Báo cáo hóa học: " Research Article Nonexpansive Matrices with Applications to Solutions of Linear Systems by Fixed Point Iterations Teck-Cheong Lim"

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 Nonexpansive Matrices with Applications to Solutions of Linear Systems by Fixed Point Iterations Teck-Cheong Lim | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010 Article ID 821928 13 pages doi 2010 821928 Research Article Nonexpansive Matrices with Applications to Solutions of Linear Systems by Fixed Point Iterations Teck-Cheong Lim Department of Mathematical Sciences George Mason University 4400 University Drive Fairfax VA 22030 USA Correspondence should be addressed to Teck-Cheong Lim tlim@ Received 28 August 2009 Accepted 19 October 2009 Academic Editor Mohamed A. Khamsi Copyright 2010 Teck-Cheong Lim. 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. We characterize i matrices which are nonexpansive with respect to some matrix norms and ii matrices whose average iterates approach zero or are bounded. Then we apply these results to iterative solutions of a system of linear equations. Throughout this paper R will denote the set of real numbers C the set of complex numbers and Mn the complex vector space of complex n X n matrices. A function II II Mn R is a matrix norm if for all A B e Mn it satisfies the following five axioms 1 AH 0 2 IIAH 0 if and only if A 0 3 cA c HAH for all complex scalars c 4 A B A B 5 IIABII AH B . Let be a norm on Cn. Define II II on Mn by IIAII rn AV 1 This norm on Mn is a matrix norm called the matrix norm induced by l . A matrix norm on Mn is called an induced matrix norm if it is induced by some norm on Cn. If II II1 is a matrix norm on Mn there exists an induced matrix norm II II2 on Mn such that A 2 IIAH 1 for all 2 Fixed Point Theory and Applications A e Mn cf. 1 page 297 . Indeed one can take II 2 to be the matrix norm induced by the norm on Cn defined by x C X H1 2 where C x is the matrix in Mn whose columns are all equal to x. For A e Mn p A denotes the spectral radius of A. Let be a norm in Cn. A matrix A e Mn is a contraction relative

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