antisymmetric mapping
Let and be a vector spaces over a field . A bilinear mappingis said to be antisymmetric if
(1) |
for all .
If is antisymmetric, then the polarization of the anti-symmetryrelation gives the condition:
(2) |
for all . If the characteristic of is not 2, thenthe two conditions are equivalent.
A multlinear mapping is said to be totally antisymmetric, or simply antisymmetric, iffor every such that
for some we have
Proposition 1
Let be a totally antisymmetric, multlinearmapping, and let be a permutation of . Then,for every we have
where according to the parity of .
Proof.Let be given. multlinearity and anti-symmetryimply that
Hence, the proposition is valid for (see cycle notation).Similarly, one can show that the proposition holds for alltranspositions
However, such transpositionsgenerate the group of permutations, and hence the proposition holds infull generality.
Note.
The determinant is an excellent example of a totallyantisymmetric, 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 |