The Drazin inverse of the sum of four matrices and its applications

ABSTRACT

Our aim is to establish relations between the Drazin inverses of the pesudo-block matrix $(P,Q,R,S)$ and the block matrix $\left[\begin{array}{cc}P& R\\ S& Q\end{array}\right]$, where $R^2=S^2=0$. Based on the relations, we give representations for the Drazin inverse of the sum P + Q + R + S under mild restrictions. As the applications, several expressions for the Drazin inverse of a $2\times 2$ block matrix are presented under some assumptions. Our results generalize several results in the literature.

KEYWORDS:

• Drazin inverse
• pesudo-block matrix
• pesudo-block decomposition

2000 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 15A09
• 15A30
• 65F20

Acknowledgments

Zhang would like to thank Prof. Xiankun Du for his suggestions that greatly improved the original manuscript and Prof. Tin-Yau Tam for his valuable comments during Zhang's visiting Auburn University. The authors are also thankful for the careful reviews of referees and the editor.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Daochang Zhang http://orcid.org/0000-0002-9648-362X

Additional information

Funding

The first author is supported by the National Natural Science Foundation of China (NSFC) (No. 11371165; No. 61672149; No. 51708091) and the Scientific and Technological Development Program Foundation of Jilin Province, China (No. 20170520052JH). The second author is supported by the Ministry of Education and Science, Republic of Serbia (No. 174007).