A note on the causality of singular linear discrete time systems

where F,G ∈ M(n × n;F), Yk ∈ M(n × 1;F) (i.e., the algebra of square matrices with elements in the field F), Xk ∈ M(m× 1;F), B ∈ M(n× l;F) and C ∈ M(m×n;F). For the sake of simplicity, we set Mn = M(n× n;F) and Mnm = M(n×m;F). We assume that the system (1) is singular, i.e. the matrix F is singular and that the input vector Vk changes only at equally space sampling instants. Many authors have studied discrete time systems, see and their applications, see [1-9, 12, 13, 16-18, 20-32]. In this article we study the causality of these systems. The results of this paper can be applied also in systems of fractional nabla difference equations, see [10, 11]. In addition they are very useful for applications in many mathematical models using systems of difference equations existing in the literature, see [14, 15, 29-32].