M Theory Lesson 228

Observe that in any dimension, the diagonal Fourier operator $D$ that generalises $\sigma_{Z}$ will have the elementary basis vectors as eigenvectors. Similarly, the basic 1-circulant $(23 \cdots n 1)$ provides a basis set that is complementary to the basis set for $D$. An example of a set of vectors complementary to the $D$ basis in three dimensions is the set $(\omega^{2}, \omega^{2}, 1)$, $(1, \omega^{2}, \omega^{2})$ and $(1, \omega, 1)$, which are the eigenvectors for the operator As with the truncated braid algebras constructed from permutations and diagonals, this equals $D \cdot (231)$. So we see that the quantum Fourier transform is very closely related to complementary basis sets.