We study the synchronizability of weighted aging scale-free networks with non-normalized and asymmetrical coupling matrices. We found that the synchronizability of such networks is improved when the couplings from older to younger nodes become dominant, where the out degrees of the nodes are heterogeneous and their in degrees are homogeneous, and that the synchronizabilty of the networks is seriously weakened or even lost when the couplings from younger to older nodes become dominant, where the out degrees of the nodes are homogeneous and their in degrees are heterogeneous. We also found that both the heterogeneity of nodes and a smaller average degree can improve the synchronizability of the networks with some appropriately chosen weighting parameter values. We finally show an example of the coupled Lorenz systems for illustration and verification of the theoretical analysis.
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