In this paper, a new type generalized Lyapunov equation for discrete singular systems is proposed. Then it is applied to study problems such as pole clustering, controllability and observability for discrete singular systems. First, some necessary and su6cient conditions for pole clustering are derived via the solution of this new type Lyapunov equation. Further, the relationship between the solution of the Lyapunov equation and structure properties of discrete singular systems will be investigated based on these results. Finally, a type of generalized Riccati equation is proposed and its solution is used to design state feedback law for discrete singular systems such that all the 8nite poles of the closed-loop systems are clustered into a speci8ed disk.