Orthonormal eigenvectors of the DFT-IV matrix by the eigenanalysis of a nearly tridiagonal matrix

Orthonormal eigenvectors are efficiently generated for the DFT-IV matrix G by a detailed eigenanalysis of a nearly tridiagonal matrix S which commutes with matrix G. Matrix S is reduced to a block diagonal form by means of a similarity transformation and the two diagonal blocks are proved to be tridiagonal matrices. Orthonormal eigenvectors of S are generated by utilizing those of the two diagonal blocks. They are rigorously proved to always be eigenvectors of matrix G irrespective of the multiplicities of the eigenvalues of S.