  In this paper we characterize the two matrix weighted boundedness of
commutators with any of the Riesz transforms (when both are matrix A${}_p$
weights) in terms of a natural two matrix weighted BMO space. Furthermore, we
identify this BMO space when $p = 2$ as the dual of a natural two matrix
weighted H${}^1$ space, and use our commutator result to provide a converse to
Bloom's matrix A${}_2$ theorem, which as a very special case proves Buckley's
summation condition for matrix A${}_2$ weights. Finally, we use our results to
prove a matrix weighted John-Nirenberg inequality, and we also briefly discuss
the challenging question of extending our results to the matrix weighted vector
BMO setting.
