Outer Inverses Research Articles

On elements whose (b,c)-inverse is idempotent in a monoid

In this paper, we investigate the elements whose (b, c)-inverse is idempotent in a monoid. Let S be a monoid and a, b, c ? S. Firstly, we give several characterizations for the idempotency of a||(b,c) as follows: a||(b,c) exists and is idempotent if and only if cab = cb, cS = cbS, Sb = Scb if and only if both a||(b,c) and 1||(b,c) exist and a||(b,c) = 1||(b,c), which establish the relationship between a||(b,c) and 1||(b,c). They imply that a||(b,c) merely depends on b, c but is independent of a when a||(b,c) exists and is idempotent. Particularly, when b = c, more characterizations which ensure the idempotency of a||b by inner and outer inverses are given. Finally, the relationship between a||b and a||bn for any n ? N+ is revealed.

Representations and geometrical properties of generalized inverses over fields

In this paper, as a generalization of Urquhart's formulas, we present a full description of the sets of inner inverses and ( B , C ) -inverses over an arbitrary field. In addition, identifying the matrix-vector space with an affine space, we analyse geometrical properties of the main generalized inverse sets. We prove that the set of inner inverses, and the set of ( B , C ) -inverses, form affine subspaces and we study their dimensions. Furthermore, under some hypotheses, we prove that the set of outer inverses is not an affine subspace, but it is an affine algebraic variety. We also provide lower and upper bounds for the dimension of the outer inverse set.

Exact solutions and convergence of gradient based dynamical systems for computing outer inverses

This paper investigates convergence properties of gradient neural network (GNN) and GNN-based dynamical systems for computing generalized inverses. The main results are exact analytical solutions for the state matrices of the corresponding GNN and GNN-based dynamical systems. The exact solutions are given in each time instant, which enables to express final results as appropriate limiting expressions. This enables more rigorous convergence analysis in terms of exact solutions. Finally, trajectories of state variables in considered dynamical systems can be generated by avoiding time-consuming numerical solving matrix differential equations inside the selected time interval. Using the fact that the stated dynamical systems are in the essence systems of linear equations, explicit solutions can be obtained using known techniques of ordinary differential equations. Main results of the paper are further transformations of obtained exact solutions using main properties of generalized inverses and linear algebra tools. It is important to mention that the convergence results are expressed in terms of expressions involving outer inverses. Several examples are presented and a closed-form expressions for the solutions are given and implemented in package Mathematica. These are compared with the numerical solutions obtained by Matlab Simulink implementation.

Weighted composite outer inverses

In order to extend and unify the definitions of W-weighted DMP, W-weighted MPD, W-weighted CMP and composite outer inverses, we present the weighted composite outer inverses. Precisely, the notions of MNOMP, MPMNO and MPMNOMP inverses are introduced as appropriate expressions involving the (M,N)-weighted (B,C)-inverse and Moore–Penrose inverse. Basic properties and a number of characterizations for the MNOMP, MPMNO or MPMNOMP inverse are discovered. Various representations and characterizations of weighted composite outer inverses are studied. General solutions for certain systems of linear equations are given in terms of weighted composite outer inverses. Numerical examples are presented on randomly generated matrices of various orders.

Representations and symbolic computation of generalized inverses over fields

This paper investigates representations of outer matrix inverses with prescribed range and/or null space in terms of inner inverses. Further, required inner inverses are computed as solutions of appropriate linear matrix equations (LME). In this way, algorithms for computing outer inverses are derived using solutions of appropriately defined LME. Using symbolic solutions to these matrix equations it is possible to derive corresponding algorithms in appropriate computer algebra systems. In addition, we give sufficient conditions to ensure the proper specialization of the presented representations. As a consequence, we derive algorithms to deal with outer inverses with prescribed range and/or null space and with meromorphic functional entries.

Weighted weak core inverse of operators

As solutions of some operator equations, we present two new outer inverses for bounded linear operators between two Hilbert spaces. Precisely, we introduce and characterize the weighted weak core inverse and the dual weighted weak core inverse. These generalized inverses are extensions of the weak core inverse and its dual which have been recently introduced for complex square matrices. Also, we give various representations for the new inverses including operator matrix representations and the most general form. Applying our results, we define and investigate the weak core inverse and the dual weak core inverse for Hilbert spaces operators.

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.

Essential crossed products for inverse semigroup actions: simplicity and pure infiniteness

We study simplicity and pure infiniteness criteria for \mathrm{C}^* -algebras associated to inverse semigroup actions by Hilbert bimodules and to Fell bundles over étale not necessarily Hausdorff groupoids. Inspired by recent work of R. Exel and D. R. Pitts ["Characterizing groupoid \mathrm{C}^* -algebras of non-Hausdorff étale groupoids", Preprint (2019); [arXiv: 1901.09683](arXiv: 1901.09683)], we introduce essential crossed products for which there are such criteria. In our approach the major role is played by a generalised expectation with values in the local multiplier algebra. We give a long list of equivalent conditions characterising when the essential and reduced \mathrm{C}^* -algebras coincide. Our most general simplicity and pure infiniteness criteria apply to aperiodic \mathrm{C}^* -inclusions equipped with supportive generalised expectations. We thoroughly discuss the relationship between aperiodicity, detection of ideals, purely outer inverse semigroup actions, and non-triviality conditions for dual groupoids.

GDMP-inverses of a matrix and their duals

ABSTRACT This paper introduces and investigates a new class of generalized inverses, called GDMP-inverses (and their duals), as a generalization of DMP-inverses. GDMP-inverses are defined from G-Drazin inverses and the Moore-Penrose inverse of a complex square matrix. In contrast to most other generalized inverses, GDMP-inverses are not only outer inverses but also inner inverses. Characterizations and representations of GDMP-inverses are obtained by means of the core-nilpotent and the Hartwig-Spindelböck decompositions.

One-sided weighted outer inverses of tensors

In this paper, for the first time in literature, we introduce one-sided weighted inverses and extend the notions of one-sided inverses, outer inverses and inverses along given elements. Although our results are new and in the matrix case, we decided to present them in tensor space with reshape operator. For this purpose, a left and right (M,N)-weighted (B,C)-inverse and the (M,N)-weighted (B,C)-inverse of a tensor are defined. Additionally, necessary and sufficient conditions for the existence of these new inverses are presented. We describe the sets of all left (or right) (M,N)-weighted (B,C)-inverses of a given tensor. As consequences of these results, we consider the one-sided (B,C)-inverse, (B,C)-inverse, one-sided inverse along a tensor and inverse along a tensor. Further, we introduce a (M,N)-weighted (B,C)-outer inverse and a W-weighted (B,C)-outer inverse of tensors with a few characterizations. Then, corresponding algorithms for computing various types of outer inverses of tensors are proposed, implemented and tested. The prowess of the proposed inverses are demonstrated for finding the solution of Poisson problem and the restoration of 3D color images.

Noncommutative Cartan \(\mathrm {C}^*\)-subalgebras

We characterise Exel's noncommutative Cartan subalgebras in several ways using uniqueness of conditional expectations, relative commutants, or purely outer inverse semigroup actions. We describe in which sense the crossed product decomposition for a noncommutative Cartan subalgebra is unique. We relate the property of being a noncommutative Cartan subalgebra to aperiodic inclusions and effectivity of dual groupoids. In particular, we extend Renault's characterisation of commutative Cartan subalgebras.

The outer generalized inverse of an even-order tensor

Necessary and sufficient conditions for the existence of the outer inverse of a tensor with the Einstein product are studied. This generalized inverse of a tensor unifies several generalized inverses of tensors introduced recently in the literature, including the weighted Moore-Penrose, the Moore-Penrose, and the Drazin inverses. The outer inverse of a tensor is expressed through the matrix unfolding of a tensor and the tensor folding. This expression is used to find a characterization of the outer inverse through group inverses, establish the behavior of outer inverse under a small perturbation, and show the existence of a full rank factorization of a tensor and obtain the expression of the outer inverse using full rank factorization. The tensor reverse rule of the weighted Moore-Penrose and Moore-Penrose inverses is examined and equivalent conditions are also developed.

Inverse complements of a matrix and applications

In this paper, the concept of “Inverse Complemented Matrix Method”, introduced by Eagambaram (2018), has been reestablished with the help of minus partial order and several new properties of complementary matrices and the inverse of complemented matrix are discovered. Class of generalized inverses and outer inverses of given matrix are characterized by identifying appropriate inverse complement. Further, in continuation, we provide a condition equivalent to the regularity condition for a matrix to have unique shorted matrix in terms of inverse complemented matrix. Also, an expression for shorted matrix in terms of inverse complemented matrix is given.

Computation of outer inverses of tensors using the QR decomposition

In this paper, we introduce new representations and characterizations of the outer inverse of tensors through QR decomposition. Derived representations are usable in generating corresponding representations of main tensor generalized inverses. Some results on reshape operation of a tensor are added to the existing theory. An effective algorithm for computing outer inverses of tensors is proposed and applied. The power of the proposed method is demonstrated by its application in 3D color image deblurring.

An efficient class of iterative methods for computing generalized outer inverse $${M_{T,S}^{(2)}}$$

In this paper, we propose a new matrix iteration scheme for computing the generalized outer inverse for a given complex matrix. The convergence analysis of the proposed scheme is established under certain necessary conditions, which indicates that the methods possess at least fourth-order convergence. The theoretical discussions show that the convergence order improves from 4 to 5 for a particular parameter choice. We prove that the sequence of approximations generated by the family satisfies the commutative property of matrices, provided the initial matrix commutes with the matrix under consideration. Some real-world and academic problems are chosen to validate our methods for solving the linear systems arising from statically determinate truss problems, steady-state analysis of a system of reactors, and elliptic partial differential equations. Moreover, we include a wide variety of large sparse test matrices obtained from the matrix market library. The performance measures used are the number of iterations, computational order of convergence, residual norm, efficiency index, and the computational time. The numerical results obtained are compared with some of the existing robust methods. It is demonstrated that our method gives improved results in terms of computational speed and efficiency.

Inclusion properties of generalized inverses in semiprime rings

Let [Formula: see text] be a semiprime ring, not necessarily with unity, with extended centroid [Formula: see text]. For [Formula: see text], let [Formula: see text] (respectively [Formula: see text], [Formula: see text]) denote the set of all outer (respectively inner, reflexive) inverses of [Formula: see text] in [Formula: see text]. In the paper, we study the inclusion properties of [Formula: see text], [Formula: see text] and [Formula: see text]. Among other results, we prove that for [Formula: see text] with [Formula: see text] von Neumann regular, [Formula: see text] (respectively [Formula: see text]) if and only if [Formula: see text] (respectively [Formula: see text]). Here, [Formula: see text] is the smallest idempotent in [Formula: see text] such that [Formula: see text]. This gives a common generalization of several known results.

Different characterizations of DMP-inverse of matrices

In this paper we discuss different properties of DMP-inverse of a square matrix introduced by Malik and Thome [On a new generalized inverse for matrices of an arbitrary index. Applied Mathematics and Computation. 2014;226: 575–580]. We characterize DMP-inverse of a square matrix as an outer inverse with prescribed range and null space, as a -inverse and as a Drazin inverse. Also we present several different characterizations based on its range and null space as well some algebraic ones.

Comparative study on metal/CFRP hybrid structures under static and dynamic loading

This study aims to explore the crushing behavior of aluminum (AL) - carbon fiber reinforced plastic (CFRP) tubes with different hybrid configurations subjected to quasi-static and dynamic loading conditions. First, a series of experimental tests are carried out to explore the crushing behaviors of hybrid tubes in comparison with the corresponding individual tubes made of single material. The experimental results indicate that the H-II hybrid tube, made of an outer aluminum circular tube and internally adhered CFRP layers, generates a unique deformation pattern; whose outer aluminum tube inverses externally and inner CFRP layers crush progressively. With these distinctive deformation features, the H-II hybrid tubes are considered to be ideal with superior crashworthiness and energy-absorbing capacity. It is also found that loading rate has little influence on deformation pattern of hybrid tubes and single material tubes, while energy-absorbing capacity of hybrid tubes and individual CFRP tubes under dynamic loading are substantially lower than those under quasi-static loading. Second, numerical simulations are performed for the H-II hybrid tubes to provide further insights into their underlying energy-absorbing mechanisms. It is found that the external inversion mode of the outer aluminum tube is the major energy-absorbing mechanism, in which the contribution of the outer aluminum tube to total energy absorption decreases with increase in thickness of CFRP layers. The internal energy of the externally inversed aluminum tube is considerably higher than internal energy of typical progressively-folded AL tube (sole aluminum tube). Third, a parametric study is further conducted, which indicates that with increasing aluminum wall thickness, the specific energy absorption (SEA) increases. Besides, it is found that varying fiber orientation of inner CFRP layers leads to no evident change in the deformation mode and SEA of the H-II hybrid tubes. When the interfacial strength in between aluminum and CFPR reaches a certain level, there is no evident increase in the total energy absorption with further increase of the interface strength, but the initial peak crushing force increases notably. These results are expected to deepen the understanding of crushing behavior of the H-II hybrid tubes, thereby providing guidance for the crashworthiness design.

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.

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.