Further characterizations and representations of the Minkowski inverse in Minkowski space

Jiale Gao; Kezheng Zuo; Qingwen Wang; Jiabao Wu; Jiale Gao; Kezheng Zuo; Qingwen Wang; Jiabao Wu

doi:10.3934/math.20231189

AIMS Mathematics

2023, Volume 8, Issue 10: 23403-23426. doi: 10.3934/math.20231189

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Research article

Further characterizations and representations of the Minkowski inverse in Minkowski space

1.
School of Mathematics and Statistics, Hubei Normal University, Huangshi 435002, China
2.
College of Science, Shanghai University, Shanghai 200444, China

Received: 23 May 2023 Revised: 25 June 2023 Accepted: 11 July 2023 Published: 25 July 2023
MSC : 15A09, 15A03, 15A24

This paper serves to identify some new characterizations and representations of the Minkowski inverse in Minkowski space. First of all, a few representations of $ \{1, 3^{\mathfrak{m}}\} $-, $ \{1, 2, 3^{\mathfrak{m}}\} $-, $ \{1, 4^{\mathfrak{m}}\} $- and $ \{1, 2, 4^{\mathfrak{m}}\} $-inverses are given in order to represent the Minkowski inverse. Second, some famous characterizations of the Moore-Penrose inverse are extended to that of the Minkowski inverse. Third, using the Hartwig-Spindelböck decomposition, we present a representation of the Minkowski inverse. And, based on this result, an interesting characterization of the Minkowski inverse is showed by a rank equation. Finally, we obtain several new representations of the Minkowski inverse in a more general form, by which the Minkowski inverse of a class of block matrices is given.
- Minkowski inverse,
- Minkowski space,
- Hartwig-Spindelböck decomposition,
- full-rank factorization
Citation: Jiale Gao, Kezheng Zuo, Qingwen Wang, Jiabao Wu. Further characterizations and representations of the Minkowski inverse in Minkowski space[J]. AIMS Mathematics, 2023, 8(10): 23403-23426. doi: 10.3934/math.20231189

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Abstract

This paper serves to identify some new characterizations and representations of the Minkowski inverse in Minkowski space. First of all, a few representations of $ \{1, 3^{\mathfrak{m}}\} $-, $ \{1, 2, 3^{\mathfrak{m}}\} $-, $ \{1, 4^{\mathfrak{m}}\} $- and $ \{1, 2, 4^{\mathfrak{m}}\} $-inverses are given in order to represent the Minkowski inverse. Second, some famous characterizations of the Moore-Penrose inverse are extended to that of the Minkowski inverse. Third, using the Hartwig-Spindelböck decomposition, we present a representation of the Minkowski inverse. And, based on this result, an interesting characterization of the Minkowski inverse is showed by a rank equation. Finally, we obtain several new representations of the Minkowski inverse in a more general form, by which the Minkowski inverse of a class of block matrices is given.

References

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