Bocci, Carlini, and Kileel have shown that the square-free Hadamard product
of a finite set of points $Z$ that all lie on a line $\ell$ in $\mathbb{P}^n$
produces a star configuration of codimension $n$. In this paper we introduce a
construction using the Hadamard product to construct star configurations of
codimension $c$. In the case that $c = n= 2$, our construction produces the
star configurations of Bocci, Carlini, and Kileel. We will call any star
configuration that can be constructed using our approach a Hadamard star
configuration. Our main result is a classification of Hadamard star
configurations.