Core Inverse Research Articles

The core inverses of linear combinations of two core invertible matrices

ABSTRACT We present the core inverses of linear combinations of two core invertible matrices. Similarly, the dual core inverses of linear combinations of two dual core invertible matrices are also given. Furthermore, sufficient conditions, which guarantee that the difference of two core invertible matrices is core invertible, are presented.

The core and dual core inverse of a morphism with factorization

Let [Formula: see text] be a category with an involution ∗. Suppose that [Formula: see text] is a morphism and [Formula: see text] is an (epic, monic) factorization of [Formula: see text] through [Formula: see text], then [Formula: see text] is core invertible if and only if [Formula: see text] and [Formula: see text] are both left invertible if and only if [Formula: see text], [Formula: see text] and [Formula: see text] are all essentially unique (epic, monic) factorizations of [Formula: see text] through [Formula: see text]. We also give the corresponding result about dual core inverse. In addition, we give some characterizations about the coexistence of core inverse and dual core inverse of an [Formula: see text]-morphism in the category of [Formula: see text]-modules of a given ring [Formula: see text].

Continuity of the core-EP inverse and its applications

ABSTRACT In this paper, firstly we study the continuity of the core-EP inverse without explicit error bounds by virtue of two methods. One is the rank equality, followed from the classical generalized inverse. The other one is matrix decomposition. The continuity of the core inverse can be derived as a particular case. Secondly, we study perturbation bounds for the core-EP inverse under prescribed conditions. Perturbation bounds for the core inverse can be derived as a particular case. Also, as corollaries, the sufficient (and necessary) conditions for the continuity of the core-EP inverse are obtained. Thirdly, a numerical example is illustrated to compare derived upper bounds. Finally, an application to semistable matrices is provided.

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

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.

Right core inverse and the related generalized inverses

In this article, we investigate some properties of right core inverses. Particularly, new characterizations and expressions for right core inverses are given, using projections and {1, 3}-inverses. Also, we introduced and investigated a new generalized right core inverse which is called right pseudo core inverse. Then, we provide the relation schema of (one-sided) core inverses, (one-sided) pseudo core inverses, and EP elements.

Characterizations and representations of outer inverse for matrices over a ring

In this paper, we investigate the existence and representations of the generalized inverse , , , , , of a matrix A over an associative ring, and give some characterizations and representations of these generalized inverses by using regularity. Some explicit expressions for and of a matrix A by means of -inverse are also given. Furthermore, we present the relationship between the generalized inverse and the -inverse of a matrix A over rings. And then using the regularity give the sufficient and necessary condition for the existence and the expression of -inverse of matrices over a ring. Some explicit expressions for -inverse of an element in semigroups by means of -inverse are also studied. As applications, some new results for the Mary inverse, the core inverse and dual core inverse are obtained.

Characterizations and representations of the core inverse and its applications

In this paper, we present the characterizations of the core inverse by the matrix equations. We give several representations for computing the core inverse. We explore the relationship between the core inverse and the nonsingularity of a bordered matrix by the Cramer rule. We also give an iterative formula for computing the core inverse and present a sufficient condition for the convergence of the iterative process.

Weighted pseudo core inverses in rings

Let R be a unital -ring and let with e,f invertible Hermitian elements. In this paper, we define two types of outer generalized inverses, called pseudo e-core inverses and pseudo f-dual core inverses. An element is pseudo e-core invertible if there exist an element and some positive integer n such that and . Dually, a is pseudo f-dual core invertible if there exist an element and some positive integer m such that xax=x, and . Moreover, we investigate both of them for their characterizations and properties. Also, the relations between the pseudo e-core inverse (resp. the pseudo f-dual core inverse) and the inverse along an element are given.

Core and dual core inverses of a sum of morphisms

Let C be an additive category with an involution *. Suppose that ? : X ? X is a morphism of C with core inverse ?# : X ? X and ? : X ? X is a morphism of C such that 1X + ?#? is invertible. Let ? = (1X+?#?)-1, ? = (1X+??#)-1, ? = (1X-??#)??(1X-?#?), ? = ?(1X-?#?)?-1??#?,? = ??#??-1(1X-??#)?,? = ?*(?#(*?*(1X-??#)?. Then f = ? + ? ? ? has a core inverse if and only if 1X-?, 1X-? and 1X-? are invertible. Moreover, the expression of the core inverse of f is presented. Let R be a unital *-ring and J(R) its Jacobson radical, if a ? R# with core inverse a # and j ? J(R), then a + j ? R# if and only if (1-aa#)j(1+a#j)-1(1-a#a) = 0. We also give the similar results for the dual core inverse.

Rank equalities related to a class of outer generalized inverse

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

Generalized core inverses of matrices

This research is supported by the National Natural Science Foundation of China (NO. 11771076 and No. 11471186). The first author is grateful to China Scholarship Council for giving him a purse for his further study in Universitat Politecnica de Valencia, Spain.

Perturbation theory for core and core-EP inverses of tensor via Einstein product

In this paper, for given tensors A,? and B = A+E, we investigate the perturbation bounds for the core inverse A# and core-EP inverse A+ under some conditions via Einstein product.

On most perfect magic squares of order four

A simple parametrization for most perfect magic (MPM) squares is derived. This parametrization permits to calculate their powers and eigenvalues as well as their Moore–Penrose, group, and core inverses in an easy fashion. Such properties as EP-ness, normality, symmetry, and partial isometriness of the MPM squares are also characterized. Moreover, their eigenspaces are identified along with some properties of such squares with prime entries.

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

ABSTRACTThis paper concerns the regular factorization and expression of the core inverse in a Hilbert space. Utilizing the regular factorization, we first give some characterizations for the existence and the expression of the group inverse and core inverse. Based on these, we prove that the core inverse of the perturbed operator has the simplest possible expression if and only if the perturbation is range-preserving, and derive an explicit expression under the rank-preserving perturbation. Thus we can conclude that both the range-preserving perturbation and the rank-preserving perturbation are all continuous perturbations. The obtained results extend and improve many recent ones in matrix theory and operator theory.

On the continuity and differentiability of the (dual) core inverse in C*-algebras

ABSTRACTThe continuity of the core inverse and the dual core inverse is studied in the setting of-algebras. Later, this study is specialized to the case of bounded Hilbert space operators and to complex matrices. In addition, the differentiability of these generalized inverses is studied in the context of -algebras.

Centralizer’s applications to the (b, c)-inverses in rings

We give several conditions in order that the absorption law for one sided (b, c)-inverses in rings holds. Also, by using centralizers, we obtain the absorption law for the (b, c)-inverse and the reverse order law of the (b, c)-inverse in rings. As applications, we obtain the related results for the inverse along an element, Moore–Penrose inverse, Drazin inverse, group inverse and core inverse.

The pseudo core inverse of a companion matrix

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.

The core invertibility of a companion matrix and a Hankel matrix

ABSTRACTIn this paper, we show that the lower companion matrix associated with the monic polynomial over a ring with unity is core invertible if and only if there exist solutions x and y to (i) , (ii) , (iii) , (iv) In addition, some methods of characterizing and computing the core inverse of a Hankel matrix and a Toeplitz matrix over a field are presented. Moreover, the expressions of core inverses of these matrices are presented.

Optimal perturbation bounds for the core inverse

In this short note, we study some perturbation properties of the core inverse. We present the closed form and perturbation bounds for the core inverse under some conditions, which extend the classical result on the perturbation of the nonsingular matrix. Our expressions for the perturbation of the core inverse are simple and perturbation bounds are sharp.

The core and dual core inverse of a morphism with kernel

ABSTRACTLet be an additive category with an involution . Suppose that is a morphism with kernel in , then is core invertible if and only if has a cokernel and both and are invertible. In this case, we give the representation of the core inverse of . We also give the corresponding result about dual core inverse.