----- I don't exactly know what you're asking for, but maybe you are trying to make a jump off Berlekamp's complex transformations by restricting to reals? I think the answer to your questions just falls out of basic representation theory of a finite cyclic group. ----- I thought it was clear I was asking for the angles theta_j. ----- In some choice of coordinates, your T commutes with a cyclic permutation of basis vectors, call that transformation T'. T' is a regular representation of the generator of a cyclic group, so we know its eigenbasis, which is shared with T. ----- The transformation T is *defined* as a cyclic permutation of (orthonormal) basis vectors. I don't know anything about representation theory, and didn't knowingly use it here. —Dan