Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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 $C$?

share|improve this question
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

1 Answer 1

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.

share|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.