  We consider the problem of constructing matrices of linear forms of constant
rank by focusing on the associated vector bundles on projective spaces.
Important examples are given by the classical Steiner bundles, as well as some
special (duals of) syzygy bundles that we call Dr{\'e}zet bundles. Using the
classification of globally generated vector bundles with small first Chern
class on projective spaces, we are able to describe completely the
indecomposable matrices of constant rank up to six; some of them come from
rigid homogeneous vector bundles, some other from Dr{\'e}zet bundles related
either to plane quartics or to instanton bundles on the three-dimensional
projective space.
