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<p>Recently, the weak Drazin inverse and its characterization have emerged as important topics in the study of matrices of index <i>k</i>. In this article, we have revisited W-weighted DMP and MPD inverses of Kyrchei [Determinantal Representations of the Weighted Core-EP, DMP, MPD, and CMP Inverses, J. Math., 9816038 (2020)] and constructed a general class of unique solutions to certain matrix equations. Moreover, we have generalized the W-weighted Drazin inverse of Meng [The DMP inverse for rectangular matrices, Filomat, 31(19), 6015–6019 (2017)] using the minimal rank W-weighted weak Drazin inverse. In addition to that, we have derived several equivalent properties of W-weighted DMP and MPD inverses for minimal rank W-weighted weak Drazin inverse of rectangular matrices. Furthermore, some projection-based results are discussed for the characterization of minimal rank W-weighted Drazin inverse, along with some new expressions that are derived for MPD and DMP inverses. Thereby, we have elaborated certain expressions of the perturbation formula for W-weighted weak MPD and DMP inverses. As an application, we establish the reverse and forward order laws using the W-weighted weak Drazin inverse and the minimal rank W-weighted weak Drazin inverse, and apply these results to solve certain matrix equation.</p>

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On Characterizations of W-weighted DMP and MPD Inverses

  • 期刊论文

  • 发表日期：

    2026年3月8日

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  • Rajesh Senapati,
  • Ashish Kumar Nandi

摘要

Recently, the weak Drazin inverse and its characterization have emerged as important topics in the study of matrices of index k. In this article, we have revisited W-weighted DMP and MPD inverses of Kyrchei [Determinantal Representations of the Weighted Core-EP, DMP, MPD, and CMP Inverses, J. Math., 9816038 (2020)] and constructed a general class of unique solutions to certain matrix equations. Moreover, we have generalized the W-weighted Drazin inverse of Meng [The DMP inverse for rectangular matrices, Filomat, 31(19), 6015–6019 (2017)] using the minimal rank W-weighted weak Drazin inverse. In addition to that, we have derived several equivalent properties of W-weighted DMP and MPD inverses for minimal rank W-weighted weak Drazin inverse of rectangular matrices. Furthermore, some projection-based results are discussed for the characterization of minimal rank W-weighted Drazin inverse, along with some new expressions that are derived for MPD and DMP inverses. Thereby, we have elaborated certain expressions of the perturbation formula for W-weighted weak MPD and DMP inverses. As an application, we establish the reverse and forward order laws using the W-weighted weak Drazin inverse and the minimal rank W-weighted weak Drazin inverse, and apply these results to solve certain matrix equation.