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Báo cáo hóa học: " A note on the Frobenius conditional number with positive definite matrices"

Tuyển tập các báo cáo nghiên cứu về hóa học được đăng trên tạp chí hóa hoc quốc tế đề tài : A note on the Frobenius conditional number with positive definite matrices | Li et al. Journal of Inequalities and Applications 2011 2011 120 http www.journalofinequalitiesandapplications.eom content 2011 1 120 RESEARCH Journal of Inequalities and Applications a SpringerOpen Journal Open Access A note on the Frobenius conditional number with positive definite matrices Hongyi Li Zongsheng Gao and Di Zhao Correspondence zdhyl2010@163. com LMIB School of Mathematics and System Science Beihang University Beijing 100191 P.R. China Abstract In this article we focus on the lower bounds of the Frobenius condition number. Using the generalized Schwarz inequality we present some lower bounds for the Frobenius condition number of a positive definite matrix depending on its trace determinant and Frobenius norm. Also we give some results on a kind of matrices with special structure the positive definite matrices with centrosymmetric structure. 1 Introduction and preliminaries In this article we use the following notations. Let cnxn and R X be the space of nxn complex and real matrices respectively. The identity matrix in cnxn is denoted by I I . Let AT Ả Ah and tr A denote the transpose the conjugate the conjugate transpose and the trace of a matrix A respectively. Re a stands for the real part of a number a. The Frobenius inner product F in cmxn is defined as A B F Re tr BhA for A B e cmxn i.e. A B is the real part of the trace of BHA. The induced matrix norm is A F v A A p V Re tr AHA ựtr AHA which is called the Frobenius Euclidean norm. The Frobenius inner product allows us to define the con-sine of the angle between two given real n X n matrices as A B p cos A B A f B f 1.1 The cosine of the angle between two real n X n depends on the Frobenius inner product and the Frobenius norms of given matrices. A matrix A e cnxn is Hermitian if AH A. An Hermitian matrix A is said to be positive semidefinite or nonnegative definite written as A 0 if see e.g. 1 p. 159 xHAx 0 Vx e C 1.2 A is further called positive definite symbolized A 0 if the strict inequality