What are the eigenvalues and eigenvectors of the Quantum Fourier Transform? 
In general, $\operatorname{QFT}_m$ is used to denote the QFT defined on basis states $|0\rangle, |1\rangle, \dotsc, |m-1\rangle$ according to
  $$
           \operatorname{QFT}_m
  \colon   |x\rangle
  \mapsto  \frac{1}{\sqrt{m}}
           \sum_{y=0}^{m-1} e^{2 \pi i \frac{x}{m} y} |y\rangle.
$$

This is the definition of QFT. I think the eigenvectors of $\operatorname{QFT}_m$ must be some linear combinations of $|0\rangle, |1\rangle, \dotsc, |m-1\rangle$. However I can't see what those eigenvectors exactly are.
Could anyone explain to me what the eigenvectors and eigenvalues of QFT are?
 A: This is asking for the eigenvalues and eigenvectors of the following matrix (with $\newcommand{\ze}{\zeta}\ze=\exp(2\pi i/m)$)
$$
A=\frac1{\sqrt m}\pmatrix{1&1&1&\cdots&1\\1&\ze&\ze^2&\cdots&\ze^{m-1}
\\1&\ze^2&\ze^4&\cdots&\ze^{2(m-1)}\\\vdots&\vdots&\vdots&\ddots&\vdots\\1&\ze^{m-1}&\ze^{2(m-1)}&\cdots&\ze^{(m-1)^2}}.$$
It is both a Vandermonde matrix and unitary.
Then
$$A^2=\pmatrix{1&0&\cdots&0&0\\0&0&\cdots&0&1
\\0&0&\cdots&1&0\\\vdots&\vdots&\ddots&\vdots&\vdots\\0&1&0&\cdots&0}.$$
Thus in general, $A^4=I$ and the eigenvalues are $\pm1$ and $\pm i$.
Each eigenspace will have dimension approximately $m/4$. The exact
figures will depend on the congruence class of $m$ modulo $4$.
A: As Lord Shark correctly showed, the eigenvalues are of the form $i^{m}$ ($m\in{1,2,3,4}$). Figuring out the eigenspaces is, as far as I know, straightforward, but time-consuming. In general, the spectral theorem gives the decomposition:
$$A = P_{1} + (-1)P_{2} + iP_{3} + (-i)P_{4},$$
where $P_{i}$ are the projectors  onto the 4 eigenspaces. I found an article that details what these eigenspaces look like exactly (the projection matrices $P_{i}$). You can take a look:
https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&ved=2ahUKEwibx77YyvriAhXNLVAKHdWyArYQFjAAegQIAhAC&url=https%3A%2F%2Fkindai.repo.nii.ac.jp%2F%3Faction%3Drepository_uri%26item_id%3D9579%26file_id%3D40%26file_no%3D1&usg=AOvVaw1Kl7DTXhKE766sbs-Ku1sD
I did not look through the whole proof in detail (the one in the article), but the results look fine. 
