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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 101957, 8 pages
http://dx.doi.org/10.1155/2012/101957
Research Article

A Note on Inclusion Intervals of Matrix Singular Values

1Xingzhi College, Zhejiang Normal University, Zhejiang, Jinhua 321004, China
2College of Mathematics, Physics, and Information Engineering, Zhejiang Normal University, Zhejiang, Jinhua 321004, China

Received 8 February 2012; Accepted 18 April 2012

Academic Editor: Song Cen

Copyright © 2012 Shu-Yu Cui and Gui-Xian Tian. 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

We establish an inclusion relation between two known inclusion intervals of matrix singular values in some special case. In addition, based on the use of positive scale vectors, a known inclusion interval of matrix singular values is also improved.

1. Introduction

The set of all -by- complex matrices is denoted by . Let . Denote the Hermitian adjoint of matrix by . Then the singular values of are the eigenvalues of . It is well known that matrix singular values play a very key role in theory and practice. The location of singular values is very important in numerical analysis and many other applied fields. For more review about singular values, readers may refer to [19] and the references therein.

Let . For a given matrix , we denote the deleted absolute row sums and column sums of by respectively. On the basis of and , the Geršgorin’s disk theorem, Brauer’s theorem and Brualdi’s theorem provide some elegant inclusion regions of the eigenvalues of (see [1012]). Recently, some authors have made efforts to establish analogues to these theorems for matrix singular values, for example, as follows.

Theorem A (Geršgorin-type [8]). Let . Then all singular values of are contained in where and for each .

Theorem B (Brauer-type [5]). Let . Then all singular values of are contained in

Let denote a nonempty subset of , and let denote its complement in . For a given matrix with , define partial absolute deleted row sums and column sums as follows: Thus, one splits each row sum and each column sum from (1.1) into two parts, depending on and , that is, Define, for each , , where For convenience, we will sometimes use (, , ) to denote (, , , resp.) unless a confusion is caused.

Theorem C (modified Brauer-type [7]). Let with . Then all singular values of are contained in where

A simple analysis shows that Theorem B improves Theorem A. On the other hand, Theorem C reduces to Theorem A if or (see Remark  2.3 in [7]).

Now it is natural to ask whether there exists an inclusion relation between Theorem B and Theorem C or not. In this note, we establish an inclusion relation between the inclusion interval of Theorem B and that of Theorem C in a particular situation. In addition, based on the use of positive scale vectors and their intersections, the inclusion interval of matrix singular values in Theorem C is also improved.

2. Main Results

In this section, we will establish an inclusion relation between the inclusion interval of Theorem B and that of Theorem C in a particular situation. We firstly remark that Theorem B and Theorem C are incomparable, for example, as follows.

Example 2.1. Consider the following matrix:

Let and . Applying Theorem C, one gets Hence, the inclusion interval of is .

Now applying Theorem B, one gets Therefore, the inclusion interval of is .

Example 2.1 shows that Theorem B and Theorem C are incomparable in the general case, but Theorem C may be better than Theorem B whenever the set is chosen suitably, for example, as follows.

Example 2.2. Take and in Example 2.1. Applying Theorem C, one gets Hence, the inclusion interval of is . However, applying Theorem B, we get that the inclusion interval of is (see Example 2.1).

Example 2.2 shows that Theorem C is an improvement on Theorem B in some cases, but Theorem C is complex in calculation. In order to simplify our calculations, we may consider the following special case that the set is a singleton, that is, for some . In this case, the associated sets from (1.6) may be defined as the following sets: By a simple analysis, and are necessarily contained in for any , we can simply write from (1.8) that, for any , This shows that is determined by sets . The associated Geršgorin-type set from (1.2) is determined by sets and the associated Brauer-type set from (1.3) is determined by sets. The following corollary is an immediate consequence of Theorem C.

Corollary 2.3. Let with . Then all singular values of are contained in

Proof. From (2.7), we get the required result.

Notice that whenever . Next, we will assume that . It is interesting to establish their relations between and , as well as between and .

Definition 2.4 (see [9]). is called a matrix with property (absolute symmetry) if for any , .

Note that a matrix with property is said as with property in [9].

Theorem 2.5. Let with . If is a matrix with property , then for each

Proof. Fix some and consider any . Then from (2.7), there exists a such that , that is, from (2.6), where the last equality holds as has the property (i.e., for any , ).
Now assume that , then for each , implying that and for above . Thus, the left part of (2.10) satisfies which contradicts the inequality (2.10). Hence, implies , that is, .
Next, we will show that . Since for any , then, from (2.8), we get . Now consider any , so that for each . Hence, for each , there exists a such that , that is, the inequality (2.10) holds. Since , there exists a such that . For this index , there exists a such that , that is, Hence, which implies . Since this is true for any . Then . This completes our proof.

Remark that the condition “the matrix has the property ” is necessary in Theorem 2.5, for example, as follows.

Example 2.6. Consider the following matrix: Let , , and . From (2.7), we get that the inclusion intervals of are , and , respectively. Hence, applying Corollary 2.3, we have . However, applying Theorem A and Theorem B, we get , which implies Theorem 2.5 is failling if the condition “the matrix has the property ” is omitted.

In the following, we will give a new inclusion interval for matrix singular values, which improves that of Theorem C. The proof of this result is based on the use of scaling techniques. It is well known that scaling techniques pay important roles in improving inclusion intervals for matrix singular values. For example, using positive scale vectors and their intersections, Qi [8] and Li et al. [6] obtained two new inclusion intervals (see Theorem  4 in [8] and Theorem  2.2 in [6], resp.), which improve these of Theorems A and B, respectively. Recently, Tian et al. [9], using this techniques, also obtained a new inclusion interval (see Theorem  2.4 in [9]), which is an improvement on these of Theorem  2.2 in [6] and Theorem B.

Theorem 2.7. Let with and be any vector with positive components. Then Theorem C remains true if one replaces the definition of and by where

Proof. Suppose that is any singular value of . Then there exist two nonzero vectors and such that (see Problem 5 of Section 7.3 in [11]).
The fundamental equation (2.17) implies that, for each ,
Let , for each . Then our fundamental equation (2.18) and become into, for each ,
Denote for each . Now using the similar technique as the proof of Theorem  2.2 in [7], one gets the required result.

Remarks 2. Write the inclusion intervals in Theorem 2.7 as . Since is any vector with positive components, then all singular values of are contained in Obviously, Theorem 2.7 reduces to Theorem C whenever , which implies that Hence, the inclusion interval (2.20) is an improvement on that of (1.8).

Acknowledgments

The authors are very grateful to the referee for much valuable, detailed comments and thoughtful suggestions, which led to a substantial improvement on the presentation and contents of this paper. This work was supported by the National Natural Science Foundation of China (no. 11126258), the Natural Science Foundation of Zhejiang Province, China (no. Y12A010011), and the Scientific Research Fund of Zhejiang Provincial Education Department (no. Y201120835).

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