# On Convergents Infinite Products and Some Generalized Inverses of Matrix Sequences

Authors: Adem Kiliçman; Zeyad Al-Zhour
Journal: Abstract and Applied Analysis
 (2011)

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## 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 inverses AT,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 AT,S(2) and A+ numerically for any arbitrary matrix Am,n of large dimension by using MATLAB and comparing the results between some of different methods.


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## Body

## 1. Introduction and Preliminaries

A scalar infinite productp=∏m=1∞bm of complex numbers is said to converge if bm is nonzero for m sufficiently large, say m≥N, and q=limm→∞∏m=1∞bm exists and is nonzero. If this is so then p is defined to be p=q=∏m=1N-1bm. With this definition, a convergent infinite product vanishes if and only if one of its factors vanishes.Let{Bm} be a sequence of k×k  matrices, then
(1.1)∏m=rsBm={BsBs-1⋯Brifr≤sIifr>s.
In [1], Daubechies and Lagarias defined the converges of an infinite product of matrices without the adverb “invertibly” as follows.(i)
An infinite product∏m=1∞Bm of k×k  matrices is said to be right converges if limm→∞B1B2⋯Bm exists, in which case
(1.2)∏m=1∞Bm=lim⁡m→∞B1B2⋯Bm.(ii)
An infinite product∏m=1∞Bm of k×k  matrices is said to be left converges if limm→∞Bm⋯B2B1 exists, in which case
(1.3)∏m=1∞Bm=lim⁡m→∞Bm⋯B2B1.The idea of invertibly convergence of sequence of matrices was introduced by Trench [2, 3] as follows. An infinite product ∏m=1∞Bm of k×k  matrices  is said to be invertibly converged if there is an integer N such that Bm is invertible for m≥N, and
(1.4)Q=lim⁡n→∞∏m=NnBm,
exists and is invertible. In this case,
(1.5)∏m=1∞Bm=Q∏m=1N-1Bm.
Let us recall some concepts that will be used below. Before starting, throughout we consider matrices over the field of complex numbers ℂ or real numbers ℝ. The set of m-by-n complex matrices is denoted by Mm,n(ℂ)=ℂm×n. For simplicity, we write Mm,n instead of Mm,n(ℂ) and when m=n, we write Mn instead  of Mn,n. The notations AT, A*, A+, AT,S(2), rank(A), rang(A), null(A), ρ(A), ∥A∥s, ∥A∥p, and σ(A) stand, respectively, for the transpose, conjugate transpose, Moore-Penrose inverse, outer inverse, rank, range, null space, spectral radius, spectrum norm, p-norm, and the set of all eigenvalues of matrix A.The Moore-Penrose and outer inverses of an arbitrary matrix (including singular and rectangular) are very useful in various applications in control system analysis, statistics, singular differential and difference equations, Markov chains, iterative methods, least-square problem, perturbation theory, neural networks problem, and many other subjects were found in the literature (see, e.g., [4–14]).It is well known thatMoore-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]):
(1.6)AXA=A,XAX=X,(AX)*=AX,(XA)*=XA
and is often denoted by X=A+∈Mn,m. In particular, when A is a square and nonsingular matrix, then A+ reduce to A-1.Forx=A+b, x′∈ℂn∖{x} arbitrary, it holds, (see, e.g., [14, 18]),
(1.7)‖b-Ax‖22=(b-Ax)*(b-Ax)≤‖b-Ax′‖22,
and ∥b-Ax∥2=∥b-Ax′∥2, then
(1.8)‖x‖22=x*x<‖x′‖22.
Thus, x=A+b  is the unique minimum least-squares solution of the following linear squares problem, (see, e.g., [14, 21, 22]),
(1.9)‖b-Ax‖2=min⁡z∈Cn⁡‖b-Az‖2.
It is well known also that the singular value decomposition of any rectangular matrix A∈Mm,n  with rank(A)=r≠0 is given by
(1.10)A=U[D000]V*:U*U=Im,V*V=In,
where D=diag(μ1,μ2,…,μr)∈Mr is a diagonal matrix with diagonal entries δi(i=1,2,…,r), and μ1≥μ2≥⋯≥μr>0 are the singular values ofA, that is, μi2(i=1,2,…,r) are the nonzero eigenvalues of A*A. This decomposition is extremely useful to represent the MPI of A∈Mm,n by [20, 23]
(1.11)A+=V[D-1000]U*,
where D-1=diag(μ1-1,μ2-1,…,μr-1)∈Mr  is a diagonal matrix with diagonal entries μi-1(i=1,…,r).Furthermore, thespectral norm of A is defined by
(1.12)‖A‖s=max⁡1≤i≤r⁡{μi}=μ1;‖A+‖s=1μr,
where μ1and μr are, respectively, the largest and smallest singular value of A.Generally speaking, the outer inverseAT,S(2) of a matrix A∈Mm,n, which is a unique matrix X∈Mn,m satisfying the following equations (see, e.g., [20, 24–27]):
(1.13)AXA=A,rang(A)=T,null(A)=S,
where T is a subspace of ℂn of s≤r, and S  is a subspace of ℂm of dimension m-s.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 AT,S(2), 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]:
(1.14)A+=Arang(A*),null(A*)(2).
Also, the representation and characterization for the outer generalized inverse AT,S(2) have been considered by many authors (see, e.g., [15, 16, 20, 27, 30, 31])Finally, given two matricesA=[aij]∈Mm,n, and B=[bkl]∈Mp,q, then the Kronecker product of A and B is defined by (see, e.g., [5, 7, 32–35])
(1.15)A⊗B=[aijB]∈Mmp,nq.
Furthermore, the Kronecker product enjoys the following well-known and important properties:(i)
The Kronecker product is associative and distributive with respect to matrix addition.(ii)
IfA∈Mm,n, B∈Mp,q, C∈Mn,r and D∈Mq,s, then
(1.16)(A⊗B)(C⊗D)=AC⊗BD(iii)
IfA∈Mm, B∈Mp are positive definite matrices, then for any real number r, we have:
(1.17)(A⊗B)r=Ar⊗Br.(iv)
IfA∈Mm,n, B∈Mp,q, then
(1.18)(A⊗B)+=A+⊗B+.


## 2. Convergent Moore-Penrose Inverse of Matrices

First, the need to computeA+ by using sequences method. The key to such results below is the following two Lemmas, due to Wei [23] and Wei and Wu [17], respectively.Lemma 2.1.
LetA∈Mm,n ba a matrix. Then
(2.1)A+=Â-1A*,
where Â=A*A⇃rang(A*) is the restriction of A*A on rang(A*).Lemma 2.2.
LetA∈Mm,n with rank(A)=r and Â=A*A⇃rang(A*).  Suppose Ω is  an open set such that σ(Â)⊂Ω⊂(0,∞).  Let {Sn(x)} be a family of continuous real valued function on Ω with limn→∞Sn(x)=1/x uniformly onσ(Â).  Then
(2.2)A+=lim⁡n→∞Sn(Â)A*.
Furthermore,
(2.3)‖Sn(Â)A*-A+‖2≤sup⁡x∈σ(Â)|Sn(x)x-1|‖A+‖2.
Moreover, for each λ∈σ(Â), we have
(2.4)μr2≤λ≤μ12.It is well known that the inverse of an invertible operator can be calculated by interpolating the function1/x, in a similar manner we will approximate the Moore-Penrose inverse by interpolating function 1/x and using Lemmas 2.1 and 2.2.One way to produce a family of functions{Sn(x)} which is suitable for use in the Lemma 2.2 is to employ the well known Euler-Knopp method. A series ∑n=0∞an is said to be Euler-Knopp summable with parameter α>0 to the value a if the sequence defined by
(2.5)Sn=α∑k=0n∑j=0k(kj)(1-α)k-jαjαj
converges to a. If ak=(1-x) for k=0,1,2,…, then we obtain as the Euler-Knopp transform of the series ∑k=0∞(1-x)k, the sequence given by
(2.6)Sn(x)=α∑k=0n(1-αx)k.
Clearly limn→∞Sn(x)=1/x  uniformly on any compact subset of the set
(2.7)Eα={x:|1-αx|<1}={x:0<x<2α}.
Another way to produce a family functions {Sn(x)} which is suitable also for use in the Lemma 2.2 is to employ the well-known Newton-Raphson method. This can be done by generating a sequence yn, where
(2.8)yn+1=yn-s(yn)s′(yn)=yn(2-xyn),
for suitable y0. Suppose that for α>0 we define a sequence of functions {Sn(x)} by
(2.9)S0(x)=α;Sn+1(x)=Sn(x)(2-xSn(x)).
In fact,
(2.10)xSn+1(x)-1=-(xSn(x)-1)2.
Iterating on this equality, it follows that if x is confined to a compact subset of Eα={x:0<x<2/α}. Then there is a constant β (defining on this compact set) with 0<β<1 and
(2.11)|xSn(x)-1|=|αx-1|2n≤β2n⟶0(n⟶∞).
According to the variational definition, A+b is the vector x∈ℂn which minimizes the functional ∥Ax-b∥2 and also has the smallest 2 norm among all such minimizing vectors. The idea of Tikhonov’s regularization [36, 37] of order zero is to approximately minimize both the functional ∥Ax-b∥2 and the norm ∥x∥2 by minimizing the functional g:ℂn→ℝ defined by
(2.12)g(x)=‖Ax-b‖22+t‖x‖22,
where t>0. The minimum of this functional will occur at the unique stationary point u of g, that is, the vector u which satisfies ∇g(u)=0. The gradient of g is given by
(2.13)∇g(x)=2(A*Ax-A*b)+2tx,
and hence the unique minimizer ut  satisfies
(2.14)ut=(A*A+tI)-1A*b.
On intuitive grounds, it seems reasonable to expect that
(2.15)lim⁡t→0+ut=(A*A)-1A*b=A+b.
Therefore, if we define a sequence of functions {Sn(x)} by using Euler-Knopp method, Newton-Raphson method and the idea of Tikhonov’s regularization that mentioned above, then we get the following nice Theorem.Theorem 2.3.
LetA∈Mm,n with rank(A)=r and 0<α<2μ1-2.  Then(i)
the sequence{An} defined by
(2.16)A0=αA*;An+1=(1-αA*A)An+αA*
converges to A+. Furthermore, the error estimate is given by
(2.17)‖An-A+‖2≤βn+1‖A+‖2,
where  0<β<1.(ii)
The sequence{An} defined by
(2.18)A0=αA*;An+1=An(2I-AAn)
converges to A+. Furthermore,  the error estimate is given by
(2.19)‖An-A+‖2≤β2n‖A+‖2.
where  0<β<1.(iii)
fort>0,
(2.20)A+=lim⁡t→0+(tI+A*A)-1A*.Thus, the error estimate is given by
(2.21)‖(tI+A*A)-1A*-A+‖2≤tμr2+t‖A+‖2.Proof.
(i) It follows fromσ(Â)⊆[μr2,μ12] that σ(Â)⊂(0,μ12], and hence we apply Lemma 2.2 if we choose the parameter α is such a way that (0,μ12]⊆Eα, where Eα is defined by (2.7). We may choose α such that 0<α<2μ1-2.  If we use the sequence defined by
(2.22)S0(x)=α;Sn+1(x)=(1-αx)Sn(x)+α,
it is easy to verify that
(2.23)lim⁡n→∞Sn(x)=1x,
uniformly on any compact subset of Eα. Hence, if 0<α<2μ1-2, then, applying Lemma 2.2, we get
(2.24)lim⁡n→∞Sn(Â)A*=A+.
But it is easy to see from (2.22) that Sn(Â)A*=An, where An is given by (2.16). This is surely the case if 0<α<2μ1-2, then, for such α, we have the representation
(2.25)A+=α∑j=0n(1-αA*A)jA*.
Note that if we set
(2.26)An=α∑j=0n(1-αA*A)jA*,
then we get (2.16).
To derive an error estimate for the Euler-Knopp method, suppose that0<α<2μ1-2. If the sequence Sn(x) is defined as in (2.22), then
(2.27)Sn+1(x)x-1=(1-αx)(Sn(x)x-1).
Therefore, since S0=α,
(2.28)|Sn(x)x-1|=|1-αx|n+1.
By μr2≤x≤μ12  for x∈σ(Â) and 0<α<2μ1-2, it follows that
(2.29)|1-αx|<β,
where β is given by
(2.30)β=max⁡⁡{|1-αμ12|,|1-αμr2|}.
Clearly,
(2.31)2>αμ12≥αμr2>0,
and therefore 0<β<1. From Lemma 2.2, we establish (2.17).
(ii) Using the Newton-Raphson iterations in (2.8)–(2.11) in conjunction with Lemma 2.2, we see that the sequence of {Sn(Â)} defined by
(2.32)S0(x)=αI;Sn+1(Â)=Sn(Â)(2I-A*ASn(Â))
has the property that limn→∞Sn(Â)A*=A+ uniformly in ℂm×n. If we set  An=Sn(Â)A*, then we get (2.18).
Ifx∈σ(Â) and 0<α<2μ1-2, then we see that |1-αx|≤β, where β is given by (2.30). It follows as in (2.11) and hence from Lemma 2.2, then we get the error bound as in (2.19).
(iii) If we set(2.33)St(x)=(t+x)-1:t>0
in Lemma 2.2 and using the idea of Tikhonov’s regularization as in (2.12)–(2.15), then it is easy to get (2.20) and (2.21).Huang and Zhang [28] presented the following sequence which is convergent to A+:
(2.34)A0=α0A*;An+1=αn+1(2An-AnAAn),n=1,2,…,
where αn+1∈[1,2] is called an acceleration parameter and chosen so as to minimize that which is bound on the maximum distance of any nonzero singular value of An+1A from 1. They chose A0 according to the first term of sequence (2.34) with α0=2/(μ1+μr), and let p0=α0μ1, then the acceleration parameters αn+1 and pn+1 have the following sequences:
(2.35)αn+1=21+(2-pn)pn,pn+1=αn+1(2-pn)pn.
We point out that the iteration (2.18) is a special case of an acceleration iteration (2.34). Further, we note that the above methods and the first-order iterative methods used by Ben-Israel and Greville [4] for computing A+ are a set of instructions for generating a sequence {An:n=1,2,3,…} converging to A+.Similarly, Liu et al. [25] introduced some necessary and sufficient conditions for iterative convergence to the generalized inverse AT,S(2) and its existence and estimated the error bounds of the iterative methods for approximating AT,S(2) by defining the sequence {An} in the following way:
(2.36)An=An-1+βX(I-AAn-1),n=1,2,…,
where β≠0 with X≠XAA0. Then the iteration (2.36) converges if and only ρ(I-βAX)<1, equivalently, ρ(I-βAX)<1. In which case, if rang(X)=T, null(X)=S, and rang(A0)⊂T. Then, AT,S(2) exists and {An} converges to AT,S(2). Furthermore, the error estimate is given by
(2.37)‖AT,S(2)-An‖≤|β|qn1-q‖X‖‖I-AA0‖=R(β,n),
where q=min{∥I-βXA∥,∥I-βAX∥}.What is the best valueβopt such that ρ(I-βAX) minimize in order to achieve good convergence? Unfortunately, it may be very difficult and still require further studies. If σ(AX) is a subset of ℝ and λmin=min{λ:λ∈σ(AX)}>0, analogous to [38, Example  4.1], we can have
(2.38)βopt=2λmin⁡+ρ(AX).


## 3. Convergent Infinite Products of Matrices

Trench [3, Definition  1] defined invertibly convergence of an infinite products matrices ∏m=1∞Bm as in the invertible of matrices Bm for all m>N  (where N  is an integer number). Here, we define the less restricted definitions of convergence of an infinite products ∏m=1∞Bm  and ∏m=1∞⊗Bm  for k×k  complex matrices such that
(3.1)∏m=rs⊗Bm={Bs⊗Bs-1⊗⋯⊗Brifr≤sIifr>s
as follows.Definition 3.1.
Let{Bm} be a sequence of k×k matrices. Then An infinite product ∏m=1∞Bm is said to be convergent if there is an integer N such that Bm≠0 (may Bm is invertible or not) for m≥N and
(3.2)Q=lim⁡n→∞∏m=NnBm
exists and is nonzero. In this case, we define
(3.3)∏m=1∞Bm=Q∏m=1N-1Bm.
Similarly, an infinite product ∏m=1∞⊗Bm converges if  there is an integer  N such that Bm≠0 for m≥N, and
(3.4)R=lim⁡n→∞∏m=Nn⊗Bm
exists and is nonzero. In this case, we define
(3.5)∏m=1∞⊗Bm=R⊗∏m=1N-1⊗Bm.
In the above Definition 3.1, the matrix R may be singular even if Bm is nonsingular  for all m≥1, but R may be singular if Bm is singular  for some m≥1. However, this definition does not require that Bm is invertible for large m.Definition 3.2.
Let{Bm} be a sequence of k×k matrices. Then an infinite product ∏m=1∞⊗Bm is said to be invertibly convergent if there is an integer N such that Bm is invertible for m≥N, and
(3.6)R=lim⁡n→∞∏m=Nn⊗Bm
exists and invertible. In this case, we define
(3.7)∏m=1∞Bm=R⊗∏m=1N-1⊗Bm.
The Definitions 3.1 and 3.2 have the following obvious consequence.Theorem 3.3.
Let{Bm}  be a sequence of k×k matrices such that the infinite products ∏m=1∞Bm and   ∏m=1∞⊗Bm are invertibly convergent. Then, both infinite products are convergent, but the converse, in general, is not true.Theorem 3.4.
Let{Bm} be a sequence of k×k matrices such that(i)
if  the infinite product∏m=1∞Bm is  convergent, then
(3.8)lim⁡m→∞Bm=QQ+,
where Q=limn→∞∏m=NnBm.(ii)
If  the infinite product∏m=1∞⊗Bm is converge, then
(3.9)lim⁡m→∞Bm=R⊗R+.Proof.
(i) Suppose that∏m=1∞Bm is convergent such that Bm≠0 when m≥N. Let Q=∏m=NnBm. Then limn→∞Qn=Q, where Q≠0. Therefore, limn→∞Qn+=Q+. Since Bn=QnQn-1+, we then have
(3.10)lim⁡n→∞Bn=(lim⁡n→∞Qn)(lim⁡n→∞Qn-1+)=QQ+.
But if Bm is invertible when m≥N, then Q is invertible and
(3.11)lim⁡n→∞Bn=(lim⁡n→∞Qn)(lim⁡n→∞Qn-1-1)=QQ-1=I.
Similarly, it is easy to prove (ii).If the infinite products∏m=1∞Bm and ∏m=1∞⊗Bm are invertibly convergent in Theorem 3.4, then we get the following corollary.Corollary 3.5.
Let{Bm}  be a sequence of k×k  matrices such that(i)
if  the infinite product∏m=1∞Bm is invertibly convergent, then
(3.12)lim⁡m→∞Bm=I(ii)
if  the infinite product∏m=1∞⊗Bm is invertibly convergent, then
(3.13)lim⁡m→∞Bm=R⊗R-1,
where R=limn→∞∏m=Nn⊗Bm.The main reason for interest in these products above is to generate matrix sequences for solving such matrix problems such as singular linear systems and singular coupled matrix equations. For example, Cao [21]  and Shi et al. [22] constructed the general stationary and nonstationary iterative process generated by  {Bm}m=1∞ for solving the singular linear system Ax=b, and Leizarowitz [39] established conditions for weak ergodicity of products, existence of optimal strategies for controlled Markov chains, and growth properties of certain linear nonautonomous differential equations based on a sequence (an infinite product) of stochastic matrices {Bm}m=1∞(∏m=1∞Bm). Also, as discussed in [2], the motivation of Definition 3.1 stems  from a question about linear systems of difference equations and coupled matrix equations, under what conditions on {Bm}m=1∞ of, for instance, the system xm=Bmxm-1,m=1,2,…, approach a finite nonzero limit whenever x0≠0? A system with property linear asymptotic equilibrium if and only if Bm is invertible for every m≥1 and ∏m=1∞Bm invertibly convergent, but a system with the so-called least-squares linear asymptotic equilibrium if Bm≠0 for every m≥1 and ∏m=1∞Bm converges.Because of Theorem3.4, we consider only infinite product of the form ∏m=1∞(I+Am), where limm→∞Am=0. We will write
(3.14)Pn=∏m=1n(I+Am);P=∏m=1∞(I+Am).
The following Theorem provides the convergence and invertibly convergence of the infinite product ∏m=1∞(I+Am), and the proof here is omitted.Theorem 3.6.
The infinite product∏m=1∞(I+Am) converges (invertibly converges) if ∑m=1∞∥Am∥p<∞,  for some p-norm ∥·∥.The following theorem relates convergence of an infinite product to the asymptotic behavior of least-square solutions of a related system of difference equations.Theorem 3.7.
The infinite product∏m=1∞(I+Am) converges if and only if for some integer N≥1 the matrix difference equation
(3.15)Xn+1=(I+An)Xn:n≥N
has a least-square solution {Xn}n=N∞ such that  limn→∞Xn=I. In this case,
(3.16)Pn=Xn+1XN+∏m=1N-1(I+Am);P=XN+∏m=1N-1(I+Am).Proof.
Suppose that∏m=1∞(I+Am) converges. Choose N so that I+Am≠0 for m≥N, and let Q=∏m=Nn(I+Am). Define
(3.17)Xn={∏m=Nn-1(I+Am)}Q+:n≥N.
Then {Xn}n=N∞ is a solution of (3.15) such that limn→∞Xn=I.
Conversely, suppose that (3.15) has a least-square solution {Xn}n=N∞ such that limn→∞Xn=I. Then(3.18)Xn={∏m=Nn-1(I+Am)}XN:n≥N,
where XN≠0. Therefore,
(3.19)∏m=Nn-1(I+Am)=XnXN+:n≥N.
Letting n→∞ shows that
(3.20)∏m=N∞(I+Am)=XN+
which implies that ∏m=1∞(I+Am) converges.
From (3.14) and (3.18), we get(3.21)Pn=∏m=1n(I+Am)=Xn+1XN+{∏m=1N-1(I+Am)},
which proves the first expression in (3.16). From (3.14) and (3.20), we get
(3.22)P=∏m=1∞(I+Am)=XN+∏m=1N-1(I+Am).
which proves the second expression in (3.16).Remark 3.8.
If the infinite product∏m=1∞(I+Am) is invertibly convergent in Theorem 3.7, then
(3.23)Pn=Xn+1XN-1∏m=1N-1(I+Am);P=XN-1∏m=1N-1(I+Am).Theorem3.7 indicates the connection between convergence of an infinite product of k×k matrices ∏m=1∞(I+Am) and the asymptotic properties of least-squares solutions of matrix difference equation as defined in (3.15). We can say that (3.15) has a least-squares linear asymptotic equilibrium if every least-squares solution of (3.15) for which XN≠0 that approaches I as n→∞.For example, Ding and Chen [5] and generalized later by Kılıçman and Al-Zhour [8]  studied the convergence of least-square solutions to the coupled Sylvester matrix equations
(3.24)AX+YB=C;DX+YE=F,
where A,D∈Mm, B,E∈Mn, C,F∈Mm,n are given constant matrices, and X,Y∈Mm,n are the unknown matrices to be solved. If the coupled Sylvester matrix equation determined by (3.24) has a unique solution X and Y, then the following iterative solution Xn+1 and Yn+1 given by [5, 8]:
(3.25)Xn+1=Xn+μG+[C-AXn-YnBF-DXn-YnE];Yn+1=Yn+μ[C-AXn-YnBF-DXn-YnE]H+,
where G=[AD] and H=[BE] are full column and full row-rank matrices, respectively;
(3.26)μ=1m+norμ=1λmax⁡[G(G*G)-1G*]+λmax⁡[H*(HH*)-1H*].
converges to X and Y for any finite initial values X0 and Y0.The convergence factorμ in (3.26) may not be the best and may be conservative. In fact, there exists a best μ such that the fast convergence rate of Xk to X and Yk to Y can be obtained as in numerical examples given by Cao [21] and Kılıçman and Al-Zhour [8]. How to find the connections between convergence of an infinite products of k×k  matrices and least-square solutions of coupled Sylvester matrix equation in (3.24) requires further research.


## 4. Numerical Examples

Here, we give some numerical example for computing outer inverseAT,S(2) and Moore-Penrose inverse AT,S(2)  by applying sequences methods which are studied and derived in Section 2. Our results are obtained in this Section by choosing Frobenius norm (∥·∥2) and using MATLAB software.Example 4.1.
Consider the matrix(4.1)A=[20.40.40.40200002000020000]∈C5×4.
Let T=ℂ4,e=[0,0,0,0,1]T∈ℂ5,S=span{e}.
Take(4.2)X=A0=[0.4000000.4000000.4000000.40]∈C4×5.HereR(β,n)=|β|qn(1-q)-1∥X∥2∥I-AA0∥2. Clearly rang(X)=T, null(X)=S, and rang(A0)⊂T. By computing, we have
(4.3)AT,S(2)=[0.5-0.1-0.1-0.1000.5000000.5000000.50]∈C4×5.
In order to satisfy q=min{∥I-βXA∥2,∥I-βAX∥2}<1, we get that β should satisfy the following 0.63474563020816<β<1.79243883581125.From the iteration (2.5) in [24, Theorem  2.2], Let A∈ℂm×r, and T and S  be given subspaces of ℂm×r such that there exists AT,S(2). Then the sequence (An)n in ℂm×r defined in the following way:
(4.4)Rn=PA(T),S-PA(T),SAAn,An+1=A0Rn+Ann=0,1,2,…
converges to AT,S(2) if and only if rang(A0)⊂T  and ρ(R0)<1 (where R(n)=(∥R0∥n+1/(1-∥R0∥))∥A0∥ and R0=PA(T),S-PA(T),SAA0).In this case, if∥R0∥=q<1, then
(4.5)‖AT,S(2)-An‖≤qn+11-q‖A0‖.Thus we have Tables1 and 2 respectively, where
(4.6)R(β,n)=|β|qn1-q‖X‖‖I-AA0‖,R(n)=‖R0‖n+11-‖R0‖‖A0‖.
Table 1 illustrates that β=1.25 is best value such that ∥AT,S(2)-An∥ reaches 2.020636405220133×10-16 iterating the least number of steps, the reason for which is that such a β is calculating by using (2.38). Thus, for an appropriate β, the iteration is better than the iteration (4.4) (cf. Tables 1 and 2). And with respect to the error bound, the iterations for almost all are also better. Let us take the error bound smaller than 10-16; for instance, the number of steps of iterations in Table 1 is smaller than that of the iterations in Table 2. But, in practice, we consider also the quantity ∥An-An-1∥ in order to cease iteration since there exist such cases as β=1.25. For example, for ∥An-An-1∥<μ∥An∥, where μ is the machine precision, the iteration for β=1.25 only needs 3 steps. Therefore, in general, the iteration of (2.36) is better than the iteration (4.4) for an appropriate β. Note that the iterations in both Tables 1 and 2 indicate a fast convergence for the quantity ∥AT,S(2)-An∥ more than the quantity R(β,n) in Table 1 and the quantity R(n) in Table 2 since each of R(β,n) and R(n) is an upper bound of the quantity ∥AT,S(2)-An∥, and, to find the best or least upper bound for the quantity, ∥AT,S(2)-An∥ requires further research.Table 1
Results for Example4.1 using the iterations (2.36) and (2.37) for β=1.25.
nE(n)=∥AT,S(2)-An∥2R(β,n)∥An-An-1∥213.464101615138×10-20.589433845824430.2433105012119322.020636405220133×10-160.204185873733733.464101615138×10-232.020636405220133×10-167.073206149893×10-2042.020636405220133×10-162.450230484805×10-2052.020636405220133×10-168.48784737987×10-3082.020636405220133×10-163.528331898118804×10-40102.020636405220133×10-164.233998277742565×10-50122.020636405220133×10-165.080797933291078×10-60142.020636405220133×10-166.096957519949294×10-70Table 2
Results for Example4.1 using the iterations (4.4) and (4.5).
nE(n)=∥AT,S(2)-An∥2R(n)∥An-An-1∥252.844222213540086×10-42.06917386097900×10-29.230211265187883×10-471.485461921416498×10-54.89980370280000×10-35.070609777923677×10-5101.609342324440632×10-75.64613461842000×10-45.734399999925342×10-7131.625817851681746×10-96.50614556479466×10-55.938877957045854×10-9161.571929418012274×10-117.497152117521204×10-65.835397257624137×10-11191.474087541668810×10-138.639107335285478×10-75.537232598086118×10-13221.295028878277394×10-159.955003497416000×10-85.130710307439707×10-15241.942890293094024×10-162.357344828188108×10-82.403703357979455×10-16252.020636405220133×10-161.147133503351594×10-84.807406715958910×10-17Example 4.2.
Consider the matrix(4.7)A=[22102371471008-113-1-113-3-213-24981-2491-75-12-6651450-22]∈C8×5.
Then by computing we have
(4.8)A+=[2.1129808×10-24.6153846×10-3-2.1073718×10-37.6041667×10-33.8060897×10-39.3108974×10-32.2115385×10-32.0528846×10-3-2.0833333×10-31.0016026×10-3-1.1097756×10-22.7403846×10-2-3.8862179×10-3-2.7604167×10-24.2067308×10-3-7.9166667×10-3-5.0000000×10-33.3750000×10-2-5.4166667×10-31.0416667×10-35.5128205×10-39.8076923×10-3-8.9743590×10-4-5.0000000×10-33.2051282×10-31.4318910×10-2-2.5961538×10-3-2.0136218×10-21.2812500×10-3-6.2099359×10-34.8958333×10-3-1.5000000×10-21.5312500×10-21.2395833×10-22.4604166×10-31.5064103×10-37.4038462×10-3-1.6987179×10-3-5.0000000×10-31.6025641×10-3]T∈C5×8.Thus, (see Tables3 and 4).Table 3
Results for Example4.2 using an acceleration iteration (2.34).
n∥XAX-X∥∥AXA-A∥∥(AX)*-AX∥∥(XA)*-XA∥10.0101348.81841.7808×10-155.5517×10-1620.00175051.15642.25×10-154.2698×10-1632.7532×10-50.0180182.2253×10-156.3027×10-1647.2707×10-94.6827×10-62.6984×10-154.1249×10-1654.4545e×10-162.9558×10-132.7911×10-155.4318×10-1667.4467×10-176.4138e×10-153.2971×10-151.0954×10-1671.4818×10-175.2442×10-152.9214×10-154.6700×10-1682.9927×10-184.3997×10-152.7295×10-157.1699×10-16Table 4
Results for  Example4.2 using the iteration (2.18).
n∥XAX-X∥∥AXA-A∥∥(AX)*-AX∥∥(XA)*-XA∥10.01305414.0886.8532×10-163.7436×10-1620.00723625.61729.7451×10-163.3446×10-1630.00132840.892079.6044×10-163.7163×10-1643.4199×10-50.0224341.2231×10-165.2064×10-1652.1567×10-81.4143×10-51.9419×10-159.9378×10-1668.5747×10-155.6179×10-122.1874×10-157.6483×10-1671.2675×10-165.8744×10-152.7634×10-158.3034×10-1682.5102×10-163.0005×10-152.5232×10-158.1248×10-16Example 4.3.
We generate a random matrix,A∈ℂ100×80 by using MATLAB, and then we obtain the results as in Tables 5 and 6.Table 5
Results for a random matrixA∈ℂ100×80 using an acceleration iteration (2.34).
n∥XAX-X∥∥AXA-A∥∥(AX)*-AX∥∥(XA)*-XA∥10.09471345.1374.1569×10-123.2203×10-1220.1232945.0321.6962e×10-121.42×10-1230.7453845.1956.3723×10-135.9666×10-1340.8565444.9852.1027×10-132.3662×10-1350.9510344.5996.4066×10-148.656×10-1461.281142.9451.7448×10-142.9065×10-1471.611836.6154.8196×10-156.8183×10-1581.484821.9841.1672×10-158.3827×10-1690.508216.03971.8502×10-155.3961×10-16100.0384450.409522.0718e×10-161.5552×10-16110.000176270.00184392.0888×10-161.222×10-16123.5682×10-93.746×10-82.0885×10-161.2319×10-17134.661×10-156.7753×10-142.0898×10-161.588×10-17145.3525×10-156.805×10-142.0897×10-161.4264×10-17154.0198×10-156.8871×10-142.0891×10-161.4191×10-17Table 6
Results for a random matrixA∈ℂ100×80 using the iteration (2.18).
n∥XAX-X∥∥AXA-A∥∥(AX)*-AX∥∥(XA)*-XA∥10.06027645.0972.05091.5239×10-420.08954245.0823.9427×10-14.0096×10-530.1011945.0818.3223×10-28.8403×10-640.09628745.1211.627×10-31.8814×10-650.109645.1853.2587××10-43.8609×10-760.1211145.2076.1535×10-57.7762×10-870.1382345.1461.1418×10-51.4853×10-980.1718244.9712.0472×10-62.9817×10-1090.2159444.6233.5587×10-75.422e×10-11100.2702243.9626.0384×10-89.337×10-12110.3478242.6889.8614×10-91.5714×10-12120.4613440.2491.5947×10-92.5064×10-13130.6216935.7662.5794×10-103.6162×10-14140.7464128.2324.0931×10-114.6385×10-15150.7359217.5886.0751×10-125.0276×10-16160.393826.82598.0089×10-133.3054×10-16170.0681221.02819.0865×10-142.3201×10-17180.00157960.0233219.2857×10-152.1781×10-17198.1324e-0071.2001e-0059.2907×10-161.7884×10-17Note that from Tables3, 4, 5, and 6, it is clear that the quantities ∥XAX-X∥, and ∥AXA-A∥, ∥(AX)*-AX∥, ∥(XA)*-XA∥ are becoming smaller and smaller and goes to zero as n  increases in both iterations (2.34) and (2.18). We can also conclude that both iterations almost have same fast of convergence when the dimension of any arbitrary matrix A is not so large, but the acceleration iteration (2.34) is better more than the iteration (2.18) when the dimension of any arbitrary matrix A is so large with an appropriate acceleration parameter αn+1∈[1,2].


## 5. Concluding Remarks

In this paper, we have studied some matrix sequences convergence to the Moore-Penrose inverseA+ and outer inverse AT,S(2) of an arbitrary matrix A∈Mm,n. The key to derive matrix sequences which are convergent to weighted Drazin and weighted Moore-Penrose inverses is the Lemma 2.2. Some sufficient conditions for infinite products ∏m=1∞Bm and ∏m=1∞⊗Bm of k×k matrices are also derived. In our opinion, it is worth establishing some connections between convergence of an infinite products of k×k  matrices and least-square solutions of such linear singular systems as well as the singular coupled matrix equations.


## 6. Acknowledgments

The authors express their sincere thanks to the referee(s) for careful reading of the manuscript and several helpful suggestions. The authors also gratefully acknowledge that this research was partially supported by Ministry of Science, Technology and Innovations (MOSTI), Malaysia under the e-Science Grant 06-01-04-SF1050.



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*Source: 536935-2011-10-19.xml*