core-EP Inverse Research Articles

On the general strong fuzzy solutions of general fuzzy matrix equation involving the Core-EP inverse

<abstract><p>The inconsistent or consistent general fuzzy matrix equation are studied in this paper. The aim of this paper is threefold. Firstly, general strong fuzzy matrix solutions of consistent general fuzzy matrix equation are derived, and an algorithm for obtaining general strong fuzzy solutions of general fuzzy matrix equation by Core-EP inverse is also established. Secondly, if inconsistent or consistent general fuzzy matrix equation satisfies $ X\in R(S^{k}) $, the unique solution or unique least squares solution of consistent or inconsistent general fuzzy matrix equation are given by Core-EP inverse. Thirdly, we present an algorithm for obtaining Core-EP inverse. Finally, we present some examples to illustrate the main results.</p></abstract>

Additive properties for the pseudo core inverse of morphisms in an additive category

In this paper, given a morhism [Formula: see text] with its pseudo core inverse [Formula: see text] and a morphism [Formula: see text] such that [Formula: see text] is invertible, a necessary and sufficient condition and two sufficient conditions are presented under which the additive property, namely [Formula: see text] holds. Several interesting results about additive properties of core inverses of bounded linear operators presented in Huang et al. are generalized to the case of pseudo core inverse of morphism. Also, many results regarding additive properties of core-EP inverses of complex matrices studied by Ma and Stanimirović are extended to the cases of morphism.

The Core-EP, Weighted Core-EP Inverse of Matrices and Constrained Systems of Linear Equations

The Core-EP, Weighted Core-EP Inverse of Matrices and Constrained Systems of Linear Equations

A Finite Method for Computing the Drazin and Core-EP Inverses of Matrices Based on Partial Full-Rank Factorization

A Finite Method for Computing the Drazin and Core-EP Inverses of Matrices Based on Partial Full-Rank Factorization

On (b,c)-inverses and (c,b)-inverses

We extend characterizations of inverses along an element to (b, c)-inverses. It is proved that if an element a in a semigroup S is both (b, c)-invertible and (c, b)-invertible for some then are group invertible, and is the (b, c)-inverse of a, is the (c, b)-inverse of a. Moreover, we establish several criteria for the existence of (b, c)-inverses and (c, b)-inverses by means of units in a ring. As applications, some new representations of weighted Moore-Penrose inverses and core-EP inverses of complex matrices are given.

Sign pattern, usability, representations and perturbation for the core-EP and weighted core-EP inverse

New properties and representations for the core-EP inverse are developed. Particularly, the core-EP inverse of an upper triangular matrix and its sign pattern are considered. Determinantal representations for the core-EP inverse and core-EP solution of linear systems are investigated. Corresponding representations of the weighted core-EP inverse are derived. The obtained results are illustrated by computational algorithms and illustrative examples. Certain results related to the perturbation constraints for the core-EP inverse are also provided. The results related with the core-EP inverse are a continuous of the results from [H. Ma and P.S. Stanimirović, Characterizations, approximation and perturbations of the Core-EP inverse, Appl. Math. Comput. 359 (2019), 404–417].

Generalization of core-EP inverse for rectangular matrices

The notion of the generalized core-EP (g-core-EP) inverse is defined as a generalization of well-known core-EP inverse to rectangular matrices. The extension is defined using the outer inverse and the Moore-Penrose inverse. Various basic properties, representations and characterizations of the g-core-EP inverse are investigated. The *g-core-EP inverse is introduced and investigated as the dual of g-core-EP inverse. Several integral and limit representations of the g-core-EP and *g-core-EP inverses are considered. Also, applications of the g-core-EP and *g-core-EP inverses in solving some restricted systems of linear equations are considered.

Representations for the weak group inverse

The aim of this paper is to provide new representations and characterizations of the weak group inverse. We give a relation between the weak group inverse and a corresponding nonsingular border matrix. Continuity of weak group inverse is studied. Several limit representations, integral representations and perturbation formulae for the weak group inverse are proposed. Various expressions for the weak group and core–EP inverses of upper block triangular matrix are established and their sign patterns are considered. The splitting method for finding weak group inverse is proved as well as the successive matrix squaring algorithm for computing weak group inverse. Also, the weak group inverse can be used in solving appropriate systems of linear equations.

Weighted MPCEP Inverse of an Operator Between Hilbert Spaces

Using compositions of the weighted core–EP inverse and Moore–Penrose inverse, we solve new systems of operator equations and define three new weighted generalized inverses of a bounded linear operator between two Hilbert spaces, which are called the W-MPCEP inverse, W-CEPMP inverse and W-MPCEPMP inverse. The W-MPCEP inverse extends the notion of the MPCEP inverse for an operator. We give operator matrix forms and some characterizations of W-MPCEP, W-CEPMP and W-MPCEPMP inverses. We also establish maximal classes of operators for which the expressions of W-MPCEP, W-CEPMP and W-MPCEPMP inverses are valid.

Solvability of New Constrained Quaternion Matrix Approximation Problems Based on Core-EP Inverses

Based on the properties of the core-EP inverse and its dual, we investigate three variants of a novel quaternion-matrix (Q-matrix) approximation problem in the Frobenius norm: $$\min \Vert \mathbf {A}\mathbf {X}\mathbf {B}-\mathbf {C}\Vert _F$$ subject to the constraints imposed to the right column space of $${\mathbf{A}}$$ and the left row space of $${\mathbf{B}}$$ . Unique solution to the considered Q-matrix problem is expressed in terms of the core inverse of $${\mathbf{A}}$$ and/or the dual core-EP inverse of $${\mathbf{B}}$$ . Thus, we propose and solve problems which generalize a well-known constrained approximation problem for complex matrices with index one to quaternion matrices with arbitrary index. Determinantal representations for solutions of proposed constrained quaternion matrix approximation problems obtained. An example is given to justify obtained theoretical results.

Solvability of some constrained matrix approximation problems using core-EP inverses

Using the core-EP inverse, we obtain the unique solution to the constrained matrix minimization problem in the Euclidean norm: $$\mathrm{Minimize }\ \Vert Mx-b\Vert _2$$ , subject to the constraint $$x\in \mathcal{R}(M^k),$$ where $$M\in {\mathbb {C}}^{n\times n}$$ , $$k=\mathrm {Ind}(M)$$ and $$b\in {\mathbb {C}}^n$$ . This problem reduces to well-known results for complex matrices of index one and for nonsingular complex matrices. We present two kinds of Cramer’s rules for finding unique solution to the above mentioned problem, applying one well-known expression and one new expression for core-EP inverse. Also, we consider a corresponding constrained matrix approximation problem and its Cramer’s rules based on the W-weighted core-EP inverse. Numerical comparison with classical strategies for solving the least squares problems with linear equality constraints is presented. Particular cases of the considered constrained optimization problem are considered as well as application in solving constrained matrix equations.

Further Results on Weighted Core-EP Inverse of Matrices

In this paper, we introduce the notation of $E$-weighted core-EP and $F$-weighted dual core-EP inverse of matrices. We then obtain a few explicit expressions for the weighted core-EP inverse of matrices through other generalized inverses. Further, we discuss the existence of generalized weighted Moore-Penrose inverse and additive properties of the weighted core-EP inverse of matrices. In addition to these, we propose the star weighted core-EP and weighted core-EP star class of matrices for solving the system of matrix equations. We further elaborate on this theory by producing a few representation and characterization of star weighted core-EP and weighted core-EP star classes of matrices.

Weighted binary relations involving core-EP inverse

We study a new binary relation defined on the set of rectangular complex matrices involving the weighted core-EP inverse and give its characterizations. This relation becomes a pre-order. Then, one-sided pre-orders associated to the weighted core-EP inverse are given from two perspectives. Finally, we make a comparison for these two sets of one-sided weighted pre-orders.

Determinantal Representations of the Weighted Core-EP, DMP, MPD, and CMP Inverses

In this paper, new notions of the weighted core-EP left inverse and the weighted MPD inverse which are dual to the weighted core-EP (right) inverse and the weighted DMP inverse, respectively, are introduced and represented. The direct methods of computing the weighted right and left core-EP, DMP, MPD, and CMP inverses by obtaining their determinantal representations are given. A numerical example to illustrate the main result is given.

Core-EP inverse in rings with involution

Core-EP inverse in rings with involution

New classes of more general weighted outer inverses

According to known definition in the literature, the W-weighted outer inverses of A is defined as the outer inverses of WAW. We introduce and study more general weighted outer inverses of rectangular complex matrices. Introduced families of outer inverses include previous research in two aspects from two approaches. In the first approach, for given complex matrices A, M, N, B, C, we define the -weighted -inverse of A and prove that it is just the outer inverse of MAN with the range and null space . Thus, the -weighted -inverse generalizes the notion of the weighted outer inverse and, finally, the notion of the outer inverse with known image and kernel. We give equivalent conditions for the existence and representations of the -weighted -inverse. Integral and limit representations of weighted generalized inverses can be derived as corollaries. Inspired by corresponding representations of the W-weighted Drazin inverse and the weighted core-EP inverse, in the second approach, we investigate the expressions , and . Particularly, conditions under which these expressions become W-weighted outer inverses of A are analysed.

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.

Perturbation bounds for the W-weighted core-EP inverse

In this paper, perturbation bounds are provided for the [Formula: see text]-weighted core-EP inverse of a rectangular matrix under reasonable conditions. Perturbation bounds for the core-EP inverse could be stated as a special case. Then, the continuity of the [Formula: see text]-weighted core-EP inverse is considered from the perspective of equations. Finally, we give an application to a semi-stable matrix involving an integral representation of the [Formula: see text]-weighted core-EP inverse of a perturbed matrix.

Perturbation of the weighted core–EP inverse

We obtain two different expressions for the weighted core–EP inverse of the sum $$A+E$$ for a given operator A and a perturbation operator E, under different conditions on the perturbing operator. As a consequence of these results, we get some relations between idempotents determined by the weighted core–EP inverses of a given operator and its perturbation and we present perturbation bounds for the weighted core–EP inverse. The application to the solution of a perturbed equation is given. We also apply these results to obtain perturbation results for the core–EP inverse.

Composite outer inverses for rectangular matrices

Various compositions of the Drazin inverse, the group inverse or the core-EP inverse with the Moore-Penrose inverse have investigated last years. Solving some type of matrix equations, we introduce three new generalized inverses of a rectangular matrix, which are called the OMP, MPO and MPOMP inverses, because the outer inverse and the Moore-Penrose inverse are incorporated in their definition. As a consequence, the notion of DMP, MPD, CMP and MPCEP inverses for a square matrix are covered by one general definition and extended to a rectangular matrix. We propose a common term, composite outer inverses, to denote such compositions of outer inverses and the Moore-Penrose inverse. Characterizations of the OMP, MPO and MPOMP inverses are derived as well as some properties of projectors determined by these new inverses. We establish maximal classes of matrices for which the representations of composite outer inverses are valid. Also, the integral and limit representations for OMP, MPO and MPOMP inverses are investigated. Possible applications of composite outer inverses are given too and interesting topics for further research are considered.