On Factorizations of Upper Triangular Nonnegative Matrices of Order Three

Abstract EN

Let T 3 ( N 0 ) denote the semigroup of 3 × 3 upper triangular matrices with nonnegative integral-valued entries.

In this paper, we investigate factorizations of upper triangular nonnegative matrices of order three.

Firstly, we characterize the atoms of the subsemigroup S of the matrices in T 3 ( N 0 ) with nonzero determinant and give some formulas.

As a consequence, problems 4a and 4c presented by Baeth et al.

(2011) are each half-answered for the case n = 3 .

And then, we consider some factorization cases of matrix A in S with ρ ( A ) = 1 and give formulas for the minimum factorization length of some special matrices in S .