As level sets of quadratic Lyapunov functions, ellipsoids have been extensively used as estimates of the domain of attraction of a linear system under saturated linear feedback. For a linear system with a single input subject to actuator saturation, based on a convex hull representation of saturation functions, a necessary and sufficient condition for an ellipsoid to be contractively invariant was previously established, which, through the solution of an LMI problem, leads to the maximal ellipsoidal invariant set. In this technical note, for a linear system with multiple inputs subject to actuator saturation, we develop a complete characterization of the maximal ellipsoidal invariant set by algebraic computation. Simulation results demonstrate the effectiveness of our results.