Permutation matrix

In mathematics, particularly in matrix theory, a permutation matrix is a square binary matrix that has exactly one entry of 1 in each row and each column with all other entries 0.^[1]^: 26 An $n \times n$ permutation matrix can represent a permutation of $n$ elements. Pre-multiplying an $n$ -row matrix $M$ by a permutation matrix $P$ , forming $PM$ , results in permuting the rows of $M$ , while post-multiplying an $n$ -column matrix $M$ , forming $MP$ , permutes the columns of $M$ .

Every permutation matrix P is orthogonal, with its inverse equal to its transpose: $P^{-1}=P^{\mathsf {T}}$ .^[1]^: 26 Indeed, permutation matrices can be characterized as the orthogonal matrices whose entries are all non-negative.^[2]

^ ^a ^b Artin, Michael (1991). Algebra. Prentice Hall. pp. 24–26, 118, 259, 322.
^ Zavlanos, Michael M.; Pappas, George J. (November 2008). "A dynamical systems approach to weighted graph matching". Automatica. 44 (11): 2817–2824. CiteSeerX 10.1.1.128.6870. doi:10.1016/j.automatica.2008.04.009. S2CID 834305. Retrieved 21 August 2022. Let $O_{n}$ denote the set of $n\times n$ orthogonal matrices and $N_{n}$ denote the set of $n\times n$ element-wise non-negative matrices. Then, $P_{n}=O_{n}\cap N_{n}$ , where $P_{n}$ is the set of $n\times n$ permutation matrices.

[Artin_Algebra-1] Artin, Michael (1991). Algebra. Prentice Hall. pp. 24–26, 118, 259, 322.

[2] Zavlanos, Michael M.; Pappas, George J. (November 2008). "A dynamical systems approach to weighted graph matching". Automatica. 44 (11): 2817–2824. CiteSeerX 10.1.1.128.6870. doi:10.1016/j.automatica.2008.04.009. S2CID 834305. Retrieved 21 August 2022. Let $O_{n}$ denote the set of $n\times n$ orthogonal matrices and $N_{n}$ denote the set of $n\times n$ element-wise non-negative matrices. Then, $P_{n}=O_{n}\cap N_{n}$ , where $P_{n}$ is the set of $n\times n$ permutation matrices.

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