1 Permutation Matrix
Let be a positive integer. A permutation matrix is any matrix which can be created by rearranging the rows and/or columns of the identity matrix. More formally, given a permutation from the symmetric group , one can define an permutation matrix by , where denotes the Kronecker delta symbol.
Premultiplying an matrix by an permutation matrix results in a rearrangement of the rows of . For example, if the matrix is obtained by swapping rows and of the identity matrix, then rows and of will be swapped in the product .
Postmultiplying an matrix by an permutation matrix results in a rearrangement of the columns of . For example, if the matrix is obtained by swapping rows and of the identity matrix, then columns and of will be swapped in the product .
Permutation matrices have the following properties:
They are orthogonal (http://planetmath.org/OrthogonalMatrices).
They are invertible.
For a fixed (http://planetmath.org/Fixed3) positive integer , the permutation matrices form a group under matrix multiplication.
Since they have a single 1 in each row and each column, they are doubly stochastic.
|Date of creation||2013-03-22 12:06:39|
|Last modified on||2013-03-22 12:06:39|
|Last modified by||Wkbj79 (1863)|