On Higher-Order Singular Discrete Linear Systems

G.I. Kalogeropoulos and D.P. Papachristopoulos (Greece)


Matrix Pencils, Canonical Forms, Singular Discrete Linear Systems


In this paper the solutions of a homogeneous higher–order singular (i.e. det A = 0) discrete linear system of the form A xk+1 + A −1xk + A −2xk−1 + · · · + A0xk− +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 sA−B is regular, and the Kronecker canonical form when sA − 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.

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