The singular values can be obtained by taking the absolute value of the fft of the 1st column of the circulant matrix. I would like to obtain matrix U by doing a simple operation on the fft result or on the dft matrix.
Thank you for your help
The circulant matrix C can indeed be diagonalized by the Fourier matrices in the form of C=F^-1*c*F, where c is the dft of the first column of C. I am looking for a similar relation for the svd decomposition.
The svd decomposition can be obtained form the discrete Hartley transform, where V is the DHT operator in matrix form, s is the absolute value of the DFT of the vector which describes the circulant matrix, and U is a permuted transformation of the DHT.
Dear Behnam, the decomposition you are talking about (Fn^-1D Fn) does not fall perfectly in the category of SVD, since singular values are real and non-negative. Here, D=diag(d1,d2,..) were d is the DFT of the first colum of the circulant matrix, which is not real and non-negative...
In fact, the decomposition you are talking about is the eigen decompostion of matrix C, as I have written in my earlier post.
I was looking for a similar relation to express the svd decomposition of circulants, which is in fact related to the discrete Hartley transform. I hope this clarifies the question.
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