Characterization of linear spaces of matrices of constant rank from syzygy bundles

In this work, we characterize matrices of linear forms and constant rank, demonstrating that, under some natural assumptions, they are always associated with a syzygy bundle that fits into a (partially linear) resolution. Furthermore, our construction offers a new perspective that unifies the approaches used in known examples from the literature and enables the creation of new ones, as well as to list all indecomposable matrices of constant rank up to 7.