We consider the optimization of a two-hop relay network based on an amplify-and-forward Multiple-Input Multiple-Output (MIMO) relay. The relay is assumed to derive the output signal by a Relay Transform Matrix (RTM) applied to the input signal. Assuming perfect channel state information about the network at the relay, the RTM is optimized according to two different criteria: {\bf\em i)} network capacity; {\bf\em ii)} network capacity based on Orthogonal Space--Time Block Codes. The two assumptions have been addressed in part in the literature. The optimization problem is reduced to a manageable convex form, whose KKT equations are explicitly solved. Then, a parametric solution is given, which yields the power constraint and the capacity achieved with uncorrelated transmitted data as functions of a positive indeterminate. The solution for a given average power constraint at the relay is amenable to a \emph{water-filling-like} algorithm, and extends earlier literature results addressing the case without the direct link. Simulation results are reported concerning a Rayleigh relay network and the role of the direct link SNR is precisely assessed.