A Note on the Dimensions of the Structural Invariant Subspaces of the Discrete-Time Singular Hamiltonian Systems

I. MAIN CONTENT In [1], the structural invariant subspaces V1 and V2, defined by (17) and (18) respectively, are used in (19), that is the analytic nonrecursive expression of the set of the admissible solutions of the discrete-time singular Hamiltonian system (10) over the time interval 0≤ k≤ kf − 1. In this note it will be shown that the dimension of the subspace V2 may be lower than n, where n denotes the dimension of the state space of the original system as defined in (1), (2). In particular, the possible loss of dimension of V2 (or, equivalently, the possible loss of rank of the matrix V2 defined by (18)) depends on the properties of the original system (1), (2) (under assumptions A.1–A.4). Let us consider system (1), (2) and perform the similarity transformation T = [T1 T2], where imT1=R, the reachable subspace of (A,B). With respect to the new basis,