The Diagonally Dominant Degree and Disc Separation for the Schur Complement of Ostrowski Matrix

Abstract

By applying the properties of Schur complement and some inequality techniques, some new estimates of diagonally and doubly diagonally dominant degree of the Schur complement of Ostrowski matrix are obtained, which improve the main results of Liu and Zhang (2005) and Liu et al. (2012). As an application, we present new inclusion regions for eigenvalues of the Schur complement of Ostrowski matrix. In addition, a new upper bound for the infinity norm on the inverse of the Schur complement of Ostrowski matrix is given. Finally, we give numerical examples to illustrate the theory results.

1. Introduction

Let ℂ^𝑛×𝑛 denote the set of all 𝑛×𝑛 complex matrices, 𝑁={1,2,…,𝑛}, and 𝐴=(𝑎_𝑖𝑗)∈ℂ^𝑛×𝑛(𝑛≥2). Denote 𝑅_𝑖(𝐴)=_𝑗≠𝑖𝑎_𝑖𝑗. (1)

We know that 𝐴 is called a strictly diagonally dominant matrix if 𝑎_𝑖𝑖>𝑅_𝑖(𝐴),∀𝑖∈𝑁. (2)

𝐴 is called a generalized Ostrowski matrix if 𝑎_𝑖𝑖𝑎_𝑗𝑗≥𝑅_𝑖(𝐴)𝑅_𝑗(𝐴),∀𝑖,𝑗∈𝑁,𝑖≠𝑗. (3)𝐴 is called Ostrowski matrix if all strict inequalities in (3) hold (see [1]).

SD_𝑛 and OS_𝑛(GOS_𝑛) will be used to denote the sets of all 𝑛×𝑛 strictly diagonally dominant matrices and the sets of all 𝑛×𝑛 (generalized) Ostrowski matrices, respectively.

As shown in [2], for all 𝑖∈𝑁, we call |𝑎_𝑖𝑖|−𝑅_𝑖(𝐴) and |𝑎_𝑖𝑖||𝑎_𝑗𝑗|−𝑅_𝑖(𝐴)𝑅_𝑗(𝐴) the 𝑖th diagonally and doubly diagonally dominant degree of 𝐴, respectively.

The infinity norm of 𝐴 is defined as ‖𝐴‖_∞=max_{1≤𝑖≤𝑛}𝑅_𝑖(𝐴)+𝑎_𝑖𝑖. (4)

For 𝛽⊆𝑁, denote by |𝛽| the cardinality of 𝛽 and 𝛽=𝑁/𝛽. If 𝛽,𝛾⊆𝑁, then 𝐴(𝛽,𝛾) is the submatrix of 𝐴 with row indices in 𝛽 and column indices in 𝛾. In particular, 𝐴(𝛽,𝛽) is abbreviated to 𝐴(𝛽). Assuming that 𝛽={𝑖₁,𝑖₂,…,𝑖_𝑘}⊂𝑁, 𝛽=𝑁/𝛽={𝑗₁,𝑗₂,…,𝑗_𝑙} and the elements of 𝛽 and 𝛽 are both conventionally arranged in an increasing order. For 1≤𝑡≤𝑙, we denote 𝐴_𝑡=𝐴𝛽∪𝑗_𝑡. (5) If 𝐴(𝛽) is nonsingular, 𝐴𝛽=𝐴𝐴𝛽=𝐴𝛽−𝐴𝛽,𝛽𝐴𝛽⁻¹𝐴𝛽,𝛽 (6) is called the Schur complement of 𝐴 with respect to 𝐴(𝛽).

The comparison matrix of 𝐴, 𝜇(𝐴)=(𝛼_𝑖𝑗), is defined by 𝛼_𝑖𝑗=𝑎_𝑖𝑗−𝑎,if𝑖=𝑗,_𝑖𝑗,if𝑖≠𝑗. (7) A matrix 𝐴=(𝑎_𝑖𝑗)∈ℂ^𝑛×𝑛 is called an 𝑀-matrix if there exist a nonnegative matrix 𝐵 and a number 𝑠>𝜌(𝐵) such that 𝐴=𝑠𝐼−𝐵, where 𝜌(𝐵) is the spectral radius of 𝐵. We know that 𝐴 is an 𝐻-matrix if and only if 𝜇(𝐴) is an 𝑀-matrix, and if 𝐴 is an 𝑀-matrix, then the Schur complement of 𝐴 is also an 𝑀-matrix and det𝐴>0 (see [3]). 𝐻_𝑛 and 𝑀_𝑛 will denote the set of all 𝑛×𝑛  𝐻-matrices and the set of all 𝑛×𝑛  𝑀-matrices, respectively.

The Schur complement has been proved to be a useful tool in many fields such as control theory, statistics, and computational mathematics. A lot of work has been done on it (see [2, 4–15]). It is well known that the Schur complements of SD_𝑛 and OS_𝑛 are SD_𝑛 and OS_𝑛, respectively. These properties have been used for the derivation of matrix inequalities in matrix analysis and for the convergence of iterations in numerical analysis (see [16–19]). Meanwhile, estimating the upper bound for the infinity norm of the inverse of the Schur complement is of great significance. We know that the upper bound of ‖𝐴⁻¹‖_∞ plays an important role in some iterations for large scale nonhomogeneous system of linear equation 𝐴𝑥=𝑏 (see [20]).

The paper is organized as follows. In Section 2, we give several new estimates of diagonally and doubly diagonally dominant degree on the Schur complement of matrices. In Section 3, new inclusion regions for eigenvalues of the Schur complement are obtained. A new upper bound of ‖(𝐴/𝛽)⁻¹‖_∞ is given in Section 4. In Section 5, we present numerical examples to illustrate the theory results.

2. The Diagonally Dominant Degree for the Schur Complement

In this section, we give several new estimates of diagonally and doubly diagonally dominant degree on the Schur complement of OS_𝑛.

Lemma 1 (see [3]). If 𝐴∈𝐻_𝑛, then [𝜇(𝐴)]⁻¹≥|𝐴⁻¹|.

Lemma 2 (see [3]). If 𝐴∈𝑆𝐷_𝑛 or 𝐴∈𝑂𝑆_𝑛, then 𝐴∈𝐻_𝑛; that is, 𝜇(𝐴)∈𝑀_𝑛.

Lemma 3 (see [6]). If 𝐴∈𝑆𝐷_𝑛 or 𝐴∈𝑂𝑆_𝑛 and 𝛽⊆𝑁, then the Schur complement of 𝐴 is in 𝑆𝐷_|𝛽| or  𝑂𝑆_|𝛽|, where 𝛽=𝑁−𝛽 is the complement of 𝛽 in 𝑁 and |𝛽| is the cardinality of 𝛽.

Lemma 4 (see [12]). Let 𝐴∈𝑆𝐷_𝑛, 𝛽={𝑖₁,𝑖₂,…,𝑖_𝑘}⊂𝑁, 𝛽={𝑗₁,𝑗₂,…,𝑗_𝑙}, and 𝑘+𝑙=𝑛. For any 𝑗_𝑡∈𝛽, denote 𝐵_𝑗𝑡≡𝑥−𝑎_{𝑗𝑡𝑖1}⋯−𝑎_{𝑗𝑡𝑖𝑘}−^𝑙_𝑣=1𝑎_{𝑖1𝑗𝑣}𝐴𝛽−⋮𝜇_𝑙𝑣=1𝑎_{𝑖𝑘𝑗𝑣},𝑥>0. (8) Then 𝐵_𝑗𝑡∈𝐺𝑂𝑆_𝑘+1 if and only if 𝑥≥max_{1≤𝑤≤𝑘}𝑅_𝑖𝑤(𝐴)𝑎_{𝑖𝑤𝑖𝑤}^𝑘_𝑣=1𝑎_{𝑗𝑡𝑖𝑣}. (9) When the strict inequality in (9) holds, 𝐵_𝑗𝑡∈𝑀_𝑘+1, and thus det𝐵_𝑗𝑡>0. If the equality in (9) occurs, then det𝐵_𝑗𝑡≥0.

Lemma 5. Let 𝐴=(𝑎_𝑖𝑗)∈𝑂𝑆_𝑛 and 𝛽={𝑖₁,𝑖₂,…,𝑖_𝑘} with an index 𝑖_𝑑(1≤𝑑≤𝑘) satisfying |𝑎_{𝑖𝑑𝑖𝑑}|≤𝑅_𝑖𝑑(𝐴), |𝑎_{𝑖𝑑𝑖𝑑}|>∑_{𝑖𝑢∈𝛽/{𝑖𝑑}}|𝑎_{𝑖𝑑𝑖𝑢}|, 𝛽={𝑗₁,𝑗₂,…,𝑗_𝑙}, 1≤𝑘<𝑛, and 𝐴/𝛽=(𝑎^′_𝑡𝑠). Then, for all 1≤𝑡≤𝑙, 𝑎^′_𝑡𝑡−𝑅_𝑡𝐴𝛽≥𝑎_{𝑗𝑡𝑗𝑡}−𝑅_𝑗𝑡(𝐴)+𝑎_{𝑖𝑑𝑖𝑑}−𝑃_𝑖𝑑(𝐴)𝑎_{𝑖𝑑𝑖𝑑}^𝑘_𝑣=1𝑎_{𝑗𝑡𝑖𝑣}≥𝑎_{𝑗𝑡𝑗𝑡}−𝑃_𝑖𝑑(𝐴)𝑎_{𝑖𝑑𝑖𝑑}𝑅_𝑗𝑡(𝐴)>0, (10) where ℎ=maxmax_{𝑖∈𝑁/{𝑖𝑑}}𝑎_𝑖𝑖𝑑𝑎_𝑖𝑖−∑_{𝑗∈𝑁/{𝑖,𝑖𝑑}}𝑎_𝑖𝑗,𝑎_{𝑖𝑑𝑖𝑑}𝑅_𝑖𝑑(𝐴),𝑃_𝑖𝑑(𝐴)=ℎ𝑅_𝑖𝑑(𝐴). (11)

Proof. From Lemmas 2 and 3, we know that 𝐴(𝛽)∈𝐻_𝑘 and 𝜇(𝐴(𝛽))∈𝑀_𝑘. Further, by Lemma 1, we have 𝜇𝐴𝛽⁻¹≥𝐴𝛽⁻¹. (12) Thus, for any 1≤𝑡≤𝑙, 𝑎^′_𝑡𝑡−𝑅_𝑡𝐴𝛽=𝑎^′_𝑡𝑡−^𝑙_{𝑠=1,≠𝑡}𝑎^′_𝑡𝑠=𝑎_{𝑗𝑡𝑗𝑡}−𝑎_{𝑗𝑡𝑖1},…,𝑎_{𝑗𝑡𝑖𝑘}𝐴𝛽⁻¹𝑎_{𝑖1𝑗𝑡}⋮𝑎_{𝑖𝑘𝑗𝑡}−^𝑙_𝑠≠𝑡𝑎_{𝑗𝑡𝑗𝑠}−𝑎_{𝑗𝑡𝑖1},…,𝑎_{𝑗𝑡𝑖𝑘}𝐴𝛽⁻¹𝑎_{𝑖1𝑗𝑠}⋮𝑎_{𝑖𝑘𝑗𝑠}≥𝑎_{𝑗𝑡𝑗𝑡}−𝑅_𝑗𝑡(𝐴)+𝑎_{𝑖𝑑𝑖𝑑}−𝑃_𝑖𝑑(𝐴)𝑎_{𝑖𝑑𝑖𝑑}^𝑘_𝑣=1𝑎_{𝑗𝑡𝑖𝑣}+𝑃_𝑖𝑑(𝐴)𝑎_{𝑖𝑑𝑖𝑑}^𝑘_𝑣=1𝑎_{𝑗𝑡𝑖𝑣}−^𝑙_𝑠=1𝑎_{𝑗𝑡𝑖1},…,𝑎_{𝑗𝑡𝑖𝑘}𝜇𝐴𝛽⁻¹𝑎_{𝑖1𝑗𝑠}⋮𝑎_{𝑖𝑘𝑗𝑠}. (13) Further, 𝑎^′_𝑡𝑡−𝑅_𝑡𝐴𝛽≥𝑎_{𝑗𝑡𝑗𝑡}−𝑅_𝑗𝑡(𝐴)+𝑎_{𝑖𝑑𝑖𝑑}−𝑃_𝑖𝑑(𝐴)𝑎_{𝑖𝑑𝑖𝑑}^𝑘_𝑣=1𝑎_{𝑗𝑡𝑖𝑣}+1𝜇𝐴𝛽𝑃det×det^𝑖𝑑(𝐴)𝑎_{𝑖𝑑𝑖𝑑}^𝑘_𝑣=1𝑎_{𝑗𝑡𝑖𝑣}−𝑎_{𝑗𝑡𝑖1}⋯−𝑎_{𝑗𝑡𝑖𝑘}−^𝑙_𝑠=1𝑎_{𝑖1𝑗𝑠}𝐴𝛽−⋮𝜇_{𝑙𝑠=1𝑖𝑘𝑗𝑠}𝑎_def.=𝑎_{𝑗𝑡𝑗𝑡}−𝑅_𝑗𝑡(𝐴)+𝑎_{𝑖𝑑𝑖𝑑}−𝑃_𝑖𝑑(𝐴)𝑎_{𝑖𝑑𝑖𝑑}^𝑘_𝑣=1𝑎_{𝑗𝑡𝑖𝑣}+1det𝜇𝐴𝛽det𝐵. (14) By Lemma 4, we can prove that det𝐵≥0. Thus, inequality (10) holds.

Remark 6. Note that 𝑃_𝑖𝑑(𝐴)𝑎_{𝑖𝑑𝑖𝑑}≤𝑅_𝑖𝑑(𝐴)𝑎_{𝑖𝑑𝑖𝑑}. (15) This shows that Lemma 5 improves Theorem 2 of [12].

Theorem 7. Let 𝐴=(𝑎_𝑖𝑗)∈𝑂𝑆_𝑛, 𝛽={𝑖₁,𝑖₂,…,𝑖_𝑘}⊂𝑁, 𝛽=𝑁/𝛽={𝑗₁,𝑗₂,…,𝑗_𝑙}, 1≤𝑘<𝑛, and 𝐴/𝛽=(𝑎^′_𝑡𝑠).(a) If there exists an 𝑖_𝑑∈𝛽(1≤𝑑≤𝑘) such that |𝑎_{𝑖𝑑𝑖𝑑}|≤𝑅_𝑖𝑑(𝐴), then, for all 1≤𝑠,  𝑡≤𝑙,  𝑡≠𝑠, 𝑎^′_𝑡𝑡𝑎^′_𝑠𝑠−𝑅_𝑡𝐴𝛽𝑅_𝑠𝐴𝛽≥𝑎_{𝑗𝑡𝑗𝑡}−max_{𝑗𝑢∈𝑁/𝑡}𝑃_𝑢(𝐴)𝑎_𝑢𝑢𝑅_𝑗𝑡(𝐴×𝑎_{𝑗𝑠𝑗𝑠})−max_{𝑖𝑣∈𝛽}𝑃_𝑖𝑣(𝐴)𝑎_{𝑖𝑣𝑖𝑣}𝑅_𝑗𝑠(𝐴,𝑎_′𝑡𝑡)𝑎^′_𝑠𝑠+𝑅_𝑡𝐴𝛽𝑅_𝑠𝐴𝛽≤𝑎_{𝑗𝑡𝑗𝑡}+max_{𝑗𝑢∈𝑁/𝑡}𝑃_𝑢(𝐴)𝑎_𝑢𝑢𝑅_𝑗𝑡(𝐴×𝑎_{𝑗𝑠𝑗𝑠})+max_{𝑖𝑣∈𝛽}𝑃_𝑖𝑣(𝐴)𝑎_{𝑖𝑣𝑖𝑣}𝑅_𝑗𝑠(𝐴), (16)(17) where 𝑃_𝑖(𝐴)=ℎ_{𝑗∈𝑁/{𝑖,𝑖𝑑}}𝑎_𝑖𝑗+𝑎_𝑖𝑖𝑑𝑖≠𝑖_𝑑, (18)  and 𝑃_𝑖𝑑(𝐴) and ℎ are such as in Lemma 5.(b) If |𝑎_{𝑖𝑑𝑖𝑑}|>𝑅_𝑖𝑑(𝐴) for any 𝑖_𝑑∈𝛽(1≤𝑑≤𝑘), then, for all 1≤𝑠,  𝑡≤𝑙,  𝑡≠𝑠, 𝑎^′_𝑡𝑡𝑎^′_𝑠𝑠−𝑅_𝑡𝐴𝛽𝑅_𝑠𝐴𝛽≥𝑎_{𝑗𝑡𝑗𝑡}−max_{𝑗𝑢∈𝑁/𝑡}𝑅_𝑢(𝐴)𝑎_𝑢𝑢𝑅_𝑗𝑡(𝐴×𝑎_{𝑗𝑠𝑗𝑠})−max_{𝑖𝑣∈𝛽}𝑄_𝑖𝑣(𝐴)𝑎_{𝑖𝑣𝑖𝑣}𝑅_𝑗𝑠(𝐴,𝑎_′𝑡𝑡)𝑎^′_𝑠𝑠+𝑅_𝑡𝐴𝛽𝑅_𝑠𝐴𝛽≤𝑎_{𝑗𝑡𝑗𝑡}+max_{𝑗𝑢∈𝑁/𝑡}𝑅_𝑢(𝐴)𝑎_𝑢𝑢𝑅_𝑗𝑡(𝐴×𝑎_{𝑗𝑠𝑗𝑠})+max_{𝑖𝑣∈𝛽}𝑄_𝑖𝑣(𝐴)𝑎_{𝑖𝑣𝑖𝑣}𝑅_𝑗𝑠(𝐴), (19)(20) where 𝜂=maxmax_{1≤𝜔≤𝑘}∑^𝑙_𝑣=1𝑎_{𝑖𝜔𝑗𝑣}𝑎_{𝑖𝜔𝑖𝜔}−∑^𝑘_𝑡≠𝜔𝑎_{𝑖𝜔𝑖𝑡},max_{1≤𝜔≤𝑘1≤𝑣≤𝑙}𝑎_{𝑖𝜔𝑗𝑣}∑^𝑙_𝑣=1𝑎_{𝑖𝜔𝑗𝑣},𝑄_𝑖𝜔(𝐴)=𝜂^𝑘_𝑡≠𝜔𝑎_{𝑖𝜔𝑖𝑡}+^𝑙_𝑣=1𝑎_{𝑖𝜔𝑗𝑣},1≤𝜔≤𝑘, (21)  and if there exists some 1≤𝜔≤𝑘 such that ∑^𝑙_𝑣=1|𝑎_{𝑖𝜔𝑗𝑣}|=0, one denotes 𝜂=1.

Proof. (a) If there exists an 𝑖_𝑑∈𝛽 such that |𝑎_{𝑖𝑑𝑖𝑑}|≤𝑅_𝑖𝑑(𝐴), then, for all 𝑗_𝑡∈𝛽, max_{𝑗𝑢∈𝑁/𝑡}𝑃_𝑢(𝐴)𝑎_𝑢𝑢=max_{𝑖𝑣∈𝛽}𝑃_𝑖𝑣(𝐴)𝑎_{𝑖𝑣𝑖𝑣}=𝑃_𝑖𝑑(𝐴)𝑎_{𝑖𝑑𝑖𝑑}. (22) By Lemma 5, for all 1≤𝑡≤𝑙, 𝑎^′_𝑡𝑡−𝑅_𝑡𝐴𝛽≥𝑎_{𝑗𝑡𝑗𝑡}−𝑃_𝑖𝑑(𝐴)𝑎_{𝑖𝑑𝑖𝑑}𝑅_𝑗𝑡(𝐴)>0. (23) Thus, for all 1≤𝑡,  𝑠≤𝑙,  𝑡≠𝑠, 𝑎^′_𝑡𝑡−𝑅_𝑡𝐴𝛽𝑎^′_𝑠𝑠−𝑅_𝑠𝐴𝛽>0. (24) From Lemma 3, 𝐴/𝛽 is in OS_|𝛽|; that is, for all 1≤𝑡,  𝑠≤𝑙,  𝑡≠𝑠, 𝑎^′_𝑡𝑡𝑎^′_𝑠𝑠−𝑅_𝑡𝐴𝛽𝑅_𝑠𝐴𝛽>0. (25) Further, for all 1≤𝑡,  𝑠≤𝑙,  𝑡≠𝑠, 𝑎^′_𝑡𝑡𝑎^′_𝑠𝑠−𝑅_𝑡𝐴𝛽𝑅_𝑠𝐴𝛽≥𝑎^′_𝑡𝑡−𝑅_𝑡𝐴𝛽×𝑎^′_𝑠𝑠−𝑅_𝑠𝐴𝛽≥𝑎_{𝑗𝑡𝑗𝑡}−max_{𝑗𝑢∈𝑁/𝑡}𝑃_𝑢(𝐴)𝑎_𝑢𝑢𝑅_𝑗𝑡(𝐴×𝑎_{𝑗𝑠𝑗𝑠})−max_{𝑖𝑣∈𝛽}𝑃_𝑖𝑣(𝐴)𝑎_{𝑖𝑣𝑖𝑣}𝑅_𝑗𝑠(𝐴). (26) Therefore, inequality (16) holds. Similarly, we can prove inequality (17).
(b) If |𝑎_{𝑖𝑑𝑖𝑑}|>𝑅_𝑖𝑑(𝐴) for any 𝑖_𝑑∈𝛽(1≤𝑑≤𝑘), then, from Lemmas 1 and 2, for all 1≤𝑡,  𝑠≤𝑙,  𝑡≠𝑠, 𝑎^′_𝑡𝑡𝑎^′_𝑠𝑠−𝑅_𝑡𝐴𝛽𝑅_𝑠𝐴𝛽=𝑎_{𝑗𝑠𝑗𝑠}−𝑎_{𝑗𝑠𝑖1},…,𝑎_{𝑗𝑠𝑖𝑘}𝐴𝛽⁻¹𝑎_{𝑖1𝑗𝑠}⋮𝑎_{𝑖𝑘𝑗𝑠}×𝑎_{𝑗𝑡𝑗𝑡}−𝑎_{𝑗𝑡𝑖1},…,𝑎_{𝑗𝑡𝑖𝑘}𝐴𝛽⁻¹𝑎_{𝑖1𝑗𝑡}⋮𝑎_{𝑖𝑘𝑗𝑡}−^𝑙_𝑣≠𝑠𝑎_{𝑗𝑠𝑗𝑣}−𝑎_{𝑗𝑠𝑖1},…,𝑎_{𝑗𝑠𝑖𝑘}×𝐴𝛽⁻¹𝑎_{𝑖1𝑗𝑣}⋮𝑎_{𝑖𝑘𝑗𝑣}×^𝑙_𝑢≠𝑡𝑎_{𝑗𝑡𝑗𝑢}−𝑎_{𝑗𝑡𝑖1},…,𝑎_{𝑗𝑡𝑖𝑘}×𝐴𝛽⁻¹𝑎_{𝑖1𝑗𝑢}⋮𝑎_{𝑖𝑘𝑗𝑢}≥𝑎_{𝑗𝑠𝑗𝑠}−𝑎_{𝑗𝑠𝑖1},…,𝑎_{𝑗𝑠𝑖𝑘}×𝜇𝐴𝛽⁻¹𝑎_{𝑖1𝑗𝑠}⋮𝑎_{𝑖𝑘𝑗𝑠}×𝑎_{𝑗𝑡𝑗𝑡}−𝑎_{𝑗𝑡𝑖1},…,𝑎_{𝑗𝑡𝑖𝑘}×𝜇𝐴𝛽⁻¹𝑎_{𝑖1𝑗𝑡}⋮𝑎_{𝑖𝑘𝑗𝑡}−^𝑙_𝑣≠𝑠𝑎_{𝑗𝑠𝑗𝑣}+𝑎_{𝑗𝑠𝑖1},…,𝑎_{𝑗𝑠𝑖𝑘}×𝜇𝐴𝛽⁻¹𝑎_{𝑖1𝑗𝑣}⋮𝑎_{𝑖𝑘𝑗𝑣}×^𝑙_𝑢≠𝑡𝑎_{𝑗𝑡𝑗𝑢}+𝑎_{𝑗𝑡𝑖1},…,𝑎_{𝑗𝑡𝑖𝑘}×𝜇𝐴𝛽⁻¹𝑎_{𝑖1𝑗𝑢}⋮_{𝑖𝑘𝑗𝑢}𝑎_def.=𝜉=𝑎_{𝑗𝑠𝑗𝑠}−𝑎_{𝑗𝑠𝑖1},…,𝑎_{𝑗𝑠𝑖𝑘}×𝜇𝐴𝛽⁻¹𝑎_{𝑖1𝑗𝑠}⋮𝑎_{𝑖𝑘𝑗𝑠}×𝑎_{𝑗𝑡𝑗𝑡}−max_{𝑗𝑢∈𝑁/𝑡}𝑅_𝑢(𝐴)𝑎_𝑢𝑢𝑅_𝑗𝑡(𝐴)+max_{𝑗𝑢∈𝑁/𝑡}𝑅_𝑢(𝐴)𝑎_𝑢𝑢𝑅_𝑗𝑡(𝐴)−𝑎_{𝑗𝑡𝑖1},…,𝑎_{𝑗𝑡𝑖𝑘}𝜇𝐴𝛽⁻¹𝑎_{𝑖1𝑗𝑡}⋮𝑎_{𝑖𝑘𝑗𝑡}−^𝑙_𝑣≠𝑠𝑎_{𝑗𝑠𝑗𝑣}−𝑎_{𝑗𝑠𝑖1},…,𝑎_{𝑗𝑠𝑖𝑘}×𝜇𝐴𝛽⁻¹𝑎_{𝑖1𝑗𝑣}⋮𝑎_{𝑖𝑘𝑗𝑣}×^𝑙_𝑢≠𝑡𝑎_{𝑗𝑡𝑗𝑢}−𝑎_{𝑗𝑡𝑖1},…,𝑎_{𝑗𝑡𝑖𝑘}×𝜇𝐴𝛽⁻¹𝑎_{𝑖1𝑗𝑢}⋮𝑎_{𝑖𝑘𝑗𝑢}. (27)
Therefore, 𝜉=𝑎_{𝑗𝑠𝑗𝑠}−𝑎_{𝑗𝑠𝑖1},…,𝑎_{𝑗𝑠𝑖𝑘}×𝜇𝐴𝛽⁻¹𝑎_{𝑖1𝑗𝑠}⋮𝑎_{𝑖𝑘𝑗𝑠}×𝑎_{𝑗𝑡𝑗𝑡}−max_{𝑗𝑢∈𝑁/𝑡}𝑅_𝑢(𝐴)𝑎_𝑢𝑢𝑅_𝑗𝑡(𝐴+𝑎_{𝑗𝑠𝑗𝑠})−𝑎_{𝑗𝑠𝑖1},…,𝑎_{𝑗𝑠𝑖𝑘}×𝜇𝐴𝛽⁻¹𝑎_{𝑖1𝑗𝑠}⋮𝑎_{𝑖𝑘𝑗𝑠}×max_{𝑗𝑢∈𝑁/𝑡}𝑅_𝑢(𝐴)𝑎_𝑢𝑢𝑅_𝑗𝑡(𝐴)−𝑎_{𝑗𝑡𝑖1},…,𝑎_{𝑗𝑡𝑖𝑘}𝜇𝐴𝛽⁻¹𝑎_{𝑖1𝑗𝑡}⋮𝑎_{𝑖𝑘𝑗𝑡}−^𝑙_𝑣≠𝑠𝑎_{𝑗𝑠𝑗𝑣}−𝑎_{𝑗𝑠𝑖1},…,𝑎_{𝑗𝑠𝑖𝑘}×𝜇𝐴𝛽⁻¹𝑎_{𝑖1𝑗𝑣}⋮𝑎_{𝑖𝑘𝑗𝑣}×^𝑙_𝑢≠𝑡𝑎_{𝑗𝑡𝑗𝑢}−𝑎_{𝑗𝑡𝑖1},…,𝑎_{𝑗𝑡𝑖𝑘}×𝜇𝐴𝛽⁻¹𝑎_{𝑖1𝑗𝑢}⋮𝑎_{𝑖𝑘𝑗𝑢}. (28) Further,𝑎_{𝑗𝑠𝑗𝑠}−𝑎_{𝑗𝑠𝑖1},…,𝑎_{𝑗𝑡𝑖𝑘}𝜇𝐴𝛽⁻¹𝑎_{𝑖1𝑗𝑠}⋮𝑎_{𝑖𝑘𝑗𝑠}=𝑎_{𝑗𝑠𝑗𝑠}−max_{𝑖𝑣∈𝛽}𝑄_𝑖𝑣(𝐴)𝑎_{𝑖𝑣𝑖𝑣}^𝑘_𝑣=1𝑎_{𝑗𝑠𝑖𝑣}+max_{𝑖𝑣∈𝛽}𝑄_𝑖𝑣(𝐴)𝑎_{𝑖𝑣𝑖𝑣}^𝑘_𝑣=1𝑎_{𝑗𝑠𝑖𝑣}−𝜂𝑎_{𝑗𝑠𝑖1},…,𝜂𝑎_{𝑗𝑠𝑖𝑘}𝜂𝜇𝐴𝛽⁻¹𝑎_{𝑖1𝑗𝑠}⋮𝑎_{𝑖𝑘𝑗𝑠}=𝑎_{𝑗𝑠𝑗𝑠}−max_{𝑖𝑣∈𝛽}𝑄_𝑖𝑣(𝐴)𝑎_{𝑖𝑣𝑖𝑣}^𝑘_𝑣=1𝑎_{𝑗𝑠𝑖𝑣}+1det𝜂𝜇𝐴𝛽×detmax_{𝑖𝑣∈𝛽}𝑄_𝑖𝑣(𝐴)𝑎_{𝑖𝑣𝑖𝑣}^𝑘_𝑣=1𝑎_{𝑗𝑠𝑖𝑣}−𝜂𝑎_{𝑗𝑠𝑖1}⋯−𝜂𝑎_{𝑗𝑠𝑖𝑘}−𝑎_{𝑖1𝑗𝑠}𝐴𝛽−𝑎⋮𝜂𝜇_{𝑖𝑘𝑗𝑠def.}=𝑎_{𝑗𝑠𝑗𝑠}−max_{𝑖𝑣∈𝛽}𝑄_𝑖𝑣(𝐴)𝑎_{𝑖𝑣𝑖𝑣}^𝑘_𝑣=1𝑎_{𝑗𝑠𝑖𝑣}+1det𝜂𝜇𝐴𝛽det𝐵₁. (29)
In 𝐵₁, for all 𝑝=1,2,3,…,𝑘, 𝜂𝑎_{𝑖𝑝𝑖𝑝}max_{𝑖𝑣∈𝛽}𝑄_𝑖𝑣(𝐴)𝑎_{𝑖𝑣𝑖𝑣}^𝑘_𝑣=1𝑎_{𝑗𝑠𝑖𝑣}≥𝜂𝑎_{𝑖𝑝𝑖𝑝}𝑄_𝑖𝑝(𝐴)𝑎_{𝑖𝑝𝑖𝑝}^𝑘_𝑣=1𝑎_{𝑗𝑠𝑖𝑣}=𝜂𝑄_𝑖𝑝(𝐴)^𝑘_𝑣=1𝑎_{𝑗𝑠𝑖𝑣}=𝜂𝜂^𝑘_𝑣≠𝑝𝑎_{𝑖𝑝𝑖𝑣}+^𝑙_𝑣=1𝑎_{𝑖𝑝𝑗𝑣}^𝑘_𝑣=1𝑎_{𝑗𝑠𝑖𝑣}≥𝜂^𝑘_𝑣≠𝑝𝑎_{𝑖𝑝𝑖𝑣}+𝑎_{𝑖𝑝𝑗𝑠}^𝑘_𝑣=1𝜂𝑎_{𝑗𝑠𝑖𝑣}. (30) And for all 𝑝,𝑞=1,2,3,…,𝑘, 𝑝≠𝑞, 𝜂𝑎_{𝑖𝑝𝑖𝑝}𝜂𝑎_{𝑖𝑞𝑖𝑞}>𝜂𝑅_𝑖𝑝(𝐴)𝜂𝑅_𝑖𝑞(𝐴)=𝜂^𝑘_𝑣≠𝑝𝑎_{𝑖𝑝𝑖𝑣}+𝜂^𝑙_𝑣=1𝑎_{𝑖𝑝𝑗𝑣}𝜂^𝑘_𝑣≠𝑞𝑎_{𝑖𝑞𝑖𝑣}+𝜂^𝑙_𝑣=1𝑎_{𝑖𝑞𝑗𝑣}≥𝜂^𝑘_𝑣≠𝑝𝑎_{𝑖𝑝𝑖𝑣}+𝑎_{𝑖𝑝𝑗𝑠}𝜂^𝑘_𝑣≠𝑞𝑎_{𝑖𝑞𝑖𝑣}+𝑎_{𝑖𝑞𝑗𝑠}. (31) Hence, by (30) and (31), we have 𝐵₁∈GOS_𝑘+1 and so det𝐵₁≥0. Further, by (29), we obtain 𝑎_{𝑗𝑠𝑗𝑠}−𝑎_{𝑗𝑠𝑖1},…,𝑎_{𝑗𝑡𝑖𝑘}𝜇𝐴𝛽⁻¹𝑎_{𝑖1𝑗𝑠}⋮𝑎_{𝑖𝑘𝑗𝑠}≥𝑎_{𝑗𝑠𝑗𝑠}−max_{𝑖𝑣∈𝛽}𝑄_𝑖𝑣(𝐴)𝑎_{𝑖𝑣𝑖𝑣}^𝑘_𝑣=1𝑎_{𝑗𝑠𝑖𝑣}≥𝑎_{𝑗𝑠𝑗𝑠}−max_{𝑖𝑣∈𝛽}𝑄_𝑖𝑣(𝐴)𝑎_{𝑖𝑣𝑖𝑣}𝑅_𝑗𝑠(𝐴). (32)
By (28) and a similar method as the proof of Theorem 2.1 in [2], we can prove 𝜉>0. Therefore, by (29) and (32), we obtain inequality (19). Similarly, we can prove inequality (20).

Remark 8. Note that 0≤ℎ,𝜂≤1. (33) This shows that Theorem 7 improves Theorem 2.1 of [2].

3. Eigenvalue Inclusion Regions of the Schur Complement

In this section, we present new inclusion regions for eigenvalues of the Schur complement of OS_𝑛.

Lemma 9 (Brauer Ovals theorem). Let 𝐴=(𝑎_𝑖𝑗)∈ℂ^𝑛×𝑛. Then the eigenvalues of 𝐴 are in the union of the following sets: 𝑈_𝑖𝑗=𝑧∈𝐶𝑧−𝑎_𝑖𝑖𝑧−𝑎_𝑗𝑗∣≤𝑅_𝑖(𝐴)𝑅_𝑗(𝐴),∀𝑖,𝑗=𝑁,𝑖≠𝑗. (34)

Theorem 10. Let 𝐴=(𝑎_𝑖𝑗)∈𝑂𝑆_𝑛, 𝛽={𝑖₁,𝑖₂,…,𝑖_𝑘}⊂𝑁, 𝛽=𝑁/𝛽={𝑗₁,𝑗₂,…,𝑗_𝑙}, 1≤𝑘<𝑛, and 𝐴/𝛽=(𝑎^′_𝑡𝑠), and let 𝜆 be eigenvalue of 𝐴/𝛽.(a) If there exists an 𝑖_𝑑∈𝛽(1≤𝑑≤𝑘) such that |𝑎_{𝑖𝑑𝑖𝑑}|≤𝑅_𝑖𝑑(𝐴), then there exist 1≤𝑡,  𝑠≤𝑙, 𝑡≠𝑠, such that 𝐴𝜆−det_𝑡𝛽𝐴det𝐴𝜆−det_𝑠𝛽𝑎det𝐴≤2_{𝑗𝑠𝑗𝑠}max_{𝑢∈𝑁/{𝑗𝑡}}𝑃_𝑢(𝐴)𝑎_𝑢𝑢𝑅_𝑗𝑡(𝐴)+𝑎_{𝑗𝑡𝑗𝑡}max_{𝑖𝑣∈𝛽}𝑃_𝑖𝑣(𝐴)𝑎_{𝑖𝑣𝑖𝑣}𝑅_𝑗𝑠(𝐴,𝐴)𝜆−det_𝑡𝛽𝐴det𝐴𝜆−det_𝑠𝛽≤𝑎det𝐴_{𝑗𝑡𝑗𝑡}+max_{𝑗𝑢∈𝑁/𝑡}𝑃_𝑢(𝐴)𝑎_𝑢𝑢𝑅_𝑗𝑡(𝐴×𝑎_{𝑗𝑠𝑗𝑠})+max_{𝑖𝑣∈𝛽}𝑃_𝑖𝑣(𝐴)𝑎_{𝑖𝑣𝑖𝑣}𝑅_𝑗𝑠(𝐴), (35)(36) where 𝑃_𝑖𝑑(𝐴) is such as in Lemma 5 and 𝑃_𝑖𝑣(𝐴)(𝑣≠𝑑) is such as in Theorem 7.(b) If |𝑎_{𝑖𝑑𝑖𝑑}|>𝑅_𝑖𝑑(𝐴) for any 𝑖_𝑑∈𝛽(1≤𝑑≤𝑘), then there exist 1≤𝑡,  𝑠≤𝑙, 𝑡≠𝑠, such that 𝐴𝜆−det_𝑡𝛽𝐴det𝐴𝜆−det_𝑠𝛽𝑎det𝐴≤2_{𝑗𝑠𝑗𝑠}max_{𝑗𝑢∈𝑁/𝑡}𝑅_𝑢(𝐴)𝑎_𝑢𝑢𝑅_𝑗𝑡(𝐴)+𝑎_{𝑗𝑡𝑗𝑡}max_{𝑖𝑣∈𝛽}𝑄_𝑖𝑣(𝐴)𝑎_{𝑖𝑣𝑖𝑣}𝑅_𝑗𝑠(𝐴,𝐴)𝜆−det_𝑡𝛽𝐴det𝐴𝜆−det_𝑠𝛽≤𝑎det𝐴_{𝑗𝑡𝑗𝑡}+max_{𝑗𝑢∈𝑁/𝑡}𝑅_𝑢(𝐴)𝑎_𝑢𝑢𝑅_𝑗𝑡(𝐴×𝑎_{𝑗𝑠𝑗𝑠})+max_{𝑖𝑣∈𝛽}𝑄_𝑖𝑣(𝐴)𝑎_{𝑖𝑣𝑖𝑣}𝑅_𝑗𝑠(𝐴), (37)(38) where 𝑄_𝑖𝑣(𝐴) is such as in Theorem 7.

Proof. By Lemma 9, we know that there exist 1≤𝑡,  𝑠≤𝑙, 𝑡≠𝑠, such that 𝜆−𝑎^′_𝑡𝑡𝜆−𝑎^′_𝑠𝑠≤𝑅_𝑡𝐴𝛽𝑅_𝑠𝐴𝛽. (39)(a) If there exists 𝑖_𝑑∈𝛽 satisfying |𝑎_{𝑖𝑑𝑖𝑑}|≤𝑅_𝑖𝑑(𝐴), by (16), we have 𝑅_𝑡𝐴𝛽𝑅_𝑠𝐴𝛽≤𝑎^′_𝑡𝑡𝑎^′_𝑠𝑠−𝑎_{𝑗𝑡𝑗𝑡}−max_{𝑗𝑢∈𝑁/𝑡}𝑃_𝑢(𝐴)𝑎_𝑢𝑢𝑅_𝑗𝑡(𝐴×𝑎_{𝑗𝑠𝑗𝑠})−max_{𝑖𝑣∈𝛽}𝑃_𝑖𝑣(𝐴)𝑎_{𝑖𝑣𝑖𝑣}𝑅_𝑗𝑠(𝐴=𝑎_{𝑗𝑠𝑗𝑠})−𝑎_{𝑗𝑠𝑖1},…,𝑎_{𝑗𝑠𝑖𝑘}𝐴𝛽⁻¹𝑎_{𝑖1𝑗𝑠}⋮𝑎_{𝑖𝑘𝑗𝑠}×𝑎_{𝑗𝑡𝑗𝑡}−𝑎_{𝑗𝑡𝑖1},…,𝑎_{𝑗𝑡𝑖𝑘}𝐴𝛽⁻¹𝑎_{𝑖1𝑗𝑡}⋮𝑎_{𝑖𝑘𝑗𝑡}−𝑎_{𝑗𝑡𝑗𝑡}−max_{𝑗𝑢∈𝑁/𝑡}𝑃_𝑢(𝐴)𝑎_𝑢𝑢𝑅_𝑗𝑡(𝐴×𝑎_{𝑗𝑠𝑗𝑠})−max_{𝑖𝑣∈𝛽}𝑃_𝑖𝑣(𝐴)𝑎_{𝑖𝑣𝑖𝑣}𝑅_𝑗𝑠(𝐴≤𝑎_{𝑗𝑡𝑗𝑡})+max_{𝑗𝑢∈𝑁/𝑡}𝑃_𝑢(𝐴)𝑎_𝑢𝑢𝑅_𝑗𝑡(𝐴×𝑎_{𝑗𝑠𝑗𝑠})+max_{𝑖𝑣∈𝛽}𝑃_𝑖𝑣(𝐴)𝑎_{𝑖𝑣𝑖𝑣}𝑅_𝑗𝑠(𝐴−𝑎_{𝑗𝑡𝑗𝑡})−max_{𝑗𝑢∈𝑁/𝑡}𝑃_𝑢(𝐴)𝑎_𝑢𝑢𝑅_𝑗𝑡(𝐴×𝑎_{𝑗𝑠𝑗𝑠})−max_{𝑖𝑣∈𝛽}𝑃_𝑖𝑣(𝐴)𝑎_{𝑖𝑣𝑖𝑣}𝑅_𝑗𝑠(𝐴𝑎)=2_{𝑗𝑠𝑗𝑠}max_{𝑗𝑢∈𝑁/𝑡}𝑃_𝑢(𝐴)𝑎_𝑢𝑢𝑅_𝑗𝑡(𝐴)+𝑎_{𝑗𝑡𝑗𝑡}max_{𝑖𝑣∈𝛽}𝑃_𝑖𝑣(𝐴)𝑎_{𝑖𝑣𝑖𝑣}𝑅_𝑗𝑠(𝐴). (40) On the other hand, for all 1≤𝑡≤𝑙, 𝜆−𝑎^′_𝑡𝑡=𝜆−𝑎_{𝑗𝑡𝑗𝑡}+𝑎_{𝑗𝑡𝑖1},…,𝑎_{𝑗𝑡𝑖𝑘}𝐴(𝛽)⁻¹𝑎_{𝑖1𝑗𝑡}⋮𝑎_{𝑖𝑘𝑗𝑡}=𝐴𝜆−det^𝑡𝐴𝛽=𝐴𝜆−det_𝑡𝛽.det𝐴 (41) Therefore, by (39), (40), and (41), we obtain inequality (35). With a Similar method, we can prove inequality (36).(b) If |𝑎_{𝑖𝑑𝑖𝑑}|>𝑅_𝑖𝑑(𝐴) for any 𝑖_𝑑∈𝛽(1≤𝑑≤𝑘), then by (19), (32), and a similar method as the part (a), we obtain inequality (37). Similarly, we can prove inequality (38).

4. Upper Bound for the Infinity Norm on the Inverse of the Schur Complement

In this section, we present a new upper bound of ‖(𝐴/𝛽)⁻¹‖_∞.

Lemma 11 (see [2]). Let 𝐴=(𝑎_𝑖𝑗)∈𝑂𝑆_𝑛 and 𝑀=(𝑚_𝑖𝑗)∈ℂ^𝑛×𝑛. Then, 𝐴⁻¹𝑀_∞≤max_{1≤𝑖,𝑗≤𝑛𝑖≠𝑗}𝑎_𝑗𝑗∑^𝑛_𝑣=1𝑚_𝑖𝑣+𝑅_𝑖(𝐴)∑^𝑛_𝑣=1𝑚_𝑗𝑣𝑎_𝑖𝑖𝑎_𝑗𝑗−𝑅_𝑖(𝐴)𝑅_𝑗(𝐴). (42)

Theorem 12. Let 𝐴=(𝑎_𝑖𝑗)∈𝑆𝐷_𝑛, 𝑀=(𝑚_𝑖𝑗)∈ℂ^𝑙×𝑙, 𝛽={𝑖₁,𝑖₂,…,𝑖_𝑘}⊂𝑁, 𝛽=𝑁/𝛽={𝑗₁,𝑗₂,…,𝑗_𝑙}, 1≤𝑘<𝑛, and 𝐴/𝛽=(𝑎^′_𝑡𝑠). Then, 𝐴𝛽⁻¹𝑀_∞≤max_{1≤𝑡,𝑠≤𝑙𝑡≠𝑠}Δ_{𝑗𝑡𝑗𝑠}×𝑎_{𝑗𝑡𝑗𝑡}−max_{𝑗𝑢∈𝑁/𝑡}𝑅_𝑢(𝐴)𝑎_𝑢𝑢𝑅_𝑗𝑡(𝐴×𝑎_{𝑗𝑠𝑗𝑠})−max_{𝑖𝑣∈𝛽}𝑄_𝑖𝑣(𝐴)𝑎_{𝑖𝑣𝑖𝑣}𝑅_𝑗𝑠(𝐴)⁻¹,𝐴𝛽⁻¹_∞≤max_{1≤𝑡,𝑠≤𝑙𝑡≠𝑠}𝑎_{𝑗𝑡𝑗𝑡}+𝑅_𝑗𝑠(𝐴)+max_{𝑖𝑣∈𝛽}𝑄_𝑖𝑣(𝐴)𝑎_{𝑖𝑣𝑖𝑣}𝑅_𝑗𝑡(𝐴)+𝑅_𝑗𝑠(𝐴×𝑎_{𝑗𝑡𝑗𝑡})−max_{𝑗𝑢∈𝑁/𝑡}𝑅_𝑢(𝐴)𝑎_𝑢𝑢𝑅_𝑗𝑡(𝐴×𝑎_{𝑗𝑠𝑗𝑠})−max_{𝑖𝑣∈𝛽}𝑄_𝑖𝑣(𝐴)𝑎_{𝑖𝑣𝑖𝑣}𝑅_𝑗𝑠(𝐴)⁻¹, (43)(44) where Δ_{𝑗𝑡𝑗𝑠}=𝑎_{𝑗𝑡𝑗𝑡}+max_{𝑖𝑣∈𝛽}𝑄_𝑖𝑣(𝐴)𝑎_{𝑖𝑣𝑖𝑣}𝑅_𝑗𝑡(𝐴)^𝑙_𝑣=1𝑚_𝑠𝑣+𝑅_𝑗𝑠(𝐴)+max_{𝑖𝑣∈𝛽}𝑄_𝑖𝑣(𝐴)𝑎_{𝑖𝑣𝑖𝑣}𝑅_𝑗𝑠(𝐴)^𝑙_𝑣=1𝑚_𝑡𝑣, (45) and 𝑄_𝑖𝑣(𝐴) is such as in Theorem 7.

Proof. By Lemma 11, we have 𝐴𝛽⁻¹𝑀_∞≤max_{1≤𝑡,𝑠≤𝑙𝑡≠𝑠}𝑎^′_𝑡𝑡∑^𝑙_𝑣=1𝑚_𝑠𝑣+𝑅_𝑠𝐴/𝛽∑^𝑙_𝑣=1𝑚_𝑡𝑣𝑎^′_𝑡𝑡𝑎^′_𝑠𝑠−𝑅_𝑡𝐴/𝛽𝑅_𝑠.𝐴/𝛽 (46) Similar to (29), we obtain 𝑎^′_𝑡𝑡=𝑎_{𝑗𝑡𝑗𝑡}−𝑎_{𝑗𝑡𝑖1},…,𝑎_{𝑗𝑡𝑖𝑘}𝐴𝛽⁻¹𝑎_{𝑖1𝑗𝑡}⋮𝑎_{𝑖𝑘𝑗𝑡}≤𝑎_{𝑗𝑡𝑗𝑡}+𝑎_{𝑗𝑡𝑖1},…,𝑎_{𝑗𝑡𝑖𝑘}𝜇𝐴𝛽⁻¹𝑎_{𝑖1𝑗𝑡}⋮𝑎_{𝑖𝑘𝑗𝑡}≤𝑎_{𝑗𝑡𝑗𝑡}+max_{𝑖𝑣∈𝛽}𝑄_𝑖𝑣(𝐴)𝑎_{𝑖𝑣𝑖𝑣}𝑅_𝑗𝑡(𝐴). (47) Thus, by Theorem 1 of [12], we have 𝑅_𝑠𝐴𝛽≤𝑎^′_𝑠𝑠−𝑎_{𝑗𝑠𝑗𝑠}+𝑅_𝑗𝑠(𝐴)≤max_{𝑖𝑣∈𝛽}𝑄_𝑖𝑣(𝐴)𝑎_{𝑖𝑣𝑖𝑣}𝑅_𝑗𝑠(𝐴)+𝑅_𝑗𝑠(𝐴). (48) Since 𝐴∈SD_𝑛, then 𝐴∈OS_𝑛. Thus, by Theorem 7, we have 𝑎^′_𝑡𝑡𝑎^′_𝑠𝑠−𝑅_𝑡𝐴𝛽𝑅_𝑠𝐴𝛽≥𝑎_{𝑗𝑡𝑗𝑡}−max_{𝑗𝑢∈𝑁/𝑡}𝑅_𝑢(𝐴)𝑎_𝑢𝑢𝑅_𝑗𝑡(𝐴×𝑎_{𝑗𝑠𝑗𝑠})−max_{𝑖𝑣∈𝛽}𝑄_𝑖𝑣(𝐴)𝑎_{𝑖𝑣𝑖𝑣}𝑅_𝑗𝑠(𝐴). (49) Further, by (46), (47), (48), and (49), we obtain inequality (43).
Let 𝑀=𝐼=diag(1,1,…,1); we can prove inequality (44).

5. Numerical Examples

In this section, we present several numerical examples to illustrate the theory results.

Example 1 (see Example 2 in [2]). Let ,{}.𝐴=1.30.20.30.40.50.220.40.50.10.30.420.10.20.40.50.130.30.50.10.20.33𝛽=1,2 (50)
By Theorem 10, the eigenvalues of 𝐴/𝛽 are in the set Γ₁={𝜆‖‖}∪{𝜆‖‖}∪{𝜆‖‖}.𝜆−1.87𝜆−2.78∣≤11.20𝜆−1.87𝜆−2.81∣≤10.40𝜆−2.78𝜆−2.81∣≤14.40 (51) From Theorem 3.1 of [2], the eigenvalues of 𝐴/𝛽 are in the set Γ^′₁={𝜆‖‖}∪{𝜆‖‖}∪{𝜆‖‖}.𝜆−1.87𝜆−2.78∣≤12.06𝜆−1.87𝜆−2.81∣≤11.20𝜆−2.78𝜆−2.81∣≤15.51 (52) Evidently, Γ₁⊂Γ^′₁, and we use Figure 1 to show this fact. And the eigenvalues of 𝐴/𝛽 are denoted by “+” in Figure 1.

Figure 1 

The red dotted line and black dashed line denote the corresponding discs Γ1 and Γ′1, respectively.

Example 2. Let ,{}.𝐴=1.60.10.50.20.20.31.50.20.20.10.20.21.80.30.40.50.30.51.00.20.50.20.20.31.9𝛽=2,4 (53)
By Theorem 10, the eigenvalues of 𝐴/𝛽 are in the set Γ₂={𝜆‖‖}∪{𝜆‖‖}∪{𝜆‖‖}.𝜆−1.49𝜆−1.64∣≤7.12𝜆−1.49𝜆−1.84∣≤7.64𝜆−1.64𝜆−1.84∣≤8.50 (54) From Theorem 3.1 of [2], the eigenvalues of 𝐴/𝛽 are in the set Γ^′₂={𝜆‖‖}∪{𝜆‖‖}∪{𝜆‖‖}.𝜆−1.49𝜆−1.64∣≤10.68𝜆−1.49𝜆−1.84∣≤11.46𝜆−1.64𝜆−1.84∣≤12.75 (55) Evidently, Γ₂⊂Γ^′₂, and we use Figure 2 to show this fact. And the eigenvalues of 𝐴/𝛽 are denoted by “+” in Figure 2.

Figure 2 

The red dotted line and black dashed line denote the corresponding discs Γ2 and Γ′2, respectively.

Example 3. Let ,{}.𝐴=1.50.20.30.20.10.20.21.20.10.30.10.40.60.21.60.10.20.10.50.20.11.80.30.20.20.10.20.31.30.10.10.20.10.31.22.5𝛽=1,3,5 (56)
By Theorem 10, the eigenvalues of 𝐴/𝛽 are in the set Γ₃={𝜆‖‖}∪{𝜆‖‖}∪{𝜆‖‖}.𝜆−1.16𝜆−1.68∣≤4.79𝜆−1.16𝜆−2.42∣≤6.78𝜆−1.68𝜆−2.42∣≤9.85 (57) From Theorem 3.1 of [2], the eigenvalues of 𝐴/𝛽 are in the set Γ^′₃={𝜆‖‖}∪{𝜆‖‖}∪{𝜆‖‖}.𝜆−1.16𝜆−1.68∣≤5.35𝜆−1.16𝜆−2.42∣≤7.60𝜆−1.68𝜆−2.42∣≤11.09 (58) Evidently, Γ₃⊂Γ^′₃, and we use Figure 3 to show this fact. And the eigenvalues of 𝐴/𝛽 are denoted by “+” in Figure 3.
Meanwhile, by Theorem 12, 𝐴𝛽⁻¹_∞≤12.22. (59) From Theorem 4.2 of [2], 𝐴𝛽⁻¹_∞≤20.60. (60)

Figure 3 

The red dotted line and black dashed line denote the corresponding discs Γ3 and Γ′3, respectively.

Remark 13. Numerical examples show that the new eigenvalue inclusion set is tighter than that in Theorem 3.1 of [2] and the new upper bound of ‖(𝐴/𝛽)⁻¹‖_∞ is sharper than that in Theorem 4.2 of [2].

Acknowledgments

The authors would like to thank the anonymous referees for their valuable suggestions and comments. This research is supported by the National Natural Science Foundation of China (71161020, 11361074), IRTSTYN, and Foundation of Yunnan University (2012CG017).