The stability analysis for a class of discrete-time linear systems with state saturation nonlinearity is studied in this paper. By introducing a free matrix whose infinity norm is less than or equal to 1 and a diagonal matrix with non-positive diagonal elements, the discrete-time state under saturation constraint is confined in a convex hull. In this way, a stability 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 using the presented iterative linear matrix inequality algorithm. The state feedback control law synthesis problem is also resolved and the corresponding iterative linear matrix inequality synthesis algorithm is given. Two numerical examples show that the presented method is applicable and effective.