core-EP Inverse Research Articles

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.

Core and core-EP inverses of tensors

Specific definitions of the core and core-EP inverses of complex tensors are introduced. Some characterizations, representations and properties of the core and core-EP inverses are investigated. The results are verified using specific algebraic approach, based on proposed definitions and previously verified properties. The approach used here is new even in the matrix case.

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.

WEIGHTED CORE–EP INVERSE AND WEIGHTED CORE–EP PRE-ORDERS IN A -ALGEBRA

We define extensions of the weighted core–EP inverse and weighted core–EP pre-orders of bounded linear operators on Hilbert spaces to elements of a $C^{\ast }$ -algebra. Some properties of the weighted core–EP inverse and weighted core–EP pre-orders are generalized and some new ones are proved. Using the weighted element, the weighted core–EP pre-order, the minus partial order and the star partial order of certain elements, new weighted pre-orders are presented on the set of all $wg$ -Drazin invertible elements of a $C^{\ast }$ -algebra. Applying these results, we introduce and characterize new partial orders which extend the core–EP pre-order to a partial order.

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.

On a new generalized inverse for Hilbert space operators

Using the Moore-Penrose inverse and the core-EP inverse, we define a new generalized inverse (called the MPCEP inverse) for a Hilbert space operator. Several equivalent conditions for a Hilbert space operator to be the MPCEP inverse are presented. We also give a canonical form for the MPCEP inverse. Various kinds of generalized inverses are integrated by MPCEP inverse and many equivalent relations among these inverses are investigated using appropriate idempotents. Also, we develop maximal classes of operators for which the representation of the MPCEP inverse is still valid. We introduce and study a new binary relation associated with the MPCEP inverse.

Characterizations, approximation and perturbations of the core-EP inverse

Several characterizations and representations for the core-EP inverse are presented and investigated. To that goal, several explicit bordering technique for characterizing and defining the core-EP inverse are analyzed. Also the matrix splitting method and the iterative successive matrix squaring algorithm for computing the core-EP inverse are investigated. The perturbation bounds related with the core-EP inverse are also estimated.

A weak group inverse for rectangular matrices

In this paper, we extend the notion of weak group inverse to rectangular matrices (called W-weighted WG inverse) by using the weighted core EP inverse recently investigated. This new generalized inverse also generalizes the well-known weighted group inverse given by Cline and Greville. In addition, we give several representations of the W-weighted WG inverse, and derive some characterizations and properties.

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.

A characterization and perturbation bounds for the weighted core-ep inverse

In this paper, we present a new characterization of the weighted core-EP inverse of a rectangular matrix. We also investigate the perturbation bounds and approximations for the weighted core-EP inverse.

Maximal classes of operators determining some weighted generalized inverses

ABSTRACT We develop maximal classes of operators for which the representations of the weighted gDMP inverse, the weighted core–EP inverse and the weighted CMP inverse are valid. Applying our results, we obtain maximal classes determining gDMP inverse, the core–EP inverse and the CMP inverse of Hilbert space operators. Thus, we present new representations and extend recent results omitting some assumptions.

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.

The new revisitation of core EP inverse of matrices

In this paper, we study some properties of core-EP inverse of square matrices. Firstly, we extend the obtained theorem proved by K.M. Prasad and K.S. Mohana. Then some properties of core-EP inverse have been given, through applying the conditions (AX)* = AX, XAk+1 = Ak and R(X) = R(X*) = R(Ak). Secondly, we get some characterizations of core-EP inverse by employing the conditions AX = PAk and XA = PR(Ak), N((Ak+1)*A). Finally, we get some properties of core-EP inverse by utilizing the condition Ak+1X = AkPAk .

Representations and properties of the W-weighted core-EP inverse

ABSTRACT In this paper, we investigate the weighted core-EP inverse introduced by Ferreyra, Levis and Thome. Several computational representations of the weighted core-EP inverse are obtained in terms of singular-value decomposition, full-rank decomposition and QR decomposition. These representations are expressed in terms of various matrix powers as well as matrix product involving the core-EP inverse, Moore-Penrose inverse and usual matrix inverse. Finally, those representations involving only Moore-Penrose inverse are compared and analyzed via computational complexity and numerical examples.

Characterizations of k-commutative equalities for some outer generalized inverses

ABSTRACTIn this paper, we present necessary and sufficient conditions for the k-commutative equality , where X is an outer generalized inverse of the square matrix A. Also, we give new representations for core EP, DMP, and CMP inverses of square matrices as outer inverses with prescribed null space and range. In addition, we characterize the core EP inverse as the solution of a new system of matrix equations.

The gDMP inverse of Hilbert space operators

We define a new generalized inverse (named the gDMP inverse) for a Hilbert space operator using its generalized Drazin inverse and its Moore–Penrose inverse. Thus, we extend the DMP inverse for a square matrix to more general case. Also, we introduce two new classes of operators, g -EP and g -normal operators which include, respectively, EP operators and normal operators. A new binary relation is associated with the gDMP inverse is presented and studied. The notion of core-EP inverse for matrices is extended to generalized Drazin invertible operators on Hilbert space.

Integral representations of two generalized core inverses

In this paper, we present integral representations for the DMP and core-EP inverse, which based on the full-rank decomposition of a given matrix. In particular, integral representations of the core and dual core inverse are given. All of these integral representations do not require any restriction on the spectrum of a certain matrix.

Maximal classes of matrices determining generalized inverses

This paper derives some further results on recent generalized inverses studied in the literature, namely core EP, DMP, and CMP inverses. Our main aim is to develop maximal classes of matrices for which their representations remain valid.

Bordering method to compute Core-EP inverse

Abstract Following the work of Kentaro Nomakuchi[10] and Manjunatha Prasad et.al., [7] which relate various generalized inverses of a given matrix with suitable bordering,we describe the explicit bordering required to obtain core-EP inverse, core-EP generalized inverse. The main result of the paper also leads to provide a characterization of Drazin index in terms of bordering.

Three limit representations of the core-EP inverse

In this paper, we present three limit representations of the core-EP inverse. The first approach is based on the full-rank decomposition of a given matrix. The second and third approaches, which depend on the explicit expression of the core-EP inverse, are established. The corresponding limit representations of the dual core-EP inverse are also given. In particular, limit representations of the core and dual core inverse are derived.