![]() For, analogous tables can be constructed.Ģ.1. ![]() We list below all the permutations of the symmetric group of n elements for, and, expressing the decomposition of the cycle in disjoint cycles and the cycle type. Two permutations are conjugate in the symmetric group if and only if they have the same cycle type.Įxample 1. ![]() , -cycles, then we will write that its cycle type is. If the cycle is a product of -cycles, -cycles. 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. Another property of permutation matrices is given below. Equivalently, the permutation matrix in which the permutation applied to the rows of the identity matrix is. We will denote by the permutation matrix associated to the permutation of M, that is to say, the permutation matrix in which the non-zero components are in columns. Its rows are a permutation of the rows of the identity matrix. Permutation matrices are monomial matrices in which all non-zero components are equal to 1. See, for example, and for further reading about this topic.Ī monomial matrix of order n is a regular -matrix which has in each row and in each column exactly one non-zero component. 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. A 1-cycle will be denoted by (i) and it means that this element remains unchanged (it is a fixed point of the permutation). The usual notation of a k-cycle means that is replaced by, by, and so on being the last replacement by. Any permutation of M can be written as a product of disjoint cycles (also called “orbits”). For any matrix, let us denote the characteristic polynomial of A and by the minimal annihilating polynomial of A. Throughout the paper, we will denote by the finite field of p elements (p is a prime number), and assume. The characteristic polynomial of permutations matrices has also been studied (see, for example, ). 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. 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). ( 1)) form a group, which means if we multiply two permutation matrices of ( 1) the result will again be a matrix of ( 1).Departament de Matemàtica Aplicada I, Universitat Politècnica de Catalunya, Barcelona, SpainĮmail: © 2015 by authors and Scientific Research Publishing Inc. Permutation matrices have the special property that their transpose is also the inverseĪll permutation matrices belonging to a certain dimension (e.g. We can do this row operations by matrices called permutation matrices $\mathbf\\ What we are supposed to do, is to exchange rows in order to get a non-zero entry at the pivot position. However, we haven't yet discussed, what to do when we encounter a zero in the pivot position. In another article we have discussed Matrix Elimination.
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