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.

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.

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 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.

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.

scholarly journals Generalizations of some conditions for Drazin inverses of the sum of two matrices

Filomat ◽  
2018 ◽  
Vol 32 (18) ◽  
pp. 6417-6430
Author(s):  
Ramesh Yousefi ◽  
Mansour Dana
Keyword(s):  
Schur Complement ◽  
Block Matrix ◽  
Generalized Schur Complement

In this article, we present some formulas of the Drazin inverses of the sum of two matrices under the conditions P2QP = 0, P2Q2 = 0, QPQ = 0 and PQP2 = 0, PQ2 = 0, QP3 = 0 respectively. These conditions are weaker than those used in some literature on this subject. Furthermore, we apply our results to give the representations for the Drazin inverses of block matrix (A B C D) (A and D are square matrices) with generalized Schur complement is zero.

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