  The capacity of a deterministic multiple-input multiple-output (MIMO) channel
under the peak and average power constraints is investigated. For the identity
channel matrix, the approach of Shamai et al. is generalized to the higher
dimension settings to derive the necessary and sufficient conditions for the
optimal input probability density function. This approach prevents the usage of
the identity theorem of the holomorphic functions of several complex variables
which seems to fail in the multi-dimensional scenarios. It is proved that the
support of the capacity-achieving distribution is a finite set of hyper-spheres
with mutual independent phases and amplitude in the spherical domain.
Subsequently, it is shown that when the average power constraint is relaxed, if
the number of antennas is large enough, the capacity has a closed form solution
and constant amplitude signaling at the peak power achieves it. Moreover, it
will be observed that in a discrete-time memoryless Gaussian channel, the
average power constrained capacity, which results from a Gaussian input
distribution, can be closely obtained by an input where the support of its
magnitude is a discrete finite set. Finally, we investigate some upper and
lower bounds for the capacity of the non-identity channel matrix and evaluate
their performance as a function of the condition number of the channel.
