We have a very-nearly-diagonal form for the tension matrix
for wavenumbers which are close to , in a scaling eigenfunction basis
We consider this matrix and its
derivative with respect to ,
The transformation between matrix representations is
Then diagonalization of the matrix can give eigenvectors which are very close to the desired rows of . In particular, an eigenvector very close to the scaling eigenfunction may be returned (if ). However the eigenvalue will equal the tension of the state resulting from a unit norm in coefficient space, which has no physical significance in the basis sets used (RPWs + EPWs). Also the null-space vectors in produce small-eigenvalue solutions (exponentially spread down to machine precision, as in Fig. 5.3) which interfere (mix) with the desired vectors. However the parabolic tension minima are still visible. The same is true if the matrix is diagonalized, only now small eigenvalues correspond to both Dirichlet (upwards-travelling) and Neumann (downwards-travelling) eigenstates. Therefore direct diagonalization is not a good way to extract the desired states. Fig. 6.4 shows the -dependence of these two diagonalizations.