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Let an n × n complex matrix A be such that I + A is invertible. The Cayley transform of A , denoted by C (A) , is defined as C (A) = (I + A) − 1 (I − A). Fallat and Tsatsomeros (2002) [5] and Mondal et al. (2024) [15] studied the Cayley transform of a matrix A in the context of P-matrices, H-matrices, M-matrices, totally positive matrices, positive definite matrices, almost skew-Hermitian matrices, and semipositive matrices. In this paper, the investigation of the Cayley transform is continued for Toeplitz matrices, circulant matrices, unipotent matrices, and dual matrices. An expression of the Cayley transform for dual matrices is established. It is shown that the Cayley transform of a dual symmetric matrix is always a dual symmetric matrix. The Cayley transform of a dual skew-symmetric matrix is discussed. The results are illustrated with examples. [ABSTRACT FROM AUTHOR]

This article establishes a few sufficient conditions of the forward-order law for the core inverse of elements in rings with involution. It also presents the forward-order law for the weighted core inverse and the triple forward-order law for the core inverse. Additionally, we discuss the hybrid forward-order law involving different generalized inverses like the Moore–Penrose inverse, the group inverse, and the core inverse. [ABSTRACT FROM AUTHOR]

The notions of C-tensor, C0-tensor and C ¯ -tensor are introduced first. Different necessary and sufficient conditions for a tensor to be a C-tensor, C0-tensor and C ¯ -tensor are provided. We next show that the sum of two C-tensors (C0-tensors) is a C-tensor (C0-tensor) while the Hadamard product of two C-tensors (C0-tensors) is not a C-tensor (C0-tensor). We also present a result that illustrates the Hadamard product of two C-tensor is again a C-tensor under some sufficient conditions. As an application of these classes of tensors, an exclusion interval for the real eigenvalues of a real tensor is proposed. Finally, we provide a necessary and sufficient condition for the exclusion interval to be nonempty. [ABSTRACT FROM AUTHOR]

The notion of the Moore–Penrose inverse of an even-order tensor and the two-term reverse-order law for the Moore–Penrose inverse of even-order tensors via the Einstein product were introduced, very recently. In this article, the Moore–Penrose inverse of an arbitrary tensor is introduced first. Various new expressions of the Moore–Penrose inverse are also proposed. A new generalized inverse of a tensor called product Moore–Penrose inverse is then introduced by extending the Moore–Penrose inverse of an arbitrary tensor. A necessary and sufficient condition for the coincidence of the Moore–Penrose inverse and the product Moore–Penrose inverse of an arbitrary tensor is also provided. Prior to these, a set of new sufficient conditions for computing the Moore–Penrose inverse of the product of two tensors via the Einstein product is illustrated. Finally, some necessary and sufficient conditions are obtained for the three-term reverse-order law of tensors via the same product. [ABSTRACT FROM AUTHOR]

Let A be a bounded linear positive operator on a complex Hilbert space H. Furthermore, let B A (H) denote the set of all bounded linear operators on H whose A-adjoint exists, and A signify a diagonal operator matrix with diagonal entries are A. Very recently, several A -numerical radius inequalities of 2 × 2 operator matrices were established. In this paper, we prove a few new A -numerical radius inequalities for 2 × 2 and n × n operator matrices. We also provide a new proof of an existing result by relaxing a sufficient condition “A is strictly positive”. Our proofs show the importance of the theory of the Moore–Penrose inverse of a bounded linear operator in this field of study. [ABSTRACT FROM AUTHOR]

This article proposes a generalized rotated shift-splitting (GRSS) iterative method for solving complex symmetric linear systems. Our analysis indicates that the proposed scheme is convergent. Spectrum features of some preconditioned matrices are also discussed. Finally, we report numerical experiments substantiating our theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a two-stage method to solve rectangular linear systems that exhibits faster convergence than typical stationary iterative methods. Under suitable conditions, we prove convergence of the new method. The number of outer iterations can be reduced by using a few significant number of inner iterations for efficient computations. Further, we perform a comparison analysis, and establish that a higher number of inner iterations ensures a smaller spectral radius of the global iteration matrix. We also discuss the uniqueness of a proper splitting, and illustrate different comparison theorems for different subclasses of proper splittings. [ABSTRACT FROM AUTHOR]

Theory of matrix splittings is a useful tool in the analysis of iterative methods for solving systems of linear equations. When two splittings are given, it is of interest to compare the spectral radii of the corresponding iteration matrices. This helps to arrive at the conclusion that which splitting should one choose so that one can reach the desired solution of accuracy or the exact solution in a faster way. In the case of many splittings are provided, the comparison of the spectral radii is time-consuming. Such a situation can be overcome by introducing another iteration scheme which converges to the same solution of interest in a much faster way. In this direction, the theory of alternating iterations for real rectangular matrices is recently proposed. In this note, some more results to the theory of alternating iterations are added. A comparison result of two different alternating iteration schemes is then presented which will help us to choose the iteration scheme that will guarantee the faster convergence of the alternating iteration scheme. In addition to these results, a comparison result for proper weak regular splittings is also obtained. [ABSTRACT FROM AUTHOR]

The notion of the Moore–Penrose inverse of tensors with the Einstein product was introduced, very recently. In this paper, we further elaborate on this theory by producing a few characterizations of different generalized inverses of tensors. A new method to compute the Moore–Penrose inverse of tensors is proposed. Reverse order laws for several generalized inverses of tensors are also presented. In addition to these, we discuss general solutions of multilinear systems of tensors using such theory. [ABSTRACT FROM AUTHOR]

Different classes of matrix splittings of semi-monotone matrices lead to many comparison results, which are useful tools in examining the convergence rate of iterative methods for solving rectangular systems of linear equations in a faster way. In this context, the theory of alternating iterations for rectangular matrices was introduced recently, in order to arrive at the desired solution of accuracy or at the exact solution. In this article, we expand convergence theory of such alternating iterations and obtain comparison results for such iterations. [ABSTRACT FROM AUTHOR]

Theory of numerical range and numerical radius for tensors is not studied much in the literature. Ke et al. [Linear Algebra Appl. 508 (2016), 100-132: MR3542984] introduced first the notion of the numerical range of a tensor via the k-mode product. However, the convexity of the numerical range via the k-mode product was not proved by them. In this paper, the notion of numerical range and numerical radius for even-order square tensors via the Einstein product are introduced first. Using the notion of the numerical radius of a tensor, we provide some sufficient conditions for a tensor to be unitary. The convexity of the numerical range is also proved. We also provide an algorithm to plot the numerical range of a tensor. Furthermore, some properties of the numerical range for the Moore–Penrose inverse of a tensor are discussed. [ABSTRACT FROM AUTHOR]

Different convergence and comparison theorems for proper regular splittings and proper weak regular splittings are discussed. The notion of double splitting is also extended to rectangular matrices. Finally, convergence and comparison theorems using this notion are presented. [ABSTRACT FROM AUTHOR]

Abstract: The extension of the nonnegative splitting for rectangular matrices called proper nonnegative splitting is proposed first. Different convergence and comparison theorems for the proper nonnegative splittings are established. The notion of double nonnegative splitting is then generalized to rectangular matrices. Finally, different convergence and comparison results are presented for this decomposition. The case for singular square matrices is also studied. [Copyright &y& Elsevier]

In this article, a convergence theorem and comparison theorems for Brou-splittings and Bran-splittings (introduced in [D. Mishra, K.C. Sivakumar, Generalizations of matrix monotonicity and their relationships with certain subclasses of proper splittings, Linear Algebra Appl. (accepted)]) are presented. We then conclude with a new characterization of range monotone matrices. [ABSTRACT FROM AUTHOR]

Abstract: In this article, a convergence theorem and several comparison theorems are presented for a subclass of proper splittings of matrices introduced recently. [Copyright &y& Elsevier]

Abstract: A new matrix decomposition of real square singular matrices called -splitting is proposed by extending the notion of B-splitting for nonsingular matrices. Then different convergence and comparison theorems for these splittings are discussed. Another convergence theorem using a generalized regular splitting is also proved. As a result, a new characterization of monotone matrices using regular splitting is obtained as a corollary. [Copyright &y& Elsevier]

A dominance rule for singular matrices using proper splittings is proposed. This extends the corresponding notion, known for nonsingular matrices. An application to the nonnegativity of the Moore–Penrose inverse is presented. [ABSTRACT FROM AUTHOR]

Abstract: In this article, a classical result of Cottle and Veinott characterizing least elements of polyhedral sets in terms of nonnegative left-inverses is extended to characterize nonnegativity of some of the most important generalized inverses. We also present generalizations of the other results of these authors for the case of the Moore–Penrose inverse. [Copyright &y& Elsevier]

Let H be a complex Hilbert space, and A be a positive bounded linear operator on H. Let B A (H) denote the set of all bounded linear operators on H whose A-adjoint exists. Let A denote a 2 × 2 operator matrix of the form A O O A . Very recently, for a strictly positive operator A, Bhunia et al. [On inequalities for A-numerical radius of operators. Electron J Linear Algebra. 2020;36:143–157] proved an important lemma (Lemma 2.4) to establish several A -numerical radius inequalities for operator matrices in B A (H ⨁ H) . In this article, we first prove an analogous result and then provide a new proof of the same lemma by dropping the assumption 'A is strictly positive'. We then establish several new upper and lower bounds for the A -numerical radius of an operator matrix whose entries are operators in B A (H). Further, we prove some refinements of earlier A-numerical radius inequalities for operators in B A (H) . [ABSTRACT FROM AUTHOR]

Iterative methods based on matrix splittings are useful tools in solving real large sparse linear systems. In this aspect, the type I double splitting approaches are straight forward from the formulation of the iteration scheme and its convergence theory is well established in the literature. However, if a double splitting is of type II, then the convergence of the iteration scheme seems not to be straight forward. In this paper, we develop convergence theory for type II double splittings to make the implementation quite simple. In this direction, we first introduce two new subclasses of double splittings and establish their convergence theory. Using this theory, we prove a new characterization of a monotone matrix. Finally, we apply our theoretical findings to the double splitting of an M-matrix in the Gauss–Seidel double SOR method to obtain a comparison result. [ABSTRACT FROM AUTHOR]

Recently, Wang (2017) has introduced the K-nonnegative double splitting using the notion of matrices that leave a cone K ⊆ ℝn invariant and studied its convergence theory by generalizing the corresponding results for the nonnegative double splitting by Song and Song (2011). However, the convergence theory for K-weak regular and K-nonnegative double splittings of type II is not yet studied. In this article, we first introduce this class of splittings and then discuss the convergence theory for these sub-classes of matrices. We then obtain the comparison results for two double splittings of a K-monotone matrix. Most of these results are completely new even for K = ℝ + n . The convergence behavior is discussed by performing numerical experiments for different matrices derived from the discretized Poisson equation. [ABSTRACT FROM AUTHOR]

In this article, we obtain some upper bounds for A-numerical radius of operators in semi-Hilbertian spaces which improve the existing ones. Furthermore, we discuss some generalizations of earlier A-numerical radius inequalities. We also establish several inequalities for B-numerical radius of n × n operator matrices, where B is an n × n diagonal matrix whose diagonal entries are positive operator A. [ABSTRACT FROM AUTHOR]

Iterative methods based on matrix splittings are useful in solving large sparse linear systems. In this direction, proper splittings and its several extensions are used to deal with singular and rectangular linear systems. In this article, we introduce a new iteration scheme called three-step alternating iterations using proper splittings and group inverses to find an approximate solution of singular linear systems, iteratively. As a special case, the same findings also work for finding an approximate solution of non-singular linear systems. A preconditioned alternating iterative scheme is also proposed to relax some sufficient conditions and to obtain faster convergence as well. We then show that our scheme converges faster than the unpreconditioned one. The theoretical findings are then validated numerically. [ABSTRACT FROM AUTHOR]

Reverse-order law for the Moore–Penrose inverses of tensors is useful in the field of multilinear algebra. In this paper, we first prove some more identities involving the Moore–Penrose inverses of tensors. We then obtain a few necessary and sufficient conditions of the reverse-order law for the Moore–Penrose inverses of tensors via the Einstein product. [ABSTRACT FROM AUTHOR]

Manufacturing industries are looking for efficient assembly planners that can swiftly develop a practically feasible assembly sequence while keeping costs and time to a minimum. Most assembly sequence planners rely on part relations in the virtual environment. Nowadays, tools and robotic grippers perform most of the assembly tasks. Ignoring the critical aspect renders solutions practically infeasible. Additionally, it is vital to test the feasibility of positioning and assembling components while employing robotic grippers and tools prior to their implementation. This paper presents a novel concept named by considering both part and tool geometry to propose "tool integrated assembly interference matrices" (TIAIMs) and a "tool integrated axis-aligned bounding box" (TIAABB) to generate practically feasible assembly sequence plans. Furthermore, the part-concatenation technique is used to determine the best assembly sequence plans for an actual mechanical component. The results show that the proposed approach effectively and efficiently deals with real-life industrial problems. [ABSTRACT FROM AUTHOR]

Matrix double splitting iterations are simple in implementation while solving real non-singular (rectangular) linear systems. In this paper, we present two Alternating Double Splitting (ADS) schemes formulated by two double splittings and then alternating the respective iterations. The convergence conditions are then discussed along with comparative analysis. The set of double splittings used in each ADS scheme induces a preconditioned system which helps in showing the convergence of the ADS schemes. We also show that the classes of matrices for which one ADS scheme is better than the other, are mutually exclusive. Numerical experiments confirm that the proposed ADS schemes have several computational advantages over the existing methods. Though the problems are considered in the rectangular matrix settings, the same problems are even new in non-singular matrix settings. [ABSTRACT FROM AUTHOR]

• Convergence theory of two-stage iterative scheme for K-weak regular splittings of both types in the proper cone setting is established. • Numerical implementation has been discussed on next generation matrix and PageRank matrix. Monotone matrices play a key role in the convergence theory of regular splittings and different types of weak regular splittings. If monotonicity fails, then it is difficult to guarantee the convergence of the above-mentioned classes of matrices. In such a case, K -monotonicity is sufficient for the convergence of K -regular and K -weak regular splittings, where K is a proper cone in R n. However, the convergence theory of a two-stage iteration scheme in a general proper cone setting is a gap in the literature. Especially, the same study for weak regular splittings of type II (even if in standard proper cone setting, i.e., K = R + n), is open. To this end, we propose convergence theory of two-stage iterative scheme for K -weak regular splittings of both types in the proper cone setting. We provide some sufficient conditions which guarantee that the induced splitting from a two-stage iterative scheme is a K -regular splitting and then establish some comparison theorems. We also study K -monotone convergence theory of the stationary two-stage iterative method in case of a K -weak regular splitting of type II. The most interesting and important part of this work is on M -matrices appearing in the Covid-19 pandemic model. Finally, numerical computations are performed using the proposed technique to compute the next generation matrix involved in the pandemic model. The computation of large PageRank matrices shows that the two-stage Gauss-Seidel method performs better than the Gauss-Seidel methods. [ABSTRACT FROM AUTHOR]