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Formulas for the Drazin inverse of matrices over skew fields

Sun Lizhu (School of Science, Harbin Institute of Technology, Harbin, PR China + City university of Hong Kong, College of Science and Engineering, Kowloon, Hong Kong)
Zheng Baodong (Harbin Institute of Technology, School of Science, Harbin, PR China)
Bai Shuyan (Harbin Engineering University, College of Science, Harbin, PR China)
Bu Changjiang (Harbin Engineering University, College of Science, Harbin, PR China + Harbin Engineering University, College of Automation, Harbin, PR China)

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. Martínez-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.

Keywords: skew field, Drazin inverse, Block matrix