Definition. Given be a linear operator such that for every then
The matrix with k-row is given by will be called circulant and denoted by
Theorem 1. , where is a primitive n-th root of unity.
We multiply the j-columm with , and add it onto the first columm, claim that , we have:
The j-element of the first columm is a product of with . We give as a polynomial with n variable on , hence is devided by . Claim that is a n-degree polynomial and the coefficient of is 1.
For , we denote and
We can easily calculate and receive that and since is a linearly independent system, we conclude that are eigenvalues of circulant matrix
ii. With non-singular matrix
(To be continued)