The core-EP inverse: a numerical approach for its acute perturbation

Abstract

This paper studies the concept of stable perturbation $B\in {\mathbb{C}}^{n\times n}$ for the core-EP inverse of a matrix $A\in {\mathbb{C}}^{n\times n}$ with index k. For a given stable perturbation B of A, explicit expressions of its core-EP inverse $B^{◯\$†\$}$ and its projection at zero $B^{\pi }$ are presented. Then, the perturbation bounds of $\parallel B^{◯\$†\$}-A^{◯\$†\$}\parallel /\parallel A^{◯\$†\$}\parallel$ and $\parallel B^{\pi }-A^{\pi }\parallel$ are given provided that B is a stable perturbation of A. In addition, we investigate the concept of acute perturbation of A. We give a perturbation analysis with respect to core-EP inverses. We provide a condition under which the acute perturbation coincides with the stable perturbation for core-EP inverses.

Keywords:

• Core inverse
• core-EP inverse
• stable perturbation
• acute perturbation

2020 Mathematics Subject Classifications:

• 15A09
• 65F20

Acknowledgments

We would like to thank the Editor and both Referees for their valuable comments and suggestions which improve the readability of the paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This research is supported by the National Natural Science Foundation of China (No. 12101315, 12101539,12171083), the China Scholarship Council (File No. 201906090122), Natural Science Foundation of Jiangsu Higher Education Institutions of China (21KJB110004), the Qing Lan Project of Jiangsu Province. The third author is partially supported by Ministerio de Economía, Industria y Competitividad of Spain (Grant Red de Excelencia RED2022-134176-T), partially supported by Universidad de Buenos Aires, Argentina (EXP-UBA: 13.019/2017, 20020170100350BA), and partially supported by Universidad Nacional de La Pampa, Argentina, Facultad de Ingeniería (Resolución del CD No 135/19); Universidad Nacional de Río Cuarto[Grant PPI 18/C559]; Universidad Nacional del Sur[Grant PGI 24/L108].