# How does $\epsilon \mu = 1/c_m^2$ change when $\mu$ or $\epsilon$ (permeability or permittivity) is a tensor?

I've read that in some materials, $\mu$ can be a tensor, not a mere scalar. (I haven't actually dealt with tensors before, but I'm assuming for my purposes here, it is synonymous with "matrix".)

I'm not sure if the same holds for $\epsilon$, but I'm assuming it might there too.

I'm fine with either of those, but then I'm wondering what the formula

$$\epsilon\,\mu = \frac{1}{c_m^2}$$

turns into when dealing with tensors? ($c_m$ being the speed of light in the medium)

If $\varepsilon$ or $\mu$ are tensors (read, matrices), then so is $c_m$: $$\overbrace{\varepsilon}^\mathrm{matrix} \underbrace{\mu}_\mathrm{matrix}=c_m^{-2}\ \leftarrow\ \text{matrix as well}$$
In other words, if the permeability and/or permittivity are matrices, then the speed of light is a matrix as well. In this case, the $\_^{\color{red}{-1}}$ is understood in the sense of matrix inverse.
In the coordiante system where $c_m$ is diagonal, we have $$c_m=\begin{pmatrix} c_1&\cdot&\cdot\\\cdot&c_2&\cdot\\\cdot&\cdot&c_3\end{pmatrix}$$ where $c_i$ is the speed of light along the $x_i$ axis.