Abstract:
Given a zero/nonzero pattern or a sign pattern for a matrix, there is a range of possible ranks among the real matrices that exhibit this pattern. To know all ranks, it is generally sufficient to know the minimum rank (mr). But the mr is notoriously difficult to calculate. An interesting lower bound is the triangle minimum rank (tmr). Often tmr = mr, but not always. We discuss what can happen and give a scheme for computing mr in many cases. Several classical (such as projective planes) and new concepts come into play.