  The fundamental diversity-multiplexing tradeoff of the three-node,
multi-input, multi-output (MIMO), quasi-static, Rayleigh faded, half-duplex
relay channel is characterized for an arbitrary number of antennas at each node
and in which opportunistic scheduling (or dynamic operation) of the relay is
allowed, i.e., the relay can switch between receive and transmit modes at a
channel dependent time. In this most general case, the diversity-multiplexing
tradeoff is characterized as a solution to a simple, two-variable optimization
problem. This problem is then solved in closed form for special classes of
channels defined by certain restrictions on the numbers of antennas at the
three nodes. The key mathematical tool developed here that enables the explicit
characterization of the diversity-multiplexing tradeoff is the joint eigenvalue
distribution of three mutually correlated random Wishart matrices. Previously,
without actually characterizing the diversity-multiplexing tradeoff, the
optimality in this tradeoff metric of the dynamic compress-and-forward (DCF)
protocol based on the classical compress-and-forward scheme of Cover and El
Gamal was shown by Yuksel and Erkip. However, this scheme requires global
channel state information (CSI) at the relay. In this work, the so-called
quantize-map and forward (QMF) coding scheme due to Avestimehr {\em et} {\em
al} is adopted as the achievability scheme with the added benefit that it
achieves optimal tradeoff with only the knowledge of the (channel dependent)
switching time at the relay node. Moreover, in special classes of the MIMO
half-duplex relay channel, the optimal tradeoff is shown to be attainable even
without this knowledge. Such a result was previously known only for the
half-duplex relay channel with a single antenna at each node, also via the QMF
scheme.
