Analytical approximation of the maximal invariant ellipsoids for linear systems with actuator saturation is established. Simultaneously, a necessary and sufficient condition for such approximation equalling to the exact maximal invariant ellipsoid is presented. In addition, a geometric property of the maximal invariant ellipsoid in "minimal energy control with guaranteed convergence rate" problem is proposed. All results are given in explicit forms and easy to implement in system analysis and design. Two numerical examples are used to show the effectiveness of the proposed approach.