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In this article, we study the stability of a class of singular linear matrix difference equations whose coefficients are square constant matrices and the leading coefficient matrix is singular. Speciffically we analyze the stability, the asymptotic stability and the Lyapunov stability of the equilibrium states of an homogeneous singular linear discrete time system and we define the set of all equilibrium states. After we prove that if every equilibrium state of the homogeneous system is stable in the Lyapounov's sense, then all solutions of the non homogeneous system are continuously depending on the initial conditions and are bounded provided that the input vector is also bounded. Moreover, we consider the case where the equilibrium states of the system are not stable. For this case we provide necessary and sufficient conditions for stabilization.

On stability and state feedback stabilization of singular linear matrix difference equations