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 rank2 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).