Vertex implications for totally nonnegative matrices

Considered are matrix intervals, i.e. families of real matrices which are known to lie between two given matrices, the so-called corner matrices. Here the underlying partial ordering is the chequerboard partial ordering. We present conditions under which total nonnegativity can be ascertained for a matrix interval by checking only a subset of the vertex matrices, i.e. matrices with entries from the corner matrices. It turns out that in many cases inspection of only the two corner matric:es is sufficient.