Linear and multilinear algebra; matrix theory Definition

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.

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Matrix definition
Square matrix definition
Permutation matrix definition

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Tag	`0Q5SM60HTH`
URI	`m2://0Q5SM60HTH`
Kind	definition
Writing	W1 stated
MSC	15A99, 05A05
Defines	Permutation matrix
Uses	Square matrix

Permutation matrix

Definition

Remarks

Reading path

Where this leads

Further reading

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