Core Inverse Research Articles

Jacobson’s lemma and Cline’s formula for generalized inverses in a ring with involution

In a ring with involution, we prove that a Drazin invertible element is pseudo core invertible if and only if its spectral idempotent is {1, 4}-invertible. As its applications, we obtain necessary and sufficient conditions for 1 − ba (resp., ba) being pseudo core invertible while 1 − ab (resp., ab) has pseudo core inverse, and the pseudo core inverse of 1 − ba (resp., ba) is given in terms of 1 − ab (resp., ab). Inspired by the above idea, Jacobson’s lemma for Moore-Penrose inverse is considered.

Characterizations and maximal classes of elements related to pseudo core inverses

We present some new characterizations for the pseudo core inverse in rings with involution. Firstly, we characterize the pseudo core inverse by a unique projection. Secondly, we prove that the pseudo core inverse is EP and obtain another expression of the pseudo core inverse. Furthermore, we establish the relation between the EP element and the pseudo core inverse. Finally, maximal classes of elements in rings determining the pseudo core inverse are investigated.

Reverse-order law for core inverse of tensors

The notion of the core inverse of tensors with the Einstein product was introduced, very recently (Sahoo et al. Comp Appl Math 39:9, 2020). In this paper we establish a few sufficient and necessary conditions for reverse-order law of this inverse. Further, we discuss mixed-type reverse-order law for core inverse. In addition to these, we discuss core inverse solutions of multilinear systems via the Einstein product. The prowess of the inverse is demonstrated to solve the Poisson problem in the multilinear system framework.

Displacement structure of the core inverse

A matrix A is said to possess an UV-displacement structure if rank is small compared with the rank of A. Estimates for the rank are presented, where is the core inverse of A. The computation of the displacement is also considered.

Projections generated by Moore–Penrose inverses and core inverses

Let [Formula: see text] be a unital ∗-ring. As is well known, idempotents and projections can be constructed by the Moore–Penrose inverse and the core inverse of an element in [Formula: see text]. In this paper, we mainly investigate characterizations and properties of these types of idempotents and projections. Also, several results in [Hartwig and Spindelböck, Matrices for which [Formula: see text] and [Formula: see text] commute, Linear Multilinear Algebra 14 (1984) 241–256] are extended to a general ∗-ring without ∗-cancellable conditions. As applications, the characterization of EP elements is given.

Group inverse and core inverse in Banach and C*-algebras

In this paper, we have focused our study on the acute perturbation of the group inverse for the elements of Banach algebra with respect to the spectral radius. We also give perturbation analysis for the core inverse in C*- algebra. The perturbation bounds for the core inverse under some conditions are presented. Additionally, this paper extends the results obtained in [11, 14].

Core inverse in Banach algebras

We define and characterize the core inverse in the context of Banach algebras. The Banach space operator case is also considered. Using the core inverse, we present new characterizations of EP Banach space operators and EP Banach algebra elements. The dual core inverse for Banach algebra elements is presented too. Some new characterizations of co–EP Banach algebra elements are given by means the core inverse and dual core inverse.

Idempotents generated by weighted generalized inverses in rings with involution

Let R be an associate ring with unity 1 and involution. In this paper, we investigate several properties and characterizations of idempotents generated by weighted Moore-penrose inverses and weighted pseudo core inverses in R. Moreover, several new characterizations about weighted EP elements and existence criteria of weighted pseudo core inverses are also given.

Revisiting of the BT-inverse of matrices

<abstract> <p>In this paper, we discuss different characteristics of the BT-inverse of a square matrix introduced by Baksalary and Trenkler [On a generalized core inverse, Appl. Math. Comput., <bold>236</bold> (2014), 450-457]. While the BT-inverse is defined by a expression, we present some necessary and sufficient conditions for a matrix to be the BT-inverse. Then we give a canonical form of BT-inverse and investigate the relationships between BT-inverse and other generalized inverses by Core-EP decomposition. Some properties of BT-inverse concerned with some classes of special matrix are identified by Core-EP decomposition. Furthermore new representations of BT-inverse are given by the maximal classes of matrices.</p> </abstract>

Core–EP inverses in Banach algebras

We introduce the e–core–EP inverse and the core–EP inverse for elements of Banach algebras. Several characterizations and representations for these inverses are presented using idempotents and Drazin inverses. Then, we define the e–core inverse of elements in Banach algebras and give its characterizations, generalizing the related results of the core inverse.

Core invertibility of triangular matrices over a ring

We obtained several equivalent conditions for the existence of core inverses and dual core inverses of triangular matrices over a ring with involution. As applications, some necessary and sufficient conditions for the (2,2,0) core inverse problem are given.

Complex Varying-Parameter Zhang Neural Networks for Computing Core and Core-EP Inverse

An improved complex varying-parameter Zhang neural network (CVPZNN) for computing outer inverses is established in this paper. As a consequence, three types of complex Zhang functions (ZFs) which are used for computing the time-varying core-EP inverse and core inverse are given. The convergence rate of the proposed complex varying-parameter Zhang neural networks (CVPZNNs) is accelerated. The super-exponential performance of the proposed CVPZNNs with linear activation is proved. Also, the upper bounds of a finite time convergence which correspond to the proposed CVPZNN with underlying Li and tunable activation functions are estimated. The simulation results, which relate the CVPZNNs with different activation functions, are presented.

Determinantal Representations of the Quaternion Core Inverse and Its Generalizations

In this paper, we extend notions of the core inverse, core-EP inverse, DMP inverse, and CMP inverse over the quaternion skew field $${\mathbb {H}}$$ that have some features in comparison to complex matrices. We give the direct method of their computing, namely, their determinantal representations by using column and row noncommutative determinants previously introduced by the author. As the special case, we give their determinantal representations for complex matrices as well. A numerical example to illustrate the main result is given.

The 𝔪-core inverse and its applications

ABSTRACT In this paper, we introduce the -core inverse in the Minkowski space, and get a sufficient and necessary condition for the existence of the inverse and some other related properties. Furthermore, by using the inverse, we introduce the -core partial ordering and obtain solutions (or restricted least squares solutions) of some matrix equations in the Minkowski space.

The one-sided inverse along two elements in rings

Let be a ring and . In this paper, it is shown that if a is left -invertible and cab is regular, then a has at least one regular left -inverse. And the expression of the -inverse of a is given in terms of the inner inverse of cab. Moreover, the strongly left (or right) -inverse is introduced in terms of the regularity of cab. It is shown when , a is left -invertible implies that a is strongly left -invertible, and left -invertibility coincides with right -invertibility. As applications, we consider when the one-sided core inverse is core invertible. It is shown that left core invertibility coincides with right core invertibility in every strongly π-regular ring.

Determinantal Representations of the Core Inverse and Its Generalizations with Applications

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.

The group inverse and core inverse of sums of two elements in a ring

In Dedekind-finite ring, we present the group inverse of sum of two group invertible elements under different conditions. Then, the core inverse of a sum of two core invertible elements is investigated. Furthermore, the core inverse of the difference of two core invertible elements is presented. These results generalized the corresponding results of complex matrices.

Right core inverses of a product and a companion matrix

In this paper, characterizations of right core inverse are given by one-sided invertibility. The necessary and sufficient conditions, which guarantee that paq has right core inverses, are investigated. Furthermore, characterizations of right core inverses of triangular matrices, matrices and a companion matrix are considered.

Hua’s Identity for the $$\{1, 2\}$$-Inverse

In this paper, we generalized Hua’s identity to the $$\{1, 2\}$$-inverse case. In addition, theconditions that make Hua’s identities true for the Moore–Penrose inverse, group inverse, core inverse and dual core inverse are given, respectively.

The core inverses of differences and products of projections in rings with involution

Let R be a unital ring with involution. The core inverses of differences and products of projections in R are investigated. Let $$p, q\in R$$ be two projections, we obtained equivalent conditions such that $$p-q$$, $$(p-q)^2$$, $$1-qp$$, $$pq+qp$$ and $$(pq-qp)^2$$ are core invertible, respectively. Moreover, expressions for their core inverses are given.