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.
[1] Fossorier, M.P.C. (2004) Quasi-Cyclic Low-Density Parity-Check Codes from Circulant Permutation Matrices. IEEE Transactions on Information Theory, 50, 1788-1793.
http://dx.doi.org/10.1109/TIT.2004.831841
[2] Marshall, A.W., Olkin, I. and Arnold, B.C. (2011) Doubly Stochastic Matrices. Inequalities: Theory of Majorization and Its Applications. Springer, New York.
http://dx.doi.org/10.1007/978-0-387-68276-1
[3] Hamblya, B.M., Keevashc, P., O’Connella, N. and Starka, D. (2000) The Characteristic Polynomial of a Random Permutation Matrix. Stochastic Processes and Their Applications, 90, 335-346.
http://dx.doi.org/10.1016/S0304-4149(00)00046-6
[4] Skiena, S. (1990) The Cycle Structure of Permutations 1.2.4. In: Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica, Addison-Wesley, Reading, 20-24.
[5] Fripertinger, H. (2011) The Number of Invariant Subspaces under a Linear Operator on Finite Vector Spaces. Advances in Mathematics of Communications, 2, 407-416.
http://dx.doi.org/10.3934/amc.2011.5.407