Characterizations of k-commutative equalities for some outer generalized inverses

ABSTRACT

In this paper, we present necessary and sufficient conditions for the k-commutative equality $A^{k}X=X{A}^k$, where X is an outer generalized inverse of the square matrix A. Also, we give new representations for core EP, DMP, and CMP inverses of square matrices as outer inverses with prescribed null space and range. In addition, we characterize the core EP inverse as the solution of a new system of matrix equations.

KEYWORDS:

• Core inverse
• core EP inverse
• DMP inverse
• CMP inverse
• k-commutative equalities

AMS CLASSIFICATION:

• 15A09

Acknowledgements

The authors would like to thank the referees for their valuable comments which helped to improve the readability of the manuscript.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

D. E. Ferreyra F. E. Levis Partially supported by a Consejo Nacional de Investigaciones Científicas y Técnicas CONICET's Posdoctoral Research Fellowship, UNRC [grant number PPI 18/C472] and CONICET [grant number PIP 112-201501-00433CO], respectively. N. Thome Partially supported by Secretaría de Estado de Investigación, Desarrollo e Innovación Ministerio de Economía, Industria y Competitividad of Spain [grant number DGI MTM2013-43678-P and Grant Red de Excelencia PMTM2017-90682-REDT]. D. E. Ferreyra and N. Thome Partially supported by Universidad Nacional de La Pampa (UNLPam), Facultad de Ingeniería [grant Resol. No 155/14].