A dielectric material is characterized by a matrix (tensor) permittivity 7 3 -20.5 - 20.5 10 -20.5 xx Exy Exy Eyy tyz = Exz yz zz when referred to the xyz coordinate frame. If the coordinate axis is rotated into the principal axis u, u, w, the permittivity is exhibited in diagonal form: Dy D. 0 4 0 [] = 7 3 -20.5 Find the principal axis and the values of the principal dielectric constants uu/0, etc. Hint: By definition, along a principal axis a scalar equation such as Eu Eu holds. In general, if D is directed along a principal axis, then Du 7-4A/0 3 -20.5 ul 0 0 0 0 0 ww Uu 0 3 7 -20.5 3 7 -20.5 + or in words, when D is directed along a principal axis, it is related to E by a scalar constant A. The above constitutes a set of three homogeneous equations, of which the first is -2/0.5 Ex -20.5 E, A 10 70 2 0.5 4 -E-0 4 Verify that, for a solution, the following determinant must vanish: 3 7-4A/0 - 20.5 -20.5 - 20.5 = 0 104A/0 E Answer: = 30, u = 20, ww = 0. Unit vectors along the principal axis are This cubic equation gives three roots for A, which may be identified as Euur Ev ww. For any one root, say the components of a vector directed along the corresponding principal axis are proportional to the cofactors of the above determinant. This type of problem is called a matrix eigenvalue problem. The A's are the eigenvalues. 0.5 a, a = 0.5a, + 0.5a, a=0.5a, + 0.5a, + 0.5 a, a = 0.5 a, - 0.5 a,