A New Subclass of H-matrices:γ-DZT Matrices

Li Delun; Wang Shiyun; Li Changyue; Song Airu; Jiang Bo

doi:10.54254/2753-8818/2025.24951

1. Introduction

H-matrices have wide applications in numerical analysis, control theory, matrix theory and statistics [1-4]. An important method to study the properties of H-matrices is to consider its subclasses. H-matrices include many well-known subclasses, such as strictly diagonally dominant (SDD) matrices, doubly strictly diagonally dominant (DSDD) matrices, Σ-SDD matrices, γ-SDD matrices, weakly chained diagonally dominant matrices, Nekrasov matrices, Σ-Nekrasov matrices,

Dashnic Zusmanovich (DZ) matrices and eventually SDD matrices.

The class of Dashnic-Zusmanovich type (DZT) matrices were introduced by Zhao et al. in2018 [5]. They proved that DZT matrices belong to the class of H-matrices. Many results about DZT matrices have been obtained. One can refer to [6-9].

This paper introduces a new matrix subclass: $γ -$ DZT matrices. By constructing the scaling matrix, we prove that the class of $γ -$ DZT matrices belongs to H-matrices. Moreover, $γ -SDD$ matrices and DZT matrices belong to $γ -$ DZT matrices.

2. Preliminaries

We use $C^{n \times n}$ denote all the complex matrices with the order n,and denot $⟨ n ⟩ = {1,2, \dots, n}$ . For any $A \in C^{n \times n}$ ,define:

$r_{i} (A) = \sum_{j \neq i}^{n} | a_{i j} |$ (2.1)

$c_{i} (A) = \sum_{j \neq i}^{n} | a_{j i} |$ (2.2)

$r_{i}^{S} (A) = \sum_{j \neq i, j \in S}^{n} | a_{i j} | (S \subseteq ⟨ n ⟩)$ (2.3)

$c_{i}^{S} (A) = \sum_{j \neq i, j \in S}^{n} | a_{j i} | (S \subseteq ⟨ n ⟩)$ (2.4)

$Γ_{i} (A) = {j \in ⟨ n ⟩ - {i} : (| a_{i i} | - r_{i}^{⟨ n ⟩ - {j}} (A)) | a_{j j} | > | a_{i j} | r_{j} (A)}, i \in ⟨ n ⟩$ (2.5)

If $S = ⟨ n ⟩ - {j}$ ,

$r_{i}^{S} (A) and c_{i}^{S} (A) are simplified as r_{i}^{{j}} (A) and c_{i}^{{j}} (A), respectively.$

Definition 2.1. Let $A = (a_{i j}) \in C^{n \times n}$ ,We say that A is an SDD matrix if for all $i \in ⟨ n ⟩$ , it holds that $| a_{i i} | > r_{i} (A)$ .

Definition 2.2.Let $A = (a_{i j}) \in C^{n \times n}$ ,if there exists $X = d i a g (x_{1} x_{2} \dots x_{n})$ such that $A X$ is an SDD matrix,we call $A$ is a nonsingular H-matrix. X is called scaling matrix of A.

Definition 2.3. Let $A = (a_{i j}) \in C^{n \times n}$ ,if for all $i \in ⟨ n ⟩$ ,either $i \in N^{+} (A)$ ,or $Γ_{i} (A) \neq \emptyset$ ,we call $A$ is a DZT matrix.

Definition 2.4.Let $A = (a_{i j}) \in C^{n \times n}$ ,we say that A is a $γ -SDD$ matrix if there exists an $α \in [0,1]$ such that A is an $α -SDD$ matrix, i.e.,

| a_{i i} | > α r_{i} (A) + (1 - α) c_{i} (A), \forall i \in ⟨ n ⟩ .

3. Main results

Definition 3.1. A matrix $A = [a_{i j}] \in C^{n \times n}, n \geq 2$ , is called an $α - D Z T$ matrix if there exists $α \in [0,1]$ , such that for all $i \in ⟨ n ⟩$ , there exists $j \neq i$ satisfying

$(| a_{i i} | - α \cdot r_{i}^{{j}} (A) - (1 - α) c_{i}^{{j}} (A)) | a_{j j} | > (α | a_{i j} | + (1 - α) | a_{j i} |) (α \cdot r_{j} (A) - (1 - α) c_{j} (A)),$ (3.1)

We say that A is a $γ -$ DZT matrix if there exists an $α \in [0,1]$ such that A is an $α - D Z T$ matrix.

It is easy to see that

$| a_{i i} | > \frac{α \cdot r_{j} (A) + (1 - α) c_{j} (A)}{| a_{j j} |} (α | a_{i j} | + (1 - α) | a_{j i} |) + α \cdot r_{i}^{{j}} (A) + (1 - α) c_{i}^{{j}} (A) .$ (3.2)

Let

N_{1} = {i : | a_{i i} | \leq α \cdot r_{i} (A) + (1 - α) c_{i} (A)},

N_{2} = {i : | a_{i i} | > α \cdot r_{i} (A) + (1 - α) c_{i} (A)} .

Theorem 3.1. Let $A = [a_{i j}] \in C^{n \times n}, n \geq 2$ be an $α - D Z T$ matrix, there exists a diagonal matrix $D = d i a g (d_{1}, d_{2}, \dots, d_{i})$ such that DAD is an SDD matrix, where

d_{i} = {\begin{matrix} 1, i \in N_{1} \\ \frac{α \cdot r_{i} (A) + (1 - α) c_{i} (A)}{| a_{i i} |} + ε, i \in N_{2} \end{matrix}

Proof. Let $B = D A D$ . We will show that $| b_{i i} | > α \cdot r_{i} (B) + (1 - α) c_{i} (B)$ for all $i \in ⟨ n ⟩$ .

Case 1. $i \in N_{1}$ . There must be $j_{0} \in N_{2}$ , that is $\frac{α \cdot r_{j_{0}} (A) + (1 - α) c_{j_{0}} (A)}{| a_{j_{0} j_{0}} |} < 1$ ,and $d_{i} = 1$ , then $| b_{i i} | = | a_{i i} |$ .

Because

$α \sum_{j \in N_{1}} | a_{i j} | + α \sum_{j \in N_{2} \ {j_{0}}} | a_{i j} | = α \cdot r_{i}^{{j_{0}}} (A)$ ,

then

$α \cdot r_{i} (B) < α \cdot r_{i}^{{j_{0}}} (A) + α | a_{i j_{0}} | \cdot \frac{α \cdot r_{j_{0}} (A) + (1 - α) c_{j_{0}} (A)}{| a_{j_{0} j_{0}} |}$ .

Because

$(1 - α) \sum_{j \in N_{1}} | a_{j i} | + (1 - α) \sum_{j \in N_{2} \ {j_{0}}} | a_{j i} | = (1 - α) \cdot c_{i}^{{j_{0}}} (A)$

then

$(1 - α) \cdot c_{i} (B) < (1 - α) \cdot c_{i}^{{j_{0}}} (A) + (1 - α) | a_{j_{0} i} | \cdot \frac{α \cdot r_{j_{0}} (A) + (1 - α) c_{j_{0}} (A)}{| a_{j_{0} j_{0}} |}$ ,

and because of formula (3.2),

we proved $| b_{i i} | > α \cdot r_{i} (B) + (1 - α) c_{i} (B)$ .

Case 2. $i \in N_{2}$ . There must be $\frac{α \cdot r_{i} (A) + (1 - α) c_{i} (A)}{| a_{i i} |} < 1$ ,

and $d_{i} = \frac{α \cdot r_{i} (A) + (1 - α) c_{i} (A)}{| a_{i i} |} + ε$ , then $| b_{i i} | = d_{i}^{2} | a_{i i} |$ .

The above formula can be expanded to get

| b_{i i} | = (α \cdot r_{i} (A) + (1 - α) c_{i} (A) + ε | a_{i i} |) (\frac{α \cdot r_{i} (A) + (1 - α) c_{i} (A)}{| a_{i i} |} + ε)

= (\frac{α \cdot r_{i} (A) + (1 - α) c_{i} (A)}{| a_{i i} |}) (α \cdot r_{i} (A) + α ε | a_{i i} |) + (\frac{α \cdot r_{i} (A) + (1 - α) c_{i} (A)}{| a_{i i} |}) ((1 - α) c_{i} (A) + (1 - α) ε | a_{i i} |)

= d_{i} (α \cdot r_{i} (A) + α ε | a_{i i} |) + d_{i} ((1 - α) c_{i} (A) + (1 - α) ε | a_{i i} |)

α \cdot r_{i} (B) = d_{i} α (\sum_{j \in N_{1}} | a_{i j} | + \sum_{j \in N_{2}} | a_{i j} | \cdot d_{j})

= d_{i} α (\sum_{j \in N_{1}} | a_{i j} | + \sum_{j \in N_{2}} | a_{i j} | (\frac{α \cdot r_{i} (A) + (1 - α) c_{i} (A)}{| a_{i i} |} + ε))

= d_{i} α (\sum_{j \in N_{1}} | a_{i j} | + \sum_{j \in N_{2}} | a_{i j} | \frac{α \cdot r_{i} (A) + (1 - α) c_{i} (A)}{| a_{i i} |} + ε \sum_{j \in N_{2}} | a_{i j} |)

\leq d_{i} α (\sum_{j \in N_{1}} | a_{i j} | + \sum_{j \in N_{2}} | a_{i j} | + ε \sum_{j \in N_{2}} | a_{i j} |)

= d_{i} (α \cdot r_{i} (A) + α \cdot ε \sum_{j \in N_{2}} | a_{i j} |)

< d_{i} (α \cdot r_{i} (A) + α \cdot ε | a_{i i} |)

And,

(1 - α) \cdot c_{i} (B) = (1 - α) (\sum_{j \in N_{1}} | a_{j i} | + \sum_{j \in N_{2}} d_{j} | a_{j i} |) \cdot d_{i}

= d_{i} (1 - α) (\sum_{j \in N_{1}} | a_{j i} | + \sum_{j \in N_{2}} (\frac{α \cdot r_{i} (A) + (1 - α) c_{i} (A)}{| a_{i i} |} + ε) | a_{j i} |)

= d_{i} (1 - α) (\sum_{j \in N_{1}} | a_{j i} | + \sum_{j \in N_{2}} \frac{α \cdot r_{i} (A) + (1 - α) c_{i} (A)}{| a_{i i} |} | a_{j i} | + ε \sum_{j \in N_{2}} | a_{j i} |)

\leq d_{i} (1 - α) (\sum_{j \in N_{1}} | a_{j i} | + \sum_{j \in N_{2}} | a_{j i} | + ε \sum_{j \in N_{2}} | a_{j i} |)

= d_{i} ((1 - α) c_{i} (A) + (1 - α) ε \sum_{j \in N_{2}} | a_{j i} |)

< d_{i} ((1 - α) c_{i} (A) + (1 - α) ε | a_{i i} |)

Solving simultaneously, we get

$α \cdot r_{i} (B) + (1 - α) \cdot c_{i} (B) < d_{i} (α \cdot r_{i} (A) + α \cdot ε | a_{i i} |) + d_{i} ((1 - α) c_{i} (A) + (1 - α) ε | a_{i i} |)$ ,

then $| b_{i i} | > α \cdot r_{i} (B) + (1 - α) c_{i} (B)$ .we compete the proof.

Theorem 3.2. A matrix $A = [a_{i j}] \in C^{n \times n}, n \geq 2$ , then the matrix A satisfies the following relation:

{\begin{matrix} A \in S D D \Rightarrow A \in D Z T \Rightarrow A \in γ - D Z T \\ A \in S D D \Rightarrow A \in γ - S D D \Rightarrow A \in γ - D Z T \end{matrix}

Proof. If A is a DZT matrix, then A is a 1-DZT matrix. Now we show that if A is an $α - S D D$ matrix, then A is an $α - D Z T$ matrix.

First of all, by definition, it is obvious that if matrix A belongs to SDD matrix, then A must belong to $α - S D D$ matrix. In the same way, if matrix A belongs to DZT matrix, then A must belong to the $α - D Z T$ matrix. So now it is only necessary to prove that when A belongs to SDD matrix and $α - S D D$ matrix, A must belong to DZT matrix and $α - D Z T$ matrix respectively.

By definition, if the hypothesis $A \in S D D \subseteq A \in D Z T$ is true, then

r_{i} (A) > \frac{r_{j_{0}} (A)}{| a_{j_{0} j_{0}} |} | a_{i j_{0}} | + r_{i}^{{j_{0}}} (A)

r_{i}^{{j_{0}}} (A) + | a_{i j_{0}} | > \frac{r_{j_{0}} (A)}{| a_{j_{0} j_{0}} |} | a_{i j_{0}} | + r_{i}^{{j_{0}}} (A)

| a_{i j_{0}} | > \frac{r_{j_{0}} (A)}{| a_{j_{0} j_{0}} |} | a_{i j_{0}} |

Because $\frac{r_{j_{0}} (A)}{| a_{j_{0} j_{0}} |} < 1$ , therefore, the above inequality is true, that is, the hypothesis is true.

For the argument, we also use the reverse order method, assuming that the conclusion $A \in α - S D D \subseteq A \in α - D Z T$ is true, then by definition,

α \cdot r_{i} (A) + (1 - α) c_{i} (A) > \frac{α \cdot r_{j_{0}} (A) + (1 - α) c_{j_{0}} (A)}{| a_{j_{0} j_{0}} |} (α | a_{i j_{0}} | + (1 - α) | a_{j_{0} i} |)

+ α \cdot r_{i}^{{j_{0}}} (A) + (1 - α) c_{i}^{{j_{0}}} (A)

α \cdot [r_{i}^{{j_{0}}} (A) + | a_{i j_{0}} |] + (1 - α) [c_{i}^{{j_{0}}} (A) + | a_{j_{0} i} |] > \frac{α \cdot r_{j_{0}} (A) + (1 - α) c_{j_{0}} (A)}{| a_{j_{0} j_{0}} |} (α | a_{i j_{0}} | + (1 - α) | a_{j_{0} i} |)

+ α \cdot r_{i}^{{j_{0}}} (A) + (1 - α) c_{i}^{{j_{0}}} (A)

$α | a_{i j_{0}} | + (1 - α) | a_{j_{0} i} | > \frac{α \cdot r_{j_{0}} (A) + (1 - α) c_{j_{0}} (A)}{| a_{j_{0} j_{0}} |} (α | a_{i j_{0}} | + (1 - α) | a_{j_{0} i} |)$

Eliminate the common terms on both sides of the inequality at the same time.

In the same way, because $\frac{α \cdot r_{j_{0}} (A) + (1 - α) c_{j_{0}} (A)}{| a_{j_{0} j_{0}} |} < 1$ , therefore the hypothesis is true. In summary, theorem 2 is proved.

For the proof of theorem 2, we mainly use the core idea of reverse order. We start with the result we want to prove, analyze what conditions are needed to get that result, and then work our way forward to see how we can obtain these conditions from what we already know. Thus, we prove the relationship between SDD, DZT, $γ - S D D$ and $γ - D Z T$ matrix.

4. Conclusion

This paper successfully introduces a new subclass of matrices, the $γ - D Z T$ matrix, and proves that the $γ - D Z T$ matrix belongs to the H-matrix by constructing a scaling matrix. The inclusion relationships among the four types of matrices have been clarified, and H-matrices is clarified. These conclusions provide new theoretical basis and research directions for further studies on the properties and applications of H-matrix, and are of great significance in the development of matrix theory and related application fields.

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Cite this article

Delun,L.;Shiyun,W.;Changyue,L.;Airu,S.;Bo,J. (2025). A New Subclass of H-matrices:γ-DZT Matrices. Theoretical and Natural Science,104,1-5.

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Volume title: Proceedings of CONF-MPCS 2025 Symposium: Mastering Optimization: Strategies for Maximum Efficiency

ISBN：978-1-80590-165-5(Print) / 978-1-80590-166-2(Online)

Editor：Marwan Omar

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Conference date: 21 March 2025

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Volume number: Vol.104

ISSN：2753-8818(Print) / 2753-8826(Online)

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