  In this paper we refine the recent sparse domination of the integrated $p =
2$ matrix weighted dyadic square function by T. Hytonen, S. Petermichl, and A.
Volberg to prove a pointwise sparse domination of general matrix weighted
dyadic square functions. We then use this to prove sharp two matrix weighted
strong type inequalities for matrix weighted dyadic square functions when $1 <
p \leq 2$.
