 Author
 Date
 1102016
 Title
 Geometric aspects of the symmetric inverse Mmatrix problem
 Journal
 Linear Algebra and its Applications
 Volume
 506
 Pages (fromto)
 33–81
 Document type
 Article
 Faculty
 Faculty of Science (FNWI)
 Institute
 Kortewegde Vries Institute for Mathematics (KdVI)
 Abstract

We investigate the symmetric inverse Mmatrix problem from a geometric perspective. The central question in this geometric context is, which conditions on the kdimensional facets of an nsimplex S guarantee that S has no obtuse dihedral angles. The simplest of such conditions is that if all triangular facets of S are equilateral, then S is regular and thus nonobtuse. First we study the properties of an nsimplex S whose kfacets are all nonobtuse, and generalize some classical results by Fiedler [14]. We prove that if all (n−1)facets of an nsimplex 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 suborthocentric, with the property that if all tetrahedral facets of S are suborthocentric, 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 [28] is the inverse of a weakly diagonally dominant Mmatrix. The geometric proof provides valuable insights that supplement the discrete measure theoretic [24] and the linear algebraic [25] proof.
The review papers [20,21] support our believe that the linear algebraic perspective on the inverse Mmatrix 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 kfacets of an nsimplex S that may render S nonobtuse also for k>3. For this, we generalize the class of suborthocentric tetrahedra that gives rise to the class of ultrametric matrices, to suborthocentric simplices that define symmetric positive definite matrices A with special types of k×k principal submatrices for k>3. Each suborthocentric simplices is nonobtuse, and we conjecture that any simplex with suborthocentric facets only, is suborthocentric itself.
Along the way, several additional new concepts will be introduced, such as vertex Gramians, simplicial matrix classes, the dual hull and the suborthocentric 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.
 URL
 go to publisher's site
 Language
 English
 Note
 In memory of Professor Miroslav Fiedler (1926–2015)
 Permalink
 http://hdl.handle.net/11245.1/080da4cc7bb64c2b9fa5e67808c435ee
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