- Research Article
- Open Access
A Note on Kantorovich Inequality for Hermite Matrices
© Zhibing Liu et al. 2011
- Received: 10 November 2010
- Accepted: 19 February 2011
- Published: 13 March 2011
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.
- Real Number
- Applied Mathematic
- Classical Result
- Error Bound
- Equivalent Form
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 [2–4]). 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 be an invertible Hermite matrix with real eigenvalues , and the corresponding orthonormal eigenvectors with , where denotes 2-norm of the vector of .
To simplify the proof, we first introduce some lemmas.
The other inequality can be obtained similarly, the proof is completed.
We are now ready to prove the theorem.
The proof of Theorem 2.1 is completed.
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.
we get a sharpen upper bound.
In this paper, we introduce a new Kantorovich type inequality for the invertible Hermite matrices. In Theorem 2.1, if , , 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.
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.
- Horn RA, Johnson CR: Matrix Analysis. Cambridge University Press, Cambridge, UK; 1985:xiii+561.MATHView ArticleGoogle Scholar
- Householder AS: The Theory of Matrices in Numerical Analysis. Blaisdell, New York, NY, USA; 1964:xi+257.MATHGoogle Scholar
- 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.MATHMathSciNetView ArticleGoogle Scholar
- Nocedal J: Theory of algorithms for unconstrained optimization. Acta Numerica 1992, 1: 199–242.MathSciNetView ArticleGoogle Scholar
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