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Circulant linear systems and the Discrete Fourier Transform

  1. Jun 28, 2013 #1
    1. The problem statement, all variables and given/known data
    Hi, this is not a homework question per se, but something I'm wondering. Let C be a circulant n x n matrix, let x, b, be vectors such that

    C x = b.

    We would like to find a solution x. One way is to use the DFT: According to section 5, In Linear Equations, in the wikipedia article available at

    http://en.wikipedia.org/wiki/Circulant_matrix#In_linear_equations

    if we let c be the 1st column of C, we have a convolution c * x = b from which the Circular Convolution Theorem gives

    F_k(c) F_k(x) = F_k(b),

    where F_k(x) denotes the k-th component of the Fourier transformation of the vector x. My question is, as I'm fairly new to all this, what is F_k(x) if F_k(c) = 0 for some k? Do we choose F_k(x) arbitrarily in this case? Thank you.
     
  2. jcsd
  3. Jun 28, 2013 #2
  4. Jun 28, 2013 #3
    It's probably a yes I'm thinking implying more than one solution for x.
     
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