Transient response in matrix discrete-time linear systems

The behavior of trajectories of multidimensional linear discrete-time systems with nonzero initial conditions is considered in two cases as follows. The first case is the systems with infinite degree of stability (the processes of a finite duration); the second case is the stable systems with a spectral radius close to 1. It is demonstrated that in both cases, large deviations of the trajectories from the equilibrium may occur. These results are applied to accelerated unconstrained optimization methods (such as the Heavy-ball method) for explaining the nonmonotonic behavior of the methods, which is observed in practice.