  In this paper we extend the theory of two weight, $A_p$ bump conditions to
the setting of matrix weights. We prove two matrix weight inequalities for
fractional maximal operators, fractional and singular integrals, sparse
operators and averaging operators. As applications we prove quantitative, one
weight estimates, in terms of the matrix $A_p$ constant, for singular
integrals, and prove a Poincar\'e inequality related to those that appear in
the study of degenerate elliptic PDEs.
