  It is well-known that dyadic martingale transforms are a good model for
Calder\'on-Zygmund singular integral operators. In this paper we extend some
results on weighted norm inequalities to vector-valued functions. We prove
that, if $W$ is an $A_2$ matrix weight, then the weighted $L^2$-norm of a
Calder\'on-Zygmund operator with cancellation has the same dependence on the
$A_2$ characteristic of $W$ as the weighted $L^2$-norm of the martingale
transform. Thus the question of the dependence of the norm of matrix-weighted
Calder\'on-Zygmund operators on the $A_2$ characteristic of the weight is
reduced to the case of dyadic martingales and paraproducts. We also show a
slightly different proof for the special case of Calder\'on-Zygmund operators
with even kernel. We conclude the paper by proving a version of the
matrix-weighted Carleson Embedding Theorem.
  Our method uses the Bellman function technique to obtain the right estimates
for the norm of dyadic Haar shift operators. We then apply the representation
theorem of T. Hyt\"onen to extend the result to general Calder\'on-Zygmund
operators.
