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, this construction allows us to list all indecomposable matrices of constant rank up to 7, as well as describing the moduli spaces of simple vector bundles naturally defined by families of constant rank matrices.