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antisymmetric
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(Definition)
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Let $U$ and $V$ be a vector spaces over a field $K$ . A bilinear mapping $B:U\times U\rightarrow V$ is said to be antisymmetric if \begin{equation} B(u,u)=0 \end{equation}for all $u\in U$ .
If $B$ is antisymmetric, then the polarization of the anti-symmetry relation gives the condition: \begin{equation} B(u,v) + B(v,u) = 0 \end{equation}for all $u,v \in U$ . If the characteristic of $K$ is not 2, then the two conditions are equivalent.
A multlinear mapping $M:U^k\rightarrow V$ is said to be totally antisymmetric, or simply antisymmetric, if for every $u_1,\ldots,u_k\in U$ such that $$u_{i+1} = u_i$$ for some $i=1,\ldots,k-1$ we have $$M(u_1,\ldots,u_k)=0.$$
Proposition 1 Let $M:U^k\rightarrow V$ be a totally antisymmetric, multlinear mapping, and let $\pi$ be a permutation of $\{1,\ldots,k\}$ . Then, for every $u_1,\ldots,u_k\in U$ we have $$M(u_{\pi_1},\ldots,u_{\pi_k}) = \mathrm{sgn}(\pi) M(u_1,\ldots,u_k),$$ where $\mathrm{sgn}(\pi)=\pm1$ according to the parity of $\pi$ .
Proof. Let $u_1,\ldots,u_k\in U$ be given. multlinearity and anti-symmetry imply that
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Hence, the proposition is valid for $\pi=(12)$ (see cycle notation). Similarly, one can show that the proposition holds for all transpositions $$\pi=(i,i+1),\quad i=1,\ldots,k-1.$$ 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.
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"antisymmetric" is owned by rmilson. [ full author list (2) ]
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Cross-references: determinant, group, generate, transpositions, cycle notation, valid, proposition, imply, proof, parity, permutation, mapping, equivalent, characteristic, relation, polarization, bilinear mapping, field, vector spaces
There are 32 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 13713 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) |
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Pending Errata and Addenda
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