Schur Complement-Based Infinity Norm Bounds for the Inverse of GDSDD Matrices

Abstract

Based on the Schur complement, some upper bounds for the infinity norm of the inverse of generalized doubly strictly diagonally dominant matrices are obtained. In addition, it is shown that the new bound improves the previous bounds. Numerical examples are given to illustrate our results. By using the infinity norm bound, a lower bound for the smallest singular value is given.

1. Introduction

Throughout the paper, let n be an integer number, $N:=\{1,2,\dots ,n\}$ be the set of all indices, $C^{n\times n}$ (resp.$R^{n\times n}$) denote the set of all $n\times n$ complex (resp.real) matrices, $I\in R^{n\times n}$ be an identity matrix, $|A|=[|a_{ij}|]\in R^{n\times n}$, $r_i(A):={\displaystyle \sum _{j=1,j\ne i}^{j=n}}|a_{ij}|$ denote deleted ith row sum, and $J:=\{i\in N:|a_{ii}|>r_i(A)\}$.

Definition 1

([1]). Let $A=\left[a_{ij}\right]\in {\mathbb{C}}^{n\times n}$ ($n\ge 2$), then A is called:

(i)

A row diagonally dominant ($DD$) if for all $i\in N$,

$$\begin{array}{c}\hfill |a_{ii}|\ge r_i(A);\end{array}$$

(1)

(ii)

A strictly diagonally dominant ($SDD$) if all the strict inequalities in (1) hold;

(iii)

A doubly strictly diagonally dominant ($DSDD$) matrix, if:

$$\begin{array}{c}\hfill |a_{ii}||a_{jj}|>r_i(A)r_j(A),i,j\in N,i\ne j.\end{array}$$

(2)

The class of generalized doubly strictly diagonally dominant (GDSDD) matrices was presented by Gao and Wang in [2].

Definition 2

([2]). A matrix A is called a GDSDD matrix if $J\ne \varnothing$ and there exist proper subsets $N_1,N_2$ of N such that $N_1\cap N_2=\varnothing ,N_1\cup N_2=N$ and:

$$\begin{array}{c}\hfill (|a_{ii}|-{\alpha }_i)(|a_{jj}|-{\beta }_j)>{\beta }_i{\alpha }_j\end{array}$$

(3)

for any $i\in N_1$ and $j\in N_2$, where with $s=i$ or j,

$$\begin{gathered} {\alpha }_s=\sum _{t\in N_1 \\ t\ne s}|a_{st}|,{\beta }_s=\sum _{t\in N_2 \\ t\ne s}|a_{st}|. \end{gathered}$$

Here the ${\alpha }_s$ and ${\beta }_s$ may be interpreted as the sums of the absolute values of the nondiagonal elements in row s that fall in the columns $N_1$ and $N_2$, respectively. When $n=2$, a GDSDD matrix is nothing but a $DSDD$ matrix. A matrix A satisfying (3) may not be generalized doubly diagonally dominant for another pair subsets $N_1$ and $N_2$. Assuming that the matrix order is $n\ge 2$, we adopt the notation: $GDSD{D}_n^{N_1,N_2}$ for generalized doubly strictly diagonally dominant.

Definition 3

([3]). A matrix A is called an H-matrix if its comparison matrix $\mu (A)=\left[{\mu }_{ij}\right]$ defined by:

$$\begin{array}{c}\hfill {\mu }_{ij}=\left\{\begin{array}{c}\hfill |a_{ij}|ifi=j,\\ \hfill -|a_{ij}|ifi\ne j,\end{array}\right.\end{array}$$

is an M-matrix, i.e., ${(\mu (A))}^{-1}\ge 0$.

It is shown in [3] that if A is an H-matrix, then:

$$\begin{array}{c}\hfill {(\mu (A))}^{-1}\ge |A^{-1}|.\end{array}$$

(4)

In addition, it was shown that $SDD$ matrices, $DSDD$ matrices, and GDSDD matrices are subclass of H-matrices [4].

Upper bounds for the infinity norm of the inverse of nonsingular matrices can be used to the convergence analysis of matrix splitting and matrix multi-splitting iterative methods for solving the large sparse of linear equations. Moreover, the upper bounds can also be applied to the smallest singular value of matrices [5,6].

One traditional method for finding upper bounds for the infinity norm of the inverse of nonsingular matrices is to use the definition and properties of a given matrix class; see [7,8,9,10] for details. The first work may due to Varah [10], who in 1975 presented the following upper bound for the infinity norm of the inverse for $SDD$ matrices.

Theorem 1

([10]). If $A=\left[a_{ij}\right]\in C^{n\times n}$ is $SDD$, then:

$$\begin{array}{c}\hfill {∥A^{-1}∥}_{\infty }\le \underset{i\in N}{max}\frac{1}{|a_{ii}|-r_i(A)}.\end{array}$$

However, if the diagonal dominance of A (i.e. $|a_{ii}|-r_i(A)$) is weak, Varah’s bound may yield a large value. In 2020, based on the Schur complement, Li [11] obtained two upper bounds for the infinity norm of inverse of $SDD$ matrices.

Using the definition and properties of $DSDD$ matrices, Liu, Zhang and Liu in [9] obtained an upper bound of ${∥A^{-1}∥}_{\infty }$ for a $DSDD$ matrix A as follows.

Theorem 2

([7]). If $A=\left[a_{ij}\right]\in C^{n\times n}$ is $DSDD$, then:

$$\begin{array}{c}\hfill {∥A^{-1}∥}_{\infty }\le \underset{\begin{array}{c}i,j\in N\\ i\ne j\end{array}}{max}\frac{|a_{jj}|+r_i(A)}{|a_{ii}||a_{jj}|-r_i(A)r_j(A)}.\end{array}$$

In 2020, based on the Schur complement, Sang [12] obtained two upper bounds for the infinity norm of $DSDD$ matrices.

And Moraa in [13] give an upper bound for the infinity norm of GDSDD matrices as follows.

Theorem 3.

If $A=\left[a_{ij}\right]\in C^{n\times n}$ is $GDSD{D}_n^{N_1,N_2}$, then:

$$\begin{array}{c}\hfill {∥A^{-1}∥}_{\infty }\le max\left\{\underset{i\in N_1,j\in N_2}{max}\frac{|a_{jj}|-{\beta }_j+{\beta }_i}{(|a_{ii}|-{\alpha }_i)(|a_{jj}|-{\beta }_j)-{\alpha }_j{\beta }_i},\underset{i\in N_1,j\in N_2}{max}\frac{|a_{ii}|-{\alpha }_i+{\alpha }_j}{(|a_{ii}|-{\alpha }_i)(|a_{jj}|-{\beta }_j)-{\alpha }_j{\beta }_i}\right\}.\end{array}$$

(5)

In this paper, based on the Schur complement, we present some upper bounds for the infinity norm of the inverse of GDSDD matrices, and numerical examples are given to show the effectiveness of the obtained results. In addition, applying these new bounds, a lower bound for the smallest singular value of GDSDD matrices is obtained.

2. Schur Complement-Based Infinity Bounds for the Inverse of GDSDD Matrices

Firstly, we recall the Schur complement of matrices.

Let $\alpha$ be a proper subset of N and denote by $|\alpha |$ the cardinality of $\alpha$ and by ${\alpha }^{\prime }=N-\alpha$ the complement to $\alpha$ in N. For nonempty index sets $\alpha ,\beta \subseteq N$, we write $A(\alpha ,\beta )$ to mean the submatrix of $A\in C^{n\times n}$ lying in the rows indexed by $\alpha$ and the columns indexed by $\beta$. $A(\alpha ,\alpha )$ is abbreviated to $A(\alpha )$. Throughout this paper, supposing $\alpha =\{i_1,i_2,\dots ,i_l\}\subset N$, the complement ${\alpha }^{\prime }=N-\alpha =\{j_1,j_2,\dots ,j_m\}$ and the elements of $\alpha$ and ${\alpha }^{\prime }$ are both conventionally arranged in increasing order. Furthermore, if $A(\alpha )$ is nonsingular, we define the Schur complement of A with respect to $A(\alpha )$, which is denoted by $A/A(\alpha )$ or simply $A/\alpha$, to be:

$$\begin{array}{c}\hfill A/\alpha =A({\alpha }^{\prime })-A({\alpha }^{\prime },\alpha )A{(\alpha )}^{-1}A(\alpha ,{\alpha }^{\prime }).\end{array}$$

Liu, Huang, and Zhang in [14] proved that the Schur complements of GDSDD matrices are GDSDD matrices. Many similar or related results had been obtained; see [9,15,16] for details.

For a given nonempty proper subset $\alpha$, there is always a permutation matrix P such that:

$$P^{\mathrm{T}}AP=\left[\begin{array}{cc}A(\alpha )& A(\alpha ,{\alpha }^{\prime })\\ A({\alpha }^{\prime },\alpha )& A({\alpha }^{\prime })\end{array}\right].$$

It is well known that the inverse of a permutation matrix is also a permutation matrix and the infinity norm for the inverse of a permutation matrix equals to 1 (${\parallel P\parallel }_{\infty }=1$). Hence, if A is nonsingular, then:

$$\begin{array}{c}\hfill \parallel A^{-1}{\parallel }_{\infty }=\parallel P{(P^{\mathrm{T}}AP)}^{-1}{P}^{\mathrm{T}}{\parallel }_{\infty }\le {\parallel ({(P^{\mathrm{T}}AP)}^{-1})\parallel }_{\infty }.\end{array}$$

So we next only consider the upper bound for $\parallel ({(P^{\mathrm{T}}AP)}^{-1}){\parallel }_{\infty }$. For the sake of simplicity, we consider that:

$$\begin{array}{c}\hfill P^{\mathrm{T}}AP=\left[\begin{array}{cc}{A}_{11}& A_{12}\\ A_{21}& A_{22}\end{array}\right],\end{array}$$

(6)

where $A_{11}=A(\alpha )$ and $A_{22}=A({\alpha }^{\prime })$.

Lemma 1

([17]). Let $A\in C^{n\times n}$ with the form (6), and $I_{11}$ (resp. $I_{22}$) be the identity matrix of order l (resp. m). If $A_{11}$ is nonsingular, then:

$$\begin{array}{c}\hfill S(P^{\mathrm{T}}AP)Q=\left[\begin{array}{cc}{A}_{11}& 0\\ 0& A_{22}-A_{21}{A}_{11}^{-1}{A}_{12}\end{array}\right],\end{array}$$

where

$$S=\left[\begin{array}{cc}{I}_{11}& 0\\ -A_{21}{A}_{11}^{-1}& I_{22}\end{array}\right]andQ=\left[\begin{array}{cc}{I}_{11}& -A_{11}^{-1}{A}_{12}\\ 0& I_{22}\end{array}\right].$$

From Lemma 1, we obtain:

$$\begin{array}{c}\hfill A={(P^{\mathrm{T}})}^{-1}{S}^{-1}S(P^{\mathrm{T}}AP)Q{Q}^{-1}{P}^{-1},\end{array}$$

and hence:

$$\begin{array}{c}\hfill A^{-1}=PQ{(S(P^{\mathrm{T}}AP)Q)}^{-1}S{P}^{\mathrm{T}},\end{array}$$

which implies that:

$$\begin{array}{c}\hfill {∥A^{-1}∥}_{\infty }={∥PQ{(S{P}^{\mathrm{T}}APQ)}^{-1}S{P}^{\mathrm{T}}∥}_{\infty }\le {∥Q∥}_{\infty }{∥{(S{P}^{\mathrm{T}}APQ)}^{-1}∥}_{\infty }{∥S∥}_{\infty }.\end{array}$$

(7)

This implies that if upper bounds for ${∥S∥}_{\infty }$, ${∥Q∥}_{\infty }$, ${∥A_{11}^{-1}∥}_{\infty }$, and ${∥{(A/\alpha )}^{-1}∥}_{\infty }$ are given, then the product of these bounds could be regarded as an upper bound for ${∥A^{-1}∥}_{\infty }$ by (7). It is not difficult to compute that:

$$\begin{array}{c}\hfill {∥S∥}_{\infty }=1+{∥A_{21}{A}_{11}^{-1}∥}_{\infty },\\ \hfill {∥Q∥}_{\infty }=1+{∥A_{11}^{-1}{A}_{12}∥}_{\infty },\end{array}$$

and

$$\begin{array}{c}\hfill {∥{(S{P}^{\mathrm{T}}APQ)}^{-1}∥}_{\infty }=max\left\{{∥A_{11}^{-1}∥}_{\infty },{∥{(A_{22}-A_{21}{A}_{11}^{-1}{A}_{12})}^{-1}∥}_{\infty }\right\}.\end{array}$$

In 2020, Li [11] gave an upper bound for ${∥S∥}_{\infty }$ as follows:

Lemma 2

([11]). Let $A=\left[a_{ij}\right]\in C^{n\times n}$ be nonsingular with $a_{ii}\ne 0$ for $i\in N$, and $\varnothing \ne \alpha \subset N$. If $A_{11}$ is nonsingular, and:

$$\begin{array}{c}\hfill 1>\underset{i\in \alpha }{max}\frac{\underset{j\in \alpha ,j\ne i}{max}|a_{ji}|}{|a_{ii}|}·(l-1),\end{array}$$

(8)

then

$$\begin{array}{c}\hfill {∥S∥}_{\infty }\le \gamma (\alpha )=1+l·\underset{i\in \alpha }{max}\frac{\underset{j\in {\alpha }^{\prime }}{max}|a_{ji}|}{|a_{ii}|}·{\left(1-\underset{i\in \alpha }{max}\frac{\underset{j\in \alpha ,j\ne i}{max}|a_{ji}|}{|a_{ii}|}·(l-1)\right)}^{-1}.\end{array}$$

(9)

Next, we only consider upper bounds for ${∥Q∥}_{\infty }$, ${∥A_{11}^{-1}∥}_{\infty }$ and ${∥{(A_{22}-A_{21}{A}_{11}^{-1}{A}_{12})}^{-1}∥}_{\infty }$ when A is $GDSD{D}_n^{N_1,N_2}$.

Lemma 3

([14]). Let A be $GDSD{D}_n^{N_1,N_2}$. Then $A(N_1)$ and $A(N_2)$ are $SDD$.

Lemma 4

([14]). Let A be $GDSD{D}_n^{N_1,N_2}$. Then $N_1\subseteq J$ or $N_2\subseteq J$.

In this paper, we assume $N_1\subseteq J$.

Lemma 5

([14]). Let A be $GDSD{D}_n^{N_1,N_2}$ and $\alpha \subset N$. Then $A(\alpha )$ is $GDSD{D}_{|\alpha |}^{N_1\cap \alpha ,N_2\cap \alpha }$.

Lemma 6

([14]). Let A be $GDSD{D}_n^{N_1,N_2}$. Then for any proper subset α of N, $A/\alpha$ is $GDSD{D}_{|N-\alpha |}^{N_1-\alpha ,N_2-\alpha }$.

Lemma 7

([18]). Let $A=\left[a_{ij}\right]\in C^{n\times n}$ be $SDD$, and $B=\left[b_{ij}\right]\in C^{n\times m}$. Then:

$$\begin{array}{c}\hfill {∥A^{-1}B∥}_{\infty }\le \underset{i\in N}{max}\frac{R_i(B)}{|a_{ii}|-r_i(A)},\end{array}$$

(10)

where $R_i(B)={\displaystyle \sum _{k=1}^m}|b_{ik}|$.

Lemma 8.

Let $A=\left[a_{ij}\right]\in C^{n\times n}$ be $GDSD{D}_n^{N_1,N_2}$, and $B=\left[b_{ij}\right]\in C^{n\times m}$. Then:

$$\begin{gathered} \begin{array}{c}\hfill {∥A^{-1}B∥}_{\infty }\le max\left\{\underset{i\in N_1 \\ j\in N_2}{max}\frac{(|a_{jj}|-{\beta }_j)R_i(B)+{\beta }_i{R}_j(B)}{(|a_{ii}|-{\alpha }_i)(|a_{jj}|-{\beta }_j)-{\beta }_i{\alpha }_j},\underset{i\in N_1 \\ j\in N_2}{max}\frac{(|a_{ii}|-{\alpha }_i)R_j(B)+{\alpha }_j{R}_i(B)}{(|a_{ii}|-{\alpha }_i)(|a_{jj}|-{\beta }_j)-{\beta }_i{\alpha }_j}\right\},\end{array} \end{gathered}$$

where $R_i(B)=\sum _{k=1}^m|b_{ik}|$.

Proof. 

Since there exists an m-dimensional vector $x={(x_1,x_2,\dots ,x_m)}^{\mathrm{T}}$ such that ${∥x∥}_{\infty }=1$ and:

$$\begin{array}{c}\hfill {∥A^{-1}B∥}_{\infty }={∥A^{-1}Bx∥}_{\infty }={∥y∥}_{\infty }=\underset{i\in N}{max}|y_i|,\end{array}$$

(11)

where $y=A^{-1}Bx={(y_1,y_2,\dots ,y_n)}^{\mathrm{T}}$, we have $Ay=Bx$, and let $|y_p|=\underset{i\in N_1}{max}|y_i|,|y_q|=\underset{j\in N_2}{max}|y_j|$, which implies that:

$$\begin{gathered} \begin{array}{c}\hfill a_{ii}{y}_i+\sum _{j\in N \\ j\ne i}{a}_{ij}{y}_j=\sum _{k=1}^m{b}_{ik}{x}_k,i\in N.\end{array} \end{gathered}$$

Then,

$$\begin{gathered} \begin{array}{c}\hfill \sum _{k=1}^m{b}_{pk}{x}_k=a_{pp}{y}_p+\sum _{j\in N \\ j\ne p}{a}_{pj}{y}_j=a_{pp}{y}_p+\sum _{j\in N_1 \\ j\ne p}{a}_{pj}{y}_j+\sum _{j\in N_2}{a}_{pj}{y}_j,\end{array} \end{gathered}$$

$$\begin{gathered} \begin{array}{c}\hfill \sum _{k=1}^m{b}_{qk}{x}_k=a_{qq}{y}_q+\sum _{j\in N \\ j\ne q}{a}_{qj}{y}_j=a_{qq}{y}_q+\sum _{j\in N_1}{a}_{qj}{y}_j+\sum _{\begin{array}{c}j\in N_2,j\ne q\end{array}}{a}_{qj}{y}_j.\end{array} \end{gathered}$$

We can obtain:

$$\begin{gathered} \begin{array}{ccc}\hfill \sum _{k=1}^m|b_{pk}|\underset{1\le k\le m}{max}|x_k|& \ge & \sum _{k=1}^m|b_{pk}||x_k|\ge |\sum _{k=1}^m{b}_{pk}{x}_k|\hfill \\ & =& |a_{pp}{y}_p+\sum _{j\in N \\ j\ne p}{a}_{pj}{y}_j|\hfill \\ & =& |a_{pp}{y}_p+\sum _{j\in N_1 \\ j\ne p}{a}_{pj}{y}_j+\sum _{j\in N_2}{a}_{pj}{y}_j|\hfill \\ & \ge & |a_{pp}{y}_p|-|\sum _{j\in N_1 \\ j\ne p}{a}_{pj}{y}_j|-|\sum _{j\in N_2}{a}_{pj}{y}_j|\hfill \\ & \ge & |a_{pp}||y_p|-\sum _{j\in N_1 \\ j\ne p}|a_{pj}||y_j|-\sum _{j\in N_2}|a_{pj}||y_j|\hfill \\ & \ge & |a_{pp}||y_p|-\sum _{j\in N_1 \\ j\ne p}|a_{pj}||y_p|-\sum _{j\in N_2}|a_{pj}||y_q|\hfill \\ & =& |a_{pp}||y_p|-{\alpha }_p|y_p|-{\beta }_p|y_q|\hfill \end{array} \end{gathered}$$

and

$$\begin{gathered} \begin{array}{ccc}\hfill \sum _{k=1}^m|b_{qk}|\underset{1\le k\le m}{max}|x_k|& \ge & \sum _{k=1}^m|b_{qk}||x_k|\ge |\sum _{k=1}^m{b}_{qk}{x}_k|\hfill \\ & =& |a_{qq}{y}_q+\sum _{j\in N \\ j\ne q}{a}_{qj}{y}_j|\hfill \end{array} \end{gathered}$$

$$\begin{gathered} \begin{array}{ccc}\hfill & =& |a_{qq}{y}_q+\sum _{j\in N_1}{a}_{qj}{y}_j+\sum _{j\in N_2 \\ j\ne q}{a}_{qj}{y}_j|\hfill \\ & \ge & |a_{qq}{y}_q|-|\sum _{j\in N_1}{a}_{qj}{y}_j|-|\sum _{j\in N_2 \\ j\ne q}{a}_{qj}{y}_j|\hfill \\ & \ge & |a_{qq}||y_q|-\sum _{j\in N_1}|a_{qj}||y_j|-\sum _{\begin{array}{c}j\in N_2\\ j\ne q\end{array}}|a_{qj}||y_j|\hfill \\ & \ge & |a_{qq}||y_q|-\sum _{j\in N_1}|a_{qj}||y_p|-\sum _{j\in N_2 \\ j\ne q}|a_{qj}||y_q|\hfill \\ & =& |a_{qq}||y_p|-{\alpha }_q|y_p|-{\beta }_q|y_q|.\hfill \end{array} \end{gathered}$$

That is to say,

$$\begin{array}{c}\hfill \sum _{k=1}^m|b_{pk}|\underset{1\le k\le m}{max}|x_k|\ge |a_{pp}||y_p|-{\alpha }_p|y_p|-{\beta }_p|y_q|\end{array}$$

(12)

and

$$\begin{array}{c}\hfill \sum _{k=1}^m|b_{qk}|\underset{1\le k\le m}{max}|x_k|\ge |a_{qq}||y_p|-{\alpha }_q|y_p|-{\beta }_q|y_q|.\end{array}$$

(13)

The proof is divided into two cases.

Case 1: $|y_p|\ge |y_q|$. Then, $(12)\times (|a_{qq}|-{\beta }_q)+(13)\times {\beta }_p$ yields

$$\begin{array}{c}\hfill (|a_{qq}|-{\beta }_q)\sum _{k=1}^m|b_{pk}|\underset{1\le k\le m}{max}|x_k|+{\beta }_p\sum _{k=1}^m|b_{qk}|\underset{1\le k\le m}{max}|x_k|\ge |y_p|[(|a_{pp}|-{\alpha }_p)(|a_{qq}|-{\beta }_q)-{\beta }_p{\alpha }_q],\end{array}$$

which leads to:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {∥A^{-1}B∥}_{\infty }& =\underset{i\in N}{max}|y_i|=|y_p|\hfill \\ & \le \frac{(|a_{qq}|-{\beta }_q){\displaystyle \sum _{k=1}^m}|b_{pk}|\underset{1\le k\le m}{max}|x_k|+{\beta }_p{\displaystyle \sum _{k=1}^m}|b_{qk}|\underset{1\le k\le m}{max}|x_k|}{(|a_{pp}|-{\alpha }_p)(|a_{qq}|-{\beta }_q)-{\alpha }_q{\beta }_p}\hfill \\ & =\frac{(|a_{qq}|-{\beta }_q){\displaystyle \sum _{k=1}^m}|b_{pk}|+{\beta }_p{\displaystyle \sum _{k=1}^m}|b_{qk}|}{(|a_{pp}|-{\alpha }_p)(|a_{qq}|-{\beta }_q)-{\alpha }_q{\beta }_p}\hfill \\ & \le \underset{i\in N_1,j\in N_2}{max}\frac{(|a_{jj}|-{\beta }_j){\displaystyle \sum _{k=1}^m}|b_{ik}|+{\beta }_i{\displaystyle \sum _{k=1}^m}|b_{jk}|}{(|a_{ii}|-{\alpha }_i)(|a_{jj}|-{\beta }_j)-{\alpha }_j{\beta }_i}\hfill \\ & =\underset{i\in N_1,j\in N_2}{max}\frac{(|a_{jj}|-{\beta }_j)R_i(B)+{\beta }_i{R}_j(B)}{(|a_{ii}|-{\alpha }_i)(|a_{jj}|-{\beta }_j)-{\alpha }_j{\beta }_i}.\hfill \end{array}\end{array}$$

(14)

Case 2: $|y_p|<|y_q|$. Then, $(12)\times {\alpha }_q+(13)\times (|a_{pp}|-{\alpha }_p)$ yields:

$$\begin{array}{c}\hfill {\alpha }_q\sum _{k=1}^m|b_{pk}|\underset{1\le k\le m}{max}|x_k|+(|a_{pp}|-{\alpha }_p)\sum _{k=1}^m|b_{qk}|\underset{1\le k\le m}{max}|x_k|\ge |y_q|[(|a_{pp}|-{\alpha }_p)(|a_{qq}|-{\beta }_q)-{\beta }_p{\alpha }_q],\end{array}$$

which leads to:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {∥A^{-1}B∥}_{\infty }& =\underset{i\in N}{max}|y_i|=|y_q|\hfill \\ & \le \frac{{\alpha }_q{\displaystyle \sum _{k=1}^m}|b_{pk}|\underset{1\le k\le m}{max}|x_k|+(|a_{pp}|-{\alpha }_p){\displaystyle \sum _{k=1}^m}|b_{qk}|\underset{1\le k\le m}{max}|x_k|}{(|a_{pp}|-{\alpha }_p)(|a_{qq}|-{\beta }_q)-{\alpha }_q{\beta }_p}\hfill \\ & =\frac{{\alpha }_q{\displaystyle \sum _{k=1}^m}|b_{pk}|+(|a_{pp}|-{\alpha }_p){\displaystyle \sum _{k=1}^m}|b_{qk}|}{(|a_{pp}|-{\alpha }_p)(|a_{qq}|-{\beta }_q)-{\alpha }_q{\beta }_p}\hfill \\ & \le \underset{i\in N_1,j\in N_2}{max}\frac{{\alpha }_j{\displaystyle \sum _{k=1}^m}|b_{ik}|+(|a_{ii}|-{\alpha }_i){\displaystyle \sum _{k=1}^m}|b_{jk}|}{(|a_{ii}|-{\alpha }_i)(|a_{jj}|-{\beta }_j)-{\alpha }_j{\beta }_i}\hfill \\ & =\underset{i\in N_1,j\in N_2}{max}\frac{(|a_{ii}|-{\alpha }_i)R_j(B)+{\alpha }_j{R}_i(B)}{(|a_{ii}|-{\alpha }_i)(|a_{jj}|-{\beta }_j)-{\alpha }_j{\beta }_i}.\hfill \end{array}\end{array}$$

(15)

The conclusion follows from equality (11) and inequalities (14) and (15). □

Corollary 1.

Let $A=\left[a_{ij}\right]\in C^{n\times n}$ be $GDSD{D}_n^{N_1,N_2}$, and $\varnothing \ne \alpha \subset N$. Then:

$$\begin{array}{c}\hfill {∥Q∥}_{\infty }=1+{∥A_{11}^{-1}{A}_{12}∥}_{\infty }\le 1+\phi (\alpha ),\end{array}$$

(16)

where

$$\begin{gathered} \begin{array}{c}\hfill \phi (\alpha )=max\left\{\underset{i\in N_1\cap \alpha \\ j\in N_2\cap \alpha }{max}\frac{(|a_{jj}|-{\beta }_j)R_i(A_{12})+{\beta }_i{R}_j(A_{12})}{(|a_{ii}|-{\alpha }_i)(|a_{jj}|-{\beta }_j)-{\beta }_i{\alpha }_j},\underset{i\in N_1\cap \alpha \\ j\in N_2\cap \alpha }{max}\frac{(|a_{ii}|-{\alpha }_i)R_j(A_{12})+{\alpha }_j{R}_i(A_{12})}{(|a_{ii}|-{\alpha }_i)(|a_{jj}|-{\beta }_j)-{\beta }_i{\alpha }_j}\right\}.\end{array} \end{gathered}$$

Specially, if $\alpha \subset N_1$ or $\alpha \subset N_2$, then $A_{11}$ is $SDD$, and

$$\begin{array}{c}\hfill {∥Q∥}_{\infty }=1+{∥A_{11}^{-1}{A}_{12}∥}_{\infty }\le 1+\underset{i\in \alpha }{max}\frac{R_i(A_{12})}{|a_{ii}|-r_i(A_{11})}.\end{array}$$

(17)

For the sake of convenience, denote:

$$\begin{array}{c}\hfill \left[l\right]=\{1,2,\dots ,l\},\end{array}$$

$$\begin{array}{c}\hfill \left[m\right]=\{1,2,\dots ,m\},\end{array}$$

$$\begin{array}{c}\hfill v_{j_k}:=(a_{j_k{i}_1},\dots ,a_{j_k{i}_l}),\end{array}$$

$$\begin{array}{c}\hfill w_{j_k}:={(a_{i_1{j}_k},\dots ,a_{i_l{j}_k})}^{\mathrm{T}},\end{array}$$

$$\begin{array}{c}\hfill |v_{j_k}|:=(|a_{j_k{i}_1}|,\dots ,|a_{j_k{i}_l}|),\end{array}$$

$$\begin{array}{c}\hfill |w_{j_k}|:=(|a_{i_1{j}_k}|,\dots ,|a_{i_l{j}_k}{|)}^{\mathrm{T}},\end{array}$$

for any $k=1,2,\dots ,m$. And thus:

$$\begin{array}{c}\hfill \sum _{k\in \left[m\right]}|w_{j_k}|={\left(\sum _{k\in \left[m\right]}|a_{i_1{j}_k}|,\dots ,\sum _{k\in \left[m\right]}|a_{i_l{j}_k}|\right)}^{\mathrm{T}}=\sum _{j_k\in {\alpha }^{\prime }}|w_{j_k}|.\end{array}$$

(18)

Lemma 9.

Let $x={\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}{\displaystyle \sum _{k\in \left[m\right]}}|w_{j_k}|={(x_1,\dots ,x_l)}^{\mathrm{T}}$, $x_g=\underset{k\in \left[l\right]}{max}{x}_k$. If $i_g\in \alpha$ is a strictly diagonally dominant row, then:

$$\begin{array}{c}\hfill 0\le x_k<1,k\in \left[l\right].\end{array}$$

(19)

Proof. 

$$\begin{array}{c}\hfill x={\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}\sum _{k\in \left[m\right]}|w_{j_k}|={(x_1,\dots ,x_l)}^{\mathrm{T}},i.e.,\mu \left[A(\alpha )\right]x=\sum _{k\in \left[m\right]}|w_{j_k}|.\end{array}$$

(20)

For any $\alpha \subset N$, $A_{11}$ is GDSDD, so is $\mu (A(\alpha ))$, and consequently, $\mu (A(\alpha ))$ is an H-matrix, and ${\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}\ge 0$ by (4). Hence, x is nonnegative. Let $x_g=\underset{k\in \left[l\right]}{max}{x}_k$. By $i_g$ is a strictly diagonally dominant row, then:

$$\begin{array}{c}\hfill |a_{i_g{i}_g}|>\sum _{v\in \left[l\right],v\ne g}|a_{i_g{i}_v}|+\sum _{k\in \left[m\right]}|a_{i_g{j}_k}|.\end{array}$$

(21)

By the gth equality of (20) and inequality (21), we have:

$$\begin{array}{c}\hfill \sum _{k\in \left[m\right]}|a_{i_g{j}_k}|=|a_{i_g{i}_g}|x_g-\sum _{v\in \left[l\right],v\ne g}|a_{i_g{i}_v}|x_v\ge x_g\left(|a_{i_g{i}_g}|-\sum _{v\in \left[l\right],v\ne g}|a_{i_g{i}_v}|\right),\end{array}$$

and hence:

$$\begin{array}{c}\hfill x_g\le \frac{{\displaystyle \sum _{k\in \left[m\right]}}|a_{i_g{j}_k}|}{|a_{i_g{i}_g}|-{\displaystyle \sum _{v\in \left[l\right],v\ne g}}|a_{i_g{i}_v}|}<1,\end{array}$$

which implies that:

$$\begin{array}{c}\hfill 0\le x_k<1,k\in \left[l\right].\end{array}$$

The proof is completed. □

Theorem 4.

Let $A=\left[a_{ij}\right]$ be $GDSD{D}_n^{N_1,N_2}$ with $a_{ii}\ne 0,i\in N$, and $\varnothing \ne \alpha \subset N$. Assume that (8) holds, if $\alpha \subset N_1$, then:

$$\begin{array}{c}\hfill {∥A^{-1}∥}_{\infty }\le \gamma (\alpha )·(1+\underset{i\in \alpha }{max}\frac{R_i(A_{12})}{|a_{ii}|-r_i(A_{11})})·\theta (\alpha ),\end{array}$$

where

$$\begin{array}{c}\hfill \theta (\alpha )=max\left\{\underset{i\in \alpha }{max}\frac{1}{|a_{ii}|-r_i(A_{11})},\eta (\alpha )\right\},\end{array}$$

$$\begin{gathered} \begin{array}{c}\hfill \eta (\alpha )=max\left\{\underset{\hspace{1em}\hspace{1em}i,j\in {\alpha }^{\prime } \\ i\in N_1-\alpha ,j\in N_2}{max}\frac{|a_{jj}|-{\beta }_j(A)+{\beta }_i(A)+{\displaystyle \sum _{k\in \alpha }}|a_{jk}|+{\displaystyle \sum _{k\in \alpha }}|a_{ik}|}{y_{i,j}-z_{i,j}},\right.\\ \hfill \left.\underset{\hspace{1em}\hspace{1em}i,j\in {\alpha }^{\prime } \\ i\in N_1-\alpha ,j\in N_2}{max}\frac{|a_{ii}|+{\alpha }_j(A)-{\displaystyle \sum _{k\in N_1-\alpha }}|a_{ik}|+{\displaystyle \sum _{k\in \alpha }}|a_{ik}|}{y_{i,j}-z_{i,j}}\right\},\end{array} \end{gathered}$$

$$\begin{array}{c}\hfill y_{i,j}=(|a_{ii}|-\sum _{k\in N_1-\alpha }|a_{ik}|-|v_i{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}\sum _{k\in N_1-\alpha }|w_k|)(|a_{jj}|-{\beta }_j(A)-|v_j{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}\sum _{k\in N_2}|w_k|),\end{array}$$

$$\begin{array}{c}\hfill z_{i,j}=(\sum _{k\in N_1-\alpha }|a_{jk}|+|v_j{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}\sum _{k\in N_1-\alpha }|w_k|)({\beta }_i(A)+|v_i{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}\sum _{k\in N_2}|w_k|).\end{array}$$

Proof. 

Denote the $(s,t)$-entry of $A/\alpha$ by $(a_{st}^{\prime })$. Then for any $s,t\in \left[m\right]$,

$$\begin{array}{c}\hfill a_{st}^{\prime }=a_{j_s{j}_t}-v_{j_s}{\left[A(\alpha )\right]}^{-1}{w}_{j_t},\end{array}$$

(22)

and by inequality (4), we have:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |a_{st}^{\prime }|& =|a_{j_s{j}_t}-v_{j_s}{\left[A(\alpha )\right]}^{-1}{w}_{j_t}|\hfill \\ & \le |a_{j_s{j}_t}|+|v_{j_s}{||\left[A(\alpha )\right]|}^{-1}|w_{j_t}|\hfill \\ & \le |a_{j_s{j}_t}|+|v_{j_s}{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}|w_{j_t}|,\hfill \end{array}\end{array}$$

(23)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |a_{st}^{\prime }|& =|a_{j_s{j}_t}-v_{j_s}{\left[A(\alpha )\right]}^{-1}{w}_{j_t}|\hfill \\ & \ge |a_{j_s{j}_t}|-|v_{j_s}{||\left[A(\alpha )\right]|}^{-1}|w_{j_t}|\hfill \\ & \ge |a_{j_s{j}_t}|-|v_{j_s}{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}|w_{j_t}|.\hfill \end{array}\end{array}$$

(24)

Since A is $GDSD{D}_n^{N_1,N_2}$, $\alpha \subset N_1$, by Lemmas 5 and 6, we have $A_{11}$ is $SDD$, and $A/\alpha$ is $GDSD{D}_{|N-\alpha |}^{N_1-\alpha ,N_2-\alpha }$. Applying the Varah’s bound to $A_{11}$, we get:

$$\begin{array}{c}\hfill {∥A_{11}^{-1}∥}_{\infty }\le \underset{i\in \alpha }{max}\frac{1}{|a_{ii}|-r_i(A_{11})}.\end{array}$$

(25)

Applying Theorem 3 to $A/\alpha$ yields:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {∥{(A/\alpha )}^{-1}∥}_{\infty }\le max\left\{\underset{\begin{array}{c}s,t\in \left[m\right]\\ j_s\in N_1-\alpha ,j_t\in N_2\end{array}}{max}\frac{|a_{tt}^{\prime }|-{\beta }_t(A/\alpha )+{\beta }_s(A/\alpha )}{(|a_{ss}^{\prime }|-{\alpha }_s(A/\alpha ))(|a_{tt}^{\prime }|-{\beta }_t(A/\alpha ))-{\alpha }_t(A/\alpha ){\beta }_s(A/\alpha )},\right.\\ \hfill \left.\underset{\begin{array}{c}s,t\in \left[m\right]\\ j_s\in N_1-\alpha ,j_t\in N_2\end{array}}{max}\frac{|a_{ss}^{\prime }|-{\alpha }_s(A/\alpha )+{\alpha }_t(A/\alpha )}{(|a_{ss}^{\prime }|-{\alpha }_s(A/\alpha ))(|a_{tt}^{\prime }|-{\beta }_t(A/\alpha ))-{\alpha }_t(A/\alpha ){\beta }_s(A/\alpha )}\right\}.\end{array}\end{array}$$

(26)

By Theorem 2 in [14], we have:

$$\begin{gathered} \begin{array}{c}\hfill \begin{array}{cc}& (|a_{ss}^{\prime }|-{\alpha }_s(A/\alpha ))(|a_{tt}^{\prime }|-{\beta }_t(A/\alpha ))-{\alpha }_t(A/\alpha ){\beta }_s(A/\alpha )\hfill \\ & \ge (|a_{j_s{j}_s}|-\sum _{j_k\in N_1-\alpha \\ \hspace{1em}k\ne s}|a_{j_s{j}_k}|-|v_{j_s}{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}\sum _{j_k\in N_1-\alpha }|w_{j_k}|)(|a_{j_t{j}_t}|-\sum _{\begin{array}{c}{j}_k\in N_2\\ k\ne t\end{array}}|a_{j_t{j}_k}|\hfill \\ & -|v_{j_t}{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}\sum _{\begin{array}{c}{j}_k\in N_2\end{array}}|w_{j_k}|)-(\sum _{j_k\in N_1-\alpha }|a_{j_t{j}_k}|+|v_{j_t}{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}\sum _{j_k\in N_1-\alpha }|w_{j_k}|)(\sum _{j_k\in N_2}|a_{j_s{j}_k}|\hfill \\ & +|v_{j_s}{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}\sum _{j_k\in N_2}|w_{j_k}|)\hfill \\ & >0.\hfill \end{array}\end{array} \end{gathered}$$

(27)

Now, the upper bounds of $|a_{tt}^{\prime }|-{\beta }_t(A/\alpha )+{\beta }_s(A/\alpha )$ and $|a_{ss}^{\prime }|-{\alpha }_s(A/\alpha )+{\alpha }_t(A/\alpha )$ are considered. Since $\alpha \subset N_1\subseteq J$, it satisfies Lemma 9, which implies that (19) holds.

By (19), (20), (23) and (24), we have:

$$\begin{gathered} \begin{array}{c}\hfill \begin{array}{cc}& |a_{tt}^{\prime }|-{\beta }_t(A/\alpha )+{\beta }_s(A/\alpha )\hfill \\ & =|a_{tt}^{\prime }|-\sum _{j_k\in N_2 \\ k\ne t}|a_{tk}^{\prime }|+\sum _{j_k\in N_2}|a_{sk}^{\prime }|\hfill \\ & \le |a_{j_t{j}_t}|+|v_{j_t}{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}|w_{j_t}|-\sum _{j_k\in N_2 \\ k\ne t}(|a_{j_t{j}_k}|-|v_{j_t}{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}|w_{j_k}|)\hfill \\ & +\sum _{j_k\in N_2}(|a_{j_s{j}_k}|+|v_{j_s}{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}|w_{j_k}|)(byinequalities(23)and(24))\hfill \\ & =|a_{j_t{j}_t}|+|v_{j_t}{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}|w_{j_t}|-\sum _{j_k\in N_2 \\ \hspace{1em}k\ne t}|a_{j_t{j}_k}|+|v_{j_t}{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}\sum _{j_k\in N_2 \\ \hspace{1em}k\ne t}|w_{j_k}|\hfill \end{array}\end{array} \end{gathered}$$

$$\begin{gathered} \begin{array}{c}\hfill \begin{array}{cc}& +\sum _{j_k\in N_2}|a_{j_s{j}_k}|+|v_{j_s}{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}\sum _{j_k\in N_2}|w_{j_k}|\hfill \\ & =|a_{j_t{j}_t}|-\sum _{j_k\in N_2 \\ k\ne t}|a_{j_t{j}_k}|+\sum _{j_k\in N_2}|a_{j_s{j}_k}|+|v_{j_t}{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}\sum _{\begin{array}{c}{j}_k\in N_2\end{array}}|w_{j_k}|\hfill \\ & +|v_{j_s}{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}\sum _{j_k\in N_2}|w_{j_k}|\hfill \\ & =|a_{j_t{j}_t}|-\sum _{j_k\in N_2 \\ k\ne t}|a_{j_t{j}_k}|+\sum _{j_k\in N_2}|a_{j_s{j}_k}|+(|v_{j_t}|+|v_{j_s}{|)\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}\sum _{j_k\in N_2}|w_{j_k}|\hfill \\ & \le |a_{j_t{j}_t}|-\sum _{j_k\in N_2 \\ k\ne t}|a_{j_t{j}_k}|+\sum _{j_k\in N_2}|a_{j_s{j}_k}|+(|v_{j_t}|+|v_{j_s}{|)\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}\sum _{k\in \left[m\right]}|w_{j_k}|(by(18)and{N}_2\subset {\alpha }^{\prime })\hfill \\ & =|a_{j_t{j}_t}|-\sum _{j_k\in N_2 \\ k\ne t}|a_{j_t{j}_k}|+\sum _{j_k\in N_2}|a_{j_s{j}_k}|+\sum _{k\in \left[l\right]}(|a_{j_t{i}_k}|+|a_{j_s{i}_k}|)x_k\hfill \\ & \le |a_{j_t{j}_t}|-\sum _{j_k\in N_2 \\ k\ne t}|a_{j_t{j}_k}|+\sum _{j_k\in N_2}|a_{j_s{j}_k}|+\sum _{k\in \left[l\right]}(|a_{j_t{i}_k}|+|a_{j_s{i}_k}|)(byinequality(19))\hfill \\ & =|a_{j_t{j}_t}|-{\beta }_{j_t}(A)+{\beta }_{j_s}(A)+\sum _{k\in \left[l\right]}|a_{j_t{i}_k}|+\sum _{k\in \left[l\right]}|a_{j_s{i}_k}|.\hfill \end{array}\end{array} \end{gathered}$$

(28)

Similarly,

$$\begin{gathered} \begin{array}{c}\hfill \begin{array}{cc}& |a_{ss}^{\prime }|-{\alpha }_s(A/\alpha )+{\alpha }_t(A/\alpha )\hfill \\ & =|a_{ss}^{\prime }|-\sum _{j_k\in N_1-\alpha \\ k\ne s}|a_{sk}^{\prime }|+\sum _{j_k\in N_1-\alpha }|a_{tk}^{\prime }|\hfill \\ & \le |a_{j_s{j}_s}|+|v_{j_s}{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}|w_{j_s}|-\sum _{j_k\in N_1-\alpha \\ k\ne s}(|a_{j_s{j}_k}|-|v_{j_s}{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}|w_{j_k}|)\hfill \\ & +\sum _{j_k\in N_1-\alpha }(|a_{j_t{j}_k}|+|v_{j_t}{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}|w_{j_k}|)(byinequalities(23)and(24))\hfill \\ & =|a_{j_s{j}_s}|+|v_{j_s}{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}|w_{j_s}|-\sum _{j_k\in N_1-\alpha \\ k\ne s}|a_{j_s{j}_k}|+|v_{j_s}{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}\sum _{j_k\in N_1-\alpha \\ k\ne s}|w_{j_k}|\hfill \\ & +\sum _{j_k\in N_1-\alpha }|a_{j_t{j}_k}|+|v_{j_t}{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}\sum _{j_k\in N_1-\alpha }|w_{j_k}|\hfill \\ & =|a_{j_s{j}_s}|-\sum _{j_k\in N_1-\alpha \\ k\ne s}|a_{j_s{j}_k}|+\sum _{j_k\in N_1-\alpha }|a_{j_t{j}_k}|+|v_{j_s}{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}\sum _{\begin{array}{c}{j}_k\in N_1-\alpha \end{array}}|w_{j_k}|\hfill \\ & +|v_{j_t}{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}\sum _{j_k\in N_1-\alpha }|w_{j_k}|\hfill \\ & =|a_{j_s{j}_s}|-\sum _{j_k\in N_1-\alpha \\ k\ne s}|a_{j_s{j}_k}|+\sum _{j_k\in N_1-\alpha }|a_{j_t{j}_k}|+(|v_{j_s}|+|v_{j_t}{|)\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}\sum _{j_k\in N_1-\alpha }|w_{j_k}|\hfill \\ & \le |a_{j_s{j}_s}|-\sum _{j_k\in N_1-\alpha \\ k\ne s}|a_{j_s{j}_k}|+\sum _{j_k\in N_1-\alpha }|a_{j_t{j}_k}|+(|v_{j_s}|+|v_{j_t}{|)\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}\sum _{k\in \left[m\right]}|w_{j_k}|(by(18)and{N}_1-\alpha \subset {\alpha }^{\prime })\hfill \\ & =|a_{j_s{j}_s}|-\sum _{j_k\in N_1-\alpha \\ k\ne s}|a_{j_s{j}_k}|+\sum _{j_k\in N_1-\alpha }|a_{j_t{j}_k}|+\sum _{k\in \left[l\right]}(|a_{j_s{i}_k}|+|a_{j_t{i}_k}|)x_k\hfill \\ & \le |a_{j_s{j}_s}|-\sum _{j_k\in N_1-\alpha \\ k\ne s}|a_{j_s{j}_k}|+\sum _{j_k\in N_1-\alpha }|a_{j_t{j}_k}|+\sum _{k\in \left[l\right]}(|a_{j_s{i}_k}|+|a_{j_t{i}_k}|)(byinequality(19))\hfill \\ & =|a_{j_s{j}_s}|+{\alpha }_{j_t}(A)-\sum _{j_k\in N_1-\alpha \\ k\ne s}|a_{j_s{j}_k}|+\sum _{k\in \left[l\right]}|a_{j_s{i}_k}|.\hfill \end{array}\end{array} \end{gathered}$$

(29)

Furthermore, by (26)–(29), we have:

$$\begin{gathered} \begin{array}{c}\hfill \begin{array}{cc}\hfill {∥{(A/\alpha )}^{-1}∥}_{\infty }& \le max\left\{\underset{ s,t\in \left[m\right] \\ {j}_s\in N_1-\alpha \\ {j}_t\in N_2-\alpha }{max}\frac{|a_{j_t{j}_t}|-{\beta }_{j_t}(A)+{\beta }_{j_s}(A)+{\displaystyle \sum _{k\in \left[l\right]}}|a_{j_t{i}_k}|+{\displaystyle \sum _{k\in \left[l\right]}}|a_{j_s{i}_k}|}{y_{j_s,j_t}-z_{j_s,j_t}},\right.\hfill \\ & \left.\underset{ s,t\in \left[m\right] \\ {j}_s\in N_1-\alpha \\ {j}_t\in N_2-\alpha }{max}\frac{|a_{j_s{j}_s}|+{\alpha }_{j_t}(A)-{\displaystyle \sum _{j_k\in N_1-\alpha \\ \hspace{1em}k\ne s}}|a_{j_s{j}_k}|+{\displaystyle \sum _{k\in \left[l\right]}}|a_{j_s{i}_k}|}{y_{j_s,j_t}-z_{j_s,j_t}}\right\}\hfill \\ & =max\left\{\underset{\hspace{1em}\hspace{1em}i,j\in {\alpha }^{\prime } \\ i\in N_1-\alpha ,j\in N_2}{max}\frac{|a_{jj}|-{\beta }_j(A)+{\beta }_i(A)+{\displaystyle \sum _{k\in \alpha }}|a_{jk}|+{\displaystyle \sum _{k\in \alpha }}|a_{ik}|}{y_{i,j}-z_{i.j}},\right.\hfill \\ & \left.\underset{\hspace{1em}\hspace{1em}i,j\in {\alpha }^{\prime } \\ i\in N_1-\alpha ,j\in N_2}{max}\frac{|a_{ii}|+{\alpha }_j(A)-{\displaystyle \sum _{k\in N_1-\alpha \\ \hspace{1em}k\ne i}}|a_{ik}|+{\displaystyle \sum _{k\in \alpha }}|a_{ik}|}{y_{i,j}-z_{i,j}}\right\},\hfill \end{array}\end{array} \end{gathered}$$

(30)

where

$$\begin{array}{cc}\hfill y_{j_s,j_t}& =(|a_{j_s,j_s}|-\sum _{\begin{array}{c}{j}_k\in N_1-\alpha \\ k\ne s\end{array}}|a_{j_s{j}_k}|-|v_{j_s}{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}\sum _{j_k\in N_1-\alpha }|w_{j_k}|)(|a_{j_t{j}_t}|\\ & -\sum _{\begin{array}{c}{j}_k\in N_2\\ k\ne t\end{array}}|a_{j_t{j}_k}|-|v_{j_t}{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}\sum _{j_k\in N_2}|w_{j_k}|),\end{array}$$

$$\begin{array}{cc}\hfill z_{j_s,j_t}& =(\sum _{j_k\in N_1-\alpha }|a_{j_t{j}_k}|+|v_{j_t}{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}\sum _{j_k\in N_1-\alpha }|w_{j_k}|)(\sum _{j_k\in N_2}|a_{j_s{j}_k}|\\ & +|v_{j_s}{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}\sum _{j_k\in N_2}|w_{j_k}|).\end{array}$$

By Corollary 1, we have:

$$\begin{array}{c}\hfill {∥Q∥}_{\infty }\le 1+\underset{i\in \alpha }{max}\frac{R_i(A_{12})}{|a_{ii}|-r_i(A_{11})}.\end{array}$$

(31)

Furthermore, by (7), (9), (25), (30), and (31), the conclusion follows. □

Theorem 5.

Let $A=\left[a_{ij}\right]$ be $GDSD{D}_n^{N_1,N_2}$ with $a_{ii}\ne 0,i\in N$, and $\varnothing \ne \alpha \subset N$. Assume that (8) and (19) hold, if $\alpha \subset N_2$, then:

$$\begin{array}{c}\hfill {∥A^{-1}∥}_{\infty }\le \gamma (\alpha )·(1+\underset{i\in \alpha }{max}\frac{R_i(A_{12})}{|a_{ii}|-r_i(A_{11})})·\delta (\alpha ),\end{array}$$

where

$$\begin{array}{c}\hfill \delta (\alpha )=max\left\{\underset{i\in \alpha }{max}\frac{1}{|a_{ii}|-r_i(A_{11})},\xi (\alpha )\right\},\end{array}$$

$$\begin{array}{cc}\hfill \xi (\alpha )=max& \left\{\underset{\begin{array}{c}i,j\in {\alpha }^{\prime }\\ i\in N_1,j\in N_2-\alpha \end{array}}{max}\frac{|a_{jj}|+{\beta }_i(A)-{\displaystyle \sum _{k\in N_2-\alpha ,k\ne j}}|a_{jk}|+{\displaystyle \sum _{k\in \alpha }}|a_{jk}|}{f_{i,j}-g_{i,j}},\right.\\ & \left.\underset{\begin{array}{c}i,j\in {\alpha }^{\prime }\\ i\in N_1,j\in N_2-\alpha \end{array}}{max}\frac{|a_{ii}|-{\alpha }_i(A)+{\alpha }_j(A)+{\displaystyle \sum _{k\in \alpha }}|a_{ik}|+{\displaystyle \sum _{k\in \alpha }}|a_{jk}|}{f_{i,j}-g_{i,j}}\right\},\end{array}$$

$$\begin{array}{c}\hfill f_{i,j}=(|a_{ii}|-{\alpha }_i(A)-|v_i{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}\sum _{k\in N_1}|w_k|)(|a_{jj}|-\sum _{\begin{array}{c}k\in N_2-\alpha \\ k\ne j\end{array}}|a_{jk}|-|v_j{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}\sum _{k\in N_2-\alpha }|w_k|),\end{array}$$

$$\begin{array}{c}\hfill g_{i,j}=({\beta }_j(A)+|v_j{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}\sum _{k\in N_1}|w_k|)(\sum _{k\in N_2-\alpha }|a_{ik}|+|v_i{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}\sum _{k\in N_2-\alpha }|w_k|).\end{array}$$

Proof. 

Since A is $GDSD{D}_n^{N_1,N_2}$, $\alpha \subset N_2$, by Lemmas 5 and 6, we have $A_{11}$ is $SDD$, and $A/\alpha$ is $GDSD{D}_{|N-\alpha |}^{N_1,N_2-\alpha }$. Applying the Varah’s bound to $A_{11}$, we get:

$$\begin{array}{c}\hfill {∥A_{11}^{-1}∥}_{\infty }\le \underset{i\in \alpha }{max}\frac{1}{|a_{ii}|-r_i(A_{11})}.\end{array}$$

(32)

Applying Theorem 3 to $A/\alpha$ yields:

$$\begin{gathered} \begin{array}{c}\hfill \begin{array}{c}\hfill {∥{(A/\alpha )}^{-1}∥}_{\infty }\le max\left\{\underset{\hspace{1em}\hspace{1em}s,t\in \left[m\right], \\ {j}_s\in N_1-\alpha ,j_t\in N_2}{max}\frac{|a_{tt}^{\prime }|-{\beta }_t(A/\alpha )+{\beta }_s(A/\alpha )}{(|a_{ss}^{\prime }|-{\alpha }_s(A/\alpha ))(|a_{tt}^{\prime }|-{\beta }_t(A/\alpha ))-{\alpha }_t(A/\alpha ){\beta }_s(A/\alpha )},\right.\\ \hfill \left.\underset{\hspace{1em}\hspace{1em}s,t\in \left[m\right], \\ {j}_s\in N_1-\alpha ,j_t\in N_2}{max}\frac{|a_{ss}^{\prime }|-{\alpha }_s(A/\alpha )+{\alpha }_t(A/\alpha )}{(|a_{ss}^{\prime }|-{\alpha }_s(A/\alpha ))(|a_{tt}^{\prime }|-{\beta }_t(A/\alpha ))-{\alpha }_t(A/\alpha ){\beta }_s(A/\alpha )}\right\}.\end{array}\end{array} \end{gathered}$$

(33)

By Theorem 2 in [14], we have:

$$\begin{gathered} \begin{array}{c}\hfill \begin{array}{cc}& (|a_{ss}^{\prime }|-{\alpha }_s(A/\alpha ))(|a_{tt}^{\prime }|-{\beta }_t(A/\alpha ))-{\alpha }_t(A/\alpha ){\beta }_s(A/\alpha )\hfill \\ & \ge (|a_{j_s{j}_s}|-\sum _{j_k\in N_1 \\ k\ne s}|a_{j_s{j}_k}|-|v_{j_s}{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}\sum _{\begin{array}{c}{j}_k\in N_1\end{array}}|w_{j_k}|)(|a_{j_t{j}_t}|-\sum _{j_k\in N_2-\alpha \\ \hspace{1em}k\ne t}|a_{j_t{j}_k}|\hfill \\ & -|v_{j_t}{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}\sum _{\begin{array}{c}{j}_k\in N_2-\alpha \end{array}}|w_{j_k}|)-(\sum _{j_k\in N_1}|a_{j_t{j}_k}|+|v_{j_t}{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}\sum _{j_k\in N_1}|w_{j_k}|)(\sum _{j_k\in N_2-\alpha }|a_{j_s{j}_k}|\hfill \\ & +|v_{j_s}{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}\sum _{j_k\in N_2-\alpha }|w_{j_k}|)\hfill \\ & >0.\hfill \end{array}\end{array} \end{gathered}$$

(34)

Now, the upper bounds of $|a_{tt}^{\prime }|-{\beta }_t(A/\alpha )+{\beta }_s(A/\alpha )$ and $|a_{ss}^{\prime }|-{\alpha }_s(A/\alpha )+{\alpha }_t(A/\alpha )$ are considered.

By (19), (20), (23), and (24), we have:

$$\begin{array}{c}\hfill \begin{array}{cc}& |a_{tt}^{\prime }|-{\beta }_t(A/\alpha )+{\beta }_s(A/\alpha )\hfill \\ & =|a_{tt}^{\prime }|-\sum _{\begin{array}{c}{j}_k\in N_2-\alpha \\ k\ne t\end{array}}|a_{tk}^{\prime }|+\sum _{j_k\in N_2-\alpha }|a_{sk}^{\prime }|\hfill \\ & \le |a_{j_t{j}_t}|+|v_{j_t}{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}|w_{j_t}|-\sum _{\begin{array}{c}{j}_k\in N_2-\alpha \\ k\ne t\end{array}}(|a_{j_t{j}_k}|-|v_{j_t}{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}|w_{j_k}|)\hfill \end{array}\end{array}$$

$$\begin{gathered} \begin{array}{c}\hfill \begin{array}{cc}& +\sum _{j_k\in N_2-\alpha }(|a_{j_s{j}_k}|+|v_{j_s}{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}|w_{j_k}|)(byinequalities(23)and(24))\hfill \\ & =|a_{j_t{j}_t}|+|v_{j_t}{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}|w_{j_t}|-\sum _{j_k\in N_2-\alpha \\ k\ne t}|a_{j_t{j}_k}|+|v_{j_t}{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}\sum _{\begin{array}{c}{j}_k\in N_2-\alpha \\ k\ne t\end{array}}|w_{j_k}|\hfill \\ & +\sum _{j_k\in N_2-\alpha }|a_{j_s{j}_k}|+|v_{j_s}{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}\sum _{j_k\in N_2-\alpha }|w_{j_k}|\hfill \\ & =|a_{j_t{j}_t}|-\sum _{j_k\in N_2-\alpha \\ k\ne t}|a_{j_t{j}_k}|+\sum _{j_k\in N_2-\alpha }|a_{j_s{j}_k}|+|v_{j_t}{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}\sum _{\begin{array}{c}{j}_k\in N_2-\alpha \end{array}}|w_{j_k}|\hfill \\ & +|v_{j_s}{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}\sum _{j_k\in N_2-\alpha }|w_{j_k}|\hfill \\ & =|a_{j_t{j}_t}|-\sum _{\begin{array}{c}{j}_k\in N_2-\alpha \\ k\ne t\end{array}}|a_{j_t{j}_k}|+\sum _{j_k\in N_2-\alpha }|a_{j_s{j}_k}|+(|v_{j_t}|+|v_{j_s}{|)\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}\sum _{j_k\in N_2-\alpha }|w_{j_k}|\hfill \\ & \le |a_{j_t{j}_t}|-\sum _{\begin{array}{c}{j}_k\in N_2-\alpha \\ k\ne t\end{array}}|a_{j_t{j}_k}|+\sum _{j_k\in N_2-\alpha }|a_{j_s{j}_k}|+(|v_{j_t}|+|v_{j_s}{|)\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}\sum _{k\in \left[m\right]}|w_{j_k}|(by(18)and{N}_2-\alpha \subset {\alpha }^{\prime })\hfill \\ & =|a_{j_t{j}_t}|-\sum _{\begin{array}{c}{j}_k\in N_2-\alpha \\ k\ne t\end{array}}|a_{j_t{j}_k}|+\sum _{j_k\in N_2-\alpha }|a_{j_s{j}_k}|+\sum _{k\in \left[l\right]}(|a_{j_t{i}_k}|+|a_{j_s{i}_k}|)x_k\hfill \\ & \le |a_{j_t{j}_t}|-\sum _{\begin{array}{c}{j}_k\in N_2-\alpha \\ k\ne t\end{array}}|a_{j_t{j}_k}|+\sum _{j_k\in N_2-\alpha }|a_{j_s{j}_k}|+\sum _{k\in \left[l\right]}(|a_{j_t{i}_k}|+|a_{j_s{i}_k}|)(byinequlaity(19))\hfill \\ & =|a_{j_t{j}_t}|+{\beta }_{j_s}(A)-\sum _{\begin{array}{c}{j}_k\in N_2-\alpha \\ k\ne t\end{array}}|a_{j_t{j}_k}|+\sum _{k\in \left[l\right]}|a_{j_t{i}_k}|.\hfill \end{array}\end{array} \end{gathered}$$

(35)

Similarly,

$$\begin{gathered} \begin{array}{c}\hfill \begin{array}{cc}& |a_{ss}^{\prime }|-{\alpha }_s(A/\alpha )+{\alpha }_t(A/\alpha )\hfill \\ & =|a_{ss}^{\prime }|-\sum _{\begin{array}{c}{j}_k\in N_1\\ k\ne s\end{array}}|a_{sk}^{\prime }|+\sum _{j_k\in N_1}|a_{tk}^{\prime }|\hfill \\ & \le |a_{j_s{j}_s}|+|v_{j_s}{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}|w_{j_s}|-\sum _{\begin{array}{c}{j}_k\in N_1\\ k\ne s\end{array}}(|a_{j_s{j}_k}|-|v_{j_s}{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}|w_{j_k}|)\hfill \\ & +\sum _{j_k\in N_1}(|a_{j_t{j}_k}|+|v_{j_t}{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}|w_{j_k}|)(byinequalities(23)and(24))\hfill \\ & =|a_{j_s{j}_s}|+|v_{j_s}{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}|w_{j_s}|-\sum _{j_k\in N_1 \\ k\ne s}|a_{j_s{j}_k}|+|v_{j_s}{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}\sum _{\begin{array}{c}{j}_k\in N_1\\ k\ne s\end{array}}|w_{j_k}|\hfill \\ & +\sum _{j_k\in N_1}|a_{j_t{j}_k}|+|v_{j_t}{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}\sum _{j_k\in N_1}|w_{j_k}|\hfill \\ & =|a_{j_s{j}_s}|-\sum _{j_k\in N_1 \\ k\ne s}|a_{j_s{j}_k}|+\sum _{j_k\in N_1}|a_{j_t{j}_k}|+|v_{j_s}{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}\sum _{\begin{array}{c}{j}_k\in N_1\end{array}}|w_{j_k}|\hfill \\ & +|v_{j_t}{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}\sum _{j_k\in N_1}|w_{j_k}|\hfill \\ & =|a_{j_s{j}_s}|-\sum _{\begin{array}{c}{j}_k\in N_1\\ k\ne s\end{array}}|a_{j_s{j}_k}|+\sum _{j_k\in N_1}|a_{j_t{j}_k}|+(|v_{j_s}|+|v_{j_t}{|)\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}\sum _{j_k\in N_1}|w_{j_k}|\hfill \\ & \le |a_{j_s{j}_s}|-\sum _{\begin{array}{c}{j}_k\in N_1\\ k\ne s\end{array}}|a_{j_s{j}_k}|+\sum _{j_k\in N_1}|a_{j_t{j}_k}|+(|v_{j_s}|+|v_{j_t}{|)\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}\sum _{k\in \left[m\right]}|w_{j_k}|(by(18)and{N}_1\subset {\alpha }^{\prime })\hfill \end{array}\end{array} \end{gathered}$$

$$\begin{array}{c}\hfill \begin{array}{cc}& =|a_{j_s{j}_s}|-\sum _{\begin{array}{c}{j}_k\in N_1\\ k\ne s\end{array}}|a_{j_s{j}_k}|+\sum _{j_k\in N_1}|a_{j_t{j}_k}|+\sum _{k\in \left[l\right]}(|a_{j_s{i}_k}|+|a_{j_t{i}_k}|)x_k\hfill \\ & \le |a_{j_s{j}_s}|-\sum _{\begin{array}{c}{j}_k\in N_1\\ k\ne s\end{array}}|a_{j_s{j}_k}|+\sum _{j_k\in N_1}|a_{j_t{j}_k}|+\sum _{k\in \left[l\right]}(|a_{j_s{i}_k}|+|a_{j_t{i}_k}|)(byinequalities(19))\hfill \\ & =|a_{j_s{j}_s}|-{\alpha }_{j_s}(A)+{\alpha }_{j_t}(A)+\sum _{k\in \left[l\right]}|a_{j_s{i}_k}|+\sum _{k\in \left[l\right]}|a_{j_t{i}_k}|.\hfill \end{array}\end{array}$$

(36)

Furthermore, by (33)–(36), we have:

$$\begin{gathered} \begin{array}{c}\hfill \begin{array}{cc}\hfill {∥{(A/\alpha )}^{-1}∥}_{\infty }\le max& \left\{\underset{\hspace{1em}\hspace{1em}s,t\in \left[m\right] \\ {j}_s\in N_1,j_t\in N_2-\alpha }{max}\frac{|a_{j_t{j}_t}|+{\beta }_{j_s}(A)-{\displaystyle \sum _{\begin{array}{c}{j}_k\in N_2-\alpha \\ k\ne t\end{array}}}|a_{j_t{j}_k}|+{\displaystyle \sum _{k\in \left[l\right]}}|a_{j_t{i}_k}|}{f_{j_s,j_t}-g_{j_s,j_t}},\right.\hfill \\ & \left.\underset{\begin{array}{c}s,t\in \left[m\right]\\ j_s\in N_1,j_t\in N_2-\alpha \end{array}}{max}\frac{|a_{j_s{j}_s}|-{\alpha }_{j_s}(A)+{\alpha }_{j_t}(A)+{\displaystyle \sum _{k\in \left[l\right]}}|a_{j_s{i}_k}|+{\displaystyle \sum _{k\in \left[l\right]}}|a_{j_t{i}_k}|}{f_{j_s,j_t}-g_{j_s,j_t}}\right\}\hfill \\ \hfill =max& \left\{\underset{\hspace{1em}\hspace{1em}i,j\in {\alpha }^{\prime } \\ i\in N_1,j\in N_2-\alpha }{max}\frac{|a_{jj}|+{\beta }_i(A)-{\displaystyle \sum _{\begin{array}{c}k\in N_2-\alpha \\ k\ne j\end{array}}}|a_{jk}|+{\displaystyle \sum _{k\in \alpha }}|a_{jk}|}{f_{i,j}-g_{i,j}},\right.\hfill \\ & \left.\underset{\hspace{1em}\hspace{1em}i,j\in {\alpha }^{\prime } \\ i\in N_1,j\in N_2-\alpha }{max}\frac{|a_{ii}|-{\alpha }_i(A)+{\alpha }_j(A)+{\displaystyle \sum _{\begin{array}{c}k\in \alpha \end{array}}}|a_{ik}|+{\displaystyle \sum _{k\in \alpha }}|a_{jk}|}{f_{i,j}-g_{i,j}}\right\},\hfill \end{array}\end{array} \end{gathered}$$

(37)

where

$$\begin{array}{cc}\hfill f_{j_s,j_t}& =(|a_{j_s{j}_s}|-\sum _{\begin{array}{c}{j}_k\in N_1\\ k\ne s\end{array}}|a_{j_s{j}_k}|-|v_{j_s}{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}\sum _{j_k\in N_1}|w_{j_k}|)(|a_{j_t{j}_t}|\\ & -\sum _{\begin{array}{c}{j}_k\in N_2-\alpha \\ k\ne t\end{array}}|a_{j_t{j}_k}|-|v_{j_t}{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}\sum _{j_k\in N_2-\alpha }|w_{j_k}|),\end{array}$$

$$\begin{array}{cc}\hfill g_{j_s,j_t}& =(\sum _{j_k\in N_1}|a_{j_t{j}_k}|+|v_{j_t}{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}\sum _{j_k\in N_1}|w_{j_k}|)(\sum _{j_k\in N_2-\alpha }|a_{j_s{j}_k}|\\ & +|v_{j_s}{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}\sum _{j_k\in N_2-\alpha }|w_{j_k}|).\end{array}$$

By Corollary 1, we have:

$$\begin{array}{c}\hfill {∥Q∥}_{\infty }\le 1+\underset{i\in \alpha }{max}\frac{R_i(A_{12})}{|a_{ii}|-r_i(A_{11})}.\end{array}$$

(38)

Furthermore, by (7), (9), (32), (37), and (38), the conclusion follows. □

Theorem 6.

Let $A=\left[a_{ij}\right]$ be $GDSD{D}_n^{N_1,N_2}$ with $a_{ii}\ne 0,i\in N$, and $\varnothing \ne \alpha \subset N$. Assume that (8) and (19) hold, if $\alpha \subseteq N_1,\alpha \subseteq N_2,N_1\subseteq \alpha$, and $N_2\subseteq \alpha$, then:

$$\begin{array}{c}\hfill {∥A^{-1}∥}_{\infty }\le \gamma (\alpha )·(1+\phi (\alpha ))·\zeta (\alpha ),\end{array}$$

where

$$\begin{gathered} \begin{array}{c}\hfill \phi (\alpha )=max\left\{\underset{i\in N_1\cap \alpha \\ j\in N_2\cap \alpha }{max}\frac{(|a_{jj}|-{\beta }_j)R_i(A_{12})+{\beta }_i{R}_j(A_{12})}{(|a_{ii}|-{\alpha }_i)(|a_{jj}|-{\beta }_j)-{\beta }_i{\alpha }_j},\underset{\begin{array}{c}i\in N_1\cap \alpha \\ j\in N_2\cap \alpha \end{array}}{max}\frac{(|a_{ii}|-{\alpha }_i)R_j(A_{12})+{\alpha }_j{R}_i(A_{12})}{(|a_{ii}|-{\alpha }_i)(|a_{jj}|-{\beta }_j)-{\beta }_i{\alpha }_j}\right\},\end{array} \end{gathered}$$

$$\begin{array}{c}\hfill \zeta (\alpha )=max\{\omega (\alpha ),\tau (\alpha )\},\end{array}$$

$$\begin{gathered} \begin{array}{c}\hfill \omega (\alpha )=max\left\{\underset{i\in N_1\cap \alpha \\ j\in N_2\cap \alpha }{max}\frac{|a_{jj}|-{\beta }_j+{\beta }_i}{(|a_{ii}|-{\alpha }_i)(|a_{jj}|-{\beta }_j)-{\beta }_i{\alpha }_j},\underset{\begin{array}{c}i\in N_1\cap \alpha \\ j\in N_2\cap \alpha \end{array}}{max}\frac{|a_{ii}|-{\alpha }_i+{\alpha }_j}{(|a_{ii}|-{\alpha }_i)(|a_{jj}|-{\beta }_j)-{\beta }_i{\alpha }_j}\right\},\end{array} \end{gathered}$$

$$\begin{array}{cc}\hfill \tau (\alpha )=max& \left\{\underset{\begin{array}{c}i,j\in {\alpha }^{\prime }\\ i\in N_1-\alpha ,j\in N_2-\alpha \end{array}}{max}\frac{|a_{jj}|+{\beta }_i(A)-{\displaystyle \sum _{k\in N_2-\alpha }}|a_{jk}|+{\displaystyle \sum _{k\in N_1\cap \alpha }}|a_{ik}|+{\displaystyle \sum _{k\in \alpha }}|a_{jk}|}{p_{i,j}-q_{i,j}},\right.\\ & \left.\underset{\begin{array}{c}i,j\in {\alpha }^{\prime }\\ i\in N_1-\alpha ,j\in N_2-\alpha \end{array}}{max}\frac{|a_{ii}|+{\alpha }_j(A)-{\displaystyle \sum _{k\in N_2-\alpha }}|a_{ik}|+{\displaystyle \sum _{k\in N_2\cap \alpha }}|a_{jk}|+{\displaystyle \sum _{k\in \alpha }}|a_{ik}|}{p_{i,j}-q_{i,j}}\right\},\end{array}$$

$$\begin{gathered} \begin{array}{c}\hfill p_{i,j}=(|a_{ii}|-\sum _{k\in N_1-\alpha \\ \hspace{1em}k\ne s}|a_{ik}|-|v_i{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}\sum _{k\in N_1-\alpha }|w_k|)(|a_{jj}|-\sum _{\begin{array}{c}k\in N_2-\alpha \\ k\ne t\end{array}}|a_{jk}|-|v_j{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}\sum _{k\in N_2-\alpha }|w_k|),\end{array} \end{gathered}$$

$$\begin{array}{c}\hfill q_{i,j}=(\sum _{k\in N_1-\alpha }|a_{jk}|+|v_j{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}\sum _{k\in N_1-\alpha }|w_k|)(\sum _{k\in N_2-\alpha }|a_{ik}|+|v_i{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}\sum _{k\in N_2-\alpha }|w_k|).\end{array}$$

Proof. 

Since A is $GDSD{D}_n^{N_1,N_2}$, $\alpha \subseteq N_1,\alpha \subseteq N_2,N_1\subseteq \alpha$ and $N_2\subseteq \alpha$, by Lemmas 5 and 6, we have $A_{11}$ is $GDSD{D}_{|\alpha |}^{N_1\cap \alpha ,N_2\cap \alpha }$, and $A/\alpha$ is $GDSD{D}_{|N-\alpha |}^{N_1-\alpha ,N_2-\alpha }$. Applying Theorem 3 to $A_{11}$, we get:

$$\begin{gathered} \begin{array}{c}\hfill {∥A_{11}^{-1}∥}_{\infty }\le max\left\{\underset{i\in N_1\cap \alpha \\ j\in N_2\cap \alpha }{max}\frac{|a_{jj}|-{\beta }_j+{\beta }_i}{(|a_{ii}|-{\alpha }_i)(|a_{jj}|-{\beta }_j)-{\beta }_i{\alpha }_j},\underset{\begin{array}{c}i\in N_1\cap \alpha \\ j\in N_2\cap \alpha \end{array}}{max}\frac{|a_{ii}|-{\alpha }_i+{\alpha }_j}{(|a_{ii}|-{\alpha }_i)(|a_{jj}|-{\beta }_j)-{\beta }_i{\alpha }_j}\right\}.\end{array} \end{gathered}$$

(39)

Applying Theorem 3 to $A/\alpha$ yields:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {∥{(A/\alpha )}^{-1}∥}_{\infty }\le max\left\{\underset{\begin{array}{c}s,t\in \left[m\right]\\ j_s\in N_1-\alpha ,j_t\in N_2-\alpha \end{array}}{max}\frac{|a_{tt}^{\prime }|-{\beta }_t(A/\alpha )+{\beta }_s(A/\alpha )}{(|a_{ss}^{\prime }|-{\alpha }_s(A/\alpha ))(|a_{tt}^{\prime }|-{\beta }_t(A/\alpha ))-{\alpha }_t(A/\alpha ){\beta }_s(A/\alpha )},\right.\\ \hfill \left.\underset{\begin{array}{c}s,t\in \left[m\right]\\ j_s\in N_1-\alpha ,j_t\in N_2-\alpha \end{array}}{max}\frac{|a_{ss}^{\prime }|-{\alpha }_s(A/\alpha )+{\alpha }_t(A/\alpha )}{(|a_{ss}^{\prime }|-{\alpha }_s(A/\alpha ))(|a_{tt}^{\prime }|-{\beta }_t(A/\alpha ))-{\alpha }_t(A/\alpha ){\beta }_s(A/\alpha )}\right\}.\end{array}\end{array}$$

(40)

By Theorem 2 in [14], we have:

$$\begin{gathered} \begin{array}{c}\hfill \begin{array}{cc}& (|a_{ss}^{\prime }|-{\alpha }_s(A/\alpha ))(|a_{tt}^{\prime }|-{\beta }_t(A/\alpha ))-{\alpha }_t(A/\alpha ){\beta }_s(A/\alpha )\hfill \\ & \ge (|a_{j_s{j}_s}|-\sum _{j_k\in N_1-\alpha \\ \hspace{1em}k\ne s}|a_{j_s{j}_k}|-|v_{j_s}{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}\sum _{j_k\in N_1-\alpha }|w_{j_k}|)(|a_{j_t{j}_t}|-\sum _{\begin{array}{c}{j}_k\in N_2-\alpha \\ k\ne t\end{array}}|a_{j_t{j}_k}|\hfill \\ & -|v_{j_t}{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}\sum _{\begin{array}{c}{j}_k\in N_2-\alpha \end{array}}|w_{j_k}|)-(\sum _{j_k\in N_1-\alpha }|a_{j_t{j}_k}|+|v_{j_t}{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}\sum _{j_k\in N_1-\alpha }|w_{j_k}|)(\sum _{j_k\in N_2-\alpha }|a_{j_s{j}_k}|\hfill \\ & +|v_{j_s}{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}\sum _{j_k\in N_2-\alpha }|w_{j_k}|)\hfill \\ & >0.\hfill \end{array}\end{array} \end{gathered}$$

(41)

Now, the upper bounds of $|a_{tt}^{\prime }|-{\beta }_t(A/\alpha )+{\beta }_s(A/\alpha )$ and $|a_{ss}^{\prime }|-{\alpha }_s(A/\alpha )+{\alpha }_t(A/\alpha )$ are considered.

By (19), (20), (23), and (24), we have:

$$\begin{gathered} \begin{array}{c}\hfill \begin{array}{cc}& |a_{tt}^{\prime }|-{\beta }_t(A/\alpha )+{\beta }_s(A/\alpha )\hfill \\ & =|a_{tt}^{\prime }|-\sum _{\begin{array}{c}{j}_k\in N_2-\alpha \\ k\ne t\end{array}}|a_{tk}^{\prime }|+\sum _{j_k\in N_2-\alpha }|a_{sk}^{\prime }|\hfill \\ & \le |a_{j_t{j}_t}|+|v_{j_t}{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}|w_{j_t}|-\sum _{\begin{array}{c}{j}_k\in N_2-\alpha \\ k\ne t\end{array}}(|a_{j_t{j}_k}|-|v_{j_t}{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}|w_{j_k}|)\hfill \\ & +\sum _{j_k\in N_2-\alpha }(|a_{j_s{j}_k}|+|v_{j_s}{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}|w_{j_k}|)(byinequalities(23)and(24))\hfill \\ & =|a_{j_t{j}_t}|+|v_{j_t}{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}|w_{j_t}|-\sum _{j_k\in N_2-\alpha \\ \hspace{1em}k\ne t}|a_{j_t{j}_k}|+|v_{j_t}{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}\sum _{\begin{array}{c}{j}_k\in N_2-\alpha \\ k\ne t\end{array}}|w_{j_k}|\hfill \\ & +\sum _{j_k\in N_2-\alpha }|a_{j_s{j}_k}|+|v_{j_s}{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}\sum _{j_k\in N_2-\alpha }|w_{j_k}|\hfill \\ & =|a_{j_t{j}_t}|-\sum _{j_k\in N_2-\alpha \\ \hspace{1em}k\ne t}|a_{j_t{j}_k}|+\sum _{j_k\in N_2-\alpha }|a_{j_s{j}_k}|+|v_{j_t}{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}\sum _{\begin{array}{c}{j}_k\in N_2-\alpha \end{array}}|w_{j_k}|\hfill \\ & +|v_{j_s}{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}\sum _{j_k\in N_2-\alpha }|w_{j_k}|\hfill \\ & =|a_{j_t{j}_t}|-\sum _{\begin{array}{c}{j}_k\in N_2-\alpha \\ k\ne t\end{array}}|a_{j_t{j}_k}|+\sum _{j_k\in N_2-\alpha }|a_{j_s{j}_k}|+(|v_{j_t}|+|v_{j_s}{|)\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}\sum _{j_k\in N_2-\alpha }|w_{j_k}|\hfill \\ & \le |a_{j_t{j}_t}|-\sum _{\begin{array}{c}{j}_k\in N_2-\alpha \\ k\ne t\end{array}}|a_{j_t{j}_k}|+\sum _{j_k\in N_2-\alpha }|a_{j_s{j}_k}|+(|v_{j_t}|+|v_{j_s}{|)\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}\sum _{k\in \left[m\right]}|w_{j_k}|(by(18)and{N}_2-\alpha \subset {\alpha }^{\prime })\hfill \\ & =|a_{j_t{j}_t}|-\sum _{\begin{array}{c}{j}_k\in N_2-\alpha \\ k\ne t\end{array}}|a_{j_t{j}_k}|+\sum _{j_k\in N_2-\alpha }|a_{j_s{j}_k}|+\sum _{k\in \left[l\right]}(|a_{j_t{i}_k}|+|a_{j_s{i}_k}|)x_k\hfill \\ & \le |a_{j_t{j}_t}|-\sum _{\begin{array}{c}{j}_k\in N_2-\alpha \\ k\ne t\end{array}}|a_{j_t{j}_k}|+\sum _{j_k\in N_2-\alpha }|a_{j_s{j}_k}|+\sum _{k\in \left[l\right]}(|a_{j_t{i}_k}|+|a_{j_s{i}_k}|)(byinequality(19))\hfill \\ & =|a_{j_t{j}_t}|-\sum _{\begin{array}{c}{j}_k\in N_2-\alpha \\ k\ne t\end{array}}|a_{j_t{j}_k}|+\sum _{j_k\in N_2-\alpha }|a_{j_s{j}_k}|+\sum _{k\in \left[l\right]}|a_{j_t{i}_k}|+\sum _{k\in \left[l\right]}|a_{j_s{i}_k}|.\hfill \end{array}\end{array} \end{gathered}$$

(42)

Similarly,

$$\begin{gathered} \begin{array}{c}\hfill \begin{array}{cc}& |a_{ss}^{\prime }|-{\alpha }_s(A/\alpha )+{\alpha }_t(A/\alpha )\hfill \\ & =|a_{ss}^{\prime }|-\sum _{\begin{array}{c}{j}_k\in N_1-\alpha \\ k\ne s\end{array}}|a_{sk}^{\prime }|+\sum _{j_k\in N_1-\alpha }|a_{tk}^{\prime }|\hfill \\ & \le |a_{j_s{j}_s}|+|v_{j_s}{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}|w_{j_s}|-\sum _{\begin{array}{c}{j}_k\in N_1-\alpha \\ k\ne s\end{array}}(|a_{j_s{j}_k}|-|v_{j_s}{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}|w_{j_k}|)\hfill \\ & +\sum _{j_k\in N_1-\alpha }(|a_{j_t{j}_k}|+|v_{j_t}{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}|w_{j_k}|)(byinequalities(23)and(24))\hfill \\ & =|a_{j_s{j}_s}|+|v_{j_s}{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}|w_{j_s}|-\sum _{j_k\in N_1-\alpha \\ \hspace{1em}k\ne s}|a_{j_s{j}_k}|+|v_{j_s}{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}\sum _{\begin{array}{c}{j}_k\in N_1-\alpha \\ k\ne s\end{array}}|w_{j_k}|\hfill \\ & +\sum _{j_k\in N_1-\alpha }|a_{j_t{j}_k}|+|v_{j_t}{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}\sum _{j_k\in N_1-\alpha }|w_{j_k}|\hfill \\ & =|a_{j_s{j}_s}|-\sum _{j_k\in N_1-\alpha \\ \hspace{1em}k\ne s}|a_{j_s{j}_k}|+\sum _{j_k\in N_1-\alpha }|a_{j_t{j}_k}|+|v_{j_s}{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}\sum _{\begin{array}{c}{j}_k\in N_1-\alpha \end{array}}|w_{j_k}|\hfill \\ & +|v_{j_t}{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}\sum _{j_k\in N_1-\alpha }|w_{j_k}|\hfill \\ & =|a_{j_s{j}_s}|-\sum _{\begin{array}{c}{j}_k\in N_1-\alpha \\ k\ne s\end{array}}|a_{j_s{j}_k}|+\sum _{j_k\in N_1-\alpha }|a_{j_t{j}_k}|+(|v_{j_s}|+|v_{j_t}{|)\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}\sum _{j_k\in N_1-\alpha }|w_{j_k}|\hfill \\ & \le |a_{j_s{j}_s}|-\sum _{\begin{array}{c}{j}_k\in N_1-\alpha \\ k\ne s\end{array}}|a_{j_s{j}_k}|+\sum _{j_k\in N_1-\alpha }|a_{j_t{j}_k}|+(|v_{j_s}|+|v_{j_t}{|)\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}\sum _{k\in \left[m\right]}|w_{j_k}|(by(18)and{N}_1-\alpha \subset {\alpha }^{\prime })\hfill \\ & =|a_{j_s{j}_s}|-\sum _{\begin{array}{c}{j}_k\in N_1-\alpha \\ k\ne s\end{array}}|a_{j_s{j}_k}|+\sum _{j_k\in N_1-\alpha }|a_{j_t{j}_k}|+\sum _{k\in \left[l\right]}(|a_{j_s{i}_k}|+|a_{j_t{i}_k}|)x_k\hfill \\ & \le |a_{j_s{j}_s}|-\sum _{\begin{array}{c}{j}_k\in N_1-\alpha \\ k\ne s\end{array}}|a_{j_s{j}_k}|+\sum _{j_k\in N_1-\alpha }|a_{j_t{j}_k}|+\sum _{k\in \left[l\right]}(|a_{j_s{i}_k}|+|a_{j_t{i}_k}|)(byinquality(19))\hfill \\ & =|a_{j_s{j}_s}|-\sum _{\begin{array}{c}{j}_k\in N_1-\alpha \\ k\ne s\end{array}}|a_{j_s{j}_k}|+\sum _{j_k\in N_1-\alpha }|a_{j_t{j}_k}|+\sum _{k\in \left[l\right]}|a_{j_s{i}_k}|+\sum _{k\in \left[l\right]}|a_{j_t{i}_k}|.\hfill \end{array}\end{array} \end{gathered}$$

(43)

Furthermore, by (40)–(43), we have:

$$\begin{gathered} \begin{array}{c}\hfill \begin{array}{cc}\hfill {∥{(A/\alpha )}^{-1}∥}_{\infty }\le max& \left\{\underset{ s,t\in \left[m\right] \\ {j}_s\in N_1-\alpha \\ {j}_t\in N_2-\alpha }{max}\frac{|a_{j_t{j}_t}|-{\displaystyle \sum _{j_k\in N_2-\alpha \\ \hspace{1em}k\ne t}}|a_{j_t{j}_k}|+{\displaystyle \sum _{j_k\in N_2-\alpha }}|a_{j_s{j}_k}|+{\displaystyle \sum _{k\in \left[l\right]}}|a_{j_t{i}_k}|+{\displaystyle \sum _{k\in \left[l\right]}}|a_{j_s{i}_k}|}{p_{j_s,j_t}-q_{j_s,j_t}},\right.\hfill \\ & \left.\underset{ s,t\in \left[m\right] \\ {j}_s\in N_1-\alpha \\ {j}_t\in N_2-\alpha }{max}\frac{|a_{j_s{j}_s}|-{\displaystyle \sum _{j_k\in N_1-\alpha ,k\ne s}}|a_{j_s{j}_k}|+{\displaystyle \sum _{j_k\in N_1-\alpha }}|a_{j_t{j}_k}|+{\displaystyle \sum _{k\in \left[l\right]}}|a_{j_s{i}_k}|+{\displaystyle \sum _{k\in \left[l\right]}}|a_{j_t{i}_k}|}{p_{j_s,j_t}-q_{j_s,j_t}}\right\}\hfill \\ \hfill =max& \left\{\underset{ i,j\in {\alpha }^{\prime } \\ i\in N_1-\alpha \\ j\in N_2-\alpha }{max}\frac{|a_{jj}|+{\beta }_i(A)-{\displaystyle \sum _{k\in N_2-\alpha \\ k\ne j}}|a_{jk}|+{\displaystyle \sum _{k\in N_1\cap \alpha }}|a_{ik}|+{\displaystyle \sum _{k\in \alpha }}|a_{jk}|}{p_{i,j}-q_{i,j}},\right.\hfill \end{array}\end{array} \end{gathered}$$

$$\begin{array}{c}\hfill \begin{array}{cc}& \left.\underset{\begin{array}{c}i,j\in {\alpha }^{\prime }\\ i\in N_1-\alpha \\ j\in N_2-\alpha \end{array}}{max}\frac{|a_{ii}|-{\displaystyle \sum _{k\in N_2-\alpha }}|a_{ik}|+{\alpha }_j(A)+{\displaystyle \sum _{k\in N_2\cap \alpha }}|a_{jk}|+{\displaystyle \sum _{k\in \alpha }}|a_{ik}|}{p_{i,j}-q_{i,j}}\right\},\hfill \end{array}\end{array}$$

(44)

where

$$\begin{array}{cc}\hfill p_{j_s,j_t}& =(|a_{j_s{j}_s}|-\sum _{\begin{array}{c}{j}_k\in N_1-\alpha \\ k\ne s\end{array}}|a_{j_s{j}_k}|-|v_{j_s}{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}\sum _{j_k\in N_1-\alpha }|w_{j_k}|)(|a_{j_t{j}_t}|\\ & -\sum _{\begin{array}{c}{j}_k\in N_2-\alpha \\ k\ne t\end{array}}|a_{j_t{j}_k}|-|v_{j_t}{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}\sum _{j_k\in N_2-\alpha }|w_{j_k}|),\end{array}$$

$$\begin{array}{cc}\hfill q_{j_s,j_t}& =(\sum _{j_k\in N_1-\alpha }|a_{j_t{j}_k}|+|v_{j_t}{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}\sum _{j_k\in N_1-\alpha }|w_{j_k}|)(\sum _{j_k\in N_2-\alpha }|a_{j_s{j}_k}|\\ & +|v_{j_s}{|\left\{\mu \left[A(\alpha )\right]\right\}}^{-1}\sum _{j_k\in N_2-\alpha }|w_{j_k}|).\end{array}$$

By Corollary 1, we have:

$$\begin{gathered} \begin{array}{c}\hfill {∥Q∥}_{\infty }\le max\left\{\underset{i\in N_1\cap \alpha \\ j\in N_2\cap \alpha }{max}\frac{(|a_{jj}|-{\beta }_j)R_i(A_{12})+{\beta }_i{R}_j(A_{12})}{(|a_{ii}|-{\alpha }_i)(|a_{jj}|-{\beta }_j)-{\beta }_i{\alpha }_j},\underset{i\in N_1\cap \alpha \\ j\in N_2\cap \alpha }{max}\frac{(|a_{ii}|-{\alpha }_i)R_j(A_{12})+{\alpha }_j{R}_i(A_{12})}{(|a_{ii}|-{\alpha }_i)(|a_{jj}|-{\beta }_j)-{\beta }_i{\alpha }_j}\right\}.\end{array} \end{gathered}$$

(45)

Furthermore, by (7), (9), (39), (44), and (45), the conclusion follows. □

Theorem 7.

Let $A=\left[a_{ij}\right]$ be $GDSD{D}_n^{N_1,N_2}$ with $a_{ii}\ne 0,i\in N$, and $\varnothing \ne \alpha \subset N$. Assume that (8) holds, if $N_1\subset \alpha$ or $N_2\subset \alpha$, then:

$$\begin{array}{c}\hfill {∥A^{-1}∥}_{\infty }\le \gamma (\alpha )·(1+\phi (\alpha ))·max\{\varphi (\alpha ),\underset{j\in {\alpha }^{\prime }}{max}\frac{1}{|a_{jj}|-{\displaystyle \sum _{\begin{array}{c}k\in {\alpha }^{\prime }\\ k\ne j\end{array}}}|a_{jk}|-w_j(A)}\},\end{array}$$

where

$$\begin{gathered} \begin{array}{c}\hfill \varphi (\alpha )=max\left\{\underset{i\in N_1\cap \alpha \\ j\in N_2\cap \alpha }{max}\frac{|a_{jj}|-{\beta }_j+{\beta }_i}{(|a_{ii}|-{\alpha }_i)(|a_{jj}|-{\beta }_j)-{\beta }_i{\alpha }_j},\underset{i\in N_1\cap \alpha \\ j\in N_2\cap \alpha }{max}\frac{|a_{ii}|-{\alpha }_i+{\alpha }_j}{(|a_{ii}|-{\alpha }_i)(|a_{jj}|-{\beta }_j)-{\beta }_i{\alpha }_j}\right\},\end{array} \end{gathered}$$

$$\begin{gathered} \begin{array}{c}\hfill w_j(A)=\underset{i\in \alpha }{max}\frac{{\displaystyle \sum _{k\in {\alpha }^{\prime }}}|a_{ik}|}{|a_{ii}|-{\displaystyle \sum _{k\in \alpha \\ k\ne i}}|a_{ik}|}\sum _{k\in \alpha }|a_{jk}|.\end{array} \end{gathered}$$

Proof. 

Since A is $GDSD{D}_n^{N_1,N_2}$, $N_1\subset \alpha$ or $N_2\subset \alpha$, by Lemmas 5 and 6, we have $A_{11}$ is $GDSD{D}_{|\alpha |}^{N_1\cap \alpha ,N_2\cap \alpha }$ and $A/\alpha$ is $SDD$, consequently, nonsingular. Let:

$$\begin{array}{c}\hfill A/\alpha =A_{22}-A_{21}{A}_{11}^{-1}{A}_{12}=(a_{st}^{\prime }).\end{array}$$

Then by Theorem 1 in [14], we have:

$$\begin{array}{c}\hfill |a_{tt}^{\prime }|-r_t(A/\alpha )\ge |a_{j_t{j}_t}|-\sum _{\begin{array}{c}k\in \left[m\right]\\ k\ne t\end{array}}|a_{j_t{j}_k}|-w_{j_t}(A).\end{array}$$

Applying the Varah’s bound to $A/\alpha$, we get:

$$\begin{array}{c}\hfill {∥{(A/\alpha )}^{-1}∥}_{\infty }\le \underset{j\in {\alpha }^{\prime }}{max}\frac{1}{|a_{jj}|-{\displaystyle \sum _{\begin{array}{c}k\in {\alpha }^{\prime }\\ k\ne j\end{array}}}|a_{jk}|-w_j(A)},\end{array}$$

(46)

where

$$\begin{array}{c}\hfill w_j(A)=\underset{i\in \alpha }{max}\frac{{\displaystyle \sum _{k\in {\alpha }^{\prime }}}|a_{ik}|}{|a_{ii}|-{\displaystyle \sum _{\begin{array}{c}k\in \alpha \\ k\ne i\end{array}}}|a_{ik}|}\sum _{k\in \alpha }|a_{jk}|.\end{array}$$

By Theorem 3,

$$\begin{gathered} \begin{array}{c}\hfill {∥A_{11}^{-1}∥}_{\infty }\le max\left\{\underset{i\in N_1\cap \alpha \\ j\in N_2\cap \alpha }{max}\frac{|a_{jj}|-{\beta }_j+{\beta }_i}{(|a_{ii}|-{\alpha }_i)(|a_{jj}|-{\beta }_j)-{\beta }_i{\alpha }_j},\underset{i\in N_1\cap \alpha \\ j\in N_2\cap \alpha }{max}\frac{|a_{ii}|-{\alpha }_i+{\alpha }_j}{(|a_{ii}|-{\alpha }_i)(|a_{jj}|-{\beta }_j)-{\beta }_i{\alpha }_j}\right\}.\end{array} \end{gathered}$$

(47)

From (7), (9), (46), (47), and Corollary 1, the conclusion follows. □

Theorem 8.

Let $A=\left[a_{ij}\right]$ be $GDSD{D}_n^{N_1,N_2}$ with $a_{ii}\ne 0,i\in N$, and $\varnothing \ne \alpha \subset N$. Assume that (8) holds, if $N_1=\alpha$ or $N_2=\alpha$, then:

$$\begin{array}{c}\hfill {∥A^{-1}∥}_{\infty }\le \gamma (\alpha )·\left(1+\underset{i\in \alpha }{max}\frac{R_i(A_{12})}{|a_{ii}|-r_i(A_{11})}\right)·max\left\{\underset{i\in \alpha }{max}\frac{1}{|a_{ii}|-r_i(A_{11})},\underset{j\in {\alpha }^{\prime }}{max}\frac{1}{|a_{jj}|-{\displaystyle \sum _{\begin{array}{c}k\in {\alpha }^{\prime }\\ k\ne j\end{array}}}|a_{jk}|-w_j(A)}\right\},\end{array}$$

where

$$\begin{array}{c}\hfill w_j(A)=\underset{i\in \alpha }{max}\frac{{\displaystyle \sum _{k\in {\alpha }^{\prime }}}|a_{ik}|}{|a_{ii}|-{\displaystyle \sum _{\begin{array}{c}k\in \alpha \\ k\ne i\end{array}}}|a_{ik}|}\sum _{k\in \alpha }|a_{jk}|.\end{array}$$

Proof. 

Since A is $GDSD{D}_n^{N_1,N_2}$, $N_1=\alpha$ or $N_2=\alpha$, by Lemmas 5 and 6, we have $A_{11}$ and $A/\alpha$ is $SDD$, consequently, nonsingular. Applying the Varah’s bound to $A_{11}$, we get:

$$\begin{array}{c}\hfill {∥A_{11}^{-1}∥}_{\infty }\le \underset{i\in \alpha }{max}\frac{1}{|a_{ii}|-r_i(A_{11})}.\end{array}$$

(48)

From Corollary 1, we can obtain:

$$\begin{array}{c}\hfill {∥Q∥}_{\infty }\le 1+\underset{i\in \alpha }{max}\frac{R_i(A_{12})}{|a_{ii}|-r_i(A_{11})}.\end{array}$$

(49)

Let,

$$\begin{array}{c}\hfill A/\alpha =A_{22}-A_{21}{A}_{11}^{-1}{A}_{12}=(a_{st}^{\prime }).\end{array}$$

Then by Theorem 1 in [14], we have:

$$\begin{array}{c}\hfill |a_{tt}^{\prime }|-r_t(A/\alpha )\ge |a_{j_t{j}_t}|-\sum _{\begin{array}{c}k\in \left[m\right]\\ k\ne t\end{array}}|a_{j_t{j}_k}|-w_{j_t}(A).\end{array}$$

Applying the Varah’s bound to $A/\alpha$, we get:

$$\begin{array}{c}\hfill {∥{(A/\alpha )}^{-1}∥}_{\infty }\le \underset{j\in {\alpha }^{\prime }}{max}\frac{1}{|a_{jj}|-{\displaystyle \sum _{\begin{array}{c}k\in {\alpha }^{\prime }\\ k\ne j\end{array}}}|a_{jk}|-w_j(A)},\end{array}$$

(50)

where

$$\begin{array}{c}\hfill w_j(A)=\underset{i\in \alpha }{max}\frac{{\displaystyle \sum _{k\in {\alpha }^{\prime }}}|a_{ik}|}{|a_{ii}|-{\displaystyle \sum _{\begin{array}{c}k\in \alpha \\ k\ne i\end{array}}}|a_{ik}|}\sum _{k\in \alpha }|a_{jk}|.\end{array}$$

From (7), (9), (48)–(50), the conclusion follows. □

Theorem 9.

Let $A=\left[a_{ij}\right]$ be $GDSD{D}_n^{N_1,N_2}$, and $n\ge 3$. Then:

$$\begin{array}{c}\hfill {∥A^{-1}∥}_{\infty }\le \Delta (A)=\underset{i\in N}{min}{\Delta }_i(A),\end{array}$$

where

$$\begin{array}{c}\hfill {\Delta }_i(A)=\left(1+\frac{\underset{j\in N,j\ne i}{max}|a_{ji}|}{|a_{ii}|}\right)\left(1+\frac{\underset{j\in N,j\ne i}{max}|a_{ij}|}{|a_{ii}|}\right){\tilde{\Delta }}_i(A),\end{array}$$

$$\begin{array}{c}\hfill {\tilde{\Delta }}_i(A)=max\left\{\frac{1}{|a_{ii}|},{\Delta }^{\prime }(A)\right\},\end{array}$$

$$\begin{gathered} \begin{array}{c}\hfill {\Delta }^{\prime }(A)=max\left\{\underset{j\in N_1-\left\{i\right\} \\ k\in N_2-\left\{i\right\}}{max}\frac{|c_{kk}|-{\displaystyle \sum _{p\in N_2 \\ p\ne k,i}}|c_{kp}|+{\displaystyle \sum _{p\in N_2,p\ne i}}|c_{jp}|}{(|c_{jj}|-{\displaystyle \sum _{p\in N_1 \\ p\ne j,i}}|c_{jp}|)(|c_{kk}|-{\displaystyle \sum _{p\in N_2 \\ p\ne k,i}}|c_{kp}|)-{\displaystyle \sum _{p\in N_1,p\ne i}}|c_{kp}|{\displaystyle \sum _{p\in N_2,p\ne i}}|c_{jp}|},\right.\\ \hfill \left.\underset{j\in N_1-\left\{i\right\} \\ k\in N_2-\left\{i\right\}}{max}\frac{|c_{jj}|-{\displaystyle \sum _{p\in N_1 \\ p\ne j,i}}|c_{jp}|+{\displaystyle \sum _{p\in N_1,p\ne i}}|c_{kp}|}{(|c_{jj}|-{\displaystyle \sum _{p\in N_1 \\ p\ne j,i}}|c_{jp}|)(|c_{kk}|-{\displaystyle \sum _{p\in N_2 \\ p\ne k,i}}|c_{kp}|)-{\displaystyle \sum _{p\in N_1,p\ne i}}|c_{kp}|{\displaystyle \sum _{p\in N_2,p\ne i}}|c_{jp}|}\right\},\end{array} \end{gathered}$$

and

$$\begin{array}{c}\hfill c_{jk}=a_{jk}-\frac{a_{ji}{a}_{ik}}{a_{ii}},j,k\in N-\left\{i\right\}.\end{array}$$

Proof. 

Since A is GDSDD, then for any $\varnothing \ne \alpha \subset N$, by Lemmas 5 and 6, we have $A_{11}$ is $GDSD{D}_{|\alpha |}^{N_1\cap \alpha ,N_2\cap \alpha }$, $A/\alpha$ is $GDSD{D}_{|N-\alpha |}^{N_1-\alpha ,N_2-\alpha }$, and consequently, they are nonsingular. Taking $\alpha =\left\{i\right\}$, then $A_{11}=a_{ii}$, ${\alpha }^{\prime }=N-\left\{i\right\}$ and:

$$\begin{array}{c}\hfill {\left[A(\alpha )\right]}^{-1}=\frac{1}{a_{ii}}.\end{array}$$

(51)

We have:

$$\begin{array}{c}\hfill {∥S∥}_{\infty }=1+\frac{\underset{s\in \left[m\right]}{max}|a_{j_{s}i}|}{|a_{ii}|}=1+\frac{\underset{j\in N,j\ne i}{max}|a_{ji}|}{|a_{ii}|},\end{array}$$

(52)

and

$$\begin{array}{c}\hfill {∥Q∥}_{\infty }=1+\frac{\underset{s\in \left[m\right]}{max}|a_{i{j}_s}|}{|a_{ii}|}=1+\frac{\underset{j\in N,j\ne i}{max}|a_{ij}|}{|a_{ii}|}.\end{array}$$

(53)

Let $A/\alpha =(a_{st}^{\prime })$. By (22), we have:

$$\begin{array}{c}\hfill |a_{st}^{\prime }|=|a_{j_s{j}_t}-\frac{a_{j_{s}i}{a}_{i{j}_t}}{a_{ii}}|=|c_{j_s{j}_t}|,s,t\in \left[m\right],j_s,j_t\in N-\left\{i\right\}.\end{array}$$

By calculation, we obtain for $j_s\in N_1-\left\{i\right\},j_t\in N_2-\left\{i\right\}$,

$$\begin{gathered} \begin{array}{c}\hfill {\alpha }_s(A/\alpha )=\sum _{j_p\in N_1-\left\{i\right\} \\ \hspace{1em}p\ne s}\left|a_{j_s{j}_p}-\frac{a_{j_{s}i}{a}_{i{j}_p}}{a_{ii}}\right|=\sum _{j_p\in N_1 \\ {j}_p\ne j_s,i}|c_{j_s{j}_p}|,\\ \hfill {\beta }_s(A/\alpha )=\sum _{j_p\in N_2-\left\{i\right\}}\left|a_{j_s{j}_p}-\frac{a_{j_{s}i}{a}_{i{j}_p}}{a_{ii}}\right|=\sum _{j_p\in N_2 \\ j_p\ne i}|c_{j_s{j}_p}|,\\ \hfill {\alpha }_t(A/\alpha )=\sum _{j_p\in N_1-\left\{i\right\}}\left|a_{j_t{j}_p}-\frac{a_{j_{t}i}{a}_{i{j}_p}}{a_{ii}}\right|=\sum _{j_p\in N_1 \\ j_p\ne i}|c_{j_t{j}_p}|,\end{array} \end{gathered}$$

and

$$\begin{gathered} \begin{array}{c}\hfill {\beta }_t(A/\alpha )=\sum _{j_p\in N_2-\left\{i\right\} \\ \hspace{1em}p\ne t}\left|a_{j_t{j}_p}-\frac{a_{j_{t}i}{a}_{i{j}_p}}{a_{ii}}\right|=\sum _{j_p\in N_2 \\ {j}_p\ne j_{,}i}|c_{j_t{j}_p}|.\end{array} \end{gathered}$$

By (40), we have:

$$\begin{gathered} \begin{array}{c}\hfill \begin{array}{c}\hfill {∥{(A/\alpha )}^{-1}∥}_{\infty }\le max\left\{\underset{j\in N_1-\left\{i\right\} \\ k\in N_2-\left\{i\right\}}{max}\frac{|c_{kk}|-{\displaystyle \sum _{p\in N_2 \\ p\ne k,i}}|c_{kp}|+{\displaystyle \sum _{p\in N_2,p\ne i}}|c_{jp}|}{(c_{jj}|-{\displaystyle \sum _{p\in N_1 \\ p\ne j,i}}|c_{jp}|)(|c_{kk}|-{\displaystyle \sum _{p\in N_2 \\ p\ne k,i}}|c_{kp}|)-{\displaystyle \sum _{p\in N_1,p\ne i}}|c_{kp}|{\displaystyle \sum _{p\in N_2,p\ne i}}|c_{jp}|},\right.\\ \hfill \left.\underset{j\in N_1-\left\{i\right\} \\ k\in N_2-\left\{i\right\}}{max}\frac{|c_{jj}|-{\displaystyle \sum _{p\in N_1 \\ p\ne j,i}}|c_{jp}|+{\displaystyle \sum _{p\in N_1,p\ne i}}|c_{kp}|}{(c_{jj}|-{\displaystyle \sum _{p\in N_1 \\ p\ne j,i}}|c_{jp}|)(|c_{kk}|-{\displaystyle \sum _{p\in N_2 \\ p\ne k,i}}|c_{kp}|)-{\displaystyle \sum _{p\in N_1,p\ne i}}|c_{kp}|{\displaystyle \sum _{p\in N_2,p\ne i}}|c_{jp}|}\right\}.\end{array}\end{array} \end{gathered}$$

(54)

By (7), (51)–(54), we have ${∥A^{-1}∥}_{\infty }\le {\Delta }_i(A)$. By the arbitrariness of i, the conclusion follows. □

In the following, we prove that the bound in Theorem 9 generally improves the bound obtained by Theorem 3 in [11] for $SDD$ matrices and Theorem 9 in [12] for $DSDD$ matrices.

Theorem 10.

Let $A=\left[a_{ij}\right]\in C^{n\times n}$ be $SDD$. Then,

$$\begin{array}{c}\hfill {∥A^{-1}∥}_{\infty }\le \Delta (A)\le B_{Schur}(A)=\underset{i\in N}{min}{B}_{Schur}(i),\end{array}$$

where $\Delta (A)$ is defined in Theorem 9 and:

$$\begin{array}{c}\hfill B_{Schur}(i)=2\left(1+\frac{\underset{j\in N,j\ne i}{max}|a_{ji}|}{|a_{ii}|}\right)max\left\{\frac{1}{|a_{ii}|},\underset{j\in N,j\ne i}{max}\frac{1}{|a_{jj}|-r_j(A)+\frac{|a_{ii}|-r_i(A)}{|a_{ii}|}|a_{ji}|}\right\}.\end{array}$$

Proof. 

Since A is $SDD$, A is GDSDD with $N_1=N,N_2=\varnothing$. From Theorem 9, we can obtain:

$$\begin{array}{c}\hfill {∥A^{-1}∥}_{\infty }\le \Delta (A)=\underset{i\in N}{min}{\Delta }_i(A),\end{array}$$

where

$$\begin{array}{c}\hfill {\Delta }_i(A)=\left(1+\frac{\underset{j\in N,j\ne i}{max}|a_{ji}|}{|a_{ii}|}\right)\left(1+\frac{\underset{j\in N,j\ne i}{max}|a_{ij}|}{|a_{ii}|}\right){\tilde{\Delta }}_i(A),\end{array}$$

$$\begin{array}{c}\hfill {\tilde{\Delta }}_i(A)=max\left\{\frac{1}{|a_{ii}|},{\Delta }^{\prime }(A)\right\},\end{array}$$

$$\begin{array}{c}\hfill {\Delta }^{\prime }(A)=\underset{j\in N,j\ne i}{max}\frac{1}{|c_{jj}|-{\displaystyle \sum _{p\in N,p\ne j,i}}|c_{jp}|},\end{array}$$

and

$$\begin{array}{c}\hfill c_{jk}=a_{jk}-\frac{a_{ji}{a}_{ik}}{a_{ii}},j,k\in N-\left\{i\right\}.\end{array}$$

From Theorem 3 of [11],

$$\begin{array}{c}\hfill {∥A^{-1}∥}_{\infty }\le B_{Schur}(A)=\underset{i\in N}{min}{B}_{Schur}(i),\end{array}$$

where

$$\begin{array}{c}\hfill B_{Schur}(i)=2\left(1+\frac{\underset{j\in N,j\ne i}{max}|a_{ji}|}{|a_{ii}|}\right)max\left\{\frac{1}{|a_{ii}|},\underset{j\in N,j\ne i}{max}\frac{1}{|a_{jj}|-r_j(A)+\frac{|a_{ii}|-r_i(A)}{|a_{ii}|}|a_{ji}|}\right\}.\end{array}$$

By calculation,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |c_{jj}|-\sum _{p\in N,p\ne j,i}|c_{jp}|& =|a_{jj}-\frac{a_{ji}{a}_{ij}}{a_{ii}}|-\sum _{p\in N,p\ne j,i}|a_{jp}-\frac{a_{ji}{a}_{ip}}{a_{ii}}|\hfill \\ & \ge |a_{jj}|-\frac{|a_{ji}||a_{ij}|}{|a_{ii}|}-\sum _{p\in N,j\ne j,i}(|a_{jp}|+\frac{|a_{ji}||a_{ip}|}{|a_{ii}|})\hfill \\ & =|a_{jj}|-\frac{|a_{ji}||a_{ij}|}{|a_{ii}|}-(\sum _{p\in N,j\ne j,i}|a_{jp}|+\sum _{p\in N,p\ne j,i}\frac{|a_{ji}||a_{ip}|}{|a_{ii}|})\hfill \\ & =|a_{jj}|-\sum _{p\in N,j\ne j,i}|a_{jp}|-\frac{|a_{ji}||a_{ij}|}{|a_{ii}|}-\sum _{p\in N,p\ne j,i}\frac{|a_{ji}||a_{ip}|}{|a_{ii}|}\hfill \\ & =|a_{jj}|-\sum _{p\in N,j\ne j,i}|a_{jp}|-|a_{ji}|+|a_{ji}|(1-\frac{|a_{ij}|}{|a_{ii}|}-\sum _{p\in N,p\ne j,i}\frac{|a_{ip}|}{|a_{ii}|})\hfill \\ & =|a_{jj}|-r_j(A)+|a_{ji}|(1-\frac{r_i(A)}{|a_{ii}|})\hfill \\ & =|a_{jj}|-r_j(A)+\frac{|a_{ii}|-r_i(A)}{|a_{ii}|}|a_{ji}|.\hfill \end{array}\end{array}$$

Then,

$$\begin{array}{c}\hfill \underset{j\in N,j\ne i}{max}\frac{1}{|c_{jj}|-{\displaystyle \sum _{p\in N,p\ne j,i}}|c_{jp}|}\le \underset{j\in N,j\ne i}{max}\frac{1}{|a_{jj}|-r_j(A)+\frac{|a_{ii}|-r_i(A)}{|a_{ii}|}|a_{ji}|}.\end{array}$$

Since A is $SDD$,

$$\begin{array}{c}\hfill |a_{ii}|>r_i(A)=\sum _{j\in N,j\ne i}|a_{ij}|,i\in N.\end{array}$$

For all $i\in N$, it is easy to get:

$$\begin{array}{c}\hfill \frac{{\displaystyle \sum _{j\in N,j\ne i}}|a_{ij}|}{|a_{ii}|}<1,\end{array}$$

$$\begin{array}{c}\hfill \left(1+\frac{\underset{j\in N,j\ne i}{max}|a_{ij}|}{|a_{ii}|}\right)<2.\end{array}$$

Therefore,

$$\begin{array}{c}\hfill {\Delta }_i(A)<B_{Schur}(i),i\in N.\end{array}$$

Furthermore,

$$\begin{array}{c}\hfill \Delta (A)<B_{Schur}(A).\end{array}$$

The proof is completed. □

Theorem 10 shows that the bound in Theorem 9 generally improves the bound obtained by Theorem 3 in [11] for $SDD$ matrices.

Theorem 11.

Let $A=\left[a_{ij}\right]\in C^{n\times n}$ be $DSDD$. Then:

$$\begin{array}{c}\hfill {∥A^{-1}∥}_{\infty }\le \Delta (A)\le \Pi (A)=\underset{i\in N}{min}{\pi }_i(A),\end{array}$$

where $\Delta (A)$ is defined in Theorem 9,

$$\begin{array}{c}\hfill {\pi }_i(A)=\left(1+\frac{\underset{i\in N,j\ne i}{max}|a_{ji}|}{|a_{ii}|}\right)\left(1+\frac{\underset{j\in N,j\ne i}{max}|a_{ij}|}{|a_{ii}|}\right){\tilde{\pi }}_i(A),\end{array}$$

$$\begin{array}{c}\hfill {\tilde{\pi }}_i(A)=max\left\{\frac{1}{|a_{ii}|},\underset{\begin{array}{c}j,k\in N-\left\{i\right\}\\ j\ne k\end{array}}{max}\frac{|c_{kk}|+{\displaystyle \sum _{p\in N,p\ne i,j}}|c_{jp}|}{|c_{jj}||c_{kk}|-{\displaystyle \sum _{p\in N,p\ne i,j}}|c_{jp}|{\displaystyle \sum _{p\in N,p\ne i,k}}|c_{kp}|}\right\},\end{array}$$

and

$$\begin{array}{c}\hfill c_{jk}=a_{jk}-\frac{a_{ji}{a}_{ik}}{a_{ii}},j,k\in N-\left\{i\right\}.\end{array}$$

Proof. 

Since A is $DSDD$, there is at most one $i_0$ such that $|a_{i_0{i}_0}|\le r_{i_0}(A)$. If there does not exist an $i_0$ such that $|a_{i_0{i}_0}|\le r_{i_0}(A)$, then A is $SDD$, the proof is the same as Theorem 10. If there exists only one $i_0$ such that $|a_{i_0{i}_0}|\le r_{i_0}(A)$, then A is GDSDD with $N_1=N-\left\{i_0\right\},N_2=\left\{i_0\right\}$. From Theorem 9, we can obtain:

$$\begin{array}{c}\hfill {∥A^{-1}∥}_{\infty }\le \Delta (A)=\underset{i\in N}{min}{\Delta }_i(A),\end{array}$$

where

$$\begin{array}{c}\hfill {\Delta }_i(A)=\left(1+\frac{\underset{j\in N,j\ne i}{max}|a_{ji}|}{|a_{ii}|}\right)\left(1+\frac{\underset{j\in N,j\ne i}{max}|a_{ij}|}{|a_{ii}|}\right){\tilde{\Delta }}_i(A),\end{array}$$

$$\begin{array}{c}\hfill {\tilde{\Delta }}_i(A)=max\left\{\frac{1}{|a_{ii}|},{\Delta }^{\prime }(A)\right\},\end{array}$$

$$\begin{array}{c}\hfill {\Delta }^{\prime }(A)=max\left\{\underset{\begin{array}{c}j\in N-\left\{i_0\right\}-\left\{i\right\}\\ k\in \left\{i_0\right\}-\left\{i\right\}\end{array}}{max}\frac{|c_{kk}|+{\displaystyle \sum _{p\in \left\{i_0\right\},p\ne i}}|c_{jp}|}{(|c_{jj}|-{\displaystyle \sum _{p\in N-\left\{i_0\right\},p\ne j,i}}|c_{jp}|)|c_{kk}|-{\displaystyle \sum _{p\in N-\left\{i_0\right\},p\ne i}}|c_{kp}|{\displaystyle \sum _{p\in \left\{i_0\right\},p\ne i}}|c_{jp}|},\right.\\ \hfill \left.\underset{\begin{array}{c}j\in N-\left\{i_0\right\}-\left\{i\right\}\\ k\in \left\{i_0\right\}-\left\{i\right\}\end{array}}{max}\frac{|c_{jj}|-\sum _{p\in N-\left\{i_0\right\},p\ne j,i}|c_{jp}|+{\displaystyle \sum _{p\in N-\left\{i_0\right\},p\ne i}}|c_{kp}|}{(|c_{jj}|-{\displaystyle \sum _{p\in N-\left\{i_0\right\},p\ne j,i}}|c_{jp}|)|c_{kk}|-{\displaystyle \sum _{p\in N-\left\{i_0\right\},p\ne i}}|c_{kp}|{\displaystyle \sum _{p\in \left\{i_0\right\},p\ne i}}|c_{jp}|}\right\}.\end{array}$$

From Theorem 9 of [12],

$$\begin{array}{c}\hfill {∥A^{-1}∥}_{\infty }\le \Pi (A)=\underset{i\in N}{min}{\pi }_i(A),\end{array}$$

where

$$\begin{array}{c}\hfill {\pi }_i(A)=\left(1+\frac{\underset{i\in N,j\ne i}{max}|a_{ji}|}{|a_{ii}|}\right)\left(1+\frac{\underset{j\in N,j\ne i}{max}|a_{ij}|}{|a_{ii}|}\right){\tilde{\pi }}_i(A),\end{array}$$

$$\begin{array}{c}\hfill {\tilde{\pi }}_i(A)=max\left\{\frac{1}{|a_{ii}|},\underset{\begin{array}{c}j,k\in N-\left\{i\right\}\\ j\ne k\end{array}}{max}\frac{|c_{kk}|+{\displaystyle \sum _{p\in N,p\ne i,j}}|c_{jp}|}{|c_{jj}||c_{kk}|-{\displaystyle \sum _{p\in N,p\ne i,j}}|c_{jp}|{\displaystyle \sum _{p\in N,p\ne i,k}}|c_{kp}|}\right\}.\end{array}$$

For $i\ne i_0$,

$$\begin{array}{c}\hfill {\Delta }^{\prime }(A)=max\left\{\underset{\begin{array}{c}j\in N-\left\{i_0\right\}-\left\{i\right\}\\ k=i_0\end{array}}{max}\frac{|c_{i_0{i}_0}|+|c_{j{i}_0}|}{|c_{jj}||c_{i_0{i}_0}|-|c_{i_0{i}_0}|{\displaystyle \sum _{p\in N-\left\{i_0\right\},p\ne j,i}}|c_{jp}|-|c_{j{i}_0}|{\displaystyle \sum _{p\in N-\left\{i_0\right\},p\ne i}}|c_{i_{0}p}|},\right.\\ \hfill \left.\underset{\begin{array}{c}j\in N-\left\{i_0\right\}-\left\{i\right\}\\ k=i_0\end{array}}{max}\frac{|c_{jj}|-{\displaystyle \sum _{p\in N-\left\{i_0\right\},p\ne j,i}}|c_{jp}|+{\displaystyle \sum _{p\in N-\left\{i_0\right\},p\ne i}}|c_{i_{0}p}|}{|c_{jj}||c_{i_0{i}_0}|-|c_{i_0{i}_0}|{\displaystyle \sum _{p\in N-\left\{i_0\right\},p\ne j,i}}|c_{jp}|-|c_{j{i}_0}|{\displaystyle \sum _{p\in N-\left\{i_0\right\},p\ne i}}|c_{i_{0}p}|}\right\}.\end{array}$$

Obviously,

$$\begin{array}{c}\hfill \begin{array}{cc}& \underset{\begin{array}{c}j,k\in N-\left\{i\right\}\\ j\ne k\end{array}}{max}\frac{|c_{kk}|+{\displaystyle \sum _{p\in N,p\ne i,j}}|c_{jp}|}{|c_{jj}||c_{kk}|-{\displaystyle \sum _{p\in N,p\ne i,j}}|c_{jp}|{\displaystyle \sum _{p\in N,p\ne i,k}}|c_{kp}|}\hfill \\ & \ge \underset{\begin{array}{c}j\in N-\left\{i_0\right\}-\left\{i\right\}\\ k=i_0\end{array}}{max}\frac{|c_{i_0{i}_0}|+{\displaystyle \sum _{p\in N,p\ne i,j}}|c_{jp}|}{|c_{jj}||c_{i_0{i}_0}|-{\displaystyle \sum _{p\in N,p\ne i,j}}|c_{jp}|{\displaystyle \sum _{p\in N,p\ne i,i_0}}|c_{i_{0}p}|},\hfill \end{array}\end{array}$$

$$\begin{array}{c}\hfill \begin{array}{cc}& \underset{\begin{array}{c}j,k\in N-\left\{i\right\}\\ j\ne k\end{array}}{max}\frac{|c_{kk}|+{\displaystyle \sum _{p\in N,p\ne i,j}}|c_{jp}|}{|c_{jj}||c_{kk}|-{\displaystyle \sum _{p\in N,p\ne i,j}}|c_{jp}|{\displaystyle \sum _{p\in N,p\ne i,k}}|c_{kp}|}\hfill \\ & \ge \underset{\begin{array}{c}j=i_0\\ k\in N-\left\{i_0\right\}-\left\{i\right\}\end{array}}{max}\frac{|c_{kk}|+{\displaystyle \sum _{p\in N,p\ne i,i_0}}|c_{i_{0}p}|}{|c_{i_0{i}_0}||c_{kk}|-{\displaystyle \sum _{p\in N,p\ne i,i_0}}|c_{i_{0}p}|{\displaystyle \sum _{p\in N,p\ne i,k}}|c_{kp}|}.\hfill \end{array}\end{array}$$

We can obtain:

$$\begin{array}{c}\hfill |c_{i_0{i}_0}|+|c_{j{i}_0}|\le |c_{i_0{i}_0}|+\sum _{p\in N,p\ne i,j}|c_{jp}|,\end{array}$$

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |c_{jj}||c_{i_0{i}_0}|-\sum _{p\in N,p\ne i,j}|c_{jp}|\sum _{p\in N,p\ne i,i_0}|c_{i_{0}p}|& =|c_{jj}||c_{i_0{i}_0}|-(|c_{j{i}_0}|+\sum _{p\in N,p\ne i,j,i_0}|c_{jp}|)\sum _{p\in N,p\ne i,i_0}|c_{i_{0}p}|\hfill \\ & =|c_{jj}||c_{i_0{i}_0}|-\sum _{p\in N,p\ne i,j,i_0}|c_{jp}|\sum _{p\in N,p\ne i,i_0}|c_{i_{0}p}|-|c_{j{i}_0}|\sum _{p\in N,p\ne i,i_0}|c_{i_{0}p}|\hfill \\ & \le |c_{jj}||c_{i_0{i}_0}|-|c_{i_0{i}_0}|\sum _{p\in N,p\ne i,j,i_0}|c_{jp}|-|c_{j{i}_0}|\sum _{p\in N,p\ne i,i_0}|c_{i_{0}p}|\hfill \\ & =|c_{jj}||c_{i_0{i}_0}|-|c_{i_0{i}_0}|\sum _{p\in N-\left\{i_0\right\},p\ne i,j}|c_{jp}|-|c_{j{i}_0}|\sum _{p\in N-\left\{i_0\right\},p\ne i}|c_{i_{0}p}|.\hfill \end{array}\end{array}$$

Therefore,

$$\begin{array}{c}\hfill \begin{array}{cc}& \underset{\begin{array}{c}j\in N-\left\{i_0\right\}-\left\{i\right\}\\ k=i_0\end{array}}{max}\frac{|c_{i_0{i}_0}|+|c_{j{i}_0}|}{|c_{jj}||c_{i_0{i}_0}|-|c_{i_0{i}_0}|{\displaystyle \sum _{p\in N-\left\{i_0\right\},p\ne j,i}}|c_{jp}|-|c_{j{i}_0}|{\displaystyle \sum _{p\in N-\left\{i_0\right\},p\ne i}}|c_{i_{0}p}|}\hfill \\ & \le \underset{\begin{array}{c}j\in N-\left\{i_0\right\}-\left\{i\right\}\\ k=i_0\end{array}}{max}\frac{|c_{i_0{i}_0}|+{\displaystyle \sum _{p\in N,p\ne i,j}}|c_{jp}|}{|c_{jj}||c_{i_0{i}_0}|-{\displaystyle \sum _{p\in N,p\ne i,j}}|c_{jp}|{\displaystyle \sum _{p\in N,p\ne i,i_0}}|c_{i_{0}p}|}.\hfill \end{array}\end{array}$$

Similarly,

$$\begin{array}{c}\hfill \begin{array}{cc}& \underset{\begin{array}{c}j\in N-\left\{i_0\right\}-\left\{i\right\}\\ k=i_0\end{array}}{max}\frac{|c_{jj}|-{\displaystyle \sum _{p\in N-\left\{i_0\right\},p\ne j,i}}|c_{jp}|+{\displaystyle \sum _{p\in N-\left\{i_0\right\},p\ne i}}|c_{i_{0}p}|}{(|c_{jj}|-{\displaystyle \sum _{p\in N-\left\{i_0\right\},p\ne j,i}}|c_{jp}|)|c_{i_0{i}_0}|-|c_{j{i}_0}|{\displaystyle \sum _{p\in N-\left\{i_0\right\},p\ne i}}|c_{i_{0}p}|}\hfill \\ & \le \underset{\begin{array}{c}j=i_0\\ k\in N-\left\{i_0\right\}-\left\{i\right\}\end{array}}{max}\frac{|c_{kk}|+{\displaystyle \sum _{p\in N,p\ne i,i_0}}|c_{i_{0}p}|}{|c_{i_0{i}_0}||c_{kk}|-{\displaystyle \sum _{p\in N,p\ne i,i_0}}|c_{i_{0}p}|{\displaystyle \sum _{p\in N,p\ne i,k}}|c_{kp}|}.\hfill \end{array}\end{array}$$

Then,

$$\begin{array}{c}\hfill \begin{array}{cc}& max\left\{\underset{\begin{array}{c}j\in N-\left\{i_0\right\}-\left\{i\right\}\\ k=i_0\end{array}}{max}\frac{|c_{i_0{i}_0}|+|c_{j{i}_0}|}{(|c_{jj}|-{\displaystyle \sum _{p\in N-\left\{i_0\right\},p\ne j,i}}|c_{jp}|)|c_{i_0{i}_0}|-{\displaystyle \sum _{p\in N-\left\{i_0\right\},p\ne i}}|c_{i_{0}p}||c_{j{i}_0}|},\right.\hfill \\ & \left.\underset{\begin{array}{c}j\in N-\left\{i_0\right\}-\left\{i\right\}\\ k=i_0\end{array}}{max}\frac{|c_{jj}|-{\displaystyle \sum _{p\in N-\left\{i_0\right\},p\ne j,i}}|c_{jp}|+{\displaystyle \sum _{p\in N-\left\{i_0\right\},p\ne i}}|c_{i_{0}p}|}{(|c_{jj}|-{\displaystyle \sum _{p\in N-\left\{i_0\right\},p\ne j,i}}|c_{jp}|)|c_{i_0{i}_0}|-{\displaystyle \sum _{p\in N-\left\{i_0\right\},p\ne i}}|c_{i_{0}p}||c_{j{i}_0}|}\right\}\hfill \\ & \le max\left\{\underset{\begin{array}{c}j\in N-\left\{i_0\right\}-\left\{i\right\}\\ k=i_0\end{array}}{max}\frac{|c_{i_0{i}_0}|+{\displaystyle \sum _{p\in N,p\ne i,j}}|c_{jp}|}{(|c_{jj}||c_{i_0{i}_0}|-{\displaystyle \sum _{p\in N,p\ne i,j}}|c_{jp}|{\displaystyle \sum _{p\in N,p\ne i,i_0}}|c_{i_{0}p}|},\right.\hfill \end{array}\end{array}$$

$$\begin{array}{c}\hfill \begin{array}{cc}& \left.\underset{\begin{array}{c}j=i_0\\ k\in N-\left\{i_0\right\}-\left\{i\right\}\end{array}}{max}\frac{|c_{kk}|+{\displaystyle \sum _{p\in N,p\ne i,i_0}}|c_{i_{0}p}|}{|c_{i_0{i}_0}||c_{kk}|-{\displaystyle \sum _{p\in N,p\ne i,i_0}}|c_{i_{0}p}|{\displaystyle \sum _{p\in N,p\ne i,k}}|c_{kp}|}\right\}\hfill \\ & \le \underset{\begin{array}{c}j,k\in N-\left\{i\right\}\\ j\ne k\end{array}}{max}\frac{|c_{kk}|+{\displaystyle \sum _{p\in N,p\ne i,j}}|c_{jp}|}{|c_{jj}||c_{kk}|-{\displaystyle \sum _{p\in N,p\ne i,j}}|c_{jp}|{\displaystyle \sum _{p\in N,p\ne i,k}}|c_{kp}|}.\hfill \end{array}\end{array}$$

Therefore,

$$\begin{array}{c}\hfill {\tilde{\Delta }}_i(A)\le {\tilde{\pi }}_i(A).\end{array}$$

For $i=i_0$,

$$\begin{array}{c}\hfill {\tilde{\Delta }}_{i_0}(A)=\frac{1}{|a_{i_0{i}_0}|},\end{array}$$

$$\begin{array}{c}\hfill {\tilde{\pi }}_{i_0}(A)=max\left\{\frac{1}{|a_{i_0{i}_0}|},\underset{\begin{array}{c}j,k\in N-\left\{i_0\right\}\\ j\ne k\end{array}}{max}\frac{|c_{kk}|+{\displaystyle \sum _{p\in N,p\ne i_0,j}}|c_{jp}|}{|c_{jj}||c_{kk}|-{\displaystyle \sum _{p\in N,p\ne i_0,j}}|c_{jp}|{\displaystyle \sum _{p\in N,p\ne i_0,k}}|c_{kp}|}\right\},\end{array}$$

$$\begin{array}{c}\hfill {\tilde{\Delta }}_{i_0}(A)\le {\tilde{\pi }}_{i_0}(A).\end{array}$$

Furthermore,

$$\begin{array}{c}\hfill {\tilde{\Delta }}_i(A)\le {\tilde{\pi }}_i(A),i\in N,\end{array}$$

$$\begin{array}{c}\hfill {\Delta }_i(A)\le {\Pi }_i(A),i\in N,\end{array}$$

$$\begin{array}{c}\hfill \Delta (A)\le \Pi (A).\end{array}$$

The proof is completed. □

Theorem 11 shows that the bound in Theorem 9 generally improves the bound obtained by Theorem 9 in [12] for $DSDD$ matrices.

We illustrate our results by the following two examples.

Example 1.

Consider the bound for ${∥A^{-1}∥}_{\infty }$ of a $SDD$ matrix A, where:

$$A=\left[\begin{array}{cccc}14& 4& 4& 5\\ 1& 19& 4& 5\\ 2& 3& 13& 5\\ 4& 5& 4& 14\end{array}\right].$$

By Theorem 1, we have:

$$\begin{array}{c}\hfill {∥A^{-1}∥}_{\infty }\le 1.\end{array}$$

By Theorem 2, we have:

$$\begin{array}{c}\hfill {∥A^{-1}∥}_{\infty }\le 1.\end{array}$$

By Theorem 3, we have:

$$\begin{array}{c}\hfill {∥A^{-1}∥}_{\infty }\le 1.\end{array}$$

By Theorem 3 in [11], we have:

$$\begin{array}{c}\hfill {∥A^{-1}∥}_{\infty }\le 0.8727.\end{array}$$

By Theorem 9 in [12], we have:

$$\begin{array}{c}\hfill {∥A^{-1}∥}_{\infty }\le 0.2200.\end{array}$$

By Theorem 9, we have:

$$\begin{array}{c}\hfill {∥A^{-1}∥}_{\infty }\le 0.2141.\end{array}$$

In fact, ${∥A^{-1}∥}_{\infty }=0.1464$. This example shows that our bound is better than those in Theorems 1–3 and Theorem 3 in [11], Theorem 9.

Example 2.

Consider the bound for ${∥A^{-1}∥}_{\infty }$ of a GDSDD matrix A for $N_1=\{1,2\},N_2=\{3,4\}$,

$$A=\left[\begin{array}{cccc}12& 5& 1& 1\\ 4& 13& 3& 1\\ 2& 0& 3& 2\\ 3& 2& 1& 6\end{array}\right],$$

$$A^{-1}=\left[\begin{array}{cccc}0.0971& -0.0354& 0.0073& -0.0127\\ -0.0177& 0.0812& -0.0808& 0.0163\\ -0.0408& 0.0336& 0.3521& -0.1162\\ -0.0358& -0.0150& -0.0354& 0.1869\end{array}\right].$$

Note that:

$$\begin{array}{c}\hfill {∥A^{-1}∥}_{\infty }=0.5426.\end{array}$$

We know that A is neither a $SDD$ matrix nor a $DSDD$ matrix, and the bound of ${∥A^{-1}∥}_{\infty }$ can not be obtained by Theorems 1 and 2, Theorem 3 in [11], or Theorem 9 in [12], but it can be estimated by Theorems 3 and 9 in this paper.

By Theorem 3, we have:

$$\begin{array}{c}\hfill {∥A^{-1}∥}_{\infty }\le 11.\end{array}$$

The bound for ${∥A^{-1}∥}_{\infty }$ can be estimated by Theorem 9 as follows:

$$\begin{array}{c}\hfill {∥A^{-1}∥}_{\infty }\le 1.0986.\end{array}$$

3. Applications to the Smallest Singular Value for GDSDD Matrices

The singular values of an $n\times m$ complex matrix A are the eigenvalues of $\sqrt{A{A}^{\ast }}$ and are denoted:

$$\begin{array}{c}\hfill {\sigma }_1(A)\ge {\sigma }_2(A)\ge \dots \ge {\sigma }_n(A)\ge 0.\end{array}$$

The smallest singular value plays a special role in expressing properties of matrices. It indicates not only whether A is nonsingular, but also how far from the singular matrices A is. In addition, the spectral condition number ${\sigma }_1(A)/{\sigma }_n(A)$ is important in studying numerical calculations involving A [6]. Hence, lower bounds for the smallest singular value of matrices are of interest.

For $SDD$ matrices case, Varah in [10] obtained a lower bound for ${\sigma }_n(A)$ which is listed as follows.

Theorem 12

([10]). If a matrix $A=\left[a_{ij}\right]\in C^{n\times n}$ and its transpose $A^{\mathrm{T}}$ are all $SDD$, then:

$$\begin{array}{c}\hfill {\sigma }_n(A)\ge \sqrt{\underset{i\in N}{min}\{|a_{ii}|-r_i(A)\}·\underset{i\in N}{min}\{|a_{ii}|-r_i(A^{\mathrm{T}})\}}.\end{array}$$

Besides the lower bound in Theorem 12, there are other existing lower bounds for ${\sigma }_n(A)$, see [5,6,11] and references therein. Based on Theorem 9, we can get a new lower bound for ${\sigma }_n(A)$.

Theorem 13.

If a matrix $A=\left[a_{ij}\right]\in C^{n\times n}$ and its transpose $A^{\mathrm{T}}$ are all GDSDD, then:

$$\begin{array}{c}\hfill {\sigma }_n(A)\ge \sqrt{\frac{1}{\Delta (A)·\Delta (A^{\mathrm{T}})}},\end{array}$$

where $\Delta (A)$ is defined as in Theorem 9.

Proof. 

Since $A^{\mathrm{T}}$ is GDSDD, then:

$$\begin{array}{c}\hfill {∥A^{-1}∥}_1={∥{(A^{-1})}^{\mathrm{T}}∥}_{\infty }={∥{(A^{\mathrm{T}})}^{-1}∥}_{\infty }\le \Delta (A^{\mathrm{T}}).\end{array}$$

By the well-known inequality (see Theorem 2 in [5]):

$$\begin{array}{c}\hfill {∥A^{-1}∥}_2^2\le {∥A^{-1}∥}_1{∥A^{-1}∥}_{\infty },\end{array}$$

we have:

$$\begin{array}{c}\hfill {∥A^{-1}∥}_2^2\le \Delta (A^{\mathrm{T}})·\Delta (A),\end{array}$$

and hence:

$$\begin{array}{c}\hfill {∥A^{-1}∥}_2^{-1}\ge \sqrt{\frac{1}{\Delta (A)·\Delta (A^{\mathrm{T}})}}.\end{array}$$

By ${∥A^{-1}∥}_2^{-1}={\sigma }_n(A)$, the conclusion follows. □

4. Conclusions

Based on the fact that the Schur complement of a GDSDD matrix is also a GDSDD matrix, we in Theorems 4–8 presented an upper bound of ${∥A^{-1}∥}_{\infty }$ for a GDSDD matrix A for any nonempty proper subset $\alpha$ of N. And then, for the case $|\alpha |=1$, we in Theorem 9 gave an upper bound for ${∥A^{-1}∥}_{\infty }$. Moreover, we show that the bound in Theorem 9 generally improves the bound obtained by Theorem 3 in [11] for $SDD$ matrices and Theorem 9 in [12] for $DSDD$ matrices.