Symmetric Tensor

A second-tensor rank symmetric tensor is defined as a tensor for which

(1)

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

			(2)
			(3)

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).

The product of a symmetric and an antisymmetric tensor is 0. This can be seen as follows. Let be antisymmetric, so

(7)

(8)

Let be symmetric, so

(9)

Then

			(10)
			(11)
			(12)

A symmetric second-tensor rank tensor has scalar invariants

			(13)
			(14)