Subdirect Sums of Doubly Strictly Diagonally Dominant Matrices

Abstract

In this paper, the question of when the subdirect sum of two doubly strictly diagonally dominant (DSDDs) matrices is addressed. Some sufficient conditions are given, and these sufficient conditions only depend on the elements of the given matrices. Moreover, examples are presented to illustrate the corresponding results.

1. Introduction

In 1999, the concept of subdirect sums of square matrices was introduced by Fallat and Johnson, which is a generalization of the usual sum of matrices [1], and arises in several contexts, such as matrix completion problems, overlapping subdomains in domain decomposition problems, and global stiffness matrices in finite element methods [2].

For a given class matrix, an important problem is that whether the -subdirect sums of matrices belong to the same class or not, which has been widely concerned for differently classes of matrices, such as nonsingular -matrices [3], -strictly diagonally dominant matrices [4], -strictly diagonally dominant matrices [5], doubly diagonally dominant matrices [6], Nekrasov matrices [7, 8], and matrices [9].

In this paper, we focus on the subdirect sum of doubly strictly diagonally dominant (shortly as ) matrices, which is a subclass of -matrices [10], and some sufficient conditions such that the -subdirect sums of matrices belong to matrices are given, and these sufficient conditions only depend on the elements of the given matrices.

Now, some notations and definitions are listed, which can also be found in [1, 11–13].

Let n be an integer number. is the set of complex matrices.

Definition 1. (see [1]). Let and be square matrices of orders and , respectively, and be an integer number such that . Let and be partitioned in a block as follows:where and are the square matrices of order . We call the following square matrix of order ,the -subdirect sum of and , and we denote it by .

In order to more explicitly express each element of in terms of the ones of and , we can define the following set of indices:

Then, can be expressed as follows:where .

Definition 2. (see [12]). Given a matrix , is called (row) diagonally dominant () ifwhereIf the inequality in (5) holds strictly for all , we say that is strictly diagonally dominant ().

Definition 3. (see [13]). The matrix is a doubly strictly diagonally dominant () matrix if

2. Subdirect Sums of Matrices

In general, the subdirect sum of two matrices is not always a matrix. We show this in the following example.

Example 1. Letbe two matrices, butis not a matrix, since .

Example 1 shows that the subdirect sum of matrices is not a matrix; then, a meaningful discussion is concerned: under what conditions, the subdirect sum of matrices is in the class of matrices?

In order to obtain the main results, we need the following lemma.

Lemma 1. Let and be square matrices of orders and partitioned as in (1), respectively. And, let be an integer number such that , be defined as in (4), and all diagonal entries of and are positive (or all negative), with , then,(a)(b)(c)(d)

Proof. For , we can writeFor , we can obtainFor , we obtainFor the rest case of , the proof is similar to the proof of .
Firstly, we study the 1-subdirect sum of matrices.

Theorem 1. Let and be matrices of orders and partitioned as in (1), respectively, and . Then, is a matrix if all diagonal entries of and are positive (or all negative) and for ,

Proof. Since and and are the matrices of orders and respectively, it is obvious that . Case 1: for , from (a) of Lemma 1, we have Since is , we obtain that for , Case 2: for , from (a) of Lemma 1, it is easy to obtain Since all diagonal entries of and are positive (or all negative), we obtain Since is , and from inequalities (13)–(15), we have From (b) of Lemma 1, it is easy to obtain that for , Case 3: for , from (a) and (d) of Lemma 1, we have Then, from inequalities (13)–(15), we have that for , Case 4: for , from (b) and (d) of Lemma 1, we obtain Since is , and from inequality (15), for , we have Case 5: for , from (d) of Lemma 1, we obtain Since is , we haveTherefore, we can draw a conclusion that for any , that is, is a matrix.

Example 2. The matricesare two matrices, and from Theorem 1, it is easy to verify thatis a matrix since
, and .
However,is not since .

Examples 2 motivates the search for other conditions such that () is also a matrix, where is a matrix and is a matrix.

Next, some sufficient conditions ensuring that the -subdirect sum of matrices is a matrix are given.

Theorem 2. Let and be matrices of orders and partitioned as in (1), respectively, and is an integer number such that . Let and be matrices, if all diagonal entries of and are positive (or all negative) and for any ,and then the -subdirect sum is .

Proof. Let and be matrices of orders and respectively; thus, it is obvious that . Case 1: for , from (a) of Lemma 1, we have Since is , we obtain Case 2: for , from (a) and (c) of Lemma 1, we obtain Since all diagonal entries of and are positive (or all negative), we obtain Since is , and from inequality (31), we have From (c) of Lemma 1, it is easy to obtain that Case 3: for , from (a) and (d) of Lemma 1, we conclude Then, from inequality (31), we have Case 4: for , from (c) of Lemma 1, we obtain Since and are , and from inequality (31), we conclude Case 5: for , from (c) and (d) of Lemma 1, we obtain Since is , and from inequality (31), we obtain Case 6: for , from (d) of Lemma 1, we obtain Since is , we obtainIn conclusion, . Therefore, is a matrix.

Example 3. Letbe two matrices. And from Definition 1, we obtain thatand . From Theorem 2, is a matrix since

From Definitions 2 and 3, it is easy to show that matrices are contained into matrices. Therefore, from Theorem 2, we obtain the following corollaries, which present sufficient conditions such that -subdirect sum is .

Corollary 1. Let and be square matrices of orders and partitioned as in (1), respectively, and is an integer number such that . We assume that is a matrix and is a matrix. If there exists an such thatAnd, all diagonal entries of and are positive (or all negative), then the -subdirect sum is a matrix.

Proof. Without loss of generality, we can assume such that and . Case 1: for , from (a) of Lemma 1, we have Since is , we obtain that for , Case 2: for , from (a) of Lemma 1, it is easy to obtain Since all diagonal entries of and are positive (or all negative), we obtain If , since is , and from inequality (49), we can obtain If , since is , and from inequality (50), we can write Case 3: for , from (a) and (d) of Lemma 1, we have If , from inequality (49), we have If , since is , and from inequality (50), we can write Case 4: for , from (b) and (c) of Lemma 1, we obtain Since is , and from inequality (50), we can obtain Case 5: for , from (c) and (d) of Lemma 1, we obtain Since is , and from inequality (50), we obtain Case 6: for , from (d) of Lemma 1, we obtain Since is , we can obtainTherefore, we can draw a conclusion that for any ; that is, is a matrix.

Corollary 2. Let and be square matrices of orders and partitioned as in (1), respectively, and is an integer number such that . We assume that is a matrix and is a matrix. If there exists a such thatAnd, all diagonal entries of and are positive (or all negative), then the -subdirect sum is a matrix.

Proof. Without loss of generality, we can assume such that and . Case 1: for , from (a) of Lemma 1, we have Since is , we obtain that for , Case 2: for , from (a) of Lemma 1, it is easy to obtain Since all diagonal entries of and are positive (or all negative), we obtain Since is and from inequality (67), we have Case 3: for , from (a) and (d) of Lemma 1, we have If , from inequality (66), we have If , since is , and from inequality (67), we can write Case 4: for , from (c) of Lemma 1, we obtain Since is , and from inequality (67), we can obtain Case 5: for , from (c) and (d) of Lemma 1, we obtain If , since is , and from inequality (66), we have If , since is , and from inequality (67), we can write Case 6: for , from (d) of Lemma 1, we obtain Since is , we can obtainIn conclusion, for any . Therefore, is a matrix.

3. Conclusions

In this paper, some sufficient conditions such that the subdirect sum of matrices is in the class of matrices are given. Moreover, numerical examples are also presented to illustrate the corresponding results.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors jointly worked on the results, and they read and approved the final manuscript.

Acknowledgments

This work was partly supported by the National Natural Science Foundations of China (31600299), Natural Science Basic Research Program of Shaanxi, China (2020JM-622), the Scientific Research Program funded by the Shaanxi Provincial Education Department (18JK0044), the Science and Technology Project of Baoji (2017JH2-21 and 2017JH2-24), the Key Project of Baoji University of Arts and Sciences (ZK2017021 and ZK16050), and the Postgraduate Innovative Research Project of Baoji University of Arts and Sciences (YJSCX20ZD05).