On the generalized Drazin inverse in Banach algebras in terms of the generalized Schur complement

2016 ◽  
Vol 284 ◽  
pp. 162-168 ◽  
Author(s):  
J. Robles ◽  
M.F. Martínez-Serrano ◽  
E. Dopazo
Keyword(s):  
Banach Algebras ◽  
Schur Complement ◽  
Drazin Inverse ◽  
Generalized Drazin Inverse ◽  
Generalized Schur Complement

scholarly journals Formulas for the Drazin inverse of matrices over skew fields

Filomat ◽  
2016 ◽  
Vol 30 (12) ◽  
pp. 3377-3388 ◽  
Author(s):  
Lizhu Sun ◽  
Baodong Zheng ◽  
Shuyan Bai ◽  
Changjiang Bu
Keyword(s):  
Schur Complement ◽  
Drazin Inverse ◽  
Explicit Formulas ◽  
Block Matrices ◽  
Skew Fields ◽  
Generalized Drazin Inverse ◽  
Generalized Schur Complement ◽  
Appl Math

For two square matrices P and Q over skew fields, the explicit formulas for the Drazin inverse of P+Q are given in the cases of (i) PQ2=0, P2QP=0, (QP)2=0; (ii) P2QP=0, P3Q=0, Q2=0, which extend the results in [M.F. Mart?nez-Serrano et al., On the Drazin inverse of block matrices and generalized Schur complement, Appl. Math. Comput.] and [C. Deng et al., New additive results for the generalized Drazin inverse, J. Math. Anal. Appl.]. By using these formulas, the representations for the Drazin inverse of 2 x 2 block matrices over skew fields are obtained, which also extend some existing results.

The drazin inverse of the sum of two matrices and its applications

Filomat ◽  
2017 ◽  
Vol 31 (16) ◽  
pp. 5151-5158 ◽  
Author(s):  
Lingling Xia ◽  
Bin Deng
Keyword(s):  
Schur Complement ◽  
Block Matrix ◽  
Drazin Inverse ◽  
Numerical Example ◽  
Generalized Schur Complement

In this paper, we give the results for the Drazin inverse of P + Q, then derive a representation for the Drazin inverse of a block matrix M = (A B C D) under some conditions. Moreover, some alternative representations for the Drazin inverse of MD where the generalized Schur complement S = D-CADB is nonsingular. Finally, the numerical example is given to illustrate our results.

Further Results on the Generalized Drazin Inverse of Block Matrices in Banach Algebras

2014 ◽  
Vol 38 (2) ◽  
pp. 483-498 ◽  
Author(s):  
Milica Z. Kolundžija ◽  
Dijana Mosić ◽  
Dragan S. Djordjević
Keyword(s):  
Banach Algebras ◽  
Drazin Inverse ◽  
Block Matrices ◽  
Generalized Drazin Inverse

scholarly journals Representations of Generalized Inverses and Drazin Inverse of Partitioned Matrix with Banachiewicz-Schur Forms

2016 ◽  
Vol 2016 ◽  
pp. 1-14 ◽  
Author(s):  
Xiaoji Liu ◽  
Hongwei Jin ◽  
Jelena Višnjić
Keyword(s):  
Sufficient Conditions ◽  
Schur Complement ◽  
Block Matrix ◽  
Necessary And Sufficient Conditions ◽  
Generalized Inverses ◽  
Drazin Inverse ◽  
Group Inverse ◽  
Necessary And Sufficient ◽  
Generalized Schur Complement ◽  
Quotient Property

Representations of 1,2,3-inverses, 1,2,4-inverses, and Drazin inverse of a partitioned matrix M=ABCD related to the generalized Schur complement are studied. First, we give the necessary and sufficient conditions under which 1,2,3-inverses, 1,2,4-inverses, and group inverse of a 2×2 block matrix can be represented in the Banachiewicz-Schur forms. Some results from the paper of Cvetković-Ilić, 2009, are generalized. Also, we expressed the quotient property and the first Sylvester identity in terms of the generalized Schur complement.

scholarly journals The perturbation theory for the Drazin inverse and its applications II

2001 ◽  
Vol 70 (2) ◽  
pp. 189-198 ◽  
Author(s):  
Vladimir Rakočevič ◽  
Yimin Wei
Keyword(s):  
Banach Space ◽  
Perturbation Theory ◽  
Banach Algebras ◽  
Linear Operators ◽  
Drazin Inverse ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Generalized Drazin Inverse

AbstractWe study the perturbation of the generalized Drazin inverse for the elements of Banach algebras and bounded linear operators on Banach space. This work, among other things, extends the results obtained by the second author and Guorong Wang on the Drazin inverse for matrices.

scholarly journals A note on the formulas for the Drazin inverse of the sum of two matrices

2019 ◽  
Vol 17 (1) ◽  
pp. 160-167
Author(s):  
Xin Liu ◽  
Xiaoying Yang ◽  
Yaqiang Wang
Keyword(s):  
Schur Complement ◽  
Block Matrix ◽  
Drazin Inverse ◽  
Numerical Example ◽  
Generalized Schur Complement ◽  
Appl Math ◽  
Complex Block

Abstract In this paper we derive the formula of (P + Q)D under the conditions Q(P + Q)P(P + Q) = 0, P(P + Q)P(P + Q) = 0 and QPQ2 = 0. Then, a corollary is given which satisfies the conditions (P + Q)P(P + Q) = 0 and QPQ2 = 0. Meanwhile, we show that the additive formula provided by Bu et al. (J. Appl. Math. Comput. 38 (2012) 631-640) is not valid for all matrices which satisfies the conditions (P + Q)P(P + Q) = 0 and QPQ2 = 0. Also, the representation can be simplified from Višnjić (Filomat 30 (2016) 125-130) which satisfies given conditions. Furthermore, we apply our result to establish a new representation for the Drazin inverse of a complex block matrix having generalized Schur complement equal to zero under some conditions. Finally, a numerical example is given to illustrate our result.

On the Drazin inverse of block matrices and generalized Schur complement

2009 ◽  
Vol 215 (7) ◽  
pp. 2733-2740 ◽  
Author(s):  
M.F. Martínez-Serrano ◽  
N. Castro-González
Keyword(s):  
Schur Complement ◽  
Drazin Inverse ◽  
Block Matrices ◽  
Generalized Schur Complement

scholarly journals Representations for the Drazin inverse of a modified matrix

Filomat ◽  
2015 ◽  
Vol 29 (4) ◽  
pp. 853-863 ◽  
Author(s):  
Daochang Zhang ◽  
Xiankun Du
Keyword(s):  
Schur Complement ◽  
Drazin Inverse ◽  
Generalized Schur Complement

In this paper expressions for the Drazin inverse of a modified matrix A - CDdB are presented in terms of the Drazin inverses of A and the generalized Schur complement D - BAdC under weaker restrictions. Our results generalize and unify several results in the literature and the Sherman-Morrison- Woodbury formula.

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