Abstract and Applied Analysis
Volume 2011 (2011), Article ID 536935, 20 pages
http://dx.doi.org/10.1155/2011/536935
Research Article

On Convergents Infinite Products and Some Generalized Inverses of Matrix Sequences

Adem Kiliçman1 and Zeyad Al-Zhour2

1Department of Mathematics and Institute of Mathematical Research, Universiti Putra Malaysia (UPM), Selangor, 43400 Serdang, Malaysia
2Department of Basic Sciences and Humanities, College of Engineering, University of Dammam (UD), P. O. Box 1982, Dammam 31451, Saudi Arabia

Received 24 January 2011; Revised 30 May 2011; Accepted 31 July 2011

Academic Editor: Alexander I. Domoshnitsky

Copyright © 2011 Adem Kiliçman and Zeyad Al-Zhour. 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.

Abstract

The definition of convergence of an infinite product of scalars is extended to the infinite usual and Kronecker products of matrices. The new definitions are less restricted invertibly convergence. Whereas the invertibly convergence is based on the invertible of matrices; in this study, we assume that matrices are not invertible. Some sufficient conditions for these kinds of convergence are studied. Further, some matrix sequences which are convergent to the Moore-Penrose inverses 𝐴 + and outer inverses 𝐴 ( 2 ) 𝑇 , 𝑆 as a general case are also studied. The results are derived here by considering the related well-known methods, namely, Euler-Knopp, Newton-Raphson, and Tikhonov methods. Finally, we provide some examples for computing both generalized inverses 𝐴 ( 2 ) 𝑇 , 𝑆 and 𝐴 + numerically for any arbitrary matrix A 𝑚 , 𝑛 of large dimension by using MATLAB and comparing the results between some of different methods.