permutation matrix
1 Permutation Matrix
Let be a positive integer. A permutation matrix is any matrix which can be created by rearranging the rows and/or columns of the identity matrix. More formally, given a permutation from the symmetric group , one can define an permutation matrix by , where denotes the Kronecker delta symbol.
Premultiplying an matrix by an permutation matrix results in a rearrangement of the rows of . For example, if the matrix is obtained by swapping rows and of the identity matrix, then rows and of will be swapped in the product .
Postmultiplying an matrix by an permutation matrix results in a rearrangement of the columns of . For example, if the matrix is obtained by swapping rows and of the identity matrix, then columns and of will be swapped in the product .
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 , the 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 |
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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 |