On generalized linear matrix difference systems

In this paper we consider Generalized Linear Matrix Difference Systems of the form: FYk+ρ = GYk, where F,G∈Rm×n or Cm×n, ρ ∈ N, and Yk are n × l matrices, for k = 0, 1, 2, .... In case sF - G is a regular matrix pencil, using the Weierstrass canonical form, the above system is decomposed in two subsystems whose solutions are obtained. Moreover the form of the so-called consistent initial condition is given. We also study the case of sF - G being a singular matrix pencil, where, using the Kronecker canonical form, we decompose the system in five subsystems and we obtain the corresponding solutions.