Graph rigidity for unitarily invariant matrix norms

TL;DR

This paper develops a rigidity theory for frameworks in matrix spaces with unitarily invariant norms, characterizing minimal rigidity and identifying smallest rigid graphs for specific matrix classes.

Contribution

It introduces a rigidity framework in matrix spaces with unitarily invariant norms, extending Maxwell's criteria and characterizing minimal rigidity in these settings.

Findings

Minimally rigid frameworks belong to (k,l)-sparse graph classes.

Characterization of infinitesimal rigidity for product norms.

Identification of smallest rigid graphs for specific matrix classes.

Abstract

A rigidity theory is developed for bar-joint frameworks in linear matrix spaces endowed with a unitarily invariant 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. A characterisation of infinitesimal rigidity is obtained for product norms and it is shown that K_6 - e (respectively, K_7) is the smallest minimally rigid graph for the class of 2 x 2 symmetric (respectively, hermitian) matrices with the trace norm.