Determinantal Representations of the Weighted Core-EP, DMP, MPD, and CMP Inverses

Kyrchei, Ivan

doi:10.1155/2020/9816038

Cited by 6 publications

(4 citation statements)

References 46 publications

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“…The main goals of this chapter are investigations of the MPCEP and CEPMP inverses, introductions and representations of new right and left MPCEPMP inverses over the quaternion skew field, and obtaining of their determinantal representations as a direct method of their constructions. The chapter develops and continues the topic raised in a number of other works [28][29][30][31][32][33], where determinantal representations of various generalized inverses were obtained.…”

Section: Introductionmentioning

confidence: 76%

Quaternion MPCEP, CEPMP, and MPCEPMP Generalized Inverses

Kyrchei¹

2023

Matrix Theory - Classics and Advances

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A generalized inverse of a matrix is an inverse in some sense for a wider class of matrices than invertible matrices. Generalized inverses exist for an arbitrary matrix and coincide with a regular inverse for invertible matrices. The most famous generalized inverses are the Moore–Penrose inverse and the Drazin inverse. Recently, new generalized inverses were introduced, namely the core inverse and its generalizations. Among them, there are compositions of the Moore–Penrose and core inverses, MPCEP (or MP–Core–EP) and EPCMP (or EP–Core–MP) inverses. In this chapter, the notions of the MPCEP inverse and CEPMP inverse are expanded to quaternion matrices and introduced new generalized inverses, the right and left MPCEPMP inverses. Direct method of their calculations, that is, their determinantal representations are obtained within the framework of theory of quaternion row-column determinants previously developed by the author. In consequence, these determinantal representations are derived in the case of complex matrices.

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Section: Introductionmentioning

confidence: 76%

Quaternion MPCEP, CEPMP, and MPCEPMP Generalized Inverses

Kyrchei¹

2023

Matrix Theory - Classics and Advances

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“…In Lemma 2.4 (iii), we have proved that the W -weighted Drazin-star matrix is an outer inverse matrix with prescribed range and null space. Since outer inverses with prescribed range and null space have a remarkable significance in Matrix Theory, the W -weighted Drazin-star matrix can provide theoretical value for future practice [11,17].…”

Section: Endmentioning

confidence: 99%

“…In literature, a variety of algorithms for computing generalized inverses were designed (e.g., [6,14,20,13]). Another papers related to weighted inverses that motivated our research are [11] by Kyrchei. It is well known that one of the most important applications of the inverse of a matrix (square and nonsingular) is its involvement to solve linear systems. This application remains being important for rectangular and singular matrices by considering outer inverses [2].…”

mentioning

confidence: 99%

The W-weighted Drazin-star matrix and its dual

Zhou

Chen

Thome

2021

ELA

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After decades studying extensively two generalized inverses, namely Moore--Penrose inverse and Drazin inverse, currently, we found immersed in a new generation of generalized inverses (core inverse, DMP inverse, etc.). The main aim of this paper is to introduce and investigate a matrix related to these new generalized inverses defined for rectangular matrices. We apply our results to the solution of linear systems.

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“…Some extensions to Minkowski spaces appear in [31] and to tensors in [27]. Weighted core-EP inverses were analyzed, for example, in [28] and determinantal applications can be seen in [14].…”

Section: Introductionmentioning

confidence: 99%