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单词 PermutationMatrix
释义

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:

  • They are orthogonalMathworldPlanetmathPlanetmath (http://planetmath.org/OrthogonalMatrices).

  • They are invertiblePlanetmathPlanetmath.

  • For a fixed (http://planetmath.org/Fixed3) positive integer n, the n×n permutation matrices form a group under matrix multiplicationMathworldPlanetmath.

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

  • They are the extreme pointsPlanetmathPlanetmath of the convex set of doubly stochastic matrices.

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更新时间:2025/5/4 12:26:42