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Báo cáo hóa học: " Some results on the partial orderings of block matrices"

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: Some results on the partial orderings of block matrices | Liu and Yang Journal of Inequalities and Applications 2011 2011 54 http www.journalofinequalitiesandapplications.eom content 2011 1 54 3 Journal of Inequalities and Applications a SpringerOpen Journal RESEARCH Open Access Some results on the partial orderings of block matrices Xifu Liu and Hu Yang Correspondence liuxifu211@hotmail.com College of Mathematics and Statistics Chongqing University Chongqing 401331 China Springer Abstract Some results relating to the block matrix partial orderings and the submatrix partial orderings are given. Special attention is paid to the star ordering of a sum of two matrices and the minus ordering of matrix product. Several equivalent conditions for the minus ordering are established. Mathematics Subject Classification 2000 15A45 15A57 Keywords Matrix partial orderings Moore-Penrose inverse Block matrix 1 Introduction Let Cmxn denote the set of all m X n matrices over the complex field C. The symbols A R A R A N A and r A denote the conjugate transpose the range orthogonal complement space the null space and the rank of a given matrix A e Cmxn. Furthermore Af will stand for the Moore-Penrose inverse of A i.e. the unique matrix satisfying the equations 1 AXA A XAX X AX AX XA XA. 1.1 Matrix partial orderings defined in CmXn are considered in this paper. First of them is the star ordering introduced by Drazin 2 which is determined by A B A A A B and AA BA 1-2 and can alternatively be specified as A B A1A A B and AA BA . 1.3 Modifying 1.2 Baksalary and Mitra 3 proposed the left-star and right-star orderings characterized as A B A A A B or A A A B and R A c R B 1.4 A B AA BA or AA BA and R A c R B . 1.5 The second partial ordering of interest is minus rank subtractivity ordering devised by Hartwig 4 and independently by Nambooripad 5 . It can be characterized as A B r B - A r B - r A 1.6 2011 Liu and Yang licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License http .