  In this paper we prove the Jones factorization theorem and the Rubio de
Francia extrapolation theorem for matrix $\mathcal A_p$ weights. These results
answer longstanding open questions in the study of matrix weights. The proof
requires the development of the theory of convex-set valued functions and
measurable seminorm functions. In particular, we define a convex-set valued
version of the Hardy Littlewood maximal operator and construct an appropriate
generalization of the Rubio de Francia iteration algorithm, which is central to
the proof of both results in the scalar case.
