related topicSymmetricBilinearForm
Related topicBilinearForm
Definesskew symmetric
Definesanti-symmetric
Definesantisymmetric
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