  We derive a general and closed-form result for the success probability in
downlink multiple-antenna (MIMO) heterogeneous cellular networks (HetNets),
utilizing a novel Toeplitz matrix representation. This main result, which is
equivalently the signal-to-interference ratio (SIR) distribution, includes
multiuser MIMO, single-user MIMO and per-tier biasing for $K$ different tiers
of randomly placed base stations (BSs), assuming zero-forcing precoding and
perfect channel state information. The large SIR limit of this result admits a
simple closed form that is accurate at moderate SIRs, e.g., above 5 dB. These
results reveal that the SIR-invariance property of SISO HetNets does not hold
for MIMO HetNets; instead the success probability may decrease as the network
density increases. We prove that the maximum success probability is achieved by
activating only one tier of BSs, while the maximum area spectral efficiency
(ASE) is achieved by activating all the BSs. This reveals a unique tradeoff
between the ASE and link reliability in multiuser MIMO HetNets. To achieve the
maximum ASE while guaranteeing a certain link reliability, we develop efficient
algorithms to find the optimal BS densities. It is shown that as the link
reliability requirement increases, more BSs and more tiers should be
deactivated.
