The　adaptive　generalized　matrix　projective　lag　synchronization　between　two　different　complex　networks　with　non-identical　nodes　and　different　dimensions　is　investigated　in　this　paper.　Based　on　Lyapunov　stability　theory　and　Barbalat＇s　lemma,　generalized　matrix　projective　lag　synchronization　criteria　are　derived　by　using　the　adaptive　control　method.　Furthermore,　each　network　can　be　undirected　or　directed,　connected　or　disconnected,　and　nodes　in　either　network　may　have　identical　or　different　dynamics.　The　proposed　strategy　is　applicable　to　almost　all　kinds　of　complex　networks.　In　addition,　numerical　simulation　results　are　presented　to　illustrate　the　effectiveness　of　this　method,　showing　that　the　synchronization　speed　is　sensitively　influenced　by　the　adaptive　law　strength,　the　network　size,　and　the　network　topological　structure.