Geometric aspects of the symmetric inverse M-matrix problem

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• 作者：Jan Brandts ; ^{J.H.Brandts@UvA.nl" class="auth_mail" title="E-mail the corresponding author} ; Apo Cihangir ^{A.Cihangir@UvA.nl" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 15B48 ; 52B11 ; secondary ; 15A23 ; 15B36 ; 51F20
• 刊名：Linear Algebra and its Applications
• 出版年：2016
• 出版时间：1 October 2016
• 年：2016
• 卷：506
• 期：Complete
• 页码：33-81
• 全文大小：986 K

文摘

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. 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 n-simplex S whose k-facets are all nonobtuse, and generalize some classical results by Fiedler [14]. We prove that if all (n−1)$(n-1)$-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 [28] is the inverse of a weakly diagonally dominant M-matrix. The geometric proof provides valuable insights that supplement the discrete measure theoretic [24] and the linear algebraic [25] proof.

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$k>3$. 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$k\times k$ principal submatrices for k>3$k>3$. 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.