Outer Inverses Research Articles

Outer inverse of reduced biquaternion matrices

Outer inverse of reduced biquaternion matrices

Efficient non-probabilistic parallel model updating based on analytical correlation propagation formula and derivative-aware deep neural network metamodel

Non-probabilistic convex models are powerful tools for structural model updating with uncertain‑but-bounded parameters. However, existing non-probabilistic model updating (NPMU) methods often struggle with detecting parameter correlation due to limited prior information. Worth still, the unique core steps of NPMU, involving nested inner layer forward uncertainty propagation and outer layer inverse parameter updating, present challenges in efficiency and convergence. In response to these challenges, a novel and flexible NPMU scheme is introduced, integrating analytical correlation propagation and parallel interval bounds prediction to enable automatic detection of parameter correlations. In the forward uncertainty propagation phase, a linear coordinate transformation is applied to map the original parameter space to a standard hypercube space, simplifying correlation-involved bounds prediction into conventional interval bounds prediction. Moreover, an analytical correlation propagation formula is derived using a second-order response approximation to sidestep the complexities of geometry-based correlation calculations. To expedite forward propagation, a derivative-aware neural network model is employed to replace the physical solver, facilitating improved fitting capabilities and automatic differentiation, including the calculation of Jacobian and Hessian matrices essential for correlation propagation. The neural network's inherent parallelism accelerates interval bounds prediction through parallel computation of samples. In the inverse parameter updating phase, the block coordinate descent algorithm is embraced to narrow the search space and boost convergence capabilities, while the perturbation method is utilized to determine the optimal starting point for optimization. Two numerical examples illustrate the efficacy of the proposed method in updating structural models while considering correlations.

Computation of Outer Inverse of Tensors Based on t‐Product

ABSTRACTTensor computations play an essential role in various fields of science and engineering, including multiway data analysis. In this study, we established a few basic properties of the range and null space of a tensor by using block circulant matrices and a discrete Fourier matrix. We then discuss the outer inverse of the tensors based on ‐product with a prescribed range and kernel of third‐order tensors. We address the relation of this outer inverse with other generalized inverses, such as the Moore–Penrose inverse, group inverse, and Drazin inverse. In addition, we present a few algorithms for computing the outer inverses of the tensors. In particular, a ‐QR decomposition based algorithm was developed to compute outer inverses. It is well known that the confidentiality of information transmitted through the virtual world grows exponentially, and color image and video security have become a significant concern when communicating over the internet. As an application, a ‐QR decomposition based algorithm was demonstrated for concealing secret color images and videos.

Extending EP matrices by means of recent generalized inverses

It is well known that a square complex matrix is called EP if it commutes with its Moore–Penrose inverse. In this paper, new classes of matrices which extend this concept are characterized. For that, we consider commutative equalities given by matrices of arbitrary index and generalized inverses recently investigated in the literature. More specifically, these classes are characterized by expressions of type AmX=XAm\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A^mX=XA^m$$\\end{document}, where X is an outer inverse of a given complex square matrix A and m is an arbitrary positive integer. The relationships between the different classes of matrices are also analyzed. Finally, a picture presents an overview of the overall studied classes.

Matrices having nonzero outer inverses

It is well known that every nonzero von Neumann regular $m\times n$-matrix $A$ over an arbitrary ring $R$ has a nonzero outer inverse $n\times m$-matrix $B$ in the sense that $B=BAB$. Generalizing previous work on von Neumann regular matrices, the matrices having nonzero outer inverses over semiperfect rings are characterized as the matrices having some entry outside the Jacobson radical of $R$. Such matrices over finite semiperfect rings and finite commutative rings are counted, and several applications are given.

Von Neumann local matrices

We use our recent results on von Neumann regular matrices, strongly regular matrices and matrices having a non-zero outer inverse to derive applications to some generalizations of these concepts, called von Neumann local, strongly von Neumann local and outer von Neumann local matrices. Among other properties, we show that the $t^{\rm th}$ compound matrix of every matrix of determinantal rank $t$ over a commutative local ring is strongly von Neumann local, and every matrix over an arbitrary semiperfect ring is outer von Neumann local.

Zeroing neural network approaches for computing time-varying minimal rank outer inverse

Generalized inverses are extremely effective in many areas of mathematics and engineering. The zeroing neural network (ZNN) technique, which is currently recognized as the state-of-the-art approach for calculating the time-varying Moore-Penrose matrix inverse, is investigated in this study as a solution to the problem of calculating the time-varying minimum rank outer inverse (TV-MROI) with prescribed range and/or TV-MROI with prescribed kernel. As a result, four novel ZNN models are introduced for computing the TV-MROI, and their efficiency is examined. Numerical tests examine and validate the effectiveness of the introduced ZNN models for calculating TV-MROI with prescribed range and/or prescribed kernel.

Outer inverses in semigroups belonging to the prescribed Green’s equivalence classes

Outer inverses in semigroups belonging to the prescribed Green’s equivalence classes

Minimal Rank Properties of Outer Inverses with Prescribed Range and Null Space

The purpose of this paper is to investigate solvability of systems of constrained matrix equations in the form of constrained minimization problems. The main novelty of this paper is the unification of solutions of considered matrix equations with corresponding minimization problems. For a particular case we extend some well-known results and give several new results for the weak Drazin inverse. The main characterizations of the Drazin inverse, group inverse and Moore–Penrose inverse are obtained as consequences.

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.

Extensions of G-outer inverses

Our first objective is to present equivalent conditions for the solvability of the system of matrix equations ADA = A, D= B and CAD = C, where D is unknown, A, B,C are of appropriate dimensions, and to obtain its general solution in terms of appropriate inner inverses. Our leading idea is to find characterizations and representations of a subclass of inner inverses that satisfy some properties of outer inverses. A G-(B,C) inverse of A is defined as a solution of this matrix system. In this way, G-(B,C) inverses are defined and investigated as an extension of G-outer inverses. One-sided versions of G-(B,C) inverse are introduced as weaker kinds of G-(B,C) inverses and generalizations of one-sided versions of G-outer inverse. Applying the G-(B,C) inverse and its one-sided versions, we propose three new partial orders on the set of complex matrices. These new partial orders extend the concepts of G-outer (T, S)-partial order and one-sided G-outer (T, S)-partial orders.

Error bounds in the computation of outer inverses with generalized Schultz iterative methods and its use in computing of Moore-Penrose inverse

An error bound in computing of outer inverses is established in each iteration of the generalized Schultz iterative methods. With this error bound, we built class of iterative method for the calculation of the Moore-Penrose inverse, the class of methods uses these error bounds to generate monotonic inclusion interval matrices which congerges to Moore-Penrose inverse, this process using intervals prevents that round-off errors cause the divergence of the method. Theorems with the error bounds as well as the convergence of the new iterative scheme are proved. Numerical examples are presented to demonstrate the efficacy of the new class of methods.

Symmetric Properties of (b,c)-Inverses

Let b and c be two elements in a semigroup S. The (b,c)-inverse is an important outer inverse because it unifies many common generalized inverses. This paper is devoted to presenting some symmetric properties of (b,c)-inverses and (c,b)-inverses. We first find that S contains a (b,c)-invertible element if and only if it contains a (c,b)-invertible element. Then, for four given elements a,b,c,d in S, we prove that a is (b,c)-invertible and d is (c,b)-invertible if and only if abd is invertible along c and dca is invertible along b. Inspired by this result, the (b,c)-invertibility is characterized by one-sided invertible elements. Furthermore, we show that a is inner (b,c)-invertible and d is inner (c,b)-invertible if and only if c is inner (a,d)-invertible and b is inner (d,a)-invertible.

Commuting Outer Inverse-Based Solutions to the Yang–Baxter-like Matrix Equation

This paper investigates new solution sets for the Yang–Baxter-like (YB-like) matrix equation involving constant entries or rational functional entries over complex numbers. Towards this aim, first, we introduce and characterize an essential class of generalized outer inverses (termed as {2,5}-inverses) of a matrix, which commute with it. This class of {2,5}-inverses is defined based on resolving appropriate matrix equations and inner inverses. In general, solutions to such matrix equations represent optimization problems and require the minimization of corresponding matrix norms. We decided to analytically extend the obtained results to the derivation of explicit formulae for solving the YB-like matrix equation. Furthermore, algorithms for computing the solutions are developed corresponding to the suggested methods in some computer algebra systems. The main features of the proposed approach are highlighted and illustrated by numerical experiments.

Reverse order law for outer inverses and Moore-Penrose inverse in the context of star order.

The reverse order law for outer inverses and the Moore-Penrose inverse is discussed in the context of associative rings. A class of pairs of outer inverses that satisfy reverse order law is determined. The notions of left-star and right-star orders have been extended to the case of arbitrary associative rings with involution and many of their interesting properties are explored. The distinct behavior of projectors in association with the star, right-star, and left-star partial orders led to several equivalent conditions for the reverse order law for the Moore-Penrose inverse.

On finding strong approximate inverses for tensors

AbstractThis article investigates a fast and highly efficient algorithm to find the strong approximation inverse of an invertible tensor. The convergence analysis shows that the proposed method is of ten order of convergence using only six tensor–tensor multiplications per iteration. Also, we obtain a bound for the perturbation error in each iteration. We show that the proposed algorithm can be used for finding the Moore–Penrose and outer inverses of tensors. We obtain the relationship between the singular values of an arbitrary tensor and eigenvalues of the . We give the computational complexity of our algorithm and prove the theoretical aspects of the article. The generalized Moore–Penrose inverse of tensors is defined. As an application, we use the iteration obtained by the algorithm as preconditioning of the Krylov subspace methods to solve the multilinear system . Several numerical experiments are proposed to show the effectiveness and accuracy of the method. Finally, we give some concluding remarks.

An efficient matrix iteration family for finding the generalized outer inverse

This paper presents a new iterative family for computing the generalized outer inverse with a prescribed range and null space of a given complex matrix. It is proved that the proposed methods achieve at least ninth-order of convergence. In general, the improved formulation of scheme uses only seven matrix multiplications at each iteration, but for the specific parameters, it uses only five matrix multiplications. The theoretical discussion on computational efficiency index is presented. Further, numerical results obtained are compared with existing robust methods to verify the theoretical analysis and higher computational efficiency. Several numerical examples, including rectangular, rank-deficient, large sparse, and non-singular matrices from the matrix computation tool-box (mctoolbox) are included. Further, the performance of methods is measured on randomly generated singular matrices and boundary value problem. It is demonstrated that the presented scheme gives improved results than the existing Schulz-type iterative methods for calculating the generalized inverse.

Generalizations of composite inverses with certain image and/or kernel

Expressions which include the Moore-Penrose (MP) inverse with various generalized inverses have been popular topic in numerical linear algebra. The most general case was the combination of the MP inverse with outer inverses AR(B),N(C)(2), known as the composite outer inverses. Starting from well-known and useful Urquhart representation of generalized inverses, in this research we consider Φ1-composite outer inverses which are based on the replacements of AR(B),N(C)(2) by the more general expressions Φ1:=B(CAB)(1)C. Algorithms for symbolic and numeric computation of Φ1-composite outer inverses are proposed and corresponding examples are developed. In addition, Φ2-composite outer inverses arising from the replacements of the term AR(B),N(C)(2) in composite outer inverses by the expressions Φ2:=B(CAB)(2)C∈A{2}, are investigated.

Unit-regularity of elements in rings

An element [Formula: see text] in a unital ring [Formula: see text] is said to have an inverse complement [Formula: see text] if [Formula: see text] is a unit of [Formula: see text] and [Formula: see text]. Unit-regular elements are studied from the viewpoint of the existence of inverse complements. As a source of unit-regular elements, we prove that if [Formula: see text] is a completely reducible submodule of [Formula: see text], then every element of [Formula: see text] is unit-regular if and only if any nonzero submodule of [Formula: see text] is not square zero. This generalizes some results due to Stopar in 2020. Finally, extending the case of real or complex matrices to the context of rings, we characterize the outer and reflexive inverses of a given unit-regular element depending only on its inverse complement.

Weighted G-Outer Inverse of Banach Spaces Operators

In order to extend the notions of G-outer inverses and weighted G-Drazin inverses, we firstly define and characterize the (M, N)-weighted outer inverses for Banach spaces operators. Using the (M, N)-weighted outer inverse, beside the (M, N)-weighted outer equivalence relation, we create and study the (M, N)-weighted G-outer inverse of an operator between two Banach spaces. This new inverse presents generalization of the G-outer inverse and the W-weighted G-Drazin inverse. By means of (M, N)-weighted G-outer inverse, we also introduce and investigate the (M, N)-weighted G-outer relation which is a preorder on corresponding set. Thus, we extend the notions of the G-outer partial order and the W-weighted G-Drazin preorder for Banach spaces operators.