Antisymmetric Tensor

An antisymmetric tensor is defined as a Tensor for which

$$A^{mn} = -A^{nm}.$$ (1)

Any Tensor can be written as a sum of Symmetric and antisymmetric parts as

$$A^{mn} = {\textstyle{1\over 2}}(A^{mn}+A^{nm})+{\textstyle{1\over 2}}(A^{mn}-A^{nm}).$$ (2)

The antisymmetric part is sometimes denoted using the special notation

$$A^{[ab]} = {\textstyle{1\over 2}}(A^{ab}-A^{ba}).$$ (3)

For a general Tensor,

$$A^{[a_1 \cdots a_n]} \equiv {1\over n!} \epsilon_{a_1 \cdots a_n} \sum_{\rm permutations} A^{a_1 \cdots a_n},$$ (4)

where $\epsilon_{a_1 \cdots a_n}$ is the Levi-Civita Symbol, a.k.a. the Permutation Symbol.

See also Symmetric Tensor