  This paper considers a multiple input multiple output (MIMO) two-way relay
channel, where two nodes want to exchange data with each other using multiple
relays. An iterative algorithm is proposed to achieve the optimal achievable
rate region, when each relay employs an amplify and forward (AF) strategy.
  The iterative algorithm solves a power minimization problem at every step,
subject to minimum signal-to-interference-and-noise ratio constraints, which is
non-convex, however, for which the Karush Kuhn Tuker conditions are sufficient
for optimality. The optimal AF strategy assumes global channel state
information (CSI) at each relay. To simplify the CSI requirements, a simple
amplify and forward strategy, called dual channel matching, is also proposed,
that requires only local channel state information, and whose achievable rate
region is close to that of the optimal AF strategy. In the asymptotic regime of
large number of relays, we show that the achievable rate region of the dual
channel matching and an upper bound differ by only a constant term and
establish the capacity scaling law of the two-way relay channel. Relay
strategies achieving optimal diversity-multiplexing tradeoff are also
considered with a single relay node. A compress and forward strategy is shown
to be optimal for achieving diversity multiplexing tradeoff for the full-duplex
case, in general, and for the half-duplex case in some cases.
