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Permutation Matrix -- from Wolfram MathWorld

  • ️Weisstein, Eric W.
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A permutation matrix is a matrix obtained by permuting the rows of an n×n identity matrix according to some permutation of the numbers 1 to n. Every row and column therefore contains precisely a single 1 with 0s everywhere else, and every permutation corresponds to a unique permutation matrix. There are therefore n! permutation matrices of size n, where n! is a factorial.

The permutation matrices of order two are given by

[1 0; 0 1],[0 1; 1 0]

(1)

and of order three are given by

[1 0 0; 0 1 0; 0 0 1],[1 0 0; 0 0 1; 0 1 0],[0 1 0; 1 0 0; 0 0 1],[0 1 0; 0 0 1; 1 0 0], [0 0 1; 1 0 0; 0 1 0],[0 0 1; 0 1 0; 1 0 0].

(2)

A permutation matrix is nonsingular, and the determinant is always +/-1. In addition, a permutation matrix A satisfies

AA^(T)=I,

(3)

where A^(T) is a transpose and I is the identity matrix.

Applied to a matrix M, A_(p)M gives M with rows interchanged according to the permutation vector p, and MA_(p) gives M with the columns interchanged according to the given permutation vector.

Interpreting the 1s in an n×n permutation matrix as rooks gives an allowable configuration of nonattacking rooks on an n×n chessboard. However, the permutation matrices provide only a subset of possible solutions.

See also

(0,1)-Matrix, Alternating Sign Matrix, Elementary Matrix, Identity, Permutation, Rook Number

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References

Bronshtein, I. N.; Semendyayev, K. A.; Musiol, G.; and Muehlig, H. Handbook of Mathematics, 4th ed. New York: Springer-Verlag, p. 889, 2004.Golub, G. H. and Van Loan, C. F. Matrix Computations, 3rd ed. Baltimore, MD: Johns Hopkins, p. 109, 1996.Horn, R. A. and Johnson, C. R. Matrix Analysis. Cambridge, England: Cambridge University Press, p. 25, 1987.

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Permutation Matrix

Cite this as:

Weisstein, Eric W. "Permutation Matrix." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PermutationMatrix.html

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