Research Article A Note on Kantorovich Inequality for Hermite Matrices

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1 Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 0, Article ID 5767, 6 pages doi:0.55/0/5767 Research Article A Note on Kantorovich Inequality for Hermite Matrices Zhibing Liu,, Kanmin Wang, and Chengfeng Xu School of Mathematical Sciences, Xiamen University, Xiamen 36005, China Department of Mathematics, Jiujiang University, Jiujiang 33005, China Correspondence should be addressed to Zhibing Liu, liuzhibingjju@6.com Received 0 November 00; Accepted 9 February 0 Academic Editor: Sin E. Takahasi Copyright q 0 Zhibing Liu et al. 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. 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.. Introduction We first state the well-known Kantorovich inequality for a positive definite Hermite matrix see,, leta M n be a positive definite Hermite matrix with real eigenvalues λ λ λ n. Then x Axx A x λ λ n λ λ n,. for any x C n, x, where A denotes the conjugate transpose of the matrix A. An equivalent form of this result is incorporated in 0 x Axx A x λ n λ λ λ n,. for any x C n, x. 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. Motivated by interests in applied mathematics outlined above, we establish in this

2 Journal of Inequalities and Applications 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 A M n be an invertible Hermite matrix with real eigenvalues λ λ λ n, and the corresponding orthonormal eigenvectors ϕ,ϕ,...,ϕ n with ϕ i i,,...,n, where ϕ i denotes -norm of the vector of C n. For A, we define the following transform C A, x x λ n I A A λ I x..3 If λ λ n > 0, then, ( ) ( )( C A,x x I A A ) I x.. λ λ n Otherwise, λ λ n < 0, then, ( ) ( )( C A,x x I A A ) I x,.5 λ k λ k where λ λ k < 0 <λ k λ n..6. New Kantorovich Inequality for Hermite Matrices Theorem.. Let A be an n n invertible Hermite matrix with real eigenvalues λ λ λ n. Then x Axx A x λ n λ λ λ n λ n λ λ k λ k λ k λ k C A, x C ( A,x ) if λ λ n > 0, C A, x C ( A,x ) if λ λ n < 0. for any x C n, x,wherec A, x, C A,x, λ k, λ k defined by.3,.,.5, and.6.

3 Journal of Inequalities and Applications 3 To simplify the proof, we first introduce some lemmas. Lemma.. With the assumptions in Theorem.,then λ x Ax λ n, 0 λ n x Ax x Ax λ λ n λ, λ n x A x λ, λ k x A x λ k, ( ) 0 x A x )(x A x λn λ n λ if λ λ λ n λ λ n > 0, ( ) 0 x A x )(x A x λk λ k λ k if λ λ k λ k λ k λ n < 0. for any x C n, x. Lemma.3. With the assumptions in Theorem.,then ( ) 0 C A,x 0 C A, x λ n λ, λ n λ λ λ n if λ λ n > 0, λ k λ k λ k λ k if λ λ n < 0.3 for any x C n, x. Proof. Let x k i ϕ i, then C A, x ( ( ) )( ) k j ϕ j λn λ j ki ϕ i λ i λ k i λ n λ i λ i λ 0. while ( ( C A, x x λn λ I A λ n λ ( λ n λ A λ n λ ))( λn λ I ) I x ( I λ n λ. A λ n λ )) I x.5 The proof about C A,x is similar.

4 Journal of Inequalities and Applications Lemma.. With the assumptions in Theorem.,then Ax x Ax λ n λ C A, x, λ n λ A x ( x A x) λ n λ C( A,x ) if λ λ n > 0, λ k λ k λ k λ k C( A,x ) if λ λ n < 0.6 for any x C n, x. Proof. Thus, Ax x Ax λ n x Ax x Ax λ x ( ) A λ λ n A λ λ n I x λ n x Ax x Ax λ x A λ I A λ n I x λ n x Ax x Ax λ x A λ I λ n I A x.7 λ n λ C A, x ( by Lemma. ). The other inequality can be obtained similarly, the proof is completed. We are now ready to prove the theorem. Proof of the Theorem.. Thus, x Axx A x x (( ) x A x I A ) x Ax I A x (( ) x A x I A ) x x Ax I A x,.8 while x Ax I A x x ( x Ax I x Ax A A ) x x A x x Ax.9 Ax x Ax. By the Lemma., we have x Ax I A x λ n λ C A, x..0

5 Journal of Inequalities and Applications 5 Similarly, we can get that, (( ) x A x I A ) x A x ( x A x) λ n λ λ n λ C( A,x ) if λ λ n > 0, λ k λ k λ k λ k C( A,x ) if λ λ n < 0.. Therefore, x Axx A x ( )( λn λ λn λ C A, x λ n λ C( A,x ) ) ( )( λn λ λk λ k C A, x λ k λ k C( A,x ) ) if λ λ n > 0, if λ λ n < 0.. From a b c d ac bd for real numbers a, b, c, d, we have x Axx A x ( λn λ λ λ n ( λn λ λ k λ k λ k λ k C A, x C ( A,x )), if λ λ n > 0 C A, x C ( A,x )), if λ λ n < 0..3 The proof of Theorem. is completed. Remark.5. Let A M n be a positive definite Hermite matrix. From Theorem., we have 0 x Axx A x λ n λ λ λ n C A, x C ( A,x ) λ n λ,. λ λ n our result improves the Kantorovich inequality., so we conclude that Theorem. gives an improvement of the Kantorovich inequality that applies all invertible Hermite matrices. Example.6. Let A 0 3, A 0, x,, 0 T

6 6 Journal of Inequalities and Applications The eigenvalues of A are: λ 7 7 /, λ, λ / by easily calculating, we have x Axx A x, λ 3 λ 7 ( ) λ λ 3 3, C A, x C A,x 3..6 Therefore, λ 3 λ λ λ 3 we get a sharpen upper bound. C A, x C ( A,x ) < λ 3 λ,.7 λ λ 3 3. Conclusion In this paper, we introduce a new Kantorovich type inequality for the invertible Hermite matrices. In Theorem., ifλ > 0, λ n > 0, 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. 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 007GZS760 and scienctific and technological project of Jiangxi education office, China No GJJ083. References R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, UK, 985. A. S. Householder, The Theory of Matrices in Numerical Analysis, Blaisdell, New York, NY, USA, S.-G. Wang, A matrix version of the Wielandt inequality and its applications to statistics, Linear Algebra and its Applications, vol. 96, no. 3, pp. 7 8, 999. J. Nocedal, Theory of algorithms for unconstrained optimization, Acta Numerica, vol., pp. 99, 99.