  Nazarov-Treil-Volberg recently proved an elegant two-weight T1 theorem for
"almost diagonal" operators that played a key role in the proof of the $A_2$
conjecture for dyadic shifts and related operators. In this paper, we obtain a
generalization of their T1 theorem to the setting of matrix weights. Our
theorem does differ slightly from the scalar results, a fact attributable
almost completely to differences between the scalar and matrix Carleson
Embedding Theorems. The main tools include a reduction to the study of
well-localized operators, a new system of Haar functions adapted to matrix
weights, and a matrix Carleson Embedding Theorem.
