core-EP Inverse Research Articles

Minimal rank weighted weak Drazin inverses

The concept of a minimal rank weak Drazin inverse for square matrices is extended to rectangular matrices. Precisely, a minimal rank weighted weak Drazin inverse is introduced and its properties are investigated. Some known generalized inverses such as the weighted Drazin inverse, the weighted core-EP inverse, and the weighted $p$-WGI are particular cases of a minimal rank weighted weak Drazin inverse. Thus, a wider class of generalized inverses is proposed. General representation forms of a minimal rank weighted weak Drazin inverse are presented as well as its canonical form. Applying the minimal rank weighted weak Drazin inverse, corresponding systems of linear matrix equations are solved and their solutions are expressed. As consequences of our results, new properties of minimal rank weak Drazin inverse are obtained.

A new extension of the core inverse

There are a number of extensions of the core inverse represented by the products of known generalized inverses, like the core-EP inverse, DMP inverse, GC inverse and so on. However, none of them is an inner inverse of the matrix, and especially for an arbitrary nilpotent matrix, such inverses are always null. Our goal is to use a new technic to introduce an extension of the core inverse that preserves the interesting property of being an inner inverse of the matrix which need not necessarily be the null matrix of a nilpotent matrix. We define the extended core inverse for square complex matrices combining the sum and the difference of three known generalized inverses. Various properties and representations of the extended core inverse are developed. The extended dual core inverse is investigated too. We apply the extended core inverse to solve some systems of linear equations and one minimization problem. A significant normal equation related to least-squares solutions can be solved using the extended core inverse.

The $w$-core–EP inverse in rings with involution

The $w$-core–EP inverse in rings with involution

Weighted generalized core–EP inverse

To solve some systems of operator equations, we generalize the concepts of the W–weighted core–EP inverse, the W-gMP inverse, the generalized core–EP inverse and their duals and we define new kinds of generalized inverses which are called the weighted generalized core–EP inverse and its dual. Thus, these new inverses present wider classes of generalized inverses. We establish a number of characterizations and representations of the weighted generalized core–EP inverse and its dual. As applications of our new inverses, we solve several systems of linear equations and a minimization problem.

Computation of tensors generalized inverses under M-product and applications

This paper introduces notions of the Drazin and core-EP inverses on tensors via M-product. We propose a few properties of the Drazin and core-EP inverses of tensors, as well as effective tensor-based algorithms for calculating these inverses. In addition, definitions of composite generalized inverses are presented in the framework of the M-product, including CMP, DMP, and MPD inverses of tensors. Tensor-based higher-order Gauss-Seidel and Gauss-Jacobi iterative methods are designed. Algorithms for these two iterative methods for solving multilinear equations have been developed. Certain multilinear systems are solved using the Drazin inverse, core-EP inverse, and composite generalized inverses, such as CMP, DMP, and MPD inverse. A tensor M-product-based regularization technique is applied to solve the color image deblurring.

(w,v)-core–EP inverse and (w,v)-pseudo-core inverse

The core and core–EP inverses are two recent generalized inverses that were first introduced and studied for complex matrices, and later for elements of rings with involution. Afterward, weighted versions (the w-core–EP inverse and the weighted w-core inverse with weight v) were introduced. Our purpose is to study all these generalized inverses in the context of monoids with involution, in a unified way. With this in mind, we introduce new, broader kinds of generalized inverses. In particular, we present the ( w , v ) -core–EP inverse and the ( w , v ) -pseudo-core inverse for elements of a monoid with involution. Necessary and sufficient conditions for the existence of these inverses are proved, and some representations of these new inverses are developed. Using these results, we obtain new properties of the weighted w-core inverse with weight v. We finally consider these inverses in the matrix setting. We notably prove that these inverses provide minimum norm or least square solutions to some matrix equations.

The core-EP inverse: a numerical approach for its acute perturbation

This paper studies the concept of stable perturbation B ∈ C n × n for the core-EP inverse of a matrix A ∈ C n × n with index k. For a given stable perturbation B of A, explicit expressions of its core-EP inverse B ◯ $ † $ and its projection at zero B π are presented. Then, the perturbation bounds of ∥ B ◯ $ † $ − A ◯ $ † $ ∥ / ∥ A ◯ $ † $ ∥ and ∥ B π − A π ∥ are given provided that B is a stable perturbation of A. In addition, we investigate the concept of acute perturbation of A. We give a perturbation analysis with respect to core-EP inverses. We provide a condition under which the acute perturbation coincides with the stable perturbation for core-EP inverses.

Alternative algebraic perturbation expressions for the core-EP inverse of a matrix

Alternative algebraic perturbation expressions for the core-EP inverse of a matrix

Existence Criteria and Related Relation of the gMP Inverse of Matrices

New characterizations of the gMP inverse are provided by the core part of the core-EP decomposition. We also answer the question as to whether X is the gMP inverse of A under the conditions of R(X)⊆R(A*Ak) or N((Ak)*)⊆N(X). We investigate the relationship between the core-EP inverse and the gMP inverse.Using the gMP inverse, the gMP relation is investigated in view of the core-EP decomposition.

Perturbation formulae for the generalized core–EP inverse

Perturbation formulae for the generalized core–EP inverse

Revisiting the m-weak core inverse

<p>The $ m $-weak core inverse of a complex matrix was introduced by D. E. Ferreyra and Saroj B. Malik in 2024. We have revisited this inverse by using the inverse along two matrices, that is, we have proved that the $ m $-weak core inverse of a complex matrix coincides with the inverse along two complex matrices. Moreover, the necessary and sufficient conditions of the $ m $-weak core inverse of a complex matrix have been obtained. The one-sided $ m $-weak core inverse has been introduced by using the core-EP (EP means Equal Prohection) inverse of $ A $.</p>

Core-EP Monotonicity Characterizations for Property-n Matrices

A square matrix is said to have property n if there exists a positive integer w such that Aw is nonnegative. In this paper, we study the core-EP monotonicity for property-n matrices. Some necessary and sufficient conditions for a property-n matrix to be core-EP monotone are given. Moreover, a necessary and sufficient condition for a real square matrix to have a nonnegative core-EP inverse is also presented.

Two Generalizations of the Core Inverse in Rings with Some Applications

In this paper, we introduce two new generalized core inverses, namely, the (p,q,m)-core inverse and the ⟨p,q,n⟩-core inverse; both extend the inverses of the ⟨i,m⟩-core inverse, the (j,m)-core inverse, the core inverse, the core-EP inverse and the DMP-inverse.

The full rank expressions for the W-weighted Drazin and core-EP inverse of a matrix and their applications

The full rank expressions for the W-weighted Drazin and core-EP inverse of a matrix and their applications

Core–EP Star and Star Core–EP Operators

In this paper, we will introduce two new classes of generalized Drazin invertible operators on Hilbert space, which are called core–EP star and star core–EP operators, using the core–EP inverse and the adjoint of a given operator. We also represent here a few characterizations of these new operators from two points of view, algebraic and geometrical, and make relations to some familiar inverses, which are studied before. Next, we will consider two decompositions of Hilbert space and give the matrix representations of these new operators, following the decompositions. As the special section, there is the case when the operator is Drazin invertible. By using a *core–EP inverse, instead of the core–EP inverse, we get another new classes called *core–EP star and star *core–EP operators.

Minimal rank weak Drazin inverses: a class of outer inverses with prescribed range

For any square matrix $A$, it is proved that minimal rank weak Drazin inverses (Campbell and Meyer, 1978) of $A$ coincide with outer inverses of $A$ with range $\mathcal{R}(A^{k})$, where $k$ is the index of $A$. It is shown that the minimal rank weak Drazin inverse behaves very much like the Drazin inverse, and many generalized inverses such as the core-EP inverse and the DMP inverse are its special cases.

Matrix inverses along the core parts of three matrix decompositions

<abstract><p>New characterizations for generalized inverses along the core parts of three matrix decompositions were investigated in this paper. Let $ A_{1} $, $ \hat{A}_{1} $ and $ \tilde{A}_{1} $ be the core parts of the core-nilpotent decomposition, the core-EP decomposition and EP-nilpotent decomposition of $ A\in \mathbb{C}^{n\times n} $, respectively, where EP denotes the EP matrix. A number of characterizations and different representations of the Drazin inverse, the weak group inverse and the core-EP inverse were given by using the core parts $ A_{1} $, $ \hat{A}_{1} $ and $ \tilde{A}_{1} $. One can prove that, the Drazin inverse is the inverse along $ A_{1} $, the weak group inverse is the inverse along $ \hat{A}_{1} $ and the core-EP inverse is the inverse along $ \tilde{A}_{1} $. A unified theory presented in this paper covers the Drazin inverse, the weak group inverse and the core-EP inverse based on the core parts of the core-nilpotent decomposition, the core-EP decomposition and EP-nilpotent decomposition of $ A\in \mathbb{C}^{n\times n} $, respectively. In addition, we proved that the Drazin inverse of $ A $ is the inverse of $ A $ along $ U $ and $ A_{1} $ for any $ U\in \{A_{1}, \hat{A}_{1}, \tilde{A}_{1}\} $; the weak group inverse of $ A $ is the inverse of $ A $ along $ U $ and $ \hat{A}_{1} $ for any $ U\in \{A_{1}, \hat{A}_{1}, \tilde{A}_{1}\} $; the core-EP inverse of $ A $ is the inverse of $ A $ along $ U $ and $ \tilde{A}_{1} $ for any $ U\in \{A_{1}, \hat{A}_{1}, \tilde{A}_{1}\} $. Let $ X_{1} $, $ X_{4} $ and $ X_{7} $ be the generalized inverses along $ A_{1} $, $ \hat{A}_{1} $ and $ \tilde{A}_{1} $, respectively. In the last section, some useful examples were given, which showed that the generalized inverses $ X_{1} $, $ X_{4} $ and $ X_{7} $ were different generalized inverses. For a certain singular complex matrix, the Drazin inverse coincides with the weak group inverse, which is different from the core-EP inverse. Moreover, we showed that the Drazin inverse, the weak group inverse and the core-EP inverse can be the same for a certain singular complex matrix.</p></abstract>

Characterizations of the Weighted Core-EP Inverses

Characterizations of the Weighted Core-EP Inverses

Characterizations and Perturbations of the Core-EP Inverse of Tensors Based on the T-Product

In this paper, we investigate the theory of the core-EP inverse of third-order tensors via the T-product (called T-core-EP inverse). A new decomposition called T-core-EP decomposition of tensors is given by the T-Schur decomposition. A canonical form and some characterizations of the T-core-EP inverse are presented. Finally, we establish the perturbation bounds for the T-core-EP inverse.

Core–EP orthogonal operators

The concept of the core orthogonality (CO) was recently introduced for two square matrices A and B of the same size and of index at most 1 as follows: A is core orthogonal to B if and , where is the core inverse of A. Using the core–EP inverse instead of the core inverse, we extend the concept of the CO and present the new concept of the core–EP orthogonality (CEPO) for two Drazin invertible bounded linear Hilbert space operators A and B as follows: A is said to be core–EP orthogonal to B if and , where is the core–EP inverse of A. A number of characterizations for CEPO are presented as well as the relation between the CEPO and core–EP additivity. Applying CEPO, the concept of strong core–EP orthogonality is defined and characterized.