Circulant linear systems and the Discrete Fourier Transform

[burritoloco]

Homework Statement

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.

 

[burritoloco]

For the definitions of circulant matrix and the DFT please see

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

 

[burritoloco]

It's probably a yes I'm thinking implying more than one solution for x.