The Drazin inverse has applications in a number of areas such as control theory, Markov chains, singular differential and difference equations, and iterative methods in numerical linear algebra. The study on representations for the Drazin inverse of block matrices stems essentially from finding the general expressions for the solutions to singular systems of differential equations, and then stimulated by a problem formulated by Campbell. In 1983, Campbell (Campbell et al. (1976)) established an explicit representation for the Drazin inverse of a 2 × 2 block matrix M in terms of the blocks of the partition, where the blocks A and D are assumed to be square matrices. Special cases of the problems have been studied. In 2009, Chunyuan Deng and Yimin Wei found an explicit representation for the Drazin inverse of an anti-triangular matrix M, where A and BC are generalized Drazin invertible, if AπAB=0 and BC (I–Aπ) =0. Afterwards, several authors have investigated this problem under some limited conditions on the blocks of M. In particular, a representation of the Drazin inverse of M, denoted by Md. In this paper, we consider the Drazin inverse of a sum of two matrices and we derive additive formulas under the conditions of ABAπ=0 and BAπ=0 respectively. Precisely, for a block matrix M, we give a new representation of Md under some conditions that AB=0 and DCAπ=0. Moreover, some particular cases of this result related to the Drazin inverse of block matrices are also considered.
| Published in | Pure and Applied Mathematics Journal (Volume 8, Issue 3) |
| DOI | 10.11648/j.pamj.20190803.12 |
| Page(s) | 54-71 |
| Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
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Copyright © The Author(s), 2024. Published by Science Publishing Group |
Drazin Inverse, Block Matrices, Drazin Index
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APA Style
Xiaolan Qin, Ricai Luo. (2019). A Note on the Formulas for the Drazin Inverse of the Sum of Two Matrices and Its Applications. Pure and Applied Mathematics Journal, 8(3), 54-71. https://doi.org/10.11648/j.pamj.20190803.12
ACS Style
Xiaolan Qin; Ricai Luo. A Note on the Formulas for the Drazin Inverse of the Sum of Two Matrices and Its Applications. Pure Appl. Math. J. 2019, 8(3), 54-71. doi: 10.11648/j.pamj.20190803.12
AMA Style
Xiaolan Qin, Ricai Luo. A Note on the Formulas for the Drazin Inverse of the Sum of Two Matrices and Its Applications. Pure Appl Math J. 2019;8(3):54-71. doi: 10.11648/j.pamj.20190803.12
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Download@article{10.11648/j.pamj.20190803.12,
author = {Xiaolan Qin and Ricai Luo},
title = {A Note on the Formulas for the Drazin Inverse of the Sum of Two Matrices and Its Applications},
journal = {Pure and Applied Mathematics Journal},
volume = {8},
number = {3},
pages = {54-71},
doi = {10.11648/j.pamj.20190803.12},
url = {https://doi.org/10.11648/j.pamj.20190803.12},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20190803.12},
abstract = {The Drazin inverse has applications in a number of areas such as control theory, Markov chains, singular differential and difference equations, and iterative methods in numerical linear algebra. The study on representations for the Drazin inverse of block matrices stems essentially from finding the general expressions for the solutions to singular systems of differential equations, and then stimulated by a problem formulated by Campbell. In 1983, Campbell (Campbell et al. (1976)) established an explicit representation for the Drazin inverse of a 2 × 2 block matrix M in terms of the blocks of the partition, where the blocks A and D are assumed to be square matrices. Special cases of the problems have been studied. In 2009, Chunyuan Deng and Yimin Wei found an explicit representation for the Drazin inverse of an anti-triangular matrix M, where A and BC are generalized Drazin invertible, if AπAB=0 and BC (I–Aπ) =0. Afterwards, several authors have investigated this problem under some limited conditions on the blocks of M. In particular, a representation of the Drazin inverse of M, denoted by Md. In this paper, we consider the Drazin inverse of a sum of two matrices and we derive additive formulas under the conditions of ABAπ=0 and BAπ=0 respectively. Precisely, for a block matrix M, we give a new representation of Md under some conditions that AB=0 and DCAπ=0. Moreover, some particular cases of this result related to the Drazin inverse of block matrices are also considered.},
year = {2019}
}
TY - JOUR T1 - A Note on the Formulas for the Drazin Inverse of the Sum of Two Matrices and Its Applications AU - Xiaolan Qin AU - Ricai Luo Y1 - 2019/08/30 PY - 2019 N1 - https://doi.org/10.11648/j.pamj.20190803.12 DO - 10.11648/j.pamj.20190803.12 T2 - Pure and Applied Mathematics Journal JF - Pure and Applied Mathematics Journal JO - Pure and Applied Mathematics Journal SP - 54 EP - 71 PB - Science Publishing Group SN - 2326-9812 UR - https://doi.org/10.11648/j.pamj.20190803.12 AB - The Drazin inverse has applications in a number of areas such as control theory, Markov chains, singular differential and difference equations, and iterative methods in numerical linear algebra. The study on representations for the Drazin inverse of block matrices stems essentially from finding the general expressions for the solutions to singular systems of differential equations, and then stimulated by a problem formulated by Campbell. In 1983, Campbell (Campbell et al. (1976)) established an explicit representation for the Drazin inverse of a 2 × 2 block matrix M in terms of the blocks of the partition, where the blocks A and D are assumed to be square matrices. Special cases of the problems have been studied. In 2009, Chunyuan Deng and Yimin Wei found an explicit representation for the Drazin inverse of an anti-triangular matrix M, where A and BC are generalized Drazin invertible, if AπAB=0 and BC (I–Aπ) =0. Afterwards, several authors have investigated this problem under some limited conditions on the blocks of M. In particular, a representation of the Drazin inverse of M, denoted by Md. In this paper, we consider the Drazin inverse of a sum of two matrices and we derive additive formulas under the conditions of ABAπ=0 and BAπ=0 respectively. Precisely, for a block matrix M, we give a new representation of Md under some conditions that AB=0 and DCAπ=0. Moreover, some particular cases of this result related to the Drazin inverse of block matrices are also considered. VL - 8 IS - 3 ER -
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