On the Wikipedia page on circulant matrices, it is clearly written that
They can be interpreted analytically as the integral kernel of a convolution operator on the cyclic group $\mathbb{Z}/n\mathbb{Z}$.
On the other hand, on the page on Toeplitz matrices, no such groups are mentioned.
My understanding is that circulant matrix are the generator of a cyclic group. Is it correct?
If so, why doesn't a Toeplitz matrix generate a more general group?
Any comments to improve my understanding would be apreciated.
Under addition the group structure is rather trivial.
Under multiplication it is clearly false, since $0$ is a Toeplitz matrix