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)
Summary
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]
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
