A note on additive results for the Drazin inverse

Abstract

For two square matrices that commute, we present some additive results for the Drazin inverse. We also give the application to relative perturbation of eigenvalues when the perturbed matrix commutes with the original matrix and perturbation bounds of the Drazin inverse.

Keywords:

• Drazin inverse
• additive result
• commutativity
• eigenvalue

AMS Subject Classifications::

• 15A09
• 65F20

Acknowledgements

The authors thank two referees for their detailed comments and valuable suggestions which greatly improved our presentation. Y. Wei thanks Prof R.E. Hartwig, C.D. Meyer, Ren-cang Li, Qiang Ye and Fu-zhen Zhang, for their useful discussions. Y. Wei is supported by the National Natural Science Foundation of China under grant 10871051, Shanghai Science & Technology Committee under grant 09DZ2272900, Shanghai Education Committee (Dawn project) and Doctoral Program of the Ministry of Education under grant 20090071110003. C. Deng is supported by a grant from the Ph.D. Programs Foundation of Ministry of Education of China (No. 20094407120001).