On Factorizations of Upper Triangular Nonnegative Matrices of Order Three

Abstract

Let denote the semigroup of 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 of the matrices in 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 . And then, we consider some factorization cases of matrix in with and give formulas for the minimum factorization length of some special matrices in .

1. Introduction and Preliminaries

Upper triangular matrices are an important class of matrices, which lead to a broader study of all integral-valued matrices. There are many papers in the literature considering these matrices and similar topics.

Factoring such matrices plays a vital role in the study of upper triangular matrices (see [1]). The problem of factoring matrices was studied by Cohn [2] as early as 1963. Later, Jacobson and Wisner [3, 4] and Ch’uan and Chuan [5, 6] investigated these factorization problems in the context of semigroups of matrices. Motivated by these results, Baeth et al. [7] applied the concepts of contemporary factorization theory to semigroups of integral-valued matrices and calculated certain important invariants to give a sense of how unique or nonunique factorization is in each of these semigroups. In [7], Baeth et al. presented six open problems.

In this paper, we will investigate factorizations of upper triangular nonnegative matrices of order three. Also, we will consider open problem 4 presented in [7].

Throughout this paper, will denote the set of all positive integers and . Also, will denote the semigroup of   upper triangular matrices with nonnegative integral-valued entries.

In the following, analogous with [7] or [8], some concepts and preliminaries are recalled.

A semigroup is a pairing where is a set and is an associative binary operation on . When the binary operation is clear from context and , we will simply write instead of . If contains an element such that for all , then is the identity of .

Let be a semigroup with identity . An element is a unit of if there exists an element such that . A nonunit is called an atom of if whenever for some elements , either or is a unit of . The semigroup is said to be atomic provided each nonunit element in can be written as a product of atoms of .

Let denote an atomic semigroup and let be a nonunit element of . The set is called the set of lengths of .

We denote by the longest (if finite) factorization length of and   the minimum factorization length of . The elasticity of , denoted by gives a coarse measure of how far away is from having unique factorization. It is not hard to see that if has a unique factorization , whence and so The elasticity of the semigroup , denoted by , is given by If with for each , then the delta set of is given by and .

This paper will be divided into two sections. In Section 2, we will consider the semigroup of upper triangular matrices with nonnegative entries and nonzero determinant. Firstly, we characterize the atoms of the subsemigroup of the matrices in with nonzero determinant and give some formulas. As a result, problems 4a and 4c presented by Baeth et al. in [7] are each half-answered for the case . And then, we consider some factorization cases of matrix in with and give formulas for the minimum factorization length of some special matrices in .

2. Upper Triangular Nonnegative Matrices of Order Three

In this section we consider the semigroup of upper triangular matrices with nonnegative entries and nonzero determinant. In this case, is the only unit of .

For each pair , let denote the matrix whose only nonzero entry is .

The following theorem gives some characterizations about the atoms of .

Theorem 1. Let denote the subsemigroup of the matrices in with nonzero determinant. The set of atoms of   consists of the matrices and, for each prime , the matrices

Proof. Suppose that for some . Since , and we can write where and if and . As a result, either or is the identity and hence is an atom.
Note that the proofs that are atoms are similar; we only prove that is an atom in the following. Suppose now that is prime and , . Since is prime, either or In either case, for . Consequently, either or is the identity and hence is an atom of .
Finally, we will show that these are the only atoms of .
For any we can write Thus, the set of atoms of consists of the matrices for each pair and with or for some prime and .

Recall that a unitriangular matrix is a matrix in whose diagonal elements are all s. Denote . From the proof of Theorem 1, we can immediately obtain the following corollary.

Corollary 2. Let denote the unitriangular matrices in and . Then is an atom if and only if .

Hereafter, for any given with nonzero determinant, we let denote the number of (not necessarily distinct) prime factors of .

Proposition 3. Let denote the subsemigroup of the matrices in with nonzero determinant. If can be factored as with each being an atom of , then , where

Proof. For each , is an atom and thus is either 1 or prime. Since we have If , then the length of this factorization of is

Lemma 4 (see [7, Theorem 4.4]). Let denote the subsemigroup of of unitriangular matrices and let . Then .

Theorem 5. Let denote the subsemigroup of the matrices in with nonzero determinant and Then the following statements hold:(1);(2)if , then and ;(3)if , , and then (4)if , , and then (5)if , , and then

Proof. (1) Suppose that with each being an atom of . By Proposition 3, where Note that the numbers of factors (, resp.) of are not more than (, resp.). Then we have and thus On the other hand, from the proof of Theorem 1, we know that and then, Thus, combining (23) and (25), we have .
(2) Suppose that is a diagonal matrix; that is, for all with , and write as in Proposition 3. If , then contains at least one factor of , , or , and then there is at least one superdiagonal entry of that is not 0. This contradicts with the fact that is a diagonal matrix. Thus, and, in this case, (3) Suppose that , . Denote as in Proposition 3. Notice that is the number of (not necessarily distinct) prime factors of and . Then , and so . Since , , can be factored as a product of one factor of and one factor of ; say where and for some positive integers and . Factor as By the above factorization of and (2), it is not hard to see that Thus, .
In this case we immediately get Hence, (3) holds.
Similarly, we can show that (4) and (5) hold.

Remark 6. Given , Theorem 5 provides a formula for . Consequently, open problems 4a and 4c in [7] are each half-answered for the case .

Theorem 7. Let denote the subsemigroup of the matrices in with nonzero determinant and If satisfies one of the following conditions,(1);(2), ;(3), ;(4), ;(5), ;(6), ;(7), ,then .

Proof. (1) If , by Theorem 5, . In this case, .
(2) Suppose that and . If , by (1), the conclusion holds. Now, assume that ; the cases of that , or , can be proved similarly. In this case, is not an atomic factor of and and are atomic factors of . Note that can commute with ; then only has the following factorization: with and the minimum factorization length .
Analogous with the proof of (2), we can show that (4) holds.
(3) Suppose that and . If , by (1), the conclusion holds. Now, assume that ; the cases of that , or , can be proved similarly. In this case, is not an atomic factor of and and are atomic factors of . Note that ; the factorizations of must satisfy that all the matrices are before matrices, and then only has the following factorization: with and the minimum factorization length .
(5) Suppose that and . In this case, is not an atomic factor of , and and are atomic factors of . Assume that can be factored as with each being an atom; then , where By Theorem 5 (1), . On the other hand, since , , are atomic factors of , and we have . Thus, . In this case, can factorize as follows: So if we give the atomic factorizations of , , and , then we will obtain the atomic factorization of with and the minimum factorization length .
Analogous with the proof of (5), we can show that (6) and (7) hold.

Example 8. Consider Then by Theorem 7 (2), we can obtain the atomic factorization of with and the minimum factorization length as follows:

Example 9. Consider Then by Theorem 7 (5), we can obtain the atomic factorization of with and the minimum factorization length as follows:

Theorem 10. Let denote the unitriangular matrices in and Then we have

Proof. It is a routine way to check that , , and . Hence the maximum length is achieved by putting all the matrices before the matrices. In this case, we have the form . The minimum length is achieved by having as few as possible matrices, that is, by putting matrices after matrices. If , we can run out of matrices, leaving a form , where these powers satisfy , , and . And, in this case, we have . If , then we can put all the matrices after all of the matrices, getting the form . Thus, in this case, we have .

Example 11. Consider Then by Theorem 10, we can obtain a factorization of with the minimum factorization length as follows:

Example 12. Consider Then by Theorem 10, we can obtain a factorization of with the minimum factorization length as follows:

Example 13. Consider Then by Theorem 10, we can obtain a factorization of with the minimum factorization length as follows:

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The author would like to extend their sincere gratitude to the referee for his instructive advice and useful suggestions that contributed to this paper. This research was supported by grants of the NNSF of China (nos. 11401246, 11426112), the NSF of Guangdong Province (nos. 2014A030310087, 2014A030310119), the NSF of Fujian Province (no. 2014J01019), the Outstanding Young Innovative Talent Training Project in Guangdong Universities (no. 2013LYM0086), and Science and Technology Plan Project of Huizhou City.