In this paper we introduce a new (extended) theory of genera of matrix solutions of linear symplectic difference systems. In comparison with the current literature (2017), we do not assume the nonoscillation of the symplectic system and drop the requirement that the solutions are conjoined bases. We derive a characterization of all genera of matrix solutions and show that they form a complete lattice. The structure of this lattice is determined by the so-called minimal genus. These results are based on an analysis of subspaces of solutions of a general linear difference system and its adjoint system and on properties of orthogonal projectors satisfying a certain Riccati-type matrix difference equation. Although this paper is motivated by the theory of genera of conjoined bases for linear Hamiltonian differential systems (2020), the results are derived by new approaches and methods, which are natural for the underlying discrete time domain.

Last change: 14.04.2026. (c) Roman Simon Hilscher