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.^: 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: $P^{-1}=P^{\mathsf {T}}$ .^: 26 Indeed, permutation matrices can be characterized as the orthogonal matrices whose entries are all non-negative.

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