I am studying circulant matrices, and I have seen that one of the properties of such matrices is the eigenvalues which are some combinations of roots of unity. I am trying to understand why it is like that. In all the places I have searched they just show that it is true, but I would like to know how come?
2026-03-25 12:16:22.1774440982
Eigenvalues of circulant matrix
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You could just check their work and try showing that the eigenvectors that they gave you are actually eigenvectors.
Another way of doing that is to note that if $A$ is circulent, then $A - xI$ is also circulent, so knowing the null space of circulent matrices in general is sufficent to know the eigenvectors of circulent matrices in general.
You can also just search "Circulant Matrix site:.edu" in Google for circulent matrices whose website url has a domain name ending in ".edu" so you tend to get professor's notes and other useful material. E.g. using that method I found in Daryl Geller, Irwin Kra, Sorin Popescu and Santiago Simanca - On Circulant Matrices.