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.
The symmetric part of a tensor is denoted using parentheses as
 | (4) |
 | (5) |
Symbols for the symmetric and antisymmetric parts of tensors can be combined, for example
![T^((ab)c)_([de])=1/4(T^(abc)_(de)+T^(bac)_(de)-T^(abc)_(ed)-T^(bac)_(ed)).](https://mathworld.wolfram.com/images/equations/SymmetricPart/NumberedEquation6.svg) | (6) |
(Wald 1984, p. 26).
