# 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

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 .

## 2. New Kantorovich Inequality for Hermite Matrices

Theorem 2.1.

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.

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

Lemma 2.3.

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.

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.

The proof of Theorem 2.1 is completed.

Remark 2.5.

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.

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

- 1.Horn RA, Johnson CR:
*Matrix Analysis*. Cambridge University Press, Cambridge, UK; 1985:xiii+561.MATHCrossRefGoogle Scholar - 2.Householder AS:
*The Theory of Matrices in Numerical Analysis*. Blaisdell, New York, NY, USA; 1964:xi+257.MATHGoogle Scholar - 3.Wang S-G:
**A matrix version of the Wielandt inequality and its applications to statistics.***Linear Algebra and its Applications*1999,**296**(1–3):171–181.MATHMathSciNetCrossRefGoogle Scholar - 4.Nocedal J:
**Theory of algorithms for unconstrained optimization.***Acta Numerica*1992,**1:**199–242.MathSciNetCrossRefGoogle Scholar

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