Editing Root of unity (section)

==Cyclic groups==
The {{mvar|n}}th roots of unity form under multiplication a [[cyclic group]] of [[order (group theory)|order]] {{mvar|n}}, and in fact these groups comprise all of the [[finite group|finite]] subgroups of the [[multiplicative group]] of the complex number field.  A [[Generating set of a group|generator]] for this cyclic group is a primitive {{mvar|n}}th root of unity.

The {{mvar|n}}th roots of unity form an irreducible [[group representation|representation]] of any cyclic group of order {{mvar|n}}. The orthogonality relationship also follows from [[group theory|group-theoretic]] principles as described in [[Character group]].

The roots of unity appear as entries of the [[eigenvector]]s of any [[circulant matrix]]; that is, matrices that are invariant under cyclic shifts, a fact that also follows from [[group representation theory]] as a variant of [[Bloch's theorem]].<ref name="yoshitaka">{{cite book
|last1 = Inui|first1 = Teturo
|last2 = Tanabe|first2 = Yukito
|last3 = Onodera|first3 = Yoshitaka
|title = Group Theory and Its Applications in Physics
|publisher = Springer
|year = 1996}}</ref>{{page needed|date=February 2023}} In particular, if a circulant [[Hermitian matrix]] is considered (for example, a discretized one-dimensional [[Laplacian]] with periodic boundaries<ref name="siam">{{cite journal
|last = Strang |first= Gilbert |author-link = Gilbert Strang
|title = The discrete cosine transform
|url = http://epubs.siam.org/sam-bin/dbq/article/33674
|journal = SIAM Review
|volume = 41
|issue = 1
|pages = 135–147
|year = 1999|doi= 10.1137/S0036144598336745 |bibcode= 1999SIAMR..41..135S }}</ref>), the orthogonality property immediately follows from the usual orthogonality of eigenvectors of Hermitian matrices.