Ultrametric Matrices and the Geometric Inverse M-Matrix Problem

We describe the symmetric inverse M-matrix problem from a geometric viewpoint. The central questioninthegeometriccontextis,whichpropertiesthelowerdimensionalfacetsofann-simplex S guarantee that S itself has no obtuse dihedral angles. The simplest but strongest of such properties is regularity of the triangular facets. Slightly weaker is to demand that all triangular facets are strongly isosceles. Even more general but also more involved is to demand ultrametricity of all threedimensional tetrahedral facets of S. As part of our exposition we show that either none, or all so-called vertex Gramians associated with an n-simplex S are ultrametric. As a result, the inverse of an ultrametric matrix is weakly diagonally dominant if and only if this inverse is a Stieltjes matrix. Thus, only one of them needs to be proved in order to obtain both.