On deriving the Drazin inverse of a modified matrix

Dopazo, Esther; Martínez-Serrano, M.F.

doi:10.1016/j.laa.2011.06.023

Cited by 6 publications

(10 citation statements)

References 6 publications

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“…The next theorem is the generalization of the Theorem 2.1 [3] which is proved for the modified matrices. We will prove the following result for the elements of Banach algebra and generalize the SMW formula to the case of generalized Drazin inverse.…”

Section: Resultsmentioning

confidence: 83%

Generalized Sherman-Morrison-Woodbury formula for the generalized Drazin inverse in Banach algebra

Kolundžija

2017

Filomat

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Section: Resultsmentioning

confidence: 83%

Generalized Sherman-Morrison-Woodbury formula for the generalized Drazin inverse in Banach algebra

Kolundžija

2017

Filomat

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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: 91%

“…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

Filomat

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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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On deriving the Drazin inverse of a modified matrix

Cited by 6 publications

References 6 publications

Generalized Sherman-Morrison-Woodbury formula for the generalized Drazin inverse in Banach algebra

Generalized Sherman-Morrison-Woodbury formula for the generalized Drazin inverse in Banach algebra

On the existence of group inverses of Peirce corner matrices

Expressions for the Drazin inverse of a modified matrix

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