Journal of Mathematics

Journal of Mathematics / 2020 / Article

Research Article | Open Access

Volume 2020 |Article ID 3810423 | https://doi.org/10.1155/2020/3810423

Xiaoyong Chen, Yaqiang Wang, "Subdirect Sums of Matrices", Journal of Mathematics, vol. 2020, Article ID 3810423, 20 pages, 2020. https://doi.org/10.1155/2020/3810423

Subdirect Sums of Matrices

Academic Editor: Feng Feng
Received16 Apr 2020
Accepted17 Jun 2020
Published16 Jul 2020

Abstract

In this paper, the question of when the subdirect sum of matrices is in the class of matrices is studied. Some sufficient conditions are given. Moreover, these sufficient conditions only depend on the elements of given matrices; thus, they provide some simple criteria for application. Numerical examples are also presented to illustrate the corresponding results.

1. Introduction

The concept of -subdirect sums of square matrices, which was introduced by Fallat and Johnson [1], is a generalization of the usual sum of matrices [2]. It is found that the subdirect sum of matrices is widely used in many subjects, such as matrix completion problems, overlapping subdomains in domain decomposition methods, analysis of matrix classes, and global stiffness matrices in finite elements [13]. Many properties of the subdirect sum were analysed [19].

For the subdirect sum, an important problem is that whether the -subdirect sums of matrices belongs to the same class or not, which has been widely concerned for different classes of matrices, such as strictly diagonally dominant matrices [5], nonsingular -matrices [3], -strictly diagonally dominant matrices [2], doubly diagonally dominant matrices [4], -strictly diagonally dominant matrices [8], QN-matrices [10], and Nekrasov matrices which are studied in [6, 7].

In this paper, we concentrate on the subdirect sum of matrices which are a subclass of -matrices [11], and some sufficient conditions ensuring that the subdirect sums of matrices is in the class of matrices are given. Numerical examples are presented to illustrate the corresponding results.

Now, some notations, definitions, and lemmas are listed.

Definition 1 (see [2]). Let and be square matrices of order and , respectively, and be an integer such that . Let and be partitioned into a block as follows:where and are square matrices of order . Following [1], we call the square matrix of order given bythe -subdirect sum of and , and denote it by . As shown in [2], if we let , , and , thenwhereObviously, ; therefore, matrix can be expressed as follows:

Definition 2 (see [2]). Given a matrix , let the th deleted absolute row sum be

Definition 3 (see [12]). Matrix is called a strictly diagonally dominant () matrix if for each ,

Definition 4 (see [11] and [13]). Matrix is called a matrix if for each ,wherewhich is the subset of indices of .

Lemma 1 (see [11]). If matrix is a matrix by rows, then must have at least one row which is strictly diagonally dominant.

Remark 1. From Definitions 3 and 4 and Lemma 1, it is easy to obtain that the class of strictly diagonally dominant matrices is a subclass of matrices, that is, .

Lemma 2 (see [12]). Matrix is an -matrix if and only if there exists a positive diagonal matrix such that is an SDD matrix.

Theorem 1 (see [11]). If matrix is a matrix by rows, then it is also an -matrix. If, in addition, has positive diagonal entries, then det .

Remark 2. From Remark 1, Lemma 2, and Theorem 1, we have the following relationship:

2. Subdirect Sums of Matrices

In the following, we give an example to show that the subdirect sum of two matrices may not be a matrix.

Example 1. We consider the following matrices and :where the 1-subdirect sum givesBy calculation,Note that . Therefore, is not a matrix.
Example 1 shows that the subdirect sum of matrices is not necessarily a matrix; then, a meaningful discussion is concerned: under what conditions, the subdirect sum of matrices is in the class of matrices?
Firstly, we study the 1-subdirect sum of matrices.

Theorem 2. Let and be square matrices of order and , respectively, which are partitioned as in (1). Let , , where . We assume that is a matrix with , and is a matrix. If all diagonal entries of and are positive (or negative), andthen the 1-subdirect sum is a matrix.

Proof. Since is a matrix with , we have for any , andSince is a matrix, we have ,According to the 1-subdirect sum , we haveIn addition, from the condition that all diagonal entries of and are positive (or negative), we haveFrom inequalities (15) and (16) and equalities (17) and (18), we havethat is, , which means that .
For any , it is easy to obtain that . Therefore, we haveFor the conditionlet us multiply by on both sides of this inequality, and then add on both sides, simultaneously; multiply both sides by , in addition, and then from (17) and (18), we haveHence,Therefore, we can draw a conclusion that, for any , that is, is a matrix.

Theorem 3. Let and be square matrices of order and , respectively, which are partitioned as in (1). Let , , where . We assume that is a matrix with , and is a matrix. If all diagonal entries of and are positive (or negative), then the 1-subdirect sum is a matrix.

Proof. Since is a matrix with , we have for any , andSince is a matrix, we have inequality (16) .
From inequalities (16) and (24) and equalities (17) and (18), we havethat is, . Therefore, the proof for can be divided into two cases as follows:Case 1: if , then , which means that .For any , we have . Therefore, we obtain thatCase 2: if , then , which means that .For any , we have . Therefore, we obtain thatFor , we have . Therefore, we obtain thatFrom Cases 1 and 2, we get that , which implies the 1-subdirect sum is a matrix.
The following Example 2 shows that Theorem 2 does not hold when .

Example 2. We consider the following matrices:where is a matrix and is a matrix. It is easy to verify that matrices and satisfy the condition of Theorem 2 by computation. Hence, is a matrix. However, is not a matrix. In fact,By calculation,Note that . Therefore, is not a matrix.
The following Example 3 shows that Theorem 3 does not hold when .

Example 3. We consider the following matrices:where is a matrix and is a matrix. It is easy to verify that matrices and satisfy the condition of Theorem 2 by computation. Therefore, is a matrix. However, is not a matrix. In fact,By calculation, is not a matrix because .
Examples 2 and 3 motivate the search for other conditions such that is a matrix, where is a matrix and is a matrix.
Next, we give some sufficient conditions ensuring that the 2-subdirect sum of matrices and matrices is a matrix.

Theorem 4. Let and be square matrices of order and , respectively, which are partitioned as in (1). Let , , where . We assume that is a matrix with , and is a matrix. If all diagonal entries of and are positive (or negative), andthen the 2-subdirect sum is a matrix.

Proof. Since is a matrix with , we getand .
Since is a matrix, we get inequality (16) .
According to the 2-subdirect sum , we getIn addition, from the condition that all diagonal entries of and are positive (or negative), we getFor any and , we haveFrom equalities (38) and (40) and inequalities (16), (36), and (42), we obtain thatSimilarly, it is easy to obtain thatThat is, , which means that .
For any , it is easy to obtain that . Therefore, we haveFrom the conditionwe haveThus,and substituting equalities (38), (39), (40), and (41) into this inequality, we obtain thatHence,Therefore, it is obvious that, for any , that is, the 2-subdirect sum is a matrix.

Corollary 1. Let and be square matrices of order and , respectively, which are partitioned as in (1). Let , , where . We assume that is a matrix with , and is a matrix. If all diagonal entries of and are positive (or negative),