A permutation matrix is a matrix obtained by permuting the rows of an identity matrix according to some permutation of the numbers 1 to . 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 permutation matrices of size , where is a factorial.
The permutation matrices of order two are given by
|
(1) |
and of order three are given by
|
(2) |
A permutation matrix is nonsingular, and the determinant is always . In addition, a permutation matrix satisfies
|
(3) |
where is a transpose and is the identity matrix.
Applied to a matrix , gives with rows interchanged according to the permutation vector , and gives with the columns interchanged according to the given permutation vector.
Interpreting the 1s in an permutation matrix as rooks gives an allowable configuration of nonattacking rooks on an 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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Cite this as:
Weisstein, Eric W. "Permutation Matrix." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PermutationMatrix.html