Abstract
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.
Highlights
As it is well known, permutations appear almost all in areas of mathematics
Permutation matrices are orthogonal matrices, and its set of eigenvalues is contained in the set of roots of unity
Taking into account that any permutation is written as a product of disjoint cycles, we can deduce that the minimal annihilating polynomials for each of the matrices associated to these disjoint cycles
Summary
The study of permutation matrices has interest 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 [1]). Many properties are known of permutation matrices. 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. The product of permutation matrices is again a permutation matrix. The characteristic polynomial of permutations matrices has been studied (see, for example, [3]). For any matrix A ∈ M n p , let us denote Q= A (t ) det ( A − tIn ) the characteristic polynomial of A and by M A (t ) the minimal annihilating polynomial of A.
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