  We provide the degrees of freedom (DoF) characterization for the $3$-user
$M_T\times M_R$ multiple-input multiple-output (MIMO) interference channel (IC)
with \emph{rank-deficient} channel matrices, where each transmitter is equipped
with $M_T$ antennas and each receiver with $M_R$ antennas, and the interfering
channel matrices from each transmitter to the other two receivers are of ranks
$D_1$ and $D_2$, respectively. One important intermediate step for both the
converse and achievability arguments is to convert the fully-connected
rank-deficient channel into an equivalent partially-connected full-rank MIMO-IC
by invertible linear transformations. As such, existing techniques developed
for full-rank MIMO-IC can be incorporated to derive the DoF outer and inner
bounds for the rank-deficient case. Our result shows that when the interfering
links are weak in terms of the channel ranks, i.e., $D_1+D_2\leq \min(M_T,
M_R)$, zero forcing is sufficient to achieve the optimal DoF. On the other
hand, when $D_1+D_2> \min(M_T, M_R)$, a combination of zero forcing and
interference alignment is in general required for DoF optimality. The DoF
characterization obtained in this paper unifies several existing results in the
literature.
