Classical procedures for the realization of transfer functions are unable to represent uncontrollable behaviors. In this paper, we use companion matrices and the Smith form to derive explicit observable realizations for a general (not necessarily controllable) linear time-invariant behavior. We then exploit the properties of companion matrices to efficiently compute trajectories, and the solutions to Lyapunov equations, for the realizations obtained. The results are motivated by the important role played by uncontrollable behaviors in the context of physical systems such as passive electrical and mechanical networks.