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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]

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]

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]

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]

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]

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]