Summary: Constrained MUSIC and beamspace MUSIC are similar algorithms in that they both require a priori information about signal directions and they both involve linear transformations on the data. Constrained MUSIC uses precise information regarding the directions of a subset of the signal directions to improve the direction estimates for the remaining signals. Beamspace MUSIC uses approximate knowledge regarding all the signal directions to reduce computational complexity and improve breakdown properties. These two methods can be combined, resulting in constrained beamspace MUSIC. We also perform asymptotic analysis of constrained and unconstrained MUSIC demonstrating that (asymptotically) improved subspace estimates always result from the use of constraints, and (asymptotically) the variance of constrained MUSIC is less than that of unconstrained MUSIC under either high coherence, large numbers of sensors, or high SNR conditions. As a part of this analysis, we study the effects of coherence on MUSIC and derive best/worst case coherences in terms of the variance of MUSIC. We also demonstrate that those conditions where the variance of MUSIC is predicted to be less than that of constrained MUSIC generally correspond to conditions where MUSIC is in breakdown (and constrained MUSIC is not). So, unconstrained MUSIC actually does not achieve its theoretically predicted advantage in those cases. While constrained MUSIC requires precise information about the known signal to improve performance when the unknown signal is very near, it can also offer performance advantages with only approximate knowledge if the unknown and known signals are not too close to each other.