  We show that the 3 user M_T x M_R MIMO interference channel has
d(M,N)=min(M/(2-1/k),N/(2+1/k)) degrees of freedom (DoF) normalized by time,
frequency, and space dimensions, where M=min(M_T,M_R), N=max(M_T,M_R),
k=ceil{M/(N-M)}. While the DoF outer bound is established for every M_T, M_R
value, the achievability is established in general subject to normalization
with respect to spatial-extensions. Given spatial-extensions, the achievability
relies only on linear beamforming based interference alignment schemes with no
need for time/frequency extensions. In the absence of spatial extensions, we
show through examples how essentially the same scheme may be applied over
time/frequency extensions. The central new insight to emerge from this work is
the notion of subspace alignment chains as DoF bottlenecks.
  The DoF value d(M,N) is a piecewise linear function of M,N, with either M or
N being the bottleneck within each linear segment. The corner points of these
piecewise linear segments correspond to A={1/2,2/3,3/4,...} and
B={1/3,3/5,5/7,...}. The set A contains all values of M/N and only those for
which there is redundancy in both M and N. The set B contains all values of M/N
and only those for which there is no redundancy in either M or N.
  Our results settle the feasibility of linear interference alignment,
introduced by Cenk et al., for the 3 user M_T x M_R MIMO interference channel,
completely for all values of M_T, M_R. Specifically, the linear interference
alignment problem (M_T x M_R, d)^3 (as defined in previous work by Cenk et al.)
is feasible if and only if d<=floor{d(M,N)}. With and only with the exception
of the values M/N\in B, we show that for every M/N value there are proper
systems that are not feasible.
  Our results show that M/N\in A are the only values for which there is no DoF
benefit of joint processing among co-located antennas at the transmitters or
receivers.
