Permutation matrices
A 
matrix 
is a permutation matrix if it is obtained by permuting the rows or columns of an identity matrix according to some permutation of the numbers 
to 
. Permutation matrices are orthogonal (hence, their inverse is their transpose: 
) and satisfy 
.
For example, the matrix
![\[P = \begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{pmatrix}.\]](https://pressbooks.pub/app/uploads/quicklatex/quicklatex.com-e92605806d7673a931a9c231cdc7c91a_l3.png "Rendered by QuickLaTeX.com")
is obtained by exchanging the columns 
and 
, and 
and 
, of the 
identity matrix.
A permutation matrix allows to exchange rows or columns of another via the matrix-matrix product. For example, if we take any 
matrix 
, then 
(with defined above) is the matrix with columns 
and 
exchanged.
