Theoretical/Practical Implications of DFT Eigenvectors

Question

Discrete Fourier transform (DFT) has only four distinct eigenvalues: $±1$ and $±i$. For large matrices , each eigenvalue $λ$ yields a multidimensional eigenspace, allowing linear combinations of eigenvectors to create new eigenvectors. Thus, using a basis of DFT's eigenvectors enables the construction of numerous additional eigenvectors

What potential theoretical/practical implications associated with deriving explicit formulations for these DFT eigenvectors? Moreover, is there a specific advantage in identifying a real eigenbasis for the DFT?

Answer 1

A method to construct a real and orthogonal eigenbasis of the DFT matrix has been developed by Dickinson and Steiglitz. There is no explicit closed-form expression, the problem is reduced to the diagonalization of a real symmetric companion matrix. This method can be used to compute fractional powers of the DFT, which has applications in signal processing.

Answer 2

If you will forgive the self-serving pointer, I wrote a little paper examining this question from the perspective of "what's the sparsest basis of eigenvectors inside the DFT?". The paper's introduction points to some earlier work on this question, which may prove more helpful than my own results, depending on the OP's interests.