  Gaussian MIMO channel under total transmit and multiple interference power
constraints (TPC and IPCs) is considered. A closed-form solution for its
optimal transmit covariance matrix is obtained in the general case (up to dual
variables). A number of more explicit closed-form solutions are obtained in
some special cases, including full-rank and rank-1 (beamforming) solutions,
which differ significantly from the well-known water-filling solutions (e.g.
signaling on the channel eigenmodes is not optimal anymore and the capacity can
be zero for non-zero transmit power). A whitening filter is shown to be an
important part of optimal precoding under interference constraints. Capacity
scaling with transmit power is studied: its qualitative behaviour is determined
by a natural linear-algebraic structure induced by MIMO channels of multiple
users. A simple rank condition is given to characterize the cases where
spectrum sharing is possible. An interplay between the TPC and IPCs is
investigated, including the transition from power-limited to
interference-limited regimes. A number of unusual properties of an optimal
covariance matrix under IPCs are pointed out and a bound on its rank is
established. Partial null forming known in the adaptive antenna array
literature is shown to be optimal from the information-theoretic perspective as
well in some cases.
