Investigating the solution properties of symmetric/skew-symmetric LTI homogeneous matrix descriptor discrete-time systems with consistent initial conditions and constant delay period

In this paper, the Thompson´s canonical form for a regular and singular matrix pencil of complex matrices with symmetric and skew symmetric structural properties is introduced for the solution of linear and time invariant (LTI) matrix homogeneous descriptor discrete time system with consistent initial conditions and time delay. Under this approach, the main equation is divided into several sub-systems whose solutions are derived. Note that the regularity or singularity of matrix pencil predetermines the number of sub-systems.