The Canonical Forms of Permutation Matrices

We address classification of permutation matrices, in terms of permutation similarity relations, which play an important role in investigating the reducible solutions of some symmetric matrix equations. We solve the three problems. First, what is the canonical form of a permutation similarity class? Second, how to obtain the standard form of arbitrary permutation matrix? Third, for any permutation matrix A, how to find the permutation matrix T, such that T(-1 )AT is in canonical form? Besides, the decomposition theorem of permutation matrices and the factorization theorem of both permutation matrices and monomial matrices are demonstrated.