Trying to show that columns of Fourier matrix are eigenvectors

Question

I posted a picture since the syntax for this one seems quite complex:

I found this: Discrete Fourier Transform - proof that columns of matrix are orthogonal which only shows that are orthogonal. In my case I need to show that they are eigenvectors. I have no idea how show this

Answer 1

Let $S$ be the cyclic shift operator. Notice that $A = -S^{-1} +2S^0 - S$. The fact that the discrete Fourier basis vectors are eigenvectors of $A$ now follows from the fact that they are eigenvectors of $S$. (And the whole point of the discrete Fourier basis vectors is that they are eigenvectors of $S$.)

A similar argument shows that the discrete Fourier basis vectors are eigenvectors for any convolution operator.