A Note on Kantorovich Inequality for Hermite Matrices

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Abstract

A new Kantorovich type inequality for Hermite matrices is proposed in this paper. It holds for the invertible Hermite matrices and provides refinements of the classical results. Elementary methods suffice to prove the inequality.

Keywords

Real Number Applied Mathematic Classical Result Error Bound Equivalent Form 

1. Introduction

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

for any Open image in new window , Open image in new window .

This famous inequality plays an important role in statistics and numerical analysis, for example, in discussions of converging rates and error bounds of solving systems of equations (see [2, 3, 4]). Motivated by interests in applied mathematics outlined above, we establish in this paper a new Kantorovich type inequality, the classical Kantorovich inequality is modified to apply not only to positive definite but also to all invertible Hermitian matrices. An elementary proof of this result is also presented.

In the next section, we will state the main theorem and its proof. Before starting, we quickly review some basic definitions and notations. Let Open image in new window be an invertible Hermite matrix with real eigenvalues Open image in new window , and the corresponding orthonormal eigenvectors Open image in new window   with Open image in new window , where Open image in new window denotes 2-norm of the vector of Open image in new window .

For Open image in new window , we define the following transform

2. New Kantorovich Inequality for Hermite Matrices

Theorem 2.1.

Let Open image in new window be an Open image in new window invertible Hermite matrix with real eigenvalues Open image in new window . Then

for any Open image in new window , Open image in new window , where Open image in new window , Open image in new window , Open image in new window , Open image in new window defined by (1.3), (1.4), (1.5), and (1.6).

To simplify the proof, we first introduce some lemmas.

Lemma 2.2.

With the assumptions in Theorem 2.1, then

for any Open image in new window , Open image in new window .

Lemma 2.3.

With the assumptions in Theorem 2.1, then

for any Open image in new window , Open image in new window .

Proof.

The proof about Open image in new window is similar.

Lemma 2.4.

With the assumptions in Theorem 2.1, then

for any Open image in new window , Open image in new window .

Proof.

The other inequality can be obtained similarly, the proof is completed.

We are now ready to prove the theorem.

Proof.

By the Lemma 2.4, we have
Similarly, we can get that,
Therefore,

The proof of Theorem 2.1 is completed.

Remark 2.5.

Let Open image in new window be a positive definite Hermite matrix. From Theorem 2.1, we have

our result improves the Kantorovich inequality (1.2), so we conclude that Theorem 2.1 gives an improvement of the Kantorovich inequality that applies all invertible Hermite matrices.

Example 2.6.

Therefore,

we get a sharpen upper bound.

3. Conclusion

In this paper, we introduce a new Kantorovich type inequality for the invertible Hermite matrices. In Theorem 2.1, if Open image in new window , Open image in new window , the result is well-known Kantorovich inequality. Moreover, it holds for negative definite Hermite matrices, even for any invertible Hermite matrix, there exists a similar inequality.

Notes

Acknowledgments

The authors would like to thank the two anonymous referees for their valuable comments which have been implemented in this revised version. This work is supported by Natural Science Foundation of Jiangxi, China No 2007GZS1760 and scienctific and technological project of Jiangxi education office, China No GJJ08432.

References

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Copyright information

© Zhibing Liu et al. 2011

This article is published under license to BioMed Central Ltd. 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.

Authors and Affiliations

  1. 1.School of Mathematical SciencesXiamen UniversityXiamenChina
  2. 2.Department of MathematicsJiujiang UniversityJiujiangChina

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