Representations For Drazin Inverses Of Modified Matrices And Generalized Drazin Inverses Of Operator Matrices

The concept of the generalized Drazin inverse (GD-inverse) in a Banach algebra was introduced by Koliha [81]. Let A be a complex unital Banach algebra. An element a of A is generalized Drazin invertible in case there is an element b∈A satisfying ab=ba, bab=b, and a-a2b is quasinilpotent. Such b, if it exists, is unique; it is called a generalized Drazin inverse of a, and will be denoted by ad. Then the spectral idempotent aπ of a corresponding to0is given by aπ=1-aad.The GD-inverse was extensively investigated for matrices over complex Banach al-gebras and matrices of bounded linear operators over complex Banach spaces. The GD-inverse of the operator matrix has various applications in singular differential equa-tions and singular difference equations, Markov chains and iterative methods and so on (see [8,19-21,28,30,52,56,113-115]), and [45-47,64,65,70,80,83,90,91,93,104,109].The generalized Drazin inverse is a generalization of Drazin inverses and group in-verses. The study on representations for the Drazin inverse of block matrices essen-tially originated from finding the general expressions for the solutions to singular sys-tems of differential equations [21-23]. Until now, there have been many formulae for the Drazin inverse of general2×2block matrices under some restrictive assumptions (see [40,52,53,61,62,66,71,88,92]).Some results of the Drazin inverse have been developed in the GD-inverse of operator matrices over Banach spaces (see [27,44,50,52,61]).The classical Sherman-Morrison-Woo dbury formula reads (A-CD-1B)-1=A-1+A-1C(D-BA-1-C)-1BA-1, where A and D are invertible matrices (not necessarily with the same size) and B and C are matrices with the appropriate size such that D-BA-1C (and so A-CD-1B) is invertible ([102,117]). The matrix A-CD-1B is called a modified matrix of A, and D-BA-1C is called the Schur complement. The Sherman-Morrison-Woodbury formula allows one to compute the inverse of a modified matrix in terms of the inverses of the original matrix and its Schur complement. Inverse matrix modification formulae of such type have been studied extensively and has numerous applications in various fields such as statistics, networks, structural analysis, numerical analysis, optimization and partial different equations, etc., see [63,68,74]. Formulae of such type have been developed in the context of generalized inverses, such as the Moore-Penrose inverse [2,89], the weighted Moore-Penrose inverse [111], the group inverse [25], the weighted Drazin inverse [36], the generalized Drazin inverse [49,87], and especially the Drazin inverse [43,54,95,101,112].The Drazin inverse of a complex square matrix A is the unique matrix Ad such that AAd=AdA, AdAAd=Ad, Ak=Ak+1Ad, where k is the smallest non-negative integer such that rank(Ak)=rank(Ak+1), called index of A and denoted by ind(A). If ind(A)=1, then Ad is called the group inverse of A and denoted by A#. The Drazin inverse is a generalization of inverses and group inverses of matrices. There are widespread applications of Drazin inverses in various fields, such as differential equations, control theory, Markov chains, iterative methods and so on (see [7,22]).Recently some years, representations for the Drazin inverse of block matrices, the GD-inverse of the operator matrix, the Drazin inverse of the sums of square matrices, the Drazin inverse of Modified matrices, and perturbations have been extensively concerned.In this paper, we focus on the representations for the generalized Drazin inverses of operator matrices, the Drazin inverses of Modified matrices, and the Drazin inverses of sums of square matrices P+Q+R+S.In section1, in the part of our research, we give a comprehensive survey on the rep-resentations for the Drazin inverses of Modified matrices, the generalized Drazin inverses of (2,2,0) operator matrices and2×2operator matrices.In section2, we derive new formulae for the GD-inverse of a (2,2,0) operator matrix under certain circumstances. Furthermore, we apply the GD-inverse of a (2,2,0) operator matrix to give representations of the GD-inverse of a2x2operator matrix under weaker restrictions, which generalizes and unifies several results of [27,44,50,52,61,96]. Theorem2.2.1Let be an operator matrix with E and F generalized Drazin invertible. If FdEFπ=0and FπFE=0, then N is generalized Drazin invertible, andTheorem2.3.1Let be an operator matrix with A and D generalized Drazin invertible. If (BC)dA(BC)π=0,(BC)πBCA=0, BDd=0, and BDiC=0for any positive integer i, then M is generalized Drazin invertible, and where In section3, first, based on an observation on Dedekind finiteness of unital matrix subalgebras, we relax and remove some restrictions in theorems in [43,54,95,101,112] and give representations of (A-CDdB)d under fewer and weaker conditions. Our results generalize and unify results of these literatures and the Sherman-Morrison-Woodbury formula. Second, We also derive a new formula for the Drazin inverse of A-CDdB, a corollary of which recovers a generalization of Jacobson’s Lemma (see [34, Theorem3.6]) for the case of matrices. Finally, we remark the conditions of existence of group inverses of Modified matrices in detail. Theorem3.1.1If AπCDdB=0and KDπZdH=KDdZπH, then with H=BAd and K=AdC, or alternatively where k=ind(A). Theorem3.1.3If AπCDdB=0, KTdHSAd=0and KTπDdH=0, then where H=BAd, K=AdC, T=HK and k=ind(A). Theorem3.1.4If AπGDdB=0,CΓdZDdB=0,GrdDπB=0and CΓπDdB=0, then where k=ind(A).Theorem3.2.1If AπCDdB=0,DπBAdC=0and DdBAAd=DdDBAd,then where π=ind(A),SA=AAdSAAd, and s=ind(Z).Theorem3.2.2If AπC=0,DdBAAd=DdDBAd and D"BAdC=0,thenSd=A(L2+N M)+C(ML+ZdM)-(A(LN+NZd)+C(MN+Z2d))DdB, where in which k=ind(A)and m=ind(z).Theorem3.2.3If AπCDdB=0,DπBAdC=0and DDdBAiAdCDDd=DdBAi+lAdC for i=0,1...,m-1,then where k=ind(A),SA=AAdsAAd, where SA=AAdSAAd,ZD=DDdZDDd and s=ind(ZD),k=s+2where Theorem3.3.3如果AπCDdB=0以及‖AdCDdB‖<1,则SA=SAAd且where k=ind(A).In section4of this paper.we focus on the representations for the Drazin inverses of SUlTIS of square matrices P+Q+R+S.Our results generalize result80f these literatures. Furthermore,we give the representations0f the Drazin inverses0f M0dmed ma七rices with idempotents.Theorem4.1.1Let P,Q,R and S be complex m-square matrices and M=P+Q+R+S.If PQ=QP=0,PS=SQ=QR=RP=0and SP=SR=0,then Md=Pd+X+Td, where for ind(T)=g, max{ind(S),ind(Q))≤l≤ind(S)+ind(Q), max{ind(R),ind(P))≤m≤ind(R)+ind(P), max{ind(S+Q),ind(Pπ(R+P))}≤n≤ind(S+Q)+ind(Pπ(R+P)).Theorem4.2.1Let A,B,C,D be complex matrices and let S=A-CDdB.If PπSPS=0,where ind(PSP)=t and ind(PπSPπ)=s.Then whereConventionally,write Ae for AAd.Theorem4.2.4Let A,B,C,D be complex matrices.If there exist idempotents P,Q such that X=(QDQ)dQBP(PAP)e=(QDQ)eQBP(PAP)d,(QDQ)eQBP(PAP)e=QBP,where ind(QDQ-XPGQ)=8,and ind(PAP))=r.Then (PAP-PCQX)d=(PAP)d-UX, where...