  In this paper, we present a novel framework that can be used to study the
capacity and power scaling properties of linear multiple-input multiple-output
(MIMO) $d\times d$ antenna amplify-and-forward (AF) relay networks. In
particular, we model these networks as random dynamical systems (RDS) and
calculate their $d$ Lyapunov exponents. Our analysis can be applied to systems
with any per-hop channel fading distribution, although in this contribution we
focus on Rayleigh fading. Our main results are twofold: 1) the total transmit
power at the $n$th node will follow a deterministic trajectory through the
network governed by the network's maximum Lyapunov exponent, 2) the capacity of
the $i$th eigenchannel at the $n$th node will follow a deterministic trajectory
through the network governed by the network's $i$th Lyapunov exponent. Before
concluding, we concentrate on some applications of our results. In particular,
we show how the Lyapunov exponents are intimately related to the rate at which
the eigenchannel capacities diverge from each other, and how this relates to
the amplification strategy and number of antennas at each relay. We also use
them to determine the extra cost in power associated with each extra
multiplexed data stream.
