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On Generalized Inverses of m × n Matrices Over a Pseudoring

Abstract

A generalized inverse of an m × n matrix A over a pseudoring means an n × m matrix G satisfying AGA = A. In this paper we give a characterization of matrices having generalized inverses. Also, we introduce and study a space decomposition of a matrix, and prove that a matrix is decomposable if and only if it has a generalized inverse. Finally, we establish necessary and sufficient conditions for a matrix to possess various types of g-inverses including Moore–Penrose inverse.

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Acknowledgements

This work is supported by the National Natural Science Foundation of China (Grant No.11861045) and the Hongliu Foundation of First-class Disciplines of Lanzhou University of Technology, China.

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Correspondence to Lin Yang.

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Yang, L., Yang, SL. On Generalized Inverses of m × n Matrices Over a Pseudoring. Vietnam J. Math. 50, 261–274 (2022). https://doi.org/10.1007/s10013-021-00502-x

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  • DOI: https://doi.org/10.1007/s10013-021-00502-x

Keywords

  • Idempotent matrix
  • Regular matrix
  • Generalized inverse
  • Space decomposition

Mathematics Subject Classification (2010)

  • 15B99
  • 06E75
  • 15A09
  • 15A23