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 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.

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 On some previous results for the Drazin inverse of block matrices

Filomat ◽  
2016 ◽  
Vol 30 (1) ◽  
pp. 125-130 ◽  
Author(s):  
Jelena Visnjic
Keyword(s):  
Block Matrix ◽  
Drazin Inverse ◽  
Short Paper ◽  
Block Matrices ◽  
Matrix Anal ◽  
Appl Math

This short paper is motivated by the paper of Bu et al. [C. Bu, C. Feng, P. Dong, A note on computational formulas for the Drazin inverse of certain block matrices, J. Appl. Math. Comput.(38) (2012) 631-640], where the authors gave additive formula for Drazin inverse for matrices under new conditions, and two representations under some specific conditions. Here is shown that the additive formula is not valid for all matrices which satisfy given conditions. Also, here is proved that the representations which were given in mentioned paper do not extend the results given by Hartwig et al. [R. Hartwig, X. Li, Y. Wei, Representations for the Drazin inverse of a 2 _ 2 block matrix, SIAM J. Matrix. Anal. Appl. (27)(2006) 757-771 ], in fact they are equivalent.

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.

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