  In this paper, we derive the optimal transmitter/ receiver beamforming
vectors and relay weighting matrix for the multiple-input multiple-output
amplify-and-forward relay channel. The analysis is accomplished in two steps.
In the first step, the direct link between the transmitter (Tx) and receiver
(Rx) is ignored and we show that the transmitter and the relay should map their
signals to the strongest right singular vectors of the Tx-relay and relay-Rx
channels. Based on the distributions of these vectors for independent
identically distributed (i.i.d.) Rayleigh channels, the Grassmannian codebooks
are used for quantizing and sending back the channel information to the
transmitter and the relay. The simulation results show that even a few number
of bits can considerably increase the link reliability in terms of bit error
rate. For the second step, the direct link is considered in the problem model
and we derive the optimization problem that identifies the optimal Tx
beamforming vector. For the i.i.d Rayleigh channels, we show that the solution
to this problem is uniformly distributed on the unit sphere and we justify the
appropriateness of the Grassmannian codebook (for determining the optimal
beamforming vector), both analytically and by simulation. Finally, a modified
quantizing scheme is presented which introduces a negligible degradation in the
system performance but significantly reduces the required number of feedback
bits.
