Journal of Applied Mathematics

Volume 2013 (2013), Article ID 973152, 10 pages

http://dx.doi.org/10.1155/2013/973152

Research Article

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

Jianxing Zhao,¹ Feng Wang,1,2 and Yaotang Li¹

¹School of Mathematics and Statistics, Yunnan University, Kunming, Yunnan 650091, China

²Department of Mathematics, Heze University, Heze, Shandong 274015, China

Received 25 April 2013; Accepted 31 August 2013

Academic Editor: Kazutake Komori

Copyright © 2013 Jianxing Zhao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 denote the set of all complex matrices, , and . Denote

We know that is called a strictly diagonally dominant matrix if

is called a generalized Ostrowski matrix if

is called Ostrowski matrix if all strict inequalities in (3) hold (see [1]).

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

As shown in [2], for all , we call and the th diagonally and doubly diagonally dominant degree of , respectively.

The infinity norm of is defined as

For , denote by the cardinality of and . If , then is the submatrix of with row indices in and column indices in . In particular, is abbreviated to . Assuming that , and the elements of and are both conventionally arranged in an increasing order. For , we denote

If is nonsingular,

is called the Schur complement of with respect to .

The comparison matrix of , , is defined by

A matrix is called an -matrix if there exist a nonnegative matrix and a number such that , where is the spectral radius of . We know that is an -matrix if and only if is an -matrix, and if is an -matrix, then the Schur complement of is also an -matrix and (see [3]). and will denote the set of all   -matrices and the set of all   -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 and are and , 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 plays an important role in some iterations for large scale nonhomogeneous system of linear equation (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 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 .

Lemma 1 (see [3]). If , then .

Lemma 2 (see [3]). If or , then ; that is, .

Lemma 3 (see [6]). If or and , then the Schur complement of is in or  , where is the complement of in and is the cardinality of .

Lemma 4 (see [12]). Let , , , and . For any , denote

Then if and only if

When the strict inequality in (9) holds, , and thus . If the equality in (9) occurs, then .

Lemma 5. Let and with an index satisfying , , , , and . Then, for all ,

where

Proof. From Lemmas 2 and 3, we know that and . Further, by Lemma 1, we have

Thus, for any ,

Further,

By Lemma 4, we can prove that . Thus, inequality (10) holds.

Remark 6. Note that

This shows that Lemma 5 improves Theorem of [12].

Theorem 7. Let , , , , and .

(a) If there exists an such that , then, for all ,  ,  ,

 where

  and and are such as in Lemma 5.(b) If for any , then, for all ,  ,  ,

 where

  and if there exists some such that , one denotes .

Proof. If there exists an such that , then, for all ,

By Lemma 5, for all ,

Thus, for all ,  ,  ,

From Lemma 3, is in ; that is, for all ,  ,  ,

Further, for all ,  ,  ,

Therefore, inequality (16) holds. Similarly, we can prove inequality (17).

If for any , then, from Lemmas 1 and 2, for all ,  ,  ,

Therefore,

Further,

In , for all ,

And for all , ,

Hence, by (30) and (31), we have and so . Further, by (29), we obtain

By (28) and a similar method as the proof of Theorem in [2], we can prove . Therefore, by (29) and (32), we obtain inequality (19). Similarly, we can prove inequality (20).

Remark 8. Note that

This shows that Theorem 7 improves Theorem 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 .

Lemma 9 (Brauer Ovals theorem). Let . Then the eigenvalues of are in the union of the following sets:

Theorem 10. Let , , , , and , and let be eigenvalue of .

(a) If there exists an such that , then there exist ,  , , such that

 where is such as in Lemma 5 and is such as in Theorem 7.(b) If for any , then there exist ,  , , such that

 where is such as in Theorem 7.

Proof. By Lemma 9, we know that there exist ,  , , such that

(a) If there exists satisfying , by (16), we have

 On the other hand, for all ,

 Therefore, by (39), (40), and (41), we obtain inequality (35). With a Similar method, we can prove inequality (36).(b) If for any , then by (19), (32), and a similar method as the part , 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 .

Lemma 11 (see [2]). Let and . Then,

Theorem 12. Let , , , , , and . Then,

where

and is such as in Theorem 7.

Proof. By Lemma 11, we have

Similar to (29), we obtain

Thus, by Theorem 1 of [12], we have

Since , then . Thus, by Theorem 7, we have

Further, by (46), (47), (48), and (49), we obtain inequality (43).

Let ; 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

By Theorem 10, the eigenvalues of are in the set

From Theorem of [2], the eigenvalues of are in the set

Evidently, , and we use Figure 1 to show this fact. And the eigenvalues of are denoted by “+” in Figure 1.

Figure 1: The red dotted line and black dashed line denote the corresponding discs and , respectively.

Example 2. Let

By Theorem 10, the eigenvalues of are in the set

From Theorem of [2], the eigenvalues of are in the set

Evidently, , and we use Figure 2 to show this fact. And the eigenvalues of are denoted by “+” in Figure 2.

Figure 2: The red dotted line and black dashed line denote the corresponding discs and , respectively.

Example 3. Let

By Theorem 10, the eigenvalues of are in the set

From Theorem of [2], the eigenvalues of are in the set

Evidently, , and we use Figure 3 to show this fact. And the eigenvalues of are denoted by “+” in Figure 3.

Meanwhile, by Theorem 12,

From Theorem 4.2 of [2],

Figure 3: The red dotted line and black dashed line denote the corresponding discs and , respectively.

Remark 13. Numerical examples show that the new eigenvalue inclusion set is tighter than that in Theorem of [2] and the new upper bound of is sharper than that in Theorem 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).

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