Matrix construction of subnormal tuples of finite type

We review Xia's analytic model for subnormal tuples of operators as well as a matricial decomposition of pure subnormal tuples of Eschmeier and Putinar. Based on these developments we create a matricial construction of pure subnormal tuples of finite type from the Xia unitary invariants. In the process we develop necessary and sufficient conditions for sets of operators {C₀^j,D₁^j} to be the Xia invariants of a subnormal tuple.