Many problems in applied mathematics can be formulated as a Sylvester matrix equation AX+XB=C. Iterative methods for solving this equation are appropriate in applications coming from numerical treatment of elliptic problems and from control and systems theory. We determine the solution of this matrix equation with the Arithmetic Mean method, which is ideally suited for implementation on parallel computers. We consider different cases: A is large and sparse with a non random sparsity pattern and B is large with a simple structure; A and B are banded; A and B are large and without any special structure. The main purpose of this paper is to develop a convergence analysis of the method, using different splittings of the matrices A and B.