  This paper considers the non-cooperative maximization of mutual information
in the vector Gaussian interference channel in a fully distributed fashion via
game theory. This problem has been widely studied in a number of works during
the past decade for frequency-selective channels, and recently for the more
general MIMO case, for which the state-of-the art results are valid only for
nonsingular square channel matrices. Surprisingly, these results do not hold
true when the channel matrices are rectangular and/or rank deficient matrices.
The goal of this paper is to provide a complete characterization of the MIMO
game for arbitrary channel matrices, in terms of conditions guaranteeing both
the uniqueness of the Nash equilibrium and the convergence of asynchronous
distributed iterative waterfilling algorithms. Our analysis hinges on new
technical intermediate results, such as a new expression for the MIMO
waterfilling projection valid (also) for singular matrices, a mean-value
theorem for complex matrix-valued functions, and a general contraction theorem
for the multiuser MIMO watefilling mapping valid for arbitrary channel
matrices. The quite surprising result is that uniqueness/convergence conditions
in the case of tall (possibly singular) channel matrices are more restrictive
than those required in the case of (full rank) fat channel matrices. We also
propose a modified game and algorithm with milder conditions for the uniqueness
of the equilibrium and convergence, and virtually the same performance (in
terms of Nash equilibria) of the original game.
