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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]