On higher-order singular discrete linear systems

In this paper the solutions of a homogeneous higher-order singular (i.e. det A(l) = 0) discrete linear system of the form A(l)x(k+1) + A(l-1)x(k) + A(l-2)x(k-1) + (...) + A(0)x(k-l+1) = 0 are investigated. By defining a new state vector, the above system is transformed to a first-order discrete linear system (A) over tildey(k+1) = (B) over tildey(k), k=0,1,2..., with suitably defined matrices Using the complex Weierstrass canonical form when the matrix pencil s (A) over tilde - (B) over tilde is regular, and the Kronecker canonical form when s (A) over tilde - (B) over tilde 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.