Some representations for the Drazin inverse of a modified matrix

Shakoor, Abdul; Yang, Hu; Ali, Ilyas

doi:10.1007/s10092-013-0098-0

Cited by 6 publications

(12 citation statements)

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“…Wei [29] derived explicit expressions of the Drazin inverse of a modified matrix A − CB under certain circumstances, which extends results of [22,26] and can be used to present a perturbation bound for the Drazin inverse studied by Campbell and Meyer [3]. Recently, Cvetković-Ilić and Ljubisavljević [10] and Dopazo and Martínez-Serrano [12] extended results of Wei [29] to the modified matrix A − CD d B. Mosić [23] and Shakoor, Yang and Ali [24] generalized results of [10,12]. These results are useful for perturbation problems and updating finite Markov chains.…”

Section: Introductionmentioning

confidence: 61%

“…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 [13,14,17]. Formulae of such type have been developed in the context of generalized inverses, such as the Moore-Penrose inverse [1,20], the weighted Moore-Penrose inverse [28], the group inverse [6], the weighted Drazin inverse [8], the generalized Drazin inverse [11,19], and especially the Drazin inverse [10,12,23,24,29].…”

Section: Introductionmentioning

confidence: 99%

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Representations for the Drazin inverse of a modified matrix

Zhang

2015

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Section: Introductionmentioning

confidence: 61%

Section: Introductionmentioning

confidence: 99%

Representations for the Drazin inverse of a modified matrix

Zhang

2015

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“…We can see from [27] how Corollary 3.3 gives and generalizes the Sherman-Morrison-Woodbury formula and some results in [11,21,22,25].…”

Section: Applications To S D and Z Dmentioning

confidence: 94%

On the existence of group inverses of Peirce corner matrices

Zhang

Mosić

Tam

2019

Linear Algebra and its Applications

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We give some statements that are equivalent to the existence of group inverses of Peirce corner matrices of a 2 ×2 block matrix and its generalized Schur complements. As applications, several new results for the Drazin inverses of the generalized Schur complements and the 2 × 2 block matrix are obtained and some of them generalize several results in the literature.2000 Mathematics Subject Classification. 15A09; 15A30; 65F20.

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“…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].…”

mentioning

confidence: 90%

“…Zhang and Du [4] relaxed and removed some assumptions of theorems proved in [3,9,10,12] and presented formulae for (A − CD D B) D under weaker conditions.…”

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confidence: 99%

Expressions for the Drazin inverse of a modified matrix

Stojkovic¹,

Mosić²

2018

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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].

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Some representations for the Drazin inverse of a modified matrix

Cited by 6 publications

References 4 publications

Representations for the Drazin inverse of a modified matrix

Representations for the Drazin inverse of a modified matrix

On the existence of group inverses of Peirce corner matrices

Expressions for the Drazin inverse of a modified matrix

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