Almost diagonal systems of linear difference equations

Linear homogeneous difference systems of equations on an infinite interval are considered. Conditions are given which allow to represent a fundamental solution as a product of two square matrices. The product is made of a diagonal matrix and a matrix which is a perturbation of the identity. The diagonal matrix being on the left in the proposed product rather than on the right as it has been traditionally represented. A self contained theorem of asymptotic approximation is provided which is applied to a special family of difference systems. A theorem, utilizing a linear transformation, which is best possible in a certain sense, is also given.