Criteria for the singularity of a pairwise $l_1$-distance matrix and their generalizations

A. G. Dyakonov

M. V. Lomonosov Moscow State University, Faculty of Computational Mathematics and Cybernetics

Abstract: We study the singularity problem for the pairwise distance matrix of a system of points, as well as generalizations of this problem that are connected with applications to interpolation theory and with an algebraic approach to recognition problems. We obtain necessary and sufficient conditions on a system under which the dimension of the range space of polynomials of bounded degree over the columns of the distance matrix is less than the number of points in the system.

Keywords: pairwise distance matrix, interpolation, metric, correctness criteria, system of points.

UDC: 519.712

MSC: Primary 15A09; Secondary 41A05, 46B85, 65D05, 94A15

Received: 07.10.2010

DOI: 10.4213/im5404

 English version: Izvestiya: Mathematics, 2012, 76:3, 517–534

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