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This paper studies vector space interference alignment for the three-userMIMO interference channel with no time or frequency diversity. The main resultis a characterization of the feasibility of interference alignment in thesymmetric case where all transmitters have M antennas and all receivers have Nantennas. If N >= M and all users desire d transmit dimensions, then alignmentis feasible if and only if (2r+1)d <= max(rN,(r+1)M) for all nonnegativeintegers 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 lengthdictates both the converse and achievability arguments.  One of the implications of our feasibility criterion is that simply countingequations and comparing to the number of variables does not predictfeasibility. Instead, a more careful investigation of the geometry of thealignment problem is required. The necessary condition obtained by countingequations is implied by our new feasibility criterion.

Geometry of the 3-user MIMO interference channel