In this note, we consider a few important issues related to the maximization of the convergence rate inside a given ellipsoid for linear systems with input saturation. For continuous-time systems, the control that maximizes the convergence rate is simply a bang-bang control. Through studying the system under the maximal convergence control, we reveal several fundamental results on set invariance. An important consequence of maximizing the convergence rate is that the maximal invariant ellipsoid is produced. We provide a simple method for finding the maximal invariant ellipsoid, and we also study the dependence of the maximal convergence rate on the Lyapunov function.