  A multi-input multi-output (MIMO) Gaussian channel with two transmit antennas
and two receive antennas is studied that is subject to an input peak-power
constraint. The capacity and the capacity-achieving input distribution are
unknown in general. The problem is shown to be equivalent to a channel with an
identity matrix but where the input lies inside and on an ellipse with
principal axis length $r_p$ and minor axis length $r_m$. If $r_p \le \sqrt{2}$,
then the capacity-achieving input has support on the ellipse. A sufficient
condition is derived under which a two-point distribution is optimal. Finally,
if $r_m < r_p \le \sqrt{2}$, then the capacity-achieving distribution is
discrete.
