  In this paper we prove two matrix weighted norm inequalities for the
commutator of a fractional integral operator and multiplication by a matrix
symbol. More precisely, we extend the recent results of the second author,
Pott, and Treil on two matrix weighted norm inequalities for commutators of
Calderon-Zygmund operators and multiplication by a matrix symbol to the
fractional integral operator setting. In particular, we completely extend the
fractional Bloom theory of Holmes, Rahm, and Spencer to the two matrix weighted
setting with a matrix
