Transforming the tensor components
You know that when we change to a different coordinate system $x'$, $y'$, and $z'$, the components $E_{x'}$, $E_{y'}$, and $E_{z'}$ of the vector will be quite different—as will also the components of $\FLPP$. So all the coefficients $\alpha_{ij}$ will be different for a different set of coordinates. You can, in fact, see how the $\alpha$’s must be changed by changing the components of $\FLPE$ and $\FLPP$ in the proper way, because if we describe the same physical electric field in the new coordinate system we should get the same polarization. For any new set of coordinates, $P_{x'}$ is a linear combination of $P_x$, $P_y$, and $P_z$: \begin{equation*} P_{x'}=aP_x+bP_y+cP_z, \end{equation*}  and similarly for the other components. If you substitute for $P_x$, $P_y$, and $P_z$ in terms of the $E$’s, using Eq. (31.4), you get \begin{alignat*}{6} P_{x'}&=&&\,a&&(\alpha_{xx}&&E_x+\alpha_{xy}&&E_y+\alpha_{xz}&&E_z)\\[.5ex] &+\,&&\,b&&(\alpha_{yx}&&E_x+\alpha_{yy}&&E_y+\alpha_{yz}&&E_z)\\[.5ex] &+\,&&\,c&&(\alpha_{zx}&&E_x+\alpha_{zy}&&E_y+\alpha_{zz}&&E_z). \end{alignat*}  Then you write $E_x$, $E_y$, and $E_z$ in terms of $E_{x'}$, $E_{y'}$, and $E_{z'}$; for instance, \begin{equation*} E_x=a'E_{x'}+b'E_{y'}+c'E_{z'}, \end{equation*}  where $a'$, $b'$, $c'$ are related to, but not equal to, $a$, $b$, $c$. So you have $P_{x'}$, expressed in terms of the components $E_{x'}$, $E_{y'}$, and $E_{z'}$; that is, you have the new $\alpha_{ij}$. It is fairly messy, but quite straightforward. When we talk about changing the axes we are assuming that the crystal stays put in space. If the crystal were rotated with the axes, the $\alpha$’s would not change. Conversely, if the orientation of the crystal were changed with respect to the axes, we would have a new set of $\alpha$’s. But if they are known for any one orientation of the crystal, they can be found for any other orientation by the transformation we have just described. In other words, the dielectric property of a crystal is described completely by giving the components of the polarization tensor $\alpha_{ij}$ with respect to any arbitrarily chosen set of axes. Just as we can associate a vector velocity $\FLPv=(v_x,v_y,v_z)$ with a particle, knowing that the three components will change in a certain definite way if we change our coordinate axes, so with a crystal we associate its polarization tensor $\alpha_{ij}$, whose nine components will transform in a certain definite way if the coordinate system is changed. The relation between $\FLPP$ and $\FLPE$ written in Eq. (31.4) can be put in the more compact notation: \begin{equation} \label{Eq:II:31:5} P_i=\sum_j\alpha_{ij}E_j, \end{equation}  where it is understood that $i$ represents either $x$, $y$, or $z$ and that the sum is taken on $j=x$, $y$, and $z$. Many special notations have been invented for dealing with tensors, but each of them is convenient only for a limited class of problems. One common convention is to omit the sum sign $(\sum)$ in Eq. (31.5), leaving it understood that whenever the same subscript occurs twice (here $j$), a sum is to be taken over that index. Since we will be using tensors so little, we will not bother to adopt any such special notations or conventions.