Permutation matrix/Cycle/C/Eigenvalues/Fact

For a permutation matrix ${}M_{\rho }$ over ${}\mathbb {C}$ for a cycle ${}\rho \in S_{n}$ with ${}\rho :i_{1}\mapsto i_{2}\mapsto \ldots \mapsto i_{k}\mapsto i_{1}$

and a ${}k$-th

root of unity ${}\zeta$, the vectors

${}v_{\zeta }:=\zeta ^{k-1}e_{i_{1}}+\zeta ^{k-2}e_{i_{2}}+\cdots +\zeta e_{i_{k-1}}+e_{i_{k}}\,$

are eigenvectors of ${}M_{\rho }$ for the eigenvalue

${}\zeta$.

In particular, a permutation matrix of a cycle over ${}\mathbb {C}$ is diagonalizable.

Proof, Write another proof