Dynamical Matrix and Eigenvectors

In the following, we confine ourselves to the longitudinal branches LA and LO which we number by an index j = 1,2. The treatment of the transverse vibrations TA and TO is completely analogous. If we substitute one of the eigen- 

The homogeneous equations (2.16) or (2.17) define the matrix e(q) to within a constant factor only. On the other hand, e(q) diagonalizes the Her-mitian matrix D(q) e (q)D(q)e(q) = A(q). Therefore, e(q) can be chosen to be a unitary matrix which fixes the undetermined factor. Thus, we require e (q) = e (q) or 

Forming the complex conjugate of (2.29a) and remembering that the eigenvalues of a Hermitian matrix are real, i.e., A (q) = A(q), one obtains 

Comparing (2.29b) with (2,17) we see that the eigenvectors e ( p and e(p are equal to within a factor of modulus unity  

Since the physical properties do not depend on the choice of the phase factor, we choose exp(iy) = 1 and obtain 

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