Irreducible Tensor

Given a general second Rank Tensor $A_{ij}$ and a Metric $g_{ij}$, define

$$\theta \equiv A_{ij}g^{ij} = A_i^i$$ (1)

$$\omega^i \equiv \epsilon^{ijk}A_{jk}$$ (2)

$$\sigma_{ij} \equiv {\textstyle{1\over 2}}(A_{ij}+A_{ji})-{\textstyle{1\over 3}}g_{ij}A^k_k,$$ (3)

where $\delta_{ij}$ is the Kronecker Delta and $\epsilon^{ijk}$ is the Levi-Civita Symbol. Then

$\sigma_{ij}+{\textstyle{1\over 3}}\theta g_{ij}+{\textstyle{1\over 2}}\epsilon_{ijk}\omega^k$
$\quad = {\textstyle{1\over 2}}(A_{ij}+A_{ji})+{\textstyle{1\over 2}}(\delta_i^\lambda \delta_j^\mu-\delta_i^\mu \delta_j^\lambda)A_{\lambda\mu}$
$\quad = {\textstyle{1\over 2}}(A_{ij}+A_{ji}) +{\textstyle{1\over 2}}(A_{ij}-A_{ji}) = A_{ij},$	(4)

where $\theta$, $\omega^i$, and $\sigma_{ij}$ are Tensors of Rank 0, 1, and 2.

See also Tensor