  We investigate the capacity scaling of MIMO systems with the system
dimensions. To that end, we quantify how the mutual information varies when the
number of antennas (at either the receiver or transmitter side) is altered. For
a system comprising $R$ receive and $T$ transmit antennas with $R>T$, we find
the following: By removing as many receive antennas as needed to obtain a
square system (provided the channel matrices before and after the removal have
full rank) the maximum resulting loss of mutual information over all
signal-to-noise ratios (SNRs) depends only on $R$, $T$ and the matrix of
left-singular vectors of the initial channel matrix, but not on its singular
values. In particular, if the latter matrix is Haar distributed the ergodic
rate loss is given by $\sum_{t=1}^{T}\sum_{r=T+1}^{R}\frac{1}{r-t}$ nats. Under
the same assumption, if $T,R\to \infty$ with the ratio $\phi\triangleq T/R$
fixed, the rate loss normalized by $R$ converges almost surely to $H(\phi)$
bits with $H(\cdot)$ denoting the binary entropy function. We also quantify and
study how the mutual information as a function of the system dimensions
deviates from the traditionally assumed linear growth in the minimum of the
system dimensions at high SNR.
