Geometric aspects of the symmetric inverse M-matrix problem

TL;DR

This paper explores the geometric properties of simplices related to symmetric inverse M-matrices, establishing conditions under which simplices are nonobtuse and connecting these to matrix properties, with implications for visualization and generalization.

Contribution

It introduces a geometric framework for understanding symmetric inverse M-matrices via simplices, generalizes sub-orthocentric tetrahedra to higher dimensions, and links geometric conditions to matrix properties.

Findings

All (n-1)-facets of an n-simplex being nonobtuse implies limited obtuse dihedral angles.

Sub-orthocentric simplices are nonobtuse and relate to ultrametric matrices.

Conjecture: simplices with only sub-orthocentric facets are themselves sub-orthocentric.

Abstract

We investigate the symmetric inverse M-matrix problem from a geometric perspective. The central question in this geometric context is, which conditions on the k-dimensional facets of an n-simplex S guarantee that S has no obtuse dihedral angles. First we study the properties of an n-simplex S whose k-facets are all nonobtuse, and generalize some classical results by Fiedler. We prove that if all (n-1)-facets of an n-simplex S are nonobtuse, each makes at most one obtuse dihedral angle with another facet. This helps to identify a special type of tetrahedron, which we will call sub-orthocentric, with the property that if all tetrahedral facets of S are sub-orthocentric, then S is nonobtuse. Rephrased in the language of linear algebra, this constitutes a purely geometric proof of the fact that each symmetric ultrametric matrix is the inverse of a weakly diagonally dominant M-matrix. Review papers support our belief 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 x 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.