|
|
|
|
antisymmetric
|
(Definition)
|
|
|
Let and be a vector spaces over a field . A bilinear mapping
is said to be antisymmetric if
 |
(1) |
for all .
If is antisymmetric, then the polarization of the anti-symmetry relation gives the condition:
 |
(2) |
for all . If the characteristic of is not 2, then the two conditions are equivalent.
A multlinear mapping
is said to be totally antisymmetric, or simply antisymmetric, if for every
such that
for some
we have
Proposition 1 Let
be a totally antisymmetric, multlinear mapping, and let be a permutation of
. Then, for every
we have
where
according to the parity of .
Proof. Let
be given. multlinearity and anti-symmetry imply that
| 0 |
 |
|
| |
 |
|
Hence, the proposition is valid for (see cycle notation). Similarly, one can show that the proposition holds for all transpositions
However, such transpositions generate the group of permutations, and hence the proposition holds in full generality.
The determinant is an excellent example of a totally antisymmetric, multlinear mapping.
|
Anyone with an account can edit this entry. Please help improve it!
"antisymmetric" is owned by rmilson. [ full author list (2) ]
|
|
(view preamble)
Cross-references: determinant, group, generate, transpositions, cycle notation, proposition, imply, parity, permutation, mapping, equivalent, characteristic, relation, polarization, bilinear mapping, field, vector spaces
There are 22 references to this entry.
This is version 5 of antisymmetric, born on 2002-04-10, modified 2006-09-14.
Object id is 2826, canonical name is AntiSymmetric.
Accessed 11296 times total.
Classification:
| AMS MSC: | 15A63 (Linear and multilinear algebra; matrix theory :: Quadratic and bilinear forms, inner products) | | | 15A69 (Linear and multilinear algebra; matrix theory :: Multilinear algebra, tensor products) |
|
|
|
|
|
|
Pending Errata and Addenda
|
|
|
|
|
|
|
|
|
|
|