New criteria for nonsingular $H$-matrices

1.   Introduction

Let $\mathbb{C}^{n\times n}$ be the set of $n$ order complex matrices and $A = (a_{ij})\in \mathbb{C}^{n\times n}$. For any $i, j\in N = \{1, 2, \cdots, n\}$, denote

Let $A = (a_{ij})\in \mathbb{C}^{n\times n}$. If $|a_{ii}|\geq R_i(A)(i\in N)$, then $A$ is called a diagonally dominant matrix, and denoted by $A\in D_0$. If $|a_{ii}| > R_i(A)(i\in N)$, then $A$ is called a strictly diagonally dominant matrix and denoted by $A\in D$.

If there is a positive diagonal matrix $X$ such that $AX\in D$, then $A$ is called a generalized strictly diagonally dominant matrix, denoted by $A\in D^{\rm *}$, and also called a nonsingular $H$-matrix.

A matrix $A$ is said to be an $H$-matrix if its comparison matrix is an $M$-matrix. Throughout this paper, we are working with $H$-matrices such that their comparison matrices are nonsingular. These matrices are called invertible class of $H$-matrices in ^[].

As a result of that a nonsingular $H$-matrix has nonzero diagonal entries, we always assume that $a_{ii}\neq 0(i\in N)$.

The nonsingular $H$-matrix is a kind of special matrix that is widely used in matrix theory. Many practical problems can usually be attributed to the problems of solving one or a group of linear algebraic equations for large sparse matrices. In the process of solving linear equations, it is often necessary to assume that the coefficient matrix is a nonsingular $H$-matrix. At the same time, nonsingular $H$-matrix has important practical value in many fields, such as economic mathematics, electric system theory, control theory and computational mathematics^[2,3]. However, it is very difficult to determine the nonsingular $H$-matrix in practice. So the determination of nonsingular $H$-matrix is a very meaningful topic in the study of matrix theory. Many scholars have conducted in-depth research on its sufficient conditions, and have further given many simple and practical results ^{[4,5,6,7,8,9,10,11,12,13,14,15,16]}.

In this paper, we introduce two different classes of $\alpha$-diagonally dominant matrices defined in ^[6,7]. In order to avoid confusion, they are called $\alpha_1$-diagonally dominant matrix and $\alpha_2$-diagonally dominant matrix respectively.

Definition 1. ^[6] Let $A = (a_{ij})\in \mathbb{C}^{n\times n}$. If $\alpha\in [0, 1]$ exists, making

then $A$ is called an $\alpha_1$-diagonally dominant matrix, and denoted by $A\in D_{\alpha_{10}}$. If $\alpha\in [0, 1]$ exists, making

then $A$ is called a strictly $\alpha_1$-diagonally dominant matrix, and denoted by $A\in D_{\alpha_1}$.

Definition 2. ^[7] Let $A = (a_{ij})\in \mathbb{C}^{n\times n}$. If $\alpha\in [0, 1]$ exists, making

then $A$ is called an $\alpha_2$-diagonally dominant matrix, and denoted by $A\in D_{\alpha_{20}}$. If $\alpha\in [0, 1]$ exists, making

then $A$ is called a strictly $\alpha_2$-diagonally dominant matrix, and denoted by $A\in D_{\alpha_2}$.

At present, many scholars have studied the properties and determination methods of $\alpha_1$-(and $\alpha_2$-) diagonally dominant matrices, see ^{[5,6,7,8,9,10,11,17]}. $\alpha_2$-diagonally dominant matrix is called geometrically $\alpha$-diagonally dominant matrix in ^[8], $\alpha$-chain diagonally dominant matrix in ^[9], and product $\alpha$-diagonally dominant matrix in ^[17].

In Definitions 1 and 2, if $\alpha = 1$, we can know $|a_{ii}| > R_i(A), \forall i\in N,$ by (1.1) and (1.2), that is, $A\in D$. If $\alpha = 0$, we can know $|a_{ii}| > C_i(A), \forall i\in N,$ by (1.1) and (1.2), that is, $A^{T}\in D$. Therefore, if $\alpha = 0$ or 1, $A$ is a nonsingular $H$-matrix, so only the case of $\alpha\in (0, 1)$ is considered in this paper.

If $A$ is an $\alpha_1$-(or $\alpha_2$-) diagonally dominant matrix, then $A\in D^{\rm *}$^[6,7]. So $\alpha_1$-(or $\alpha_2$-) diagonally dominant matrix is also a class of nonsingular $H$-matrix. These two classes are both subclasses of nonsingular $H$-matrix, and they have their equivalent theorems in the field of eigenvalue localization. It is easy to see that the class of $\alpha_1$-diagonally dominant matrix is contained in that of $\alpha_2$-diagonally dominant matrix^[18].

In this paper, by using the properties of $\alpha_1$-(or $\alpha_2$-) diagonally dominant matrix, we give some criteria for determining nonsingular $H$-matrix. Finally, numerical examples are used to compare the criteria obtained in this paper with the existing results.

2.   Preliminaries

Some relevant concepts and important conclusions are given in this section.

Definition 3. ^[9] Let $A = (a_{ij})\in \mathbb{C}^{n\times n}$. If there is a positive diagonal matrix $X$ such that $AX\in D_{\alpha_1}$, then $A$ is called a generalized $\alpha_1$-diagonally dominant matrix, which is denoted by $A\in D^{*}_{\alpha_1}$.

Definition 4. ^[7] Let $A = (a_{ij})\in \mathbb{C}^{n\times n}$. If there is a positive diagonal matrix $X$ such that $AX\in D_{\alpha_2}$, then $A$ is called a generalized $\alpha_2$-diagonally dominant matrix, which is denoted by $A\in D^{*}_{\alpha_2}$.

Definition 5. ^[10] Let $A = (a_{ij})\in \mathbb{C}^{n\times n}$ be an irreducible matrix. If there exists $\alpha\in [0, 1]$ such that $|a_{ii}|\geq\alpha [R_i(A)]+(1-\alpha)[C_i(A)], \forall i\in N$, and at least one strict inequality holds, then $A$ is said to be an irreducible $\alpha_1$-diagonally dominant matrix.

Here, similar to irreducible $\alpha_1$-diagonally dominant matrix, we give the definition of irreducible $\alpha_2$-diagonally dominant matrix.

Definition 6. Let $A = (a_{ij})\in \mathbb{C}^{n\times n}$ be an irreducible matrix. If there exists $\alpha\in [0, 1]$ such that $|a_{ii}|\geq[R_i(A)]^{\alpha}[C_i(A)]^{1-\alpha}, \forall i\in N$, and at least one strict inequality holds, then $A$ is said to be an irreducible $\alpha_2$-diagonally dominant matrix.

Lemma 1. ^[9] Let $A = (a_{ij})\in \mathbb{C}^{n\times n}$. If $A$ is a generalized $\alpha_1$-diagonally dominant matrix, then $A$ is a nonsingular $H$-matrix.

Lemma 2. ^[7] Let $A = (a_{ij})\in \mathbb{C}^{n\times n}$. Then $A$ is a generalized strictly diagonally dominant matrix if and only if $A$ is a generalized $\alpha_2$-diagonally dominant matrix.

Lemma 3. ^[10] Let $A\in D_{\alpha_{10}}$ be an irreducible matrix, and there is at least one $i\in N$ to make $|a_{ii}| > \alpha [R_i(A)]+(1-\alpha)[C_i(A)]$ hold, then $A\in D^{\rm *}$.

Lemma 4. ^[11] Let $A\in D_{\alpha_{20}}$ be an irreducible matrix, and there is at least one $i\in N$ to make $|a_{ii}| > [R_i(A)]^{\alpha}[C_i(A)]^{1-\alpha}$ hold, then $A\in D^{\rm *}$.

Lemma 5. ^[3] Suppose $A = (a_{ij})\in \mathbb{C}^{n\times n}$, if $AX$ is a nonsingular $H$-matrix, with $X = \text{diag} \; (x_1, x_2, \ldots, x_n) \; (x_i > 0, i = 1, 2, \ldots, n)$, then $A$ is a nonsingular $H$-matrix.

3.   Criteria based on $\alpha_1$-diagonally dominant matrix

Denote

It is obvious that $M_i{(\alpha)}\cap M_j{(\alpha)} = \emptyset(i\neq j)$ and $M_1{(\alpha)}\cup M_2{(\alpha)}\cup M_3{(\alpha)} = N$. We denote $\sum\limits_{i\in\emptyset} \cdot = 0$ and

Theorem 1. Let $A = (a_{ij})\in \mathbb{C}^{n\times n}$. If there is $\alpha\in(0, 1)$, such that for any $i\in M_2{(\alpha)}$,

holds, then $A$ is a nonsingular $H$-matrix.

Proof. We are going to proof the following inequality for all indices in each set $M_1(\alpha), M_2(\alpha)$ and $M_3(\alpha)$.

It can be seen from the previous denotions that $0\leq r < 1, 0 < \delta < 1$. From the definition of $T_{i, r}(A)$, we can get that for any $i\in M_3{(\alpha)}$,

holds, that is, $T_{i, r}(A)\le r|a_{ii}|, i\in M_3{(\alpha)}$. Therefore

Furthermore, according to the definition of $T_{i, r}(A)$, for any $i\in M_3{(\alpha)}$,

So

According to the definition of $h$, we can get $0\leq h < 1$, and for all $i\in M_3{(\alpha)}$,

By (3.1), for all $i\in M_2{(\alpha)}$, we can get

Let

and

In particular, if $\sum\limits_{j\in M_3{(\alpha)}}|a_{ij}| = 0$, then denote $w_i = +\infty$, according to (3.3), $w_i > 0, i\in M_2{(\alpha)}$. Notice that

Thus, take a sufficiently small positive number $\eta$ to make it meet both

and

Construct a positive diagonal matrix $X = \text{diag} \; (x_1, x_2, \ldots, x_n)$, where

And let $B = AX = (b_{i, j})$.

For any $i\in M_1{(\alpha)}$, it can be obtained from $0 < \delta < 1, 0 < \frac{\Lambda_i(A)-|a_{ii}|}{\Lambda_i(A)}\leq \delta < 1 (i\in M_2{(\alpha)})$, and $0 < \frac{T_{i, r}(A)}{|a_{ii}|}h+\eta < \delta < 1 (i\in M_3{(\alpha)})$ that

For any $i\in M_2{(\alpha)}$, if $\sum\limits_{j\in M_3{(\alpha)}}|a_{ij}| = 0$, it can be deduced from (3.1) that

If $\sum\limits_{j\in M_3{(\alpha)}}|a_{ij}|\neq 0$, it can be obtained from (3.3) that

For any $i\in M_3{(\alpha)}$, it can be deduced from $0 < \frac{\Lambda_i(A)-|a_{ii}|}{\Lambda_i(A)}\leq\delta < 1 (i\in M_2{(\alpha)})$ and (3.2) that

In conclusion, the following inequalities are always valid

By Definition 1, matrix $B$ is a strictly $\alpha_1$-diagonally dominant matrix, so matrix $A$ is a generalized $\alpha_1$-diagonally dominant matrix. According to Lemma 1, $A$ is a nonsingular $H$-matrix. □

Remark 1. If $\alpha = 1$, Theorem 1 is equivalent to Theorem 4 in ^[12]. At the same time, in Theorem 1, we improve the conditions of the theorems in ^[13,14,15]. So Theorem 1 in this paper is a further supplement to the determination methods of nonsingular $H$- matrices.

Theorem 2. Let $A = (a_{ij})\in \mathbb{C}^{n\times n}$ be an irreducible matrix. If there is $\alpha\in(0, 1)$, such that for any $i\in M_2{(\alpha)}$,

and at least one strict inequality in (3.4) holds, then matrix $A$ is a nonsingular $H$-matrix.

Proof. We are going to proof the following inequality for all indices in each set $M_1(\alpha), M_2(\alpha)$ and $M_3(\alpha)$.

Construct a positive diagonal matrix $X = \text{diag} \; (x_1, x_2, \ldots, x_n)$, where

And denote $B = AX = (b_{ij})$. Similar to the proof process of Theorem 1, for any $i\in M_1{(\alpha)}$,

For any $i\in M_2{(\alpha)}$, it can be obtained from (3.4) that

For any $i\in M_3{(\alpha)}$, by (3.2) we can obtain

To sum up, we can always get the following inequalities

Notice that there is at least one $i_0\in M_3{(\alpha)}$, such that $|b_{i_0, i_0}| > \Lambda_{i_0}(B)$, so $B$ is an irreducible $\alpha_1$-diagonally dominant matrix. According to Lemma 3, $B$ is a nonsingular $H$-matrix. Therefore, $A$ is also a nonsingular $H$-matrix by Lemma 5. □

4.   Criteria based on $\alpha_2$-diagonally dominant matrix

Let

It is obvious that $N_i{(\alpha)}\cap N_j{(\alpha)} = \emptyset(i\neq j)$ and $N_1{(\alpha)}\cup N_2{(\alpha)}\cup N_3{(\alpha)} = N$.

For any $i\in N_3{(\alpha)}$, denote

Obviously,

Theorem 3. Let $A = (a_{ij})\in \mathbb{C}^{n\times n}$. If there exists $\alpha\in(0, 1)$, such that

holds for any $i\in N_1{(\alpha)}$, then the matrix $A$ is a nonsingular $H$-matrix.

Proof. We are going to proof the following inequality for all indices in each set $N_1(\alpha), N_2(\alpha)$ and $N_3(\alpha)$.

For any $i\in N_1{(\alpha)}$, denote

It is known by (4.1) that $G_i(A) > 0, \; i\in N_1{(\alpha)}$. In particular, if $\sum\limits_{j\in N_3{(\alpha)}}|a_{ij}| = 0 (i\in N_1{(\alpha)}), G_i(A)$ = $+\infty$ is denoted. Take {a} sufficiently small positive number $\varepsilon$ to satisfy

Construct a positive diagonal matrix $X = \text{diag} \; (d_1, d_2, \ldots, d_n)$, where

It is proved below that $B = AX = (b_{ij})\in D_{\alpha_2}$. For any $i\in N_1{(\alpha)}$, according to (4.1) and (4.2),

that is, $|b_{ii}| > R_i(B)^{\alpha}(C_i(B))^{{1-\alpha}}, \; \; i\in N_1{(\alpha)}.$

For any $i\in N_2{(\alpha)}$, because $\frac{Q_i(A)-|a_{ii}|}{Q_i(A)} < 1, \; i\in N_1{(\alpha)}$, and $\frac{P_i(A)}{|a_{ii}|^{\frac{1}{\alpha}}}+\varepsilon < 1, \; i\in N_3{(\alpha)}$, obtained by (4.2), so,

For any $i\in N_3{(\alpha)}$, obviously

hence

Take the two sides of the inequality to the power of $\alpha$ respectively, we can get

Further multiply both sides of the inequality by $({\frac{P_i(A)}{|a_{ii}|^{\frac{1}{\alpha}}}}+\varepsilon)^{1-\alpha}$, then

that is, $|b_{ii}| > (R_i(B))^{\alpha}(C_i(B))^{1-\alpha}.$ To sum up, the following inequality is always true.

that is, $B\in D_{\alpha_2}.$ Therefore, we know that $A\in D^{*}_{\alpha_2}$, and according to Lemma 2, $A$ is a nonsingular $H$-matrix. □

Remark 2. According to (4.1) in Theorem 3, for any $i \in N_1{(\alpha)}$, the following inequality is always true.

Therefore, for Theorem 3 in this paper, we improve Theorem 1 in ^[10] and Theorem 1 in ^[16].

Theorem 4. Let $A = (a_{ij})\in \mathbb{C}^{n\times n}$ be an irreducible matrix. If there exists $\alpha\in (0, 1)$, such that

is true for any $i\in N_1{(\alpha)}$, then the matrix $A$ is a nonsingular $H$-matrix.

Proof. We are going to proof the following inequality for all indices in each set $N_1(\alpha), N_2(\alpha)$ and $N_3(\alpha)$.

Construct a positive diagonal matrix $X = \text{diag} \; (d_1, d_2, \ldots, d_n)$, where

Let $B = AX = (b_{ij})$. For any $i\in N_1{(\alpha)}$, it can be obtained from (4.3) that

that is, $|b_{ii}|\geq(R_i(B))^{\alpha}(C_i(B))^{1-\alpha}, i\in N_1{(\alpha)}.$

For any $i\in N_2{(\alpha)}$, because ${\frac{Q_i(A)-|a_{ii}|}{Q_i(A)}} < 1, \; i\in N_1{(\alpha)}$, and ${\frac{P_i(A)}{{|a_{ii}|^{\frac{1}{\alpha}}}}} < 1, \; i\in N_3{(\alpha)}$, we can obtain that

that is, $|b_{ii}| > (R_i(B))^{\alpha}(C_i(B))^{1-\alpha}, i\in N_2{(\alpha)}.$

For any $i\in N_3{(\alpha)}$,

Take the power $\alpha$ on both sides and multiply by $({{\frac{P_j(A)}{|a_{jj}|^{\frac{1}{\alpha}}}}})^{1-\alpha}$ at the same time, then

that is, $|b_{ii}| > (R_i(B))^{\alpha}(C_i(B))^{1-\alpha}, i\in N_3{(\alpha)}.$

In conclusion, the following inequalities are always valid.

Thus, $B$ is an irreducible $\alpha_2$-diagonally dominant matrix. According to Lemma 4, $B$ is a nonsingular $H$-matrix. Therefore, $A$ is also a nonsingular $H$-matrix by Lemma 5. □

5.   Numerical examples

Example 1. Let

Taking $\alpha = \frac{19}{20}$, we will show that

(1) The matrix $A$ satisfies the conditions of Theorem 1 in this paper, so we can determine that $A$ is a nonsingular $H$-matrix according to Theorem 1.

(2) $A$ does not meet the criteria in ^[13,14,15], so it cannot be determined by applying the methods in these papers.

In fact, for (1), it can be obtained through calculation that

So, $M_1{(\alpha)} = \{1\}, M_2{(\alpha)} = \{2, 3\}, M_3{(\alpha)} = \{4, 5\}$. And then

Therefore, $h = \max\{\frac{17}{21}, \frac{850}{1239}\} = \frac{17}{21}$. And notice that

To sum up, the conditions of Theorem 1 in this paper are satisfied. So we can determine that $A$ is a nonsingular $H$-matrix.

For (2), it is calculated that

Then the conditions of the decision theorem in ^[13] are not satisfied.

So the conditions of the decision theorem in ^[14] are also not satisfied.

Further calculation shows that

The conditions of the decision theorem in ^[15] are not satisfied.

Therefore, we know that the matrix $A$ does not meet the criteria in ^[13,14,15], so it cannot be determined by these existing methods.

Example 2. Let

Taking $\alpha = {\frac{1}{4}}$, we will show that

(1) The matrix $A$ satisfies the conditions of Theorem 3 in this paper, so we can get that $A$ is a nonsingular $H$-matrix.

(2) $A$ does not meet the criteria in ^[10,16], so it cannot be determined by applying the methods in ^[10,16].

In fact, for (1), it is calculated that

So $N_1{(\alpha)} = \{3, 4\}, \; N_2{(\alpha)} = \{2, 5\}, \; N_3{(\alpha)} = \{1, 6\}$, and then calculate

So the conditions of Theorem 3 in this paper are satisfied, thus we can determine that $A$ is a nonsingular $H$-matrix.

For (2), using Theorem 3 in ^[16], we can get

It can be obtained through calculation that

So the matrix $A$ does not satisfy the conditions of the theorem in ^[16], thus it cannot be judged using the method in ^[16].

Using Theorem 3 in ^[10], we can obtain

It is known by calculation that

Through calculation, we know that the matrix $A$ also does not meet the criteria in ^[10], so it also cannot be determined by applying the method in ^[10].

6.   Conclusions

In this paper, based on the relevant properties of two classes of $\alpha$-diagonally dominant matrices, we obtain several sufficient conditions to determine nonsingular $H$-matrix, which improves the existing results and also extends the determination theory of nonsingular $H$-matrix.

Acknowledgments

This work is supported by the Science and technology research project of education department of Jilin Province of China (JJKH20220041KJ), the Natural Sciences Program of Science and Technology of Jilin Province of China(20190201139JC) and the Graduate Inovation Project of Beihua University (2022003, 2021004).

Conflict of interest

The authors declare that there are no conflict of interest.