Synchronization of high-order discrete-time complex networks with undirected topologies is studied and the impacts of time delays are investigated. Firstly, by the state decomposition, synchronization problems are transformed into asymptotic stability ones of multiple lower dimensional time-delayed subsystems. Then, linear matrix inequality ( LMI) criteria for synchronization are given, which can guarantee the scalability of complex networks since they only include three LMI constraints independent of the number of agents. Moreover, an explicit expression of the synchronization function is presented, which can describe the synchronization behavior of all agents in complex networks. Finally, a numerical example is given to demonstrate the theoretical results, where it is shown that if the gain matrices of synchronization protocols satisfy LMI criteria for synchronization, synchronization can be achieved.