From my notes I can answer question 2:

If n is of the form 4m then the eigenvalues are:

m+1 eigenvalues that equal 1

m eigenvalues that equal -1

m eigenvalues that equal -i

m-1 eigenvalues that equal i

If n is of the form 4m+1 then the eigenvalues are:

m+1 eigenvalues that equal 1

m eigenvalues that equal -1

m eigenvalues that equal -i

m eigenvalues that equal i

If n is of the form 4m+2 then the eigenvalues are:

m+1 eigenvalues that equal 1

m+1 eigenvalues that equal -1

m eigenvalues that equal -i

m eigenvalues that equal i

If n is of the form 4m+3 then the eigenvalues are:

m+1 eigenvalues that equal 1

m+1 eigenvalues that equal -1

m+1 eigenvalues that equal -i

m eigenvalues that equal i

Unfortunately my notes are sketchy on deriving the characteristic eqtn (your question #1). So do yourself a favor and keep good notes, don't be like me. Try google for DFT and eigenvalues that may turn up an alternate reference.

]]>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): 6674" (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!!!

]]>As far as I can remember the eigenvalues for those matrices are taken from the set of the roots of unity, solutions of z^n-1=0 where n is the size of the matrix. I think there is an easy way to determine how many of each by n modulo 4. I might be mistaken, its been a long time since I have worked with a DFT or FFT. Google will probably provide what you need.

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