Simple question that I'm banging my head against the desk with,
Let $S_{ij}$ be the symmetric part of some tensor $T_{ij}$. Trying to show $S_{ij}$ stays symmetric under O(3) rotations. Let $R_{ij}$ be some O(3)/rotation matrix.
$\begin{eqnarray*} S_{ij}' &=& R_{ik}R_{jl}S_{lk} \\ &=& R_{ik}R_{jl}S_{kl} \\ &=& R_{jl}R_{ik}S_{kl} \\ &=& S_{ji}' \end{eqnarray*}$
Can someone help me go from the 2nd to 3rd line? Why do the rotation matrices commute here?