The review papers and support our believe that the linear algebraic perspective on the inverse M-matrix problem dominates the literature. The geometric perspective however connects sign properties of entries of inverses of a symmetric positive definite matrix to the dihedral angle properties of an underlying simplex, and enables an explicit visualization of how these angles and signs can be manipulated. This will serve to formulate purely geometric conditions on the k-facets of an n-simplex S that may render S nonobtuse also for k>3. For this, we generalize the class of sub-orthocentric tetrahedra that gives rise to the class of ultrametric matrices, to sub-orthocentric simplices that define symmetric positive definite matrices A with special types of k×k principal submatrices for k>3. Each sub-orthocentric simplices is nonobtuse, and we conjecture that any simplex with sub-orthocentric facets only, is sub-orthocentric itself.
Along the way, several additional new concepts will be introduced, such as vertex Gramians, simplicial matrix classes, the dual hull and the sub-orthocentric set of a nonobtuse simplex, and nonblocking matrices. These concepts may also be of use in a different linear algebraic setting, in particular in the context if completely positive matrices, for which we will prove some auxiliary results as well.