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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 483624, 11 pages
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.
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 [3–5] 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  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, 15–19] are utilized.
To simplify the proof, we first introduce some 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
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 , 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 , .
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 ,
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 , then
where , and
Then, one has the following.
If , where If where
Remark 4.8. It is easy to see that if , , our result coincides with the inequality of operator versions in . So we conclude that our results give an improvement of the Kantorovich inequality  that applies to all invertible Hermite matrices.
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 , but our proof is simple. In Theorem 4.6, if , , the results are similar to the well-known Kantorovich-type inequalities for operators in . Moreover, for any invertible Hermitian matrix, there exists a similar inequality.
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).
- R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, UK, 1985.
- A. S. Householder, The Theory of Matrices in Numerical Analysis, Blaisdell, New York, NY, USA, 1964.
- S. Liu and H. Neudecker, “Several matrix Kantorovich-type inequalities,” Journal of Mathematical Analysis and Applications, vol. 197, no. 1, pp. 23–26, 1996.
- A. W. Marshall and I. Olkin, “Matrix versions of the Cauchy and Kantorovich inequalities,” Aequationes Mathematicae, vol. 40, no. 1, pp. 89–93, 1990.
- J. K. Baksalary and S. Puntanen, “Generalized matrix versions of the Cauchy-Schwarz and Kantorovich inequalities,” Aequationes Mathematicae, vol. 41, no. 1, pp. 103–110, 1991.
- S. S. Dragomir, “New inequalities of the Kantorovich type for bounded linear operators in Hilbert spaces,” Linear Algebra and its Applications, vol. 428, no. 11-12, pp. 2750–2760, 2008.
- P. G. Spain, “Operator versions of the Kantorovich inequality,” Proceedings of the American Mathematical Society, vol. 124, no. 9, pp. 2813–2819, 1996.
- S. Liu and H. Neudecker, “Kantorovich inequalities and efficiency comparisons for several classes of estimators in linear models,” Statistica Neerlandica, vol. 51, no. 3, pp. 345–355, 1997.
- P. Bloomfield and G. S. Watson, “The inefficiency of least squares,” Biometrika, vol. 62, pp. 121–128, 1975.
- L. V. Kantorovič, “Functional analysis and applied mathematics,” Uspekhi Matematicheskikh Nauk, vol. 3, no. 6(28), pp. 89–185, 1948 (Russian).
- C. G. Khatri and C. R. Rao, “Some extensions of the Kantorovich inequality and statistical applications,” Journal of Multivariate Analysis, vol. 11, no. 4, pp. 498–505, 1981.
- C. T. Lin, “Extrema of quadratic forms and statistical applications,” Communications in Statistics A, vol. 13, no. 12, pp. 1517–1520, 1984.
- A. Galántai, “A study of Auchmuty's error estimate,” Computers & Mathematics with Applications, vol. 42, no. 8-9, pp. 1093–1102, 2001.
- P. D. Robinson and A. J. Wathen, “Variational bounds on the entries of the inverse of a matrix,” IMA Journal of Numerical Analysis, vol. 12, no. 4, pp. 463–486, 1992.
- S. S. Dragomir, “A generalization of Grüss's inequality in inner product spaces and applications,” Journal of Mathematical Analysis and Applications, vol. 237, no. 1, pp. 74–82, 1999.
- S. S. Dragomir, “Some Grüss type inequalities in inner product spaces,” Journal of Inequalities in Pure and Applied Mathematics, vol. 4, no. 2, article 42, 2003.
- S. S. Dragomir, “On Bessel and Grüss inequalities for orthonormal families in inner product spaces,” Bulletin of the Australian Mathematical Society, vol. 69, no. 2, pp. 327–340, 2004.
- S. S. Dragomir, “Reverses of Schwarz, triangle and Bessel inequalities in inner product spaces,” Journal of Inequalities in Pure and Applied Mathematics, vol. 5, no. 3, article 76, 2004.
- S. S. Dragomir, “Reverses of the Schwarz inequality generalising a Klamkin-McLenaghan result,” Bulletin of the Australian Mathematical Society, vol. 73, no. 1, pp. 69–78, 2006.
- Z. Liu, L. Lu, and K. Wang, “On several matrix Kantorovich-type inequalities,” Journal of Inequalities and Applications, vol. 2010, Article ID 571629, 5 pages, 2010.