  This paper considers the partial decode-and-forward (PDF) strategy for the
Gaussian multiple-input multiple-output (MIMO) relay channel. Unlike for the
decode-and-forward (DF) strategy or point-to-point (P2P) transmission, for
which Gaussian channel inputs are known to be optimal, the input distribution
that maximizes the achievable PDF rate for the Gaussian MIMO relay channel has
remained unknown so far. For some special cases, e.g., for relay channels where
the optimal PDF strategy reduces to DF or P2P transmission, it could be deduced
that Gaussian inputs maximize the PDF rate. For the general case, however, the
problem has remained open until now. In this work, we solve this problem by
proving that the maximum achievable PDF rate for the Gaussian MIMO relay
channel is always attained by Gaussian channel inputs. Our proof relies on the
channel enhancement technique, which was originally introduced by Weingarten et
al. to derive the (private message) capacity region of the Gaussian MIMO
broadcast channel. By combining this technique with a primal decomposition
approach, we first establish that jointly Gaussian source and relay inputs
maximize the achievable PDF rate for the aligned Gaussian MIMO relay channel.
Subsequently, we use a limiting argument to extend this result from the aligned
to the general Gaussian MIMO relay channel.
