#Permutation matrix code#
#Permutation matrix license#

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.

#Permutation 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

#Permutation matrix license#

This work is licensed under the Creative Commons Attribution International License (CC BY). In particular, since permutation matrices are orthogonal matrices with nonnegative elements, we define two gradient flows in the space of orthogonal matrices.Departament de Matemàtica Aplicada I, Universitat Politècnica de Catalunya, Barcelona, SpainĮmail: © 2015 by authors and Scientific Research Publishing Inc. 'A dynamical systems approach to weighted graph matching'. Most authors choose one representation to be consistent with other notation they have introduced, so there is generally no need to supply a name. n-queens puzzle, a permutation matrix in which there is at most one entry in each diagonal and antidiagonal.Costas array, a permutation matrix in which the displacement vectors between the entries are all distinct.So, permutation matrices do indeed permute the order of elements in vectors multiplied with them. Each such matrix, say P, represents a permutation of m elements and, when used to multiply another matrix, say A, results in permuting the rows (when pre-multiplying, to form PA) or columns (when post-multiplying, to form AP) of the matrix A. In mathematics, particularly in matrix theory, a permutation matrix is a square binary matrix that has exactly one entry of 1 in each row and each column and 0s elsewhere. ( August 2022) ( Learn how and when to remove this template message) Please help to improve this article by introducing more precise citations. This article includes a list of general references, but it lacks sufficient corresponding inline citations.