permutation matrix


1 Permutation Matrix

Let n be a positive integer. A permutation matrixMathworldPlanetmath is any n×n matrix which can be created by rearranging the rows and/or columns of the n×n identity matrixMathworldPlanetmath. More formally, given a permutationMathworldPlanetmath π from the symmetric group Sn, one can define an n×n permutation matrix Pπ by Pπ=(δiπ(j)), where δ denotes the Kronecker delta symbol.

Premultiplying an n×n matrix A by an n×n permutation matrix results in a rearrangement of the rows of A. For example, if the matrix P is obtained by swapping rows i and j of the n×n identity matrix, then rows i and j of A will be swapped in the product PA.

Postmultiplying an n×n matrix A by an n×n permutation matrix results in a rearrangement of the columns of A. For example, if the matrix P is obtained by swapping rows i and j of the n×n identity matrix, then columns i and j of A will be swapped in the product AP.

2 Properties

Permutation matrices have the following properties:

Title permutation matrix
Canonical name PermutationMatrix
Date of creation 2013-03-22 12:06:39
Last modified on 2013-03-22 12:06:39
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 20
Author Wkbj79 (1863)
Entry type Definition
Classification msc 15A36
Related topic MonomialMatrix