  This paper studies vector space interference alignment for the three-user
MIMO interference channel with no time or frequency diversity. The main result
is a characterization of the feasibility of interference alignment in the
symmetric case where all transmitters have M antennas and all receivers have N
antennas. If N >= M and all users desire d transmit dimensions, then alignment
is feasible if and only if (2r+1)d <= max(rN,(r+1)M) for all nonnegative
integers r. The analogous result holds with M and N switched if M >= N.
  It turns out that, just as for the 3-user parallel interference channel
\cite{BT09}, the length of alignment paths captures the essence of the problem.
In fact, for each feasible value of M and N the maximum alignment path length
dictates both the converse and achievability arguments.
  One of the implications of our feasibility criterion is that simply counting
equations and comparing to the number of variables does not predict
feasibility. Instead, a more careful investigation of the geometry of the
alignment problem is required. The necessary condition obtained by counting
equations is implied by our new feasibility criterion.
