The spectral properties of special matrices have been widely studied, because of their applications.
We focus on permutation matrices over a finite field and, more concretely, we compute the minimal
annihilating polynomial, and a set of linearly independent eigenvectors from the decomposition
in disjoint cycles of the permutation naturally associated to the matrix.
Garcia-Planas, M.I.; Magret, M. D. Eigenvectors of permutation matrices. "Advances in Pure Mathematics", 2015, vol. 5, p. 390-394.