Permutation matrices are also double stochastic in fact the set of doubly stochastic matrices corresponds to the convex hull of the set of permutation matrices (see ). They are invertible, and the inverse of a permutation matrix is again a permutation matrix. The product of permutation matrices is again a permutation matrix. Permutation matrices are orthogonal matrices, and therefore its set of eigenvalues is contained in the set of roots of unity. More concretely, we obtain a formula for the minimal annihilating polynomial of a permutation matrix over a finite field and obtain a set of linearly independent eigenvectors of such a matrix. In this work we focus on their spectral properties. Many properties are known of permutation matrices.
Permute a matrix code#
The study of permutation matrices has interest not only in matrix theory, but in other fields such as code theory, where they are a fundamental tool in construction of low-density parity-check codes (see ). Permutation Matrices, Eigenvalues, EigenvectorsĪs it is well known, permutations appear almost all in areas of mathematics. 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. The spectral properties of special matrices have been widely studied, because of their applications. Received 11 March 2015 accepted published This work is licensed under the Creative Commons Attribution International License (CC BY). Departament de Matemàtica Aplicada I, Universitat Politècnica de Catalunya, Barcelona, SpainĮmail: © 2015 by authors and Scientific Research Publishing Inc.
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