  In this paper, we study the dyadic Carleson Embedding Theorem in the matrix
weighted setting. We provide two new proofs of this theorem, which highlight
connections between the matrix Carleson Embedding Theorem and both maximal
functions and $H^1$-BMO duality. Along the way, we establish boundedness
results about new maximal functions associated to matrix $A_2$ weights and
duality results concerning $H^1$ and BMO sequence spaces in the matrix setting.
As an application, we then use this Carleson Embedding Theorem to show that if
$S$ is a sparse operator, then the operator norm of $S$ on $L^2(W)$ satisfies:
\[ \| S\|_{L^2(W) \rightarrow L^2(W)} \lesssim [W]_{A_2}^{\frac{3}{2}},\] for
every matrix $A_2$ weight $W$.
