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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 483624, 11 pages
http://dx.doi.org/10.1155/2012/483624
Research Article

Refinements of Kantorovich Inequality for Hermitian Matrices

School of Mathematics and Statistics, Chongqing Three Gorges University, Wanzhou, Chongqing 404000, China

Received 28 August 2012; Accepted 5 November 2012

Academic Editor: K. C. Sivakumar

Copyright © 2012 Feixiang Chen. 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

Some new Kantorovich-type inequalities for Hermitian matrix are proposed in this paper. We consider what happens to these inequalities when the positive definite matrix is allowed to be invertible and provides refinements of the classical results.

1. Introduction and Preliminaries

We first state the well-known Kantorovich inequality for a positive definite Hermite matrix (see [1, 2]), let be a positive definite Hermitian matrix with real eigenvalues . Then for any , , where denotes the conjugate transpose of matrix . A matrix is Hermitian if . An equivalent form of this result is incorporated in for any .

Attributed to Kantorovich, the inequality has built up a considerable literature. This typically comprises generalizations. Examples are [35] for matrix versions. Operator versions are developed in [6, 7]. Multivariate versions have been useful in statistics to assess the robustness of least squares, see [8, 9] and the references therein.

Due to the important applications of the original Kantorovich inequality for matrices [10] in Statistics [8, 11, 12] and Numerical Analysis [13, 14], any new inequality of this type will have a flow of consequences in the areas of applications.

Motivated by the interest in both pure and applied mathematics outlined above we establish in this paper some improvements of Kantorovich inequalities. The classical Kantorovich-type inequalities are modified to apply not only to positive definite but also to invertible Hermitian matrices. As natural tools in deriving the new results, the recent Grüss-type inequalities for vectors in inner product in [6, 1519] are utilized.

To simplify the proof, we first introduce some lemmas.

2. Lemmas

Let be a Hermitian matrix with real eigenvalues , if is positive semidefinite, we write that is, , . On , we have the standard inner product defined by , where and .

Lemma 2.1. Let , , , and be real numbers, then one has the following inequality:

Lemma 2.2. Let and be Hermitian matrices, if , then

Lemma 2.3. Let , , if , then

3. Some Results

The following lemmas can be obtained from [1619] by replacing Hilbert space with inner product spaces , so we omit the details.

Lemma 3.1. Let , , and be vectors in , and . If , , , and are real or complex numbers such that then

Lemma 3.2. With the assumptions in Lemma 3.1, one has

Lemma 3.3. With the assumptions in Lemma 3.1, if , , one has

Lemma 3.4. With the assumptions in Lemma 3.3, one has

4. New Kantorovich Inequalities for Hermitian Matrices

For a Hermitian matrix , as in [6], we define the following transform: When is invertible, if , then, Otherwise, , then, where

From Lemma 2.3 we can conclude that and .

For two Hermitian matrices and , and , , we define the following functional: When , we denote for , .

Lemma 4.1. With notations above, and for , , then if , if ,

Proof. From , then While, from Lemma 2.2, we can get Then is straightforward. The proof for is similar.

Lemma 4.2. With notations above, and for , , then

Proof. Thus, while Similarly, we can get , then we complete the proof.

Theorem 4.3. Let , be two Hermitian matrices, and , are defined as above, then for any , .
If , , then for any , .

Proof. The proof follows by Lemmas 3.1, 3.2, 3.3, and 3.4 on choosing , , and , , , , and , , , respectively.

Corollary 4.4. Let be a Hermitian matrices, and is defined as above, then for any , .
If , then for any , .

Proof. The proof follows by Theorem 4.3 on choosing , respectively.

Corollary 4.5. Let be a Hermitian matrices and is defined as above, then one has the following.
If , then for any , .
If , then where , and for any , .

Proof. The proof follows by Corollary 4.4 by replacing with , respectively.

Theorem 4.6. Let be an invertible Hermitian matrix with real eigenvalues , then one has the following.
If , then for any , .
If , then where ,

Proof. Considering When , from (4.17) and (4.21), we get From , we have then, the conclusion (4.27) holds.
Similarly, from (4.18) and (4.22), we get then, the conclusion (4.28) holds.
From (4.19) and (4.23), we get then, the conclusion (4.29) holds.
From (4.20) and (4.24), we get then, the conclusion (4.30) holds.
When , from (4.17) and (4.25), we get then, the conclusion (4.31) holds.
From (4.18) and (4.26), we get then, the conclusion (4.32) holds.

Corollary 4.7. With the notations above, for any , , one lets If , one lets If If , then where , and Then, one has the following.
If , where If where

Proof. The proof follows from that the conclusions in Corollaries 4.4 and 4.5 are independent.

Remark 4.8. It is easy to see that if , , our result coincides with the inequality of operator versions in [6]. So we conclude that our results give an improvement of the Kantorovich inequality [6] that applies to all invertible Hermite matrices.

5. Conclusion

In this paper, we introduce some new Kantorovich-type inequalities for the invertible Hermitian matrices. Inequalities (4.27) and (4.31) are the same as [4], but our proof is simple. In Theorem 4.6, if , , the results are similar to the well-known Kantorovich-type inequalities for operators in [6]. Moreover, for any invertible Hermitian matrix, there exists a similar inequality.

Acknowledgments

The authors are grateful to the associate editor and two anonymous referees whose comments have helped to improve the final version of the present work. This work was supported by the Natural Science Foundation of Chongqing Municipal Education Commission (no. KJ091104).

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