In general a permutation is a rearrangement of elements of an ordered list like the elements of a vector. Permutations can be expressed as matrix operations on a vector.
If an arrangement contains n elements n! different permutations are possible.
Permutaions as Matrix Operations
A permutation is a matrix operation that leads two the exchange of the elements of a vector, when the matrix is multiplied with the vector.
The unit matrix is a permutation matrix, that leaves the order of the vector elements unchanged when applied to the vector.
The exchange Matrix is the next example of a permutation matrix.
The exchange matrix or counterbrace matrix Jn, is a matrix where the counterdiagonal elements are all one.
All other elements are zero.
The product of the exchage matrix with a column vector will result in a reordering of the column elements.
The product of a row vector with a exchange matrix will result in a reordering of the row elements.
Square of the Exchange Matrix
J2 = I
The square of the exchange matrix is the unit matrix. Applying the exchange matrix twice will result in the original matrix.
Transpositions are permutations where only two rows of a matrix are exchanged. Every permutation can be expressed as a product of transposions.
Transpositions with a 2*2 matrix:
With two elements 2! = 2 permutations are possible.
Transpositions with a 3*3 matrix
With three elements 3! = 6 permutations are possible.