example of permutation matrix

Consider the matrix

P=\begin{pmatrix}0&1&0&0\\ 0&0&0&1\\ 1&0&0&0\\ 0&0&1&0\end{pmatrix}

that corresponds to permuting the columns of the identity matrix under the permutation $(1243)$ (i.e. (http://planetmath.org/Ie), the first column of the identity matrix is the second column of $P$ , the second column of the identity matrix is the fourth column of $P$ , etc.). Then $P$ is a permutation matrix.

We will consider what happens when we multiply a $4\times 4$ matrix by $P$ . For example, let $A$ be the matrix

A=\begin{pmatrix}4&2&6&8\\ 1&3&5&7\\ 1&0&1&0\\ -1&-2&-3&-4\end{pmatrix}.

Then

PA=\begin{pmatrix}0&1&0&0\\ 0&0&0&1\\ 1&0&0&0\\ 0&0&1&0\end{pmatrix}\begin{pmatrix}4&2&6&8\\ 1&3&5&7\\ 1&0&1&0\\ -1&-2&-3&-4\end{pmatrix}=\begin{pmatrix}1&3&5&7\\ -1&-2&-3&-4\\ 4&2&6&8\\ 1&0&1&0\end{pmatrix}.

We notice that $P A$ has the same rows as $A$ . Moreover, the rows of $A$ are the rows of $P A$ permuted under $(1243)$ : The first row of $P A$ is the second row of $A$ , the second row of $P A$ is the fourth row of $A$ , etc.

Title	example of permutation matrix
Canonical name	ExampleOfPermutationMatrix
Date of creation	2013-03-22 15:03:14
Last modified on	2013-03-22 15:03:14
Owner	Wkbj79 (1863)
Last modified by	Wkbj79 (1863)
Numerical id	8
Author	Wkbj79 (1863)
Entry type	Example
Classification	msc 15A36