Abstract

A rigidity theory is developed for bar-joint frameworks in linear matrix spaces endowed with a unitarily invariant matrix norm. Analogues of Maxwell's counting criteria are obtained and minimally rigid matrix frameworks are shown to belong to the matroidal class of (k,l)-sparse graphs for suitable k and l. An edge-colouring technique is developed to characterise infinitesimal rigidity for product norms and then applied to show that the graph of a minimally rigid bar-joint framework in the space of 2×2 symmetric (respectively, hermitian) matrices with the trace norm admits an edge-disjoint packing consisting of a (Euclidean) rigid graph and a spanning tree.

Highlights

  • A bar-joint framework is a pair (G, p) consisting of a simple undirected graph G = (V, E) and a mapping of its vertices p : V → X into a linear space X, with p(v) and p(w) distinct for each edge vw ∈ E

  • In 1864, Maxwell observed that the underlying graph G of a rigid framework in Euclidean space necessarily satisfies certain counting conditions

  • We provide analogues of the Maxwell counting criteria for Euclidean barjoint frameworks (Theorem 32) and show that the graphs of minimally rigid matrix frameworks belong to the matroidal class of (k, l)-sparse graphs for suitable values of k and l (Theorem 33)

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Summary

Introduction

These results, which may be of independent interest, are applied, where we exploit the cylindrical nature of the trace norm on the space of 2 × 2 symmetric matrices, to show that the graph of a minimally rigid matrix framework is expressible as an edge-disjoint union of a spanning tree and a spanning Laman graph (Theorem 52). We discuss sufficient conditions for the existence of a minimally rigid placement in an admissible matrix space and pose some conjectures on connectivity and packing criteria, based on recent work of Cheriyan et al [6] and Gu [9]

Preliminaries
Admissible matrix spaces
Rigid motions
Infinitesimal rigid motions
Infinitesimal rigidity for admissible matrix spaces
Support functionals
The rigidity map
Full sets
Trivial infinitesimal flexes
Infinitesimal rigidity
Product norms
Framework colours
Rigid motions of product spaces
Trivial infinitesimal flexes of frameworks
A characterisation of infinitesimal rigidity
Symmetric matrices
Hermitian matrices
Remarks on sufficient conditions
Full Text
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