Permutationlike

Permutationlike is a term used in linear algebra to describe square matrices that are similar to permutation matrices. A matrix A is permutationlike if there exists an invertible matrix S and a permutation matrix P with A = S^{-1} P S. In other words, A represents, in some basis, a permutation of the standard basis vectors.

Key properties: Since permutation matrices are diagonalizable over the complex numbers with eigenvalues that are roots of unity, any permutationlike matrix is diagonalizable over the complex field and has eigenvalues that are roots of unity. If P has order m (P^m = I), then A^m = S^{-1} P^m S = I, so A has finite order. All powers A^k are also permutationlike, being similar to P^k.

Characterizations and constraints: Permutationlike matrices are exactly those matrices that are similar to a permutation matrix. This implies that their minimal polynomial divides x^m - 1 for some m and that their Jordan form is diagonal with eigenvalues among the m-th roots of unity.

Examples and non-examples: The identity matrix is permutationlike. Any matrix similar to a cycle permutation matrix is permutationlike. A matrix with a nontrivial Jordan block cannot be permutationlike, since permutation matrices are diagonalizable. A diagonal matrix with eigenvalues not all roots of unity is not permutationlike.

See also: permutation matrix, similarity (linear algebra).