Permutation matrix

Definition

The permutation matrix of a permutation $\sigma \in S_n$ is the $n\times n$ matrix $A$ such that $A_{ij}=1$ for $j=\sigma (i)$, $A_{ij}=0$ elsewhere. (Here, $S_n$ is the symmetric group on $n$ letters.)

Example

The permuation $(12)\in S_3$ has permutation matrix $\left(\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right)$.

The standard representation of the symmetric group

Further information: standard representation of the symmetric group

The standard representation of the symmetric group is the linear representation sending each element of the symmetric group to its corresponding permutation matrix.