AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 324836 10.1155/2014/324836 324836 Letter to the Editor Extension of the GSMW Formula in Weaker Assumptions http://orcid.org/0000-0001-7709-3143 Wang Wenfeng 1,2 http://orcid.org/0000-0002-9569-728X Chen Xi 1 Kurasov Pavel 1 State Key Laboratory of Desert and Oasis Ecology, Xinjiang Institute of Ecology and Geography, Chinese Academy of Sciences, Urumqi 830011 China ucas.ac.cn 2 University of Chinese Academy of Sciences, Beijing 100093 China ucas.ac.cn 2014 1352014 2014 03 03 2014 15 04 2014 30 4 2014 2014 Copyright © 2014 Wenfeng Wang and Xi Chen. 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.

In this note, the generalized Sherman-Morrison-Woodbury (for short GSMW) formula (A+YGZ)=AAY(G+ZAY)ZA is extended under some assumptions weaker than those used by Duan, 2013.

1. Introduction

Denote by (,𝒦) (by (), when =𝒦) the set of all bounded linear operators from into 𝒦, where and 𝒦 are complex Hilbert spaces. For T(,𝒦), let T*, (T), and 𝒩(T) be the adjoint, the range, and the null space of T, respectively. Recall that the original SMW  formula (1) is only valid when A(), G(𝒦), and G-1+Z*A-1Y are invertible and the SMW formula has the form (1)(A+YGZ*)-1=A-1-A-1Y(G-1+Z*A-1Y)-1Z*A-1, where Y,Z(𝒦,).

Let I be the identity in () and let T(). Recall that the standard inverse T-1 of T must satisfy (I) TT-1=T-1T=I, while the generalized inverse S of T need only to satisfy (I)    TST=T. Note that S is unique if imposed additional conditions as (II) STS=S, (III) (TS)*=TS, (IV) (ST)*=ST, (V) TS=ST, and (VI) TkST=Tk, where S() satisfying (II) are called {2}-inverse of T, denoted by S=T-. Similarly, (I, II, V)-inverses are called group inverses, denoted by S=T#. (I, II, III, IV)-inverses are Moore-Penrose inverses, denoted by S=T+. And (II, V, VI)-inverses are called Drazin inverses, denoted by S=TD (see ), where k is the Drazin index of T. Note that the standard inverse, the group inverse, the Moore-Penrose inverse, and the Drazin inverse all belong to the 2-inverse. It is straight that the SMW formula holds for all the inverses if and only if it holds for the {2}-inverse.

Because of its wide applications in statistics, networks, structural analysis, asymptotic analysis, optimization, and partial differential equations (see ), the properties and generalizations of the SMW formula have caught mathematicians attention (see ). Duan (see ) finally generalized the SMW formula to the {2}-inverse (hence, to all the inverses, uniformly denoted by T). Under some sufficient conditions (see ), the generalized Sherman-Morrison-Woodbury (for short GSMW) formula has the form (2)(A+YGZ*)=A-AY(G+Z*AY)Z*A, where A(), G(𝒦), and Y,Z(𝒦,).

Duan questioned whether the GSMW formula can be extended in some weaker assumptions. This problem is worthy of being followed up.

2. Main Result

The following two lemmas are used to prove the main result.

Lemma 1.

If A() and P=P2(), then AP=A if and only if 𝒩(P)𝒩(A).

Lemma 2.

Let A(), G(𝒦), and Y,Z(𝒦,). Let also B=A+YGZ*, T=G+Z*AY, and X=AAY(G+Z*AY)Z*A. Then, the following three statements are equivalent:

the GSMW formula holds;

A(YGZ*-YTZ*)X=AYTZ*AYGZ*X;

X(YGZ*-YTZ*)A=XYGZ*AYTZ*A.

Proof.

The GSMW formula holds if and only if XBX=X. But it is easy to see that XAA=AAX=X. Hence, we have (3)XBX=(A-AYTZ*A)(A+YGZ*)X=(AA+AYGZ*-AYTZ*AA-AYTZ*AYGZ*)X=X+AYGZ*X-AYTZ*X-AYTZ*AYGZ*X=X+A(YGZ*-YTZ*)X-AYTZ*AYGZ*X and, meanwhile, (4)XBX=X(A+YGZ*)(A-AYTZ*A)=X(AA+YGZ*A-AAYTZ*A-YGZ*AYTZ*A)=X+XYGZ*A-XYTZ*A-XYGZ*AYTZ*A=X+X(YGZ*-YTZ*)A-XYGZ*AYTZ*A.

It is immediate that the three statements are equivalent.

Now, the first main result of this paper is given as follows.

Theorem 3.

Let A(), G(𝒦), and Y,Z(𝒦,). Let also B=A+YGZ* and T=G+Z*AY. The GSMW formula holds if one of the two following statements holds:

(Z*)(G), 𝒩(TT)𝒩(Y);

𝒩(G)𝒩(Y), (Z*)(TT).

Proof.

Note that Z*AY=T-G.

Assume that (Z*)(G), 𝒩(TT)𝒩(Y). By Lemma 1, we have Y(I-TT)=0 and (I-GG)Z*=0. Hence, (AYGZ*-AYTZ*)X-AYTZ*AYGZ*X = AY(I-TT)GZ*X-AYT(I-GG)Z*X=0.

Assume that 𝒩(G)𝒩(Y), (Z*)(TT). By Lemma 1, we have Y(I-GG)=0 and (I-TT)Z*=0. Hence, X(YGZ*-YTZ*)A-XYGZ*AYTZ*A = XYG(I-TT)Z*A--XY(I-GG)TZ*A=0.

By Lemma 2, The GSMW formula holds if one of (i) and (ii) holds.

3. Concluding Remark

According to Theorem 3 in this paper, Theorem 5 and Corollary 6 in  still hold under weaker assumptions. It must be noted that there are no assumptions on B in Theorem 3; hence, it also present more convenience than Theorem 3 and Corollary 4 in  in applications. The results are even robust for the finite dimensional case. Nevertheless, it remains undetermined whether these assumptions are the weakest. We would like to propose this unresolved issue as an open question for international research interest.

Conflict of Interests

The authors declare that there is no conflict of interests.

Acknowledgments

This research is financially supported by the CAS-SAFEA Innovation Team Project “Research on Ecological Transect in Arid Land of Central Asia.”

Bartlett M. S. An inverse matrix adjustment arising in discriminant analysis Annals of Mathematical Statistics 1951 22 107 111 MR0040068 10.1214/aoms/1177729698 ZBL0042.38203 Sherman J. Morrison W. J. Adjustment of an inverse matrix corresponding to a change in one element of a given matrix Annals of Mathematical Statistics 1950 21 124 127 MR0035118 10.1214/aoms/1177729893 ZBL0037.00901 Woodbury M. A. Inverting Modified Matrices 1950 42 Princeton, NJ, USA Statistical Research Group, Princeton University Ben-Israel A. Greville T. N. E. Generalized Inverses: Theory and Applications 2003 2nd New York, NY, USA Springer MR1987382 Hager W. W. Updating the inverse of a matrix SIAM Review 1989 31 2 221 239 10.1137/1031049 MR997457 ZBL0671.65018 Deng C. Y. A generalization of the Sherman-Morrison-Woodbury formula Applied Mathematics Letters 2011 24 9 1561 1564 10.1016/j.aml.2011.03.046 MR2803709 ZBL1241.47003 Steerneman T. van Perlo-ten Kleij F. Properties of the matrix A-XYx2a; Linear Algebra and Its Applications 2005 410 70 86 10.1016/j.laa.2004.10.028 MR2177831 ZBL1082.15011 Dou Y.-N. Du G.-C. Shao C.-F. Du H.-K. Closedness of ranges of upper-triangular operators Journal of Mathematical Analysis and Applications 2009 356 1 13 20 10.1016/j.jmaa.2009.02.014 MR2524211 ZBL1206.47001 Duan Y. T. A generalization of the SMW formula of operator A+YGZx2a; to the {2}-inverse case Abstract and Applied Analysis 2013 2013 4 10.1155/2013/694940 694940 MR3108663