For a permutation matrix M ρ {\displaystyle {}M_{\rho }} over C {\displaystyle {}\mathbb {C} } for a cycle ρ ∈ S n {\displaystyle {}\rho \in S_{n}} with ρ : 1 ↦ 2 ↦ … ↦ k ↦ 1 {\displaystyle {}\rho :1\mapsto 2\mapsto \ldots \mapsto k\mapsto 1}
root of unity ζ {\displaystyle {}\zeta } , the vectors
are eigenvectors of M ρ {\displaystyle {}M_{\rho }} for the eigenvalue
In particular, a permutation matrix of a cycle over C {\displaystyle {}\mathbb {C} } is diagonalizable.
over 
for a
cycle
with 
, the vectors
for the
eigenvalue is
diagonalizable.