Sorry for the awkwardly phrased title, I wasn't sure how to properly word it.
I want to do the following:
I have a vector $\vec J$ and a vector $\vec E$ with the following relation (with the parallel index referring to a unit-vector $\vec n = \vec e_3$: $$J_\| = i\epsilon_0\sum_s \frac{\omega_{ps}^2}{\omega}E_\|$$ $$\vec J_\bot = \epsilon_0\sum_s \omega_{ps}^2\frac{i\omega \vec E_\bot+\Omega_s \vec n \times \vec E_\bot}{\omega^2-\Omega_s^2}$$
I now want to find the tensor/matrix $\vec{\vec\sigma}$ so that:
$$\vec J = \vec{\vec{\sigma}}\vec E$$
The $\sigma_{33}$ component is simple and straightforward, but I have no idea how to get the other components of the tensor. I've tried writing both sides out component wise and relating that to my original equations but that didn't get me anywhere.
My textbook says that $\sigma_{11}$ = $\sigma_{22}$ and $\sigma_{12}$ = $-\sigma_{21}$ together with their values, but that doesn't make it click for me either.
$$J_1 = \epsilon_0\sum_s \omega_{ps}^2\frac{i\omega E_1-\Omega_s E_2}{\omega^2-\Omega_s^2}=\sigma_{11}E_1+\sigma_{12}E_2.$$
Compare and find $\sigma_{11}$ and $\sigma_{12}$. Do the same for $J_2$.