Tensor of Polarizability

E is the vector for the electric field.

P is the vector for the polarization.

The equations to relate P to E are shown below:

$P_x = \alpha_{xx} E_x + \alpha_{xy} E_y + \alpha_{xz} E_z$

$P_y = \alpha_{yx} E_x + \alpha_{yy} E_y + \alpha_{yz} E_z$

$P_z = \alpha_{zx} E_x + \alpha_{zy} E_y + \alpha_{zz} E_z$

We can get this math by multiplying a matrix on a vector:

$\begin{bmatrix} \alpha_{xx} && \alpha_{xy} && \alpha_{xz} \\ \alpha_{yx} && \alpha_{yy} && \alpha_{yz} \\ \alpha_{zx} && \alpha_{zy} && \alpha_{zz} \end{bmatrix} \begin{bmatrix} E_x \\ E_y \\ E_z \end{bmatrix} = \begin{bmatrix} P_x \\ P_y \\ P_z \end{bmatrix}$

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