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Dual number matrix decompositions have played an important role in fields such as kinematics and computer graphics in recent years. In this paper, we present a QR decomposition algorithm for dual number matrices. When dealing with large-scale problems, we present the thin QR decomposition of dual number matrices, along with its algorithm with column pivoting. In numerical experiments, we discuss the suitability of different QR algorithms when confronted with various large-scale dual matrices, providing their respective domains of applicability. Finally, we employ the QR decomposition of dual matrices to compute the DMPGI, attaining results of higher precision. [ABSTRACT FROM AUTHOR]

Abstract: For when the Sylvester matrix equation has a unique solution, this work provides a closed form solution, which is expressed as a polynomial of known matrices. In the case of non-uniqueness, the solution set of the Sylvester matrix equation is a subset of that of a deduced equation, which is a system of linear algebraic equations. [Copyright &y& Elsevier]

We present sufficient conditions for the unique solvability of the matrix equation A X B + C | X | D = E where A , B , C , D and E are given square or rectangular real matrices of appropriate size and X is unknown. We also give sufficient conditions for the non-solvability of the equation in special cases. Our sufficient conditions can be checked with a cubic complexity in terms of the size of the input matrices. [ABSTRACT FROM AUTHOR]

Various integrable systems, both continuous and discrete, having soliton solutions have been studied through the generalized Cauchy matrix method. Here we employ this approach to investigate a lattice CKP equation which is expressed by a τ function. We start from introducing a Sylvester equation and then show how to solve it. The variables in the Sylvester equation help us construct the τ function of the lattice CKP equation, which yields different types of exact solutions to the lattice CKP equation as well. [ABSTRACT FROM AUTHOR]