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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.

As it is well known, permutations appear almost all in areas of mathematics. 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 [

Many properties are known of permutation matrices. In this work we focus on their spectral properties. 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.

Permutation matrices are orthogonal matrices, and therefore its set of eigenvalues is contained in the set of roots of unity. The product of permutation matrices is again a permutation matrix. They are invertible, and the inverse of a permutation matrix is again a permutation matrix. 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 [

Throughout the paper, we will denote by

Let us consider the set

There is not an only possibility of the decomposition since being the cycles disjoint they can be written in any order and, moreover, any rotation of a given cycle specifies the same cycle.

See, for example, [

A monomial matrix of order n is a regular

The cycle type of a cycle is the data of how many cycles of each length are present in the cycle decomposition of the cycle. If the cycle is a product of

Example 1. We list below all the permutations of the symmetric group of n elements for

For

For

For

The minimal annihilating polynomial of a permutation matrix can be deduced from the permutation to which the matrix is associated. In the case where the permutation considered is the identity

In the non-trivial cases, the minimal annihilating polynomial can be determined by the decomposition of the permutation in disjoint cycles. This Section is devoted to prove this.

Lemma 1. Lep

Theorem 1. Let P be a permutation matrix associated to a permutation with cycle type

where

Proof. 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. It is straightforward to check that the permutation matrix associated to a k-cycle is annihilated by the polynomial

Example 2. We list below the minimal annihilating polynomial of the permutation matrices associated to the permutations in the cases where

For

For

For

We will determine the set of eigenvectors of a permutation matrix from the decomposition of the permutation associated to it, in disjoint cycles. The proofs (not included) are based on straightforward computations.

Theorem 2. Let P be a permutation matrix associated to a permutation which is a disjoint product of cycles. Let us assume that one of them,

Illustrative examples

We will consider the case where

1. Let us consider the 2-cycle

2. If the permutation

If we consider the

3. Let us consider now the case of

i) If the 4-cycle is

ii) If the 4-cycle is

iii) If the 4-cycle is

iv) If the 4-cycle is

v) If the 4-cycle is

vi) If the 4-cycle is

4. Let us consider the permutation matrix associated to a cycle of type 2 + 2 + 4 + 8. Then the minimal annihilating polynomial is:

Let us assume, for example, that the 2-cycles are:

We have found an easy way to write the minimal annihilating polynomial eigenvectors of a permutation matrix relating the permutation of its rows with its disjoint cycle decomposition. The results here can be generalized to monomial matrices, for example.