Symmetric 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.

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

(6)

(Wald 1984, p. 26).