# permutation matrix

## 1 Permutation Matrix

Let $n$ be a positive integer. A permutation matrix is any $n\times n$ matrix which can be created by rearranging the rows and/or columns of the $n\times n$ identity matrix. More formally, given a permutation $\pi$ from the symmetric group $S_{n}$, one can define an $n\times n$ permutation matrix $P_{\pi}$ by $P_{\pi}=(\delta_{i\,\pi(j)})$, where $\delta$ denotes the Kronecker delta symbol.

Premultiplying an $n\times n$ matrix $A$ by an $n\times n$ permutation matrix results in a rearrangement of the rows of $A$. For example, if the matrix $P$ is obtained by swapping rows $i$ and $j$ of the $n\times n$ identity matrix, then rows $i$ and $j$ of $A$ will be swapped in the product $PA$.

Postmultiplying an $n\times n$ matrix $A$ by an $n\times n$ permutation matrix results in a rearrangement of the columns of $A$. For example, if the matrix $P$ is obtained by swapping rows $i$ and $j$ of the $n\times n$ identity matrix, then columns $i$ and $j$ of $A$ will be swapped in the product $AP$.

## 2 Properties

Permutation matrices have the following properties:

Title permutation matrix PermutationMatrix 2013-03-22 12:06:39 2013-03-22 12:06:39 Wkbj79 (1863) Wkbj79 (1863) 20 Wkbj79 (1863) Definition msc 15A36 MonomialMatrix