On Convergents Infinite Products and Some Generalized Inverses of Matrix Sequences

Adem Kılıçman ,Zeyad Al–Zhour

doi:10.1155/2011/536935

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 inversesA+and outer inversesAT,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 inversesAT,S(2)andA+numerically for any arbitrary matrixAm,nof large dimension by using MATLAB and comparing the results between some of different methods.

Highlights

Introduction and Preliminaries1bm of complex numbers is said to converge if bm is nonzero for m sufficiently large, say m ≥ N, and q limm → ∞
Introduction and PreliminariesA scalar infinite product p ∞ m1bm of complex numbers is said to converge if bm is nonzero for m sufficiently large, say m ≥ N, and q limm → ∞1bm exists and is nonzero.If this is so p is defined to be p qNm−11bm
It is well known that Moore-Penrose inverse MPI of a matrix A ∈ Mm,n is defined to be the unique solution of the following four matrix equations see, e.g. 4, 11, 14–20 : AXA A, XAX X, AX ∗ AX, XA ∗ XA

Summary

Introduction and Preliminaries

1bm of complex numbers is said to converge if bm is nonzero for m sufficiently large, say m ≥ N, and q limm → ∞. In 1 , Daubechies and Lagarias defined the converges of an infinite product of matrices without the adverb “invertibly” as follows. It is well known that Moore-Penrose inverse MPI of a matrix A ∈ Mm,n is defined to be the unique solution of the following four matrix equations see, e.g. 4, 11, 14–20 : AXA A, XAX X, AX ∗ AX, XA ∗ XA. As we see in 13, 20, 24–29 , it is well-known fact that several important generalized inverses, such as the Moore-Penrose inverse A , the weighted Moore-Penrose inverse AM,N, the Drazin inverse AD, and so forth, are all the generalized inverse AT2,S, which is having the prescribed range T and null space S of outer inverse of A In this case, the Moore-Penrose inverse A can be represented in outer inverse form as follows 27 : A.

Convergent Moore-Penrose Inverse of Matrices

Convergent Infinite Products of Matrices

Am XN which implies that

A D and H

Numerical Examples

Concluding Remarks