  This paper considers reliable communications over a multiple-input
multiple-output (MIMO) Gaussian channel, where the channel matrix is within a
bounded channel uncertainty region around a nominal channel matrix, i.e., an
instance of the compound MIMO Gaussian channel. We study the optimal transmit
covariance matrix design to achieve the capacity of compound MIMO Gaussian
channels, where the channel uncertainty region is characterized by the spectral
norm. This design problem is a challenging non-convex optimization problem.
However, in this paper, we reveal that this problem has a hidden convexity
property, which can be exploited to map the problem into a convex optimization
problem. We first prove that the optimal transmit design is to diagonalize the
nominal channel, and then show that the duality gap between the capacity of the
compound MIMO Gaussian channel and the min-max channel capacity is zero, which
proves the conjecture of Loyka and Charalambous (IEEE Trans. Inf. Theory, vol.
58, no. 4, pp. 2048-2063, 2012). The key tools for showing these results are a
new matrix determinant inequality and some unitarily invariant properties.
