Go directly to derivation of polarization effects.
The response of individual electric charges to an electric field can be described with Newton's laws. The behavior of a charged fluid has a field description that will not be discussed here. The behavior of many solids in response to an electric field can be described remarkably well with the same kind of approximation that we made for stress and strain.
If we think of stress as the perturbation applied to a sample and the strain as the response of the sample to the perturbation, we can say that in our model the response is proportional to the perturbation. The response of many common solids to an applied electric field is such that the electric field inside the sample is less than the applied electric. It is as if the solid has produced its own charge distribution - become polarized so that it produces an electric field which adds, in opposition, to the applied field. Materials that do this are called dielectrics.
We take the polarization of the material to be the response in this case. It turns out that in many useful situations the polarization is proportional to the applied electric field. We need to define the polarization in a way that allows us to measure it. We follow tradition and define it as
where the constant of proportionality, 
, is called the dielectric polarizability.
This definition does not yet tell us how to measure the polarization and determine the size of . That is accomplished by relating 
to the internal electric field which reduces the applied electric field. Calling the internal electric field 
we write
where 
is the usual electric constant
With this definition, the polarization has units of electric dipole moment per unit volume. This is consistent with microscopic models of what happens inside a material to produce the observed electric field.
Switching to subscript notation again, and allowing the possibility that an electric field in one direction might produce a polarization in another (as is sometimes observed experimentally), we can write
where we use the Einstein Summation Convention, that a repeated index (in this case the repeated j index) is summed. Thus typically the ith component of polarizability is calculated as the sum of three contributions; one from each component of the applied field.
is called the polarizability tensor. It can be represented as a 3x3 array of 9 numbers.
In an isotropic material, the elements are zero unless 
. Then the array has three non-zero elements, in a diagonal line from the upper left to the lower right corner. Moreover, in an isotropic material, there are no preferred directions. This means that
For anisotropic materials, the diagonal elements are generally not equal to each other, and the off-diagonal elements are generally not zero. Two things are true, however:
1) The array is symmetric, meaning that 
. It is also said that the array is symmetric under the interchange of i and j.
2) There exists a coordinate system in which the array has no off-diagonal elements and is said to be "diagonal." (There is actually a minimum of 3 such right-handed coordinate systems.) Even in this coordinate system, for anisotropic materials, the three diagonal elements cannot all be equal.
The problem of finding the values of the three 
's in the diagonal array is called the "eigenvalue" problem. The three 's are called "eigen values," or eigenvalues, meaning unique values. The coordinate axes in which the array is diagonal are called "eigenvectors." The process of finding them is called "diagonalization" of the array (usually called a matrix).
Tables of values of polarizability will list the eigenvalues and the eigenvector directions if the material is anisotropic. Not all materials are dielectrics. In some cases, the electric field increases within the material when an external electric field is applied. Materials that do this are called para-electric or ferro-electric materials.
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