On Convergents Infinite Products and Some Generalized Inverses of Matrix Sequences

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(T,S)((2)) 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(T,S)((2)) 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.