  We study vector space interference alignment for the MIMO interference
channel with no time or frequency diversity, and no symbol extensions. We prove
both necessary and sufficient conditions for alignment. In particular, we
characterize the feasibility of alignment for the symmetric three-user channel
where all users transmit along d dimensions, all transmitters have M antennas
and all receivers have N antennas, as well as feasibility of alignment for the
fully symmetric (M=N) channel with an arbitrary number of users.
  An implication of our results is that the total degrees of freedom available
in a K-user interference channel, using only spatial diversity from the
multiple antennas, is at most 2. This is in sharp contrast to the K/2 degrees
of freedom shown to be possible by Cadambe and Jafar with arbitrarily large
time or frequency diversity.
  Moving beyond the question of feasibility, we additionally discuss
computation of the number of solutions using Schubert calculus in cases where
there are a finite number of solutions.
