permutation matrix
1 Permutation Matrix
Let $n$ be a positive integer. A permutation matrix^{} is any $n\times n$ matrix which can be created by rearranging the rows and/or columns of the $n\times n$ identity matrix^{}. More formally, given a permutation^{} $\pi $ from the symmetric group ${S}_{n}$, one can define an $n\times n$ permutation matrix ${P}_{\pi}$ by ${P}_{\pi}=({\delta}_{i\pi (j)})$, where $\delta $ denotes the Kronecker delta symbol.
Premultiplying an $n\times n$ matrix $A$ by an $n\times n$ 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 $n\times n$ identity matrix, then rows $i$ and $j$ of $A$ will be swapped in the product $PA$.
Postmultiplying an $n\times n$ matrix $A$ by an $n\times n$ 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 $n\times n$ identity matrix, then columns $i$ and $j$ of $A$ will be swapped in the product $AP$.
2 Properties
Permutation matrices have the following properties:

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They are orthogonal^{} (http://planetmath.org/OrthogonalMatrices).

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They are invertible^{}.

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For a fixed (http://planetmath.org/Fixed3) positive integer $n$, the $n\times n$ permutation matrices form a group under matrix multiplication^{}.

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Since they have a single 1 in each row and each column, they are doubly stochastic.

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They are the extreme points^{} of the convex set of doubly stochastic matrices.
Title  permutation matrix 

Canonical name  PermutationMatrix 
Date of creation  20130322 12:06:39 
Last modified on  20130322 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 