On the combinatorial structure of 0/1-matrices representing nonobtuse simplices
A 0/1-simplex is the convex hull of n+1 affinely independent vertices of the unit n-cube In. It is nonobtuse if none of its dihedral angles is obtuse, and acute if additionally none of them is right. Acute 0/1-simplices in In can be represented by 0/1-matrices P of size n× n whose Gramians G = P⊤P have an inverse that is strictly diagonally dominant, with negative off-diagonal entries.
In this paper, we will prove that the positive part D of the transposed inverse P−⊤ of P is doubly stochastic and has the same support as P. In fact, P has a fully indecomposable doubly stochastic pattern. The negative part C of P−⊤ is strictly row-substochastic and its support is complementary to that of D, showing that P−⊤ = D−C has no zero entries and has positive row sums. As a consequence, for each facet F of an acute 0/1-facet S there exists at most one other acute 0/1-simplex Ŝ in In having F as a facet. We call Ŝ the acute neighbor of S at F.
If P represents a 0/1-simplex that is merely nonobtuse, the inverse of G = PΤP is only weakly diagonally dominant and has nonpositive off-diagonal entries. These matrices play an important role in finite element approximation of elliptic and parabolic problems, since they guarantee discrete maximum and comparison principles. Consequently, P−⊤ can have entries equal to zero. We show that its positive part D is still doubly stochastic, but its support may be strictly contained in the support of P. This allows P to have no doubly stochastic pattern and to be partly decomposable. In theory, this might cause a nonobtuse 0/1-simplex S to have several nonobtuse neighbors Ŝ at each of its facets.
In this paper, we study nonobtuse 0/1-simplices S having a partly decomposable matrix representation P. We prove that if S has such a matrix representation, it also has a block diagonal matrix representation with at least two diagonal blocks. Moreover, all matrix representations of S will then be partly decomposable. This proves that the combinatorial property of having a fully indecomposable matrix representation with doubly stochastic pattern is a geometrical property of a subclass of nonobtuse 0/1-simplices, invariant under all n-cube symmetries. We will show that a nonobtuse simplex with partly decomposable matrix representation can be split in mutually orthogonal simplicial facets whose dimensions add up to n, and in which each facet has a fully indecomposable matrix representation. Using this insight, we are able to extend the one neighbor theorem for acute simplices to a larger class of nonobtuse simplices.
Keywordsacute simplex nonobtuse simplex orthogonal simplex 0/1-matrix doubly stochastic matrix fully indecomposable matrix partly decomposable matrix
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- R. A. Brualdi, H. J. Ryser: Combinatorial Matrix Theory. Encyclopedia of Mathematics and Its Applications 39, Cambridge University Press, Cambridge, 1991.Google Scholar