Permutation matrix

A square matrix with exactly one entry equal to 1 in each row and column and zeros elsewhere. Multiplying by it permutes the coordinates of a vector.

Permutation: A bijection from a finite set to itself, often represented by rearranging its elements. The locations of the matrix’s 1s encode a permutation of row or column indices.
Standard basis: The basis consisting of vectors with one coordinate equal to 1 and all others equal to 0. A permutation matrix sends each standard basis vector to another standard basis vector.
LU decomposition: A factorization of a matrix into lower- and upper-triangular factors, often with a row permutation. Pivoting is recorded by a permutation matrix in the factorization.
Signed permutation matrix: A matrix with exactly one nonzero entry in each row and column, each nonzero equal to 1 or −1. It allows sign changes in addition to the coordinate rearrangements permitted here.
Matrix multiplication: An operation that combines matrices by taking row–column dot products. Left and right multiplication apply the encoded rearrangement to rows or columns.
Symmetric group: The group of all permutations of a finite set, with composition as its operation. Permutation matrices of a fixed size represent the elements of this group.
Gaussian elimination: A sequence of row operations used to solve linear systems and reduce matrices. Row swaps can be represented and accumulated as multiplication by permutation matrices.
Doubly stochastic matrix: A nonnegative square matrix whose rows and columns each sum to 1. Permutation matrices are extreme cases, while doubly stochastic matrices can spread weight across entries.
Identity matrix: A square matrix with 1s on the main diagonal and zeros elsewhere. It is the permutation matrix for the permutation that leaves every index fixed.
Group representation: A homomorphism that represents abstract group elements as invertible linear transformations or matrices. The assignment from permutations to permutation matrices preserves composition.

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