permutationmatrix

A permutation matrix is a square matrix that has exactly one 1 in each row and each column. The 1s in the matrix are positioned so that each row and column has a single 1, and all other elements in the matrix are 0.

The name "permutation" comes from the fact that a permutation matrix represents a permutation, which is a rearrangement of the elements of a set. Specifically, if we have a permutation matrix P, then for any vector v, the product Pv will be the vector obtained by rearranging the elements of v according to the permutation represented by P.

Permutation matrices can be used to represent permutations in various ways. For example, if we have a permutation matrix P representing a permutation (a b c)(d e f), then P will have a 1 in the a row and the c column, a 1 in the b row and the e column, and a 1 in the c row and the a column. The elements d and f will be unchanged in the resulting vector Pv.

Permutation matrices have several useful properties. They are invertible, meaning that their inverse is also a permutation matrix. They also have a simple form when expressed in terms of standard basis vectors. Additionally, permutation matrices can be used to compute the cycle index of a group, which is a useful tool in group theory.

Permutation matrices are used extensively in computer science, physics, and engineering to represent and manipulate permutations. They are a fundamental tool in the study of groups and their representations.