The $ m $-weak group inverse for rectangular matrices

D. Mosić; P. S. Stanimirović; L. A. Kazakovtsev; D. Mosić; P. S. Stanimirović; L. A. Kazakovtsev

doi:10.3934/era.2024083

Electronic Research Archive

2024, Volume 32, Issue 3: 1822-1843. doi: 10.3934/era.2024083

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Research article

The $ m $-weak group inverse for rectangular matrices

1.
University of Niš, Faculty of Sciences and Mathematics, Višegradska 33, 18000 Niš, Serbia
2.
Laboratory Hybrid Methods of Modelling and Optimization in Complex Systems, Siberian Federal University, Prosp. Svobodny 79, 660041 Krasnoyarsk, Russia

Received: 06 December 2023 Revised: 29 January 2024 Accepted: 04 February 2024 Published: 29 February 2024

An extension of the $ m $-weak group inverse (or $ m $-WGI) on the set of rectangular matrices is provided to solve some systems of matrix equations. The extension is termed as the $ W $-weighted $ m $-WGI (or $ W $-$ m $-WGI). The $ W $-$ m $-WGI presents a new, wider class of generalized inverses which involves some already defined generalized inverses, such as the $ m $-WGI, $ W $-weighted weak group, and $ W $-weighted Drazin inverse. Basic properties and diverse characterizations are proved for $ W $-$ m $-WGI. Several expressions for computing $ W $-$ m $-WGI are proposed in terms of known generalized inverses and projectors, as well as its limit and integral representations. The $ W $-$ m $-WGI class is utilized to solve some linear matrix equations and express their general solutions. Some new properties of the weighted generalized group inverse and recognized properties of the $ W $-weighted Drazin inverse are obtained as corollaries. Numerical and symbolic test examples are presented to verify the obtained results.
- $ m $-weak group inverse,
- $ W $-weighted weak group inverse,
- $ W $-weighted core-EP inverse,
- $ W $-weighted Drazin inverse
Citation: D. Mosić, P. S. Stanimirović, L. A. Kazakovtsev. The $ m $-weak group inverse for rectangular matrices[J]. Electronic Research Archive, 2024, 32(3): 1822-1843. doi: 10.3934/era.2024083

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Abstract

An extension of the $ m $-weak group inverse (or $ m $-WGI) on the set of rectangular matrices is provided to solve some systems of matrix equations. The extension is termed as the $ W $-weighted $ m $-WGI (or $ W $-$ m $-WGI). The $ W $-$ m $-WGI presents a new, wider class of generalized inverses which involves some already defined generalized inverses, such as the $ m $-WGI, $ W $-weighted weak group, and $ W $-weighted Drazin inverse. Basic properties and diverse characterizations are proved for $ W $-$ m $-WGI. Several expressions for computing $ W $-$ m $-WGI are proposed in terms of known generalized inverses and projectors, as well as its limit and integral representations. The $ W $-$ m $-WGI class is utilized to solve some linear matrix equations and express their general solutions. Some new properties of the weighted generalized group inverse and recognized properties of the $ W $-weighted Drazin inverse are obtained as corollaries. Numerical and symbolic test examples are presented to verify the obtained results.

References

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