scholarly journals Rank equalities related to a class of outer generalized inverse

Filomat ◽  
2019 ◽  
Vol 33 (17) ◽  
pp. 5611-5622 ◽  
Author(s):  
Jianlong Chen ◽  
Sanzhang Xu ◽  
Julio Benítez ◽  
Xiaofeng Chen
Keyword(s):  
Generalized Inverse ◽  
Drazin Inverse ◽  
The Core ◽  
Core Inverse ◽  
Equivalent Conditions

In 2012, Drazin introduced a class of outer generalized inverse in a ring R, the (b,c)-inverse of a for a, b, c ? R and denoted by a||(b,c). In this paper, rank equalities of AkA||(B,C)- A||(B,C)Ak and (A*)kA||(B,C)- A||(B,C)(A*)k are obtained. As applications, weinvestigate equivalent conditions for the equalities (A*)kA||(B,C) = A||(B,C)(A*)k and AkA||(B,C) = A||(B,C)Ak. As corollaries we obtain rank equalities related to the Moore-Penrose inverse, the core inverse, and the Drazin inverse. The paper finishes with some rank equalities involving different e

scholarly journals Rank Equalities Related to the Generalized Inverses A‖(B1,C1), D‖(B2,C2) of Two Matrices A and D

Symmetry ◽  
2019 ◽  
Vol 11 (4) ◽  
pp. 539 ◽  
Author(s):  
Wenjie Wang ◽  
Sanzhang Xu ◽  
Julio Benítez
Keyword(s):  
Generalized Inverses ◽  
Drazin Inverse ◽  
Group Inverse ◽  
Complex Matrix ◽  
The Core ◽  
Core Inverse

Let A be an n × n complex matrix. The ( B , C ) -inverse A ∥ ( B , C ) of A was introduced by Drazin in 2012. For given matrices A and B, several rank equalities related to A ∥ ( B 1 , C 1 ) and B ∥ ( B 2 , C 2 ) of A and B are presented. As applications, several rank equalities related to the inverse along an element, the Moore-Penrose inverse, the Drazin inverse, the group inverse and the core inverse are obtained.

scholarly journals *-DMP elements in *-semigroups and *-rings

Filomat ◽  
2018 ◽  
Vol 32 (9) ◽  
pp. 3073-3085 ◽  
Author(s):  
Yuefeng Gao ◽  
Jianlong Chen ◽  
Yuanyuan Ke
Keyword(s):  
Positive Integer ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Drazin Inverse ◽  
Complex Matrices ◽  
Arbitrary Ring ◽  
The Core ◽  
Core Inverse ◽  
Necessary And Sufficient

In this paper, we investigate *-DMP elements in *-semigroups and *-rings. The notion of *-DMP element was introduced by Patr?cio and Puystjens in 2004. An element a is *-DMP if there exists a positive integer m such that am is EP. We first characterize *-DMP elements in terms of the {1,3}-inverse, Drazin inverse and pseudo core inverse, respectively. Then, we characterize the core-EP decomposition utilizing the pseudo core inverse, which extends the core-EP decomposition introduced by Wang for complex matrices to an arbitrary *-ring; and this decomposition turns to be a useful tool to characterize *-DMP elements. Further, we extend Wang?s core-EP order from complex matrices to *-rings and use it to investigate *-DMP elements. Finally, we give necessary and sufficient conditions for two elements a,b in *-rings to have aaD = bbD, which contribute to study *-DMP elements.

scholarly journals Further characterizations of the weak core inverse of matrices and the weak core matrix

2021 ◽  
Vol 7 (3) ◽  
pp. 3630-3647
Author(s):  
Zhimei Fu ◽  
◽  
Kezheng Zuo ◽  
Yang Chen
Keyword(s):  
Null Space ◽  
Matrix Equations ◽  
Range Space ◽  
The Core ◽  
Core Inverse ◽  
Core Matrix ◽  
Equivalent Conditions

<abstract><p>The present paper is devoted to characterizing the weak core inverse and the weak core matrix using the core-EP decomposition. Some new characterizations of the weak core inverse are presented by using its range space, null space and matrix equations. Additionally, we give several new representations and properties of the weak core inverse. Finally, we consider several equivalent conditions for a matrix to be a weak core matrix.</p></abstract>

scholarly journals The core inverse and constrained matrix approximation problem

2020 ◽  
Vol 18 (1) ◽  
pp. 653-661 ◽  
Author(s):  
Hongxing Wang ◽  
Xiaoyan Zhang
Keyword(s):  
Unique Solution ◽  
Approximation Problem ◽  
Frobenius Norm ◽  
Matrix Approximation ◽  
The Core ◽  
Core Inverse

Abstract In this article, we study the constrained matrix approximation problem in the Frobenius norm by using the core inverse: ||Mx-b|{|}_{F}=\hspace{.25em}\min \hspace{1em}\text{subject}\hspace{.25em}\text{to}\hspace{1em}x\in {\mathcal R} (M), where M\in {{\mathbb{C}}}_{n}^{\text{CM}} . We get the unique solution to the problem, provide two Cramer’s rules for the unique solution and establish two new expressions for the core inverse.

scholarly journals Determinantal Representations of the Core Inverse and Its Generalizations with Applications

2019 ◽  
Vol 2019 ◽  
pp. 1-13 ◽  
Author(s):  
Ivan I. Kyrchei
Keyword(s):  
Direct Method ◽  
Linear Equations ◽  
Close Relation ◽  
Numerical Example ◽  
Determinantal Representation ◽  
The Core ◽  
Core Inverse ◽  
Cramer’S Rule ◽  
The Right

In this paper, we give the direct method to find of the core inverse and its generalizations that is based on their determinantal representations. New determinantal representations of the right and left core inverses, the right and left core-EP inverses, and the DMP, MPD, and CMP inverses are derived by using determinantal representations of the Moore-Penrose and Drazin inverses previously obtained by the author. Since the Bott-Duffin inverse has close relation with the core inverse, we give its determinantal representation and its application in finding solutions of the constrained linear equations that is an analog of Cramer’s rule. A numerical example to illustrate the main result is given.

Characterizations for the n-strong Drazin invertibility in a ring

2020 ◽  
pp. 2150141
Author(s):  
Honglin Zou ◽  
Jianlong Chen ◽  
Huihui Zhu ◽  
Yujie Wei
Keyword(s):  
Positive Integer ◽  
Generalized Inverse ◽  
Product Formula ◽  
Drazin Inverse ◽  
Equivalent Definition ◽  
New Type ◽  
Definition Of ◽  
Existence Criteria

Recently, a new type of generalized inverse called the [Formula: see text]-strong Drazin inverse was introduced by Mosić in the setting of rings. Namely, let [Formula: see text] be a ring and [Formula: see text] be a positive integer, an element [Formula: see text] is called the [Formula: see text]-strong Drazin inverse of [Formula: see text] if it satisfies [Formula: see text], [Formula: see text] and [Formula: see text]. The main aim of this paper is to consider some equivalent characterizations for the [Formula: see text]-strong Drazin invertibility in a ring. Firstly, we give an equivalent definition of the [Formula: see text]-strong Drazin inverse, that is, [Formula: see text] is the [Formula: see text]-strong Drazin inverse of [Formula: see text] if and only if [Formula: see text], [Formula: see text] and [Formula: see text]. Also, we obtain some existence criteria for this inverse by means of idempotents. In particular, the [Formula: see text]-strong Drazin invertibility of the product [Formula: see text] are investigated, where [Formula: see text] is regular and [Formula: see text] are arbitrary elements in a ring.

Revisitation of the core inverse

2015 ◽  
Vol 20 (5) ◽  
pp. 381-385 ◽  
Author(s):  
Gaojun Luo ◽  
Kezheng Zuo ◽  
Liang Zhou
Keyword(s):  
The Core ◽  
Core Inverse
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