Permutation matrix

Definition

Let $n \ge 1$ and let $\sigma$ be a permutation of $\{1, \dots, n\}$. The permutation matrix of $\sigma$ is the $n \times n$ matrix $P_\sigma$ with entries $$(P_\sigma)_{ij} = \begin{cases} 1 & \text{if } i = \sigma(j), \\ 0 & \text{otherwise.} \end{cases}$$ A permutation matrix is a matrix of the form $P_\sigma$.

Remarks

The matrix $P_\sigma$ sends the standard basis vector $e_j$ to $e_{\sigma(j)}$. With this convention, $P_\sigma P_\tau = P_{\sigma \tau}$; some authors use the transposed convention. Each permutation matrix is invertible with $P_\sigma^{-1} = P_\sigma^T$, and $\det P_\sigma = \sgn \sigma$. The matrix $\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ is the permutation matrix that exchanges two coordinates.

Explanations

L1 human

A permutation matrix is the identity matrix with its columns shuffled. Multiplying a vector by it shuffles the entries of the vector in the same way. Each row and each column has exactly one $1$.