  We prove the matrix $A_2$ conjecture for the dyadic square function, that is,
a norm estimate of the matrix weighted square function, where the focus is on
the sharp linear dependence on the matrix $A_2$ constant in the estimate.
Moreover, we give a mixed estimate in terms of $A_2$ and $A_{\infty}$
constants. Key is a sparse domination of a process inspired by the integrated
form of the matrix--weighted square function.
