Abstract: This paper gave a necessary and sufficient condition for an n times n nonnegative matrix to be decomposed into a product of irreducible nonnegative matrices. It also showed that when such a decomposition is possible, the number of factors can be required to be at most three. The methods used here are constructive, and the factors can be presented.

Key words: irreducible matrix, digraph, nonnegative monomial matrix, Frobenius normal form , nonnegative matrix, irreducible matrix, digraph, nonnegative monomial matrix, Frobenius normal form