On the Infinite Products of Matrices

Yousry S. Hanna, Samya F. Ragheb

Faculty of Science, Helwan University, Helwan, Egypt.
National Research Institute of Astronomy and Geophysics, Helwan, Egypt.

DOI: 10.4236/apm.2012.25050                

Abstract

In different fields in space researches, Scientists are in need to deal with the product of matrices. In this paper, we develop conditions under which a product П_i=0^∞ of matrices chosen from a possibly infinite set of matrices M={P_j, j∈J} converges. There exists a vector norm such that all matrices in M are no expansive with respect to this norm and also a subsequence {i_k}_k=0^∞ of the sequence of nonnegative integers such that the corresponding sequence of operators {P_ik}_k=0^∞ converges to an operator which is paracontracting with respect to this norm. The continuity of the limit of the product of matrices as a function of the sequences {i_k}_k=0^∞ is deduced. The results are applied to the convergence of inner-outer iteration schemes for solving singular consistent linear systems of equations, where the outer splitting is regular and the inner splitting is weak regular.

Keywords

Matrices; Infinite Products; Iteration

Conflicts of Interest

The authors declare no conflicts of interest.

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