Permutation Matrix

Author

Eric W. Weisstein
October 13, 2005

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Combinatorics/PermutationMatrix.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/PermutationMatrix.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Initialization

PermutationMatrix[p_List]:=IdentityMatrix[Length[p]][[p]]
PermutationMatrices[n_]:=PermutationMatrix/@Permutations[Range[n]]

Enumeration

MatrixForm/@PermutationMatrices[1]

MatrixForm /@ PermutationMatrices[2]

MatrixForm /@ PermutationMatrices[3]

Det/@ PermutationMatrices[3]

Counts

Length/@Table[PermutationMatrices[n],{n,5}]

Normalization

Table[And@@(#.Transpose[#]==IdentityMatrix[n]&/@PermutationMatrices[n]),{n,5}]

Multiplication

PermutationMatrix[{1,3,2}]//MatrixForm

PermutationMatrix[{1,3,2}].Array[a,{3,3}]/.a[l__]:>Subscript[a,l]//TraditionalForm

Array[a,{3,3}].PermutationMatrix[{1,3,2}]/.a[l__]:>Subscript[a,l]//TraditionalForm