Lyapunov iteration and equation characterizations for unobservable subspace

This note is concerned with the unobservable subspace of a linear system and some Lyapunov iteration and equations. It is shown that the unobservable subspace can be characterized by the Lyapunov iteration and equations defined in the paper. The results generalize some standard results on this topic and are expected to take fundamental functions in control system theory. Both continuous-time and discrete-time systems are considered. Numerical examples show the effectiveness of the proposed results.