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I have an image with pixels that form a 1D lattice. Pixel intensities are given by `$x=(x_1,x_2,...,x_N)^T$

The mean of x is zero and variances are 1. Neighbors are correlated by covariance matrix $C_{ij}$, which is 1 for all the diagonal components i=j. All the other elements of C are 0, but when $|i-j|=1$, where |i-j| is mod N, $C_{ij} = a $ and $a \lt0.5$

$C$ is a circulant matrix, how do I find an analytical expression for the eigenvalues and eigenvectors of $C$?

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It's not a bad idea to try google and wiki first: en.wikipedia.org/wiki/Circulant_matrix You can even search this website: math.stackexchange.com/questions/297615/… –  1015 Feb 26 '13 at 16:50
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This isn't my field but I see in http://epubs.siam.org/doi/book/10.1137/1.9780898718850 that circulant matrices can be diagonalized by a Fourier matrix and their eigenvalues can be found efficiently via the FFT. See also

E. Brigham, The Fast Fourier Transform and Its Applications, Prentice–Hall, Englewood Cliffs, NJ, 1988.

I hope this points you in the right direction.

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