permutatiematrix

Permutatiematrix, or permutation matrix in English, is a square binary matrix with exactly one 1 in each row and each column and 0s elsewhere. It represents a permutation of the coordinates of a vector or the rows and columns of a matrix. Given a permutation π on {1, ..., n}, the corresponding permutation matrix P has entries P_{i,j} = 1 if j = π(i) and 0 otherwise. Then (Px)_i = x_{π(i)}, so P permutes the components of a vector x according to π.

Permutation matrices are orthogonal: P^T = P^{-1}, and det(P) equals the sign of π (either +1 or -1). The set of all such matrices forms a subgroup of GL(n) isomorphic to the symmetric group S_n; multiplication of permutation matrices corresponds to the composition of the underlying permutations.

Left-multiplication by P permutes the rows of a matrix; right-multiplication by P permutes the columns (more precisely, A P permutes columns of A according to π).

Example: for n = 3, take the permutation π with 1 → 2, 2 → 3, 3 → 1 (the cycle (1 2 3)). The corresponding permutation matrix is P = [0 1 0; 0 0 1; 1 0 0]. The identity permutation yields the identity matrix.

Applications include reordering data, implementing pivot strategies in algorithms, and rearranging components in linear algebra computations where coordinate permutation is needed.