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Báo cáo hóa học: "Research Article Several Matrix Euclidean Norm Inequalities Involving Kantorovich Inequality"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Several Matrix Euclidean Norm Inequalities Involving Kantorovich Inequality | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009 Article ID 291984 9 pages doi 10.1155 2009 291984 Research Article Several Matrix Euclidean Norm Inequalities Involving Kantorovich Inequality Litong Wang and Hu Yang College of Mathematics and Physics Chongqing University Chongqing 400030 China Correspondence should be addressed to Litong Wang wanglt80@163.com Received 21 April 2009 Accepted 4 August 2009 Recommended by Andrei Volodin Kantorovich inequality is a very useful tool to study the inefficiency of the ordinary least-squares estimate with one regressor. When regressors are more than one statisticians have to extend it. Matrix determinant and trace versions of it have been presented in the literature. In this paper we provide matrix Euclidean norm Kantorovich inequalities. Copyright 2009 L. Wang and H. Yang. 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. 1. Introduction Suppose that A is an n X n positive definite matrix and x is an n X 1 real vector then the well-known Kantorovich inequality can be expressed as x Ax fx A 1xJ 1 x x 2 4A1An 1.1 where A1 ln 0 are the eigenvalues of A. It is a very useful tool to study the inefficiency of the ordinary least-squares estimate with one regressor in the linear model. Watson 1 introduced the ratio of the variance of the best linear unbiased estimator to the variance of the ordinary least-squares estimator. Such a lower bound of this ratio was provided by Kantorovich inequality 1.1 see for example 2 3 . When regressors are more than one statisticians have to extend it. Marshall and Olkin 4 were the first to generalize Kantorovich inequality to matrices see e.g. 5 X A X 11 n 2X X X AX -1 X X 4hln 1.2 2 Journal of Inequalities and Applications where X is an n X p real matrix. If X X Ip then 1.2 becomes X A-1X 11 n 2 X AX .