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 The pseudo core inverse of a companion matrix

2018 ◽  
Vol 55 (3) ◽  
pp. 407-420
Author(s):  
Yuefeng Gao ◽  
Jianlong Chen ◽  
Pedro Patrício ◽  
Dingguo Wang
Keyword(s):  
Toeplitz Matrix ◽  
Toeplitz Matrices ◽  
Generalized Inverses ◽  
Companion Matrix ◽  
Complex Matrix ◽  
Complex Matrices ◽  
The Core ◽  
Core Inverse ◽  
Companion Matrices ◽  
Existence Criteria

The notion of core inverse was introduced by Baksalary and Trenkler for a complex matrix of index 1. Recently, the notion of pseudo core inverse extended the notion of core inverse to an element of an arbitrary index in *-rings; meanwhile, it characterized the core-EP inverse introduced by Manjunatha Prasad and Mohana for complex matrices, in terms of three equations. Many works have been done on classical generalized inverses of companion matrices and Toeplitz matrices. In this paper, we discuss existence criteria and formulae of the pseudo core inverse of a companion matrix over a *-ring. A {1,3}-inverse of a Toeplitz matrix plays an important role in that process.

scholarly journals Representations of Generalized Inverses and Drazin Inverse of Partitioned Matrix with Banachiewicz-Schur Forms

2016 ◽  
Vol 2016 ◽  
pp. 1-14 ◽  
Author(s):  
Xiaoji Liu ◽  
Hongwei Jin ◽  
Jelena Višnjić
Keyword(s):  
Sufficient Conditions ◽  
Schur Complement ◽  
Block Matrix ◽  
Necessary And Sufficient Conditions ◽  
Generalized Inverses ◽  
Drazin Inverse ◽  
Group Inverse ◽  
Necessary And Sufficient ◽  
Generalized Schur Complement ◽  
Quotient Property

Representations of 1,2,3-inverses, 1,2,4-inverses, and Drazin inverse of a partitioned matrix M=ABCD related to the generalized Schur complement are studied. First, we give the necessary and sufficient conditions under which 1,2,3-inverses, 1,2,4-inverses, and group inverse of a 2×2 block matrix can be represented in the Banachiewicz-Schur forms. Some results from the paper of Cvetković-Ilić, 2009, are generalized. Also, we expressed the quotient property and the first Sylvester identity in terms of the generalized Schur complement.

scholarly journals The W-weighted Drazin-star matrix and its dual

2021 ◽  
Vol 37 (37) ◽  
pp. 72-87
Author(s):  
Mengmeng Zhou ◽  
Jianlong Chen ◽  
Néstor Thome
Keyword(s):  
Linear Systems ◽  
Generalized Inverses ◽  
Drazin Inverse ◽  
Core Inverse ◽  
New Generation

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.

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 *-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 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.

Solving fuzzy linear systems using a block representation of generalized inverses: The group inverse

2018 ◽  
Vol 353 ◽  
pp. 66-85 ◽  
Author(s):  
Biljana Mihailović ◽  
Vera Miler Jerković ◽  
Branko Malešević
Keyword(s):  
Linear Systems ◽  
Generalized Inverses ◽  
Group Inverse ◽  
Fuzzy Linear Systems

Further results on the reverse order law for the Core inverse in C*-algebras

2021 ◽  
Vol 67 (8) ◽  
pp. 50-60
Author(s):  
Krishnaswamy D ◽  
Vijayaselvi V
Keyword(s):  
Reverse Order ◽  
C Algebras ◽  
The Core ◽  
Reverse Order Law ◽  
Core Inverse

Regular factorizations and perturbation analysis for the core inverse of linear operators in Hilbert spaces

2018 ◽  
Vol 96 (10) ◽  
pp. 1943-1956 ◽  
Author(s):  
Qianglian Huang ◽  
Saijie Chen ◽  
Zhirong Guo ◽  
Lanping Zhu
Keyword(s):  
Hilbert Spaces ◽  
Perturbation Analysis ◽  
Linear Operators ◽  
The Core ◽  
Core Inverse
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