Generalized inverses of a factorization in a ring with involution

Zhu, Huihui; Zhang, Xiaoxiang; Chen, Jianlong

doi:10.1016/j.laa.2015.01.025

Cited by 22 publications

(9 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“… Similar to generalized inverses of matrices, algebraists have defined Moore–Penrose inverses of elements in other general algebraic frameworks, such as, semigroups, rings, operator algebras, Hilbert

-modules (cf. [18] , [20] , [22] , [29] , [32] , [39] , [44] , [47] , [60] , [72] , [125] ). Without loss of generality, assume that the algebraic system is assumed to be a

-algebra

.…”

Section: Discussionmentioning

confidence: 99%

A family of 512 reverse order laws for generalized inverses of a matrix product: A review

Tian

2020

Heliyon

View full text Add to dashboard Cite

Reverse order laws for generalized inverses of matrix products are a classic object of study in the theory of generalized inverses. One of the well-known reverse order laws for a matrix product AB is , where denotes a -generalized inverse of matrix. Because -generalized inverse of a singular matrix is not unique, the relationships between both sides of the reverse order law can be divided into four situations for consideration. The aim of this paper is to give an overview of plenty of results concerning reverse order laws for -generalized inverses of the product AB , from the development of background and preliminary tools to the collection of miscellaneous formulas and facts on the reverse order laws in one place with cogent introduction and references for further study. We begin with the introduction of a linear mixed model and the presentation of two least-squares methodologies for estimating the fixed parameter vector β in the model, and the description of connections between the two types of least-squares estimators and the reverse order laws for generalized inverses of AB . We then prepare various necessary matrix study tools, including a general theory on linear or nonlinear algebraic matrix identities, a group of expansion formulas for calculating ranks of block matrices, two groups of explicit formulas for calculating the maximum and minimum ranks of , as well as necessary and sufficient conditions for to be invariant with respect to the choice of the generalized inverses, etc. Subsequently, we present a unified approach to the 512 matrix set inclusion problems associated with the above reverse order laws for the eight commonly-used types of generalized inverses of A , B , and AB by means of the definitions of generalized inverses, the block matrix methodology (BMM), the matrix equation methodology (MEM), and the matrix rank methodology (MRM).

show abstract

-modules (cf. [18] , [20] , [22] , [29] , [32] , [39] , [44] , [47] , [60] , [72] , [125] ). Without loss of generality, assume that the algebraic system is assumed to be a

-algebra

.…”

Section: Discussionmentioning

confidence: 99%

A family of 512 reverse order laws for generalized inverses of a matrix product: A review

Tian

2020

Heliyon

View full text Add to dashboard Cite

show abstract

“…Proof If a = aa * ax, then (ax) * is a {1, 4}-inverse of a according to [8,Lemma 2.2]. By the proof (ii)⇔(iii) in Theorem 2.16, it is known that a = (ax * x * a * a)a * a, and (ax * x * a * a) * is a {1, 3}-inverse of a.…”

Section: Proofmentioning

confidence: 98%

Further results on the inverse along an element in semigroups and rings

Zhu

Chen

Patrício

2015

Linear and Multilinear Algebra

Self Cite

View full text Add to dashboard Cite

In this paper, we introduce a new notion in a semigroup S as an extension of Mary's inverse. Let a, d ∈ S. An element a is called left (resp. right) invertibleAn existence criterion of this type inverse is derived. Moreover, several characterizations of left (right) regularity, left (right) π -regularity and left (right) * -regularity are given in a semigroup. Further, another existence criterion of this type inverse is given by means of a left (right) invertibility of certain elements in a ring. Finally, we study the (left, right) inverse along a product in a ring, and, as an application, Mary's inverse along a matrix is expressed.

show abstract

“…By [19,Lemma 2.2], we know that a is {1,3}-invertible and that u −1 is a {1,3}-inverse of a. Multiplying a = p+u by a * on the right gives aa * = ua * and hence a = aa * (u −1 ) * . Again, from [19,Lemma 2.2], it follows that a is {1,4}-invertible and that u −1 is a {1,4}-inverse of a. Thus, a ∈ R † and a † = a (1,4) aa (1,3)…”

Section: The Moore-penrose Inverse Of An Element and Its * -Cleannessmentioning

confidence: 99%

Generalized inverses and their relations with clean decompositions

2019

Self Cite

View full text Add to dashboard Cite

An element [Formula: see text] in a ring [Formula: see text] is called clean if it is the sum of an idempotent [Formula: see text] and a unit [Formula: see text]. Such a clean decomposition [Formula: see text] is said to be strongly clean if [Formula: see text] and special clean if [Formula: see text]. In this paper, we prove that [Formula: see text] is Drazin invertible if and only if there exists an idempotent [Formula: see text] and a unit [Formula: see text] such that [Formula: see text] is both a strongly clean decomposition and a special clean decomposition, for some positive integer [Formula: see text]. Also, the existence of the Moore–Penrose and group inverses is related to the existence of certain ∗-clean decompositions.

show abstract

Generalized inverses of a factorization in a ring with involution

Cited by 22 publications

References 11 publications

A family of 512 reverse order laws for generalized inverses of a matrix product: A review

A family of 512 reverse order laws for generalized inverses of a matrix product: A review

Further results on the inverse along an element in semigroups and rings

Generalized inverses and their relations with clean decompositions

Contact Info

Product

Resources

About