I've read this document which talks about the DFT. It describes that DFT bases are the eigenvectors of a circulant matrix. I know that every basis has a frequency in it, but I don't know what is it? And it there any relation between this frequency and eigenvalues? For example, one of the bases of DFT with $n$ points is as follow:

\begin{equation*} F^{(k)} = \begin{pmatrix} w_n^{0k} \\ w_n^{1k} \\ \vdots \\ w_n^{(n-1)k} \end{pmatrix} \end{equation*} \begin{equation*} w_n^{jk} = e^{\frac{2\pi i}{n}jk} \end{equation*} and the eigenvalue related to this eigenvector($F^{(k)}$) is $\lambda_k = \sum_{j=0}^{n-1}c_jw_n^{jk}$, where $c_i$ is the $i$-th element of the first row of circulant matrix. What is the frequency of $F^{(k)}$? Is there any relation between this frequency and $\lambda_k$? Can we talk about the frequency based on the $\lambda_k$? Since $\lambda_k$ is related to the elements of this circulant matrix but the eigenvectors are not related, so can we define special circulant matrix that we can find the frequency from its eigenvalues?


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