The authors present a survey on recent results connected to the discrete linear vector Hamiltonian system
where and are Hermitian matrix functions on the discrete interval , and it is assumed that is nonsingular on .
The paper is organized as follows. Section 2 introduces linear Hamiltonian difference systems and explains how they are special cases of symplectic difference systems. Section 3 gives an introduction to symplectic systems and introduces important notions such as conjoined bases, generalized zeros, focal points, and disconjugacy.
In Section 4 are introduced the concept and provided the motivation of trigonometric systems. An explicit transformation is also offered which transforms any symplectic system into a trigonometric system and preserves oscillatory behavior. Section 6 presents an overview of the discrete calculus of variations. In Section 7 an introduction is given to the theory of Stefan Higler in order to unify discrete and continuous analysis.