Math Is Fun Forum

I found this wikipedia page http://en.wikipedia.org/wiki/Discrete_Fourier_transform that gives a really good and correct definitions and properties of the DFT. It also gives the eigenvalues but doesn't tell exactly how to find them. This is a quote from the page:

Consider the unitary form

defined above for the DFT of length

, where

. This matrix satisfies the equation:

This can be seen from the inverse properties above: operating

twice gives the original data in reverse order, so operating

four times gives back the original data and is thus the identity matrix. This means that the eigenvalues

satisfy a characteristic equation:

Therefore, the eigenvalues of

are the four roots of unity:

is +1, -1, +i, -i

What is not clear form me is why is the characteristic equation

. How can I derive this algebraically? Also, how do I calculate the multiplicity of these eigenvalues? The calculation of the multiplicities is given in this paper "J. H. McClellan and T. W. Parks (1972). "Eigenvalues and eigenvectors of the discrete Fourier transformation". IEEE Trans. Audio Electroacoust. 20 (1): 66–74" (http://ieeexplore.ieee.org/search/wrapper.jsp?arnumber=1162342), but my university doesn't have access to this paper, so I haven't read this yet.

In summary my questions are as follows:
1) How to derive algebraically the characteristic equation

2) How to calculate the multiplicity of the eigenvalues
3) What is a good alternate reference for this paper

Thanks in advanced!!!