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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:

They are orthogonal.

They are invertible.

For a fixed positive integer $n$, the $n\times n$ permutation matrices form a group under matrix multiplication.

Since they have a single 1 in each row and each column, they are doubly stochastic.

They are the extreme points of the convex set of doubly stochastic matrices.
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