  We introduce the so called convex body valued sparse operators, which
generalize the notion of sparse operators to the case of spaces of vector
valued functions. We prove that Calder\'on--Zygmund operators as well as Haar
shifts and paraproducts can be dominated by such operators. By estimating
sparse operators we obtain weighted estimates with matrix weights. We get two
weight $A_2$-$A_\infty$ estimates, that in the one weight case give us the
estimate $$ \|T\|_{L^2(W)\to L^2 (W)} \le C [W]_{A_2}^{1/2} [W]_{A_\infty} \le
C[W]_{A_2}^{3/2} $$ where $T$ is either a Calderon--Zygmund operator (with
modulus of continuity satisfying the Dini condition), or a Haar shift or a
paraproduct.
