Two weight bump conditions for matrix weights

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.