Constructing an orthonormal set of eigenvectors for DFT matrix using Gramians and determinants
Provides a clearer exposition and implementation of an existing method for constructing orthogonal DFT eigenvectors, which is of interest to researchers in signal processing and numerical linear algebra.
The paper presents a clarified and corrected version of Matveev's method for constructing orthogonal eigenvectors of the DFT matrix, and compares its computational complexity with Gram-Schmidt, finding it more efficient. The method is implemented in Mathematica.
The problem of constructing an orthogonal set of eigenvectors for a DFT matrix is well studied. An elegant solution is mentioned by Matveev in his paper "Interwining relations between the Fourier transfom and discrete Fourier transform, the related functional identities and beyond". In this paper, we present a distilled form of his solution including some steps unexplained in his paper, along with correction of typos and errors using more consistent notation. Then we compare the computational complexity of his method with the more traditional method involving direct application of the Gram-Schmidt process. Finally, we present our implementation of Matveev's method as a Mathematica module.