Antisymmetric Part

Any square matrix can be written as a sum

(1)

where

(2)

is a symmetric matrix known as the symmetric part of and

(3)

is an antisymmetric matrix known as the antisymmetric part of . Here, is the transpose.

Any rank-2 tensor can be written as a sum of symmetric and antisymmetric parts as

(4)

The antisymmetric part of a tensor is sometimes denoted using the special notation

(5)

For a general rank- tensor,

(6)

where is the permutation symbol. Symbols for the symmetric and antisymmetric parts of tensors can be combined, for example

(7)

(Wald 1984, p. 26).