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I am trying to understand the definition of an anisotropic dielectric media. I think I understand the definition in terms of the susceptibility tensor:

$$ \begin{pmatrix} P_{x}\\ P_{y} \\ P_{z} \end{pmatrix} = \epsilon_o \, \begin{pmatrix} \chi_{1_{xx}} & \chi_{1_{xy}} & \chi_{1_{xz}}\\\chi_{1_{yx}} & \chi_{1_{yy}} & \chi_{1_{yz}} \\ \chi_{1_{zx}} & \chi_{1_{zy}} & \chi_{1_{zz}} \end{pmatrix} \, \begin{pmatrix} E_{x}\\ E_{y} \\ E_{z} \end{pmatrix} $$

But does it follow from the expression above that the polarization is never collinear with the electric field for an anisotropic media? Or on the contrary that in an anisotropic media, the polarization and the electric field are still collinear.

Thanks in advanced.

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The susceptibility tensor is a tensor, its matrix components depend on the coordinate system. Select your coordinate system so that the tensor is diagonal (the diagonal elements can be all different) and let the $\mathbf{E}$ field be parallel with one of the coordinate axes, then so will the polarization be parallel with it. See https://en.wikipedia.org/wiki/Diagonalizable_matrix

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