In this paper, we investigate joint source precoding matrices and relay processing matrix design for multi-user multiple-input multiple-output~(MU-MIMO) non-regenerative relay networks in the presence of the direct source-destination~(S-D) links. We consider both capacity and mean-squared error~(MSE) criterions subject to the distributed power constraints, which are nonconvex and apparently have no simple solutions. Therefore, we propose an optimal source precoding matrix structure based on the point-to-point MIMO channel technique, and a new relay processing matrix structure under the modified power constraint at relay node, based on which, a nested iterative algorithm of jointly optimizing sources precoding and relay processing is established. We show that the capacity based optimal source precoding matrices share the same structure with the MSE based ones. So does the optimal relay processing matrix. Simulation results demonstrate that the proposed algorithm outperforms the existing results.