$A_i$. Expanding the incident field at the position of the origin in the orthogonal eigenvectors $$\left(
\begin{array} {c}
\bm{E}_\mathrm{in}\\
\bm{H}_\mathrm{in}
\end{array} \right)=\sum_i c_i \bm{v}_i,$$ and with $
 \boldsymbol{\alpha} \bm{v}_i=A_i
\bm{v}_i$ and $\langle \bm{v}_i|\bm{v}_j\rangle=\delta_{ij}$, Eq. (\[Eq:trickyOpticalTheorem\]) reduces to $$\frac{2}{3}k^3{\sum_{i=1}^6}|c_i|^2|A_i|^2\geq{\sum_{i=1}^6}|c_i|^2\mathrm{Im}A_i,\label{Eq:almostOpticalTheorem}$$ with strict equality for lossless scatterers. Since this equation must be satisfied for any choice of incident wave (i.e., any combination of $c_i$), we find a generalized optical theorem for 6$\times$6 polarizability tensors that can be expressed in terms of the eigenpolarizabilities as $$\frac{2}{3}k^3|A_i|^2\geq \mathrm{Im} A_i \qquad \forall i=1\ldots
6, \label{Eq:TensorpticalTheorem}$$ again with strict equality for lossless scatterers. Eq. (\[Eq:TensorpticalTheorem\]) implies that the polarizability tensor represents an energy conserving scatterer, if and only if each of its 6 eigenpolarizabilities are chosen to satisfy the simple scalar optical theorem (Eq. (\[Eq:scalarOpticalTheorem\])) derived for electric scatterers. This general optical theorem highlights the importance of two new quantities: the eigenpolarizabilities, and the corresponding eigenvectors of the point scatterer polarizability.

In Eq. (\[Eq:scalarRadiationDamping\]) we reviewed the well-known addition of radiation damping required to make electrostatic polarizabilities satisfy the optical theorem. Since metamaterial scatterers are frequently treated via electrostatic circuit models, it would be extremely fruitful to generalize this method to general 6$\times$6 electrostatic polarizability tensors. It 