Interpreting FFT Coefficients from System Matrix That Is Originally Toeplitz (Not Circulant)
If I understand your question, then the answer lies in the equivalence of linear and circular convolution - the linear convolution being implemented by multiplication with a toeplitz matrix, while circular convolution is implemented by multiplication with a circulant matrix.

If you have a length N signal and length M filter impulse response, the circular and linear convolutions are equivalent if you pad the signal and filter with zeros to length N+M-1 - that is, you can implement the linear convolution using the DFT. You are padding x with zeros above, and presumably the number of zeros is such that the length N+M-1 requirement is met.  Then in matrix form, the DFT matrix would diagonalize the corresponding convolution operator.

Hope this helps - I'm short on time right now and can't provide a more detailed explanation.