This article presents the derivation of novel necessary and sufficient conditions for the synchronization of a network of arbitrary but homogeneous linear dynamic systems. We solely focus on the case of full-state coupling. Consequently, it is assumed that the agents can measure and exchange their complete state. The result is given for static but directed communication networks. It requires the agent dynamics to be stabilizable and the underlying digraph of the network to be connected. The novel conditions are fully characterized by solving a decentralized control problem. Based on this, it is possible to recast the design of the full-state coupling controller to linear matrix inequalities (LMIs). Finally, the efficacy of the approach is shown in computer simulations involving the synchronization of three aircraft with unstable dynamics.