Expressions for the Drazin inverse of a modified matrix

Abstract

We present some new expressions for the Drazin inverse of a modified matrix A − CD D B in terms of the Drazin inverse of A and its generalized Shur's complement D−BA D C under weaker conditions. Some results in recent literature are unified and generalized, and most importantly, we obtain new expressions of generalized Sherman-Morrison-Woodbury formula under weaker restrictions. IntroductionThroughout this paper, let C n×m denote the set of all n × m complex matrices. For A ∈ C n×n , the Drazin inverse of A is the unique matrix A D ∈ C n×n such thatwhere k is the least non-negative integer such that rank(A k ) = rank(A k+1 ), called index of A and denoted by ind(A). Denote by A e = AA D and A π = I − A e , where I denotes the identity matrix of proper size. When ind(A) = 1, A D = A # is called the group inverse of A. If ind(A) = 0, then A D = A −1 .If A and D are invertible matrices and B and C are matrices of appropriate size such that D − BA −1 C and A − CD −1 B are invertible, then original Sherman-Morrison-Woodbury (from now on SMW) formula gives explicit expression for the inverse of a modified matrix A − CD −1 B of A in terms of its Schur's complement D − BA −1 C [11,13]. Precisely, SMW formula is expressed asThis formula has a lot of applications in statistics, networks, optimization and partial differential equations (see [5,6,8]).Main objective of this article is to study the Drazin inverse of a modified matrix A − CD D B in terms of the Drazin inverse of A and the Drazin inverse of its generalized Schur complement, since the Drazin inverse has many applications in numerical analysis, singular differential or difference equations, Markov chains, cryptography, etc. Some of mentioned applications can be found in [1, 2].Under some assumptions, Wei [12] gave representations of the Drazin inverse of a modified matrix A − CB (in this case D = I). His results were generalized in [3,9,10].