Research Article Nonexpansive Matrices with Applications to Solutions of Linear Systems by Fixed Point Iterations

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 × n matrices. A function ‖ · ‖ : Mn → R is a matrix norm if for all A,B ∈ Mn, it satisfies the following five axioms: 1 ‖A ‖ ≥ 0; 2 ‖A ‖ 0 if and only if A 0; 3 ‖cA ‖ |c|‖A ‖ for all complex scalars c; 4 ‖A B ‖ ≤ ‖A ‖ ‖B‖; 5 ‖AB ‖ ≤ ‖A ‖ ‖B‖. Let | · | be a norm on Cn. Define ‖ · ‖ onMn by ‖A ‖ max |x|1 |Ax|. 1 This norm on Mn is a matrix norm, called the matrix norm induced by | · |. A matrix norm on Mn is called an induced matrix norm if it is induced by some norm on Cn. If ‖ · ‖1 is a matrix norm on Mn, there exists an induced matrix norm ‖ · ‖2 on Mn such that ‖A‖2 ≤ ‖A‖1 for all 2 Fixed Point Theory and Applications A ∈ Mn cf. 1, page 297. Indeed one can take ‖ · ‖2 to be the matrix norm induced by the norm | · | on Cn defined by |x | ‖Cx‖1, 2 where Cx is the matrix inMn whose columns are all equal to x. For A ∈ Mn, ρA denotes the spectral radius of A. Let | · | be a norm in Cn. A matrix A ∈ Mn is a contraction relative to | · | if it is a contraction as a transformation from Cn into Cn; that is, there exists 0 ≤ λ < 1 such that ∣Ax −Ay∣ ∣ ≤ λ∣∣x − y∣∣, x, y ∈ Cn.