Do Toeplitz matrices form a group?

Question

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.

Answer 1

Under addition the group structure is rather trivial.

Under multiplication it is clearly false, since $0$ is a Toeplitz matrix

Answer 2

the set of nxn upper (lower) triangular toeplitz matrices with 1 on the main diagonal is an abelian group under multiplication.

a similar construction holds for block-toeplitz matrices, where each block is mxm and the whole matrix is nmxnm. this group is not necessarily abelian.

Answer 3

the set of ALL the nxn toeplitz matrices (n>1) is not a multiplicative group: first, as mentioned earlier, it contains zero and other non-invertible matrices. in addition, it is not closed under multiplication. for example for n=2 E12 and E21 are toeplitz but not E12.E21 (nor E21.E12).