The stability analysis for a class of discrete-time linear systems with state saturation nonlinearity is concemed. By introducing a free matrix with infinity norm less than or equal to 1 and a diagonal matrix with nonpositive diagonal elements, the state of this discrete-time linear system under state saturation constraint is confined in a convex hull. In this way, a criterion for discrete-time linear systems with state saturation to be asymptotically stable is obtained in terms of bilinear matrix inequalities that can be resolved by using the presented iterative linear matrix inequality algorithm. Based on this criterion, the state feedback control law synthesis problem is also resolved and the corresponding iterative linear matrix inequality algorithm is given. A further study shows that the space division method can be also applied to solve this problem with less conservativeness. Numerical examples are used to illustrate the effectiveness and correctness of the proposed method.