I'm trying to figure out the proper way to write Poisson's kinematical equation in index notation. The matrix form is
$[\dot{C}] = -[\omega^\times][C]$
My first try is
$\dot{C}_{jk}=-\epsilon_{ijk}\omega_iC_{jk}$
However, this seems to leave two free indices on the left (which I want) and none on right because $j$ and $k$ are repeated, which means they're summed. Is there something I can change to make this work like it should?
There are just a few missing indices. Note first, that in a product $AB$, the $(i,j)$-entry of the product is $$ (AB)_{ij} = A_{ik}B_{kj} $$ Hence, $$ [\omega^\times][C]_{jk} = [\omega^\times]_{ji}C_{ik} $$ This gives $$ \dot C_{jk} = -\epsilon_{\ell ji}\omega_\ell C_{ik} $$