Existence and uniqueness of solutions of linear variable coefficient discrete-time descriptor systems

We consider linear discrete-time descriptor systems, i.e., systems of linear equations of the form Ekxk+1=Akxk+fk for k∈Z, where all Ek and Ak are matrices, fk are vectors and xk are the vectors of the solution we are looking for. We study the existence and uniqueness of solutions. A strangeness index is defined for such systems. Compared to the continuous-time case, see [P. Kunkel, V. Mehrmann, Differential-Algebraic Equations – Analysis and Numerical Solution, European Mathematical Society, Zürich, 2006], in the discrete-time case we have to account for the fact that it makes a difference, if one has an initial condition and one wants a solution in the future or if one has an initial condition and one wants a solution into the past and the future at the same time.