Graph rigidity for unitarily invariant matrix norms

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.