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In this paper, a dual-hop multiple-input multiple-output (MIMO) amplify-and-forward (AF) relay network, where the source, relay and destination are each equipped with multiple antennas, is studied. By deriving and maximizing the receive signal-to-noise ratio (SNR) at the destination, we first obtain the optimal beamforming (BF) weights for the relay network. In order to evaluate the performance of the relay network, we then investigate the outage probability (OP), probability density function (PDF) and moments of the receive SNR as well as the ergodic capacity of the system in a closed-form. Furthermore, the average symbol error rate (ASER) expression of the relay network with the optimal transmit-receive BF is derived for three commonly used modulation formats, namely, M-ary pulse amplitude modulation (M-PAM), M-ary phase shift keying (M-PSK), and M-ary quadrature amplitude modulation (M-QAM). Finally, computer simulations are conducted to demonstrate the validity and efficacy of the designed MIMO relay network and its performance analysis.