On higher-order singular discrete linear systems

In this paper the solutions of a homogeneous higher-order singular (i.e. det Al =0) discrete linear system of the form Alxk+1+Al-1xk+Al-2xk-1 + ... + A0xk-l+1 = 0 are investigated. By defining a new state vector, the above system is transformed to a first-order discrete linear system Ayk+1 = Byk, k =0, 1, 2, ..., with suitably defined matrices A, B. Using the complex Weierstrass canonical form when the matrix pencil s A - B is regular, and the Kronecker canonical form when s A - B is singular, the above system can be splitted into two or five subsystems respectively, whose solutions are obtained. Finally, the uniqueness of the solution (only for the regular case) is proved.