Graph rigidity for unitarily invariant matrix norms

2017/09/26 by Kitson, Derek, Levene, Rupert H. · 2 citations
#05C50 #15A60 #52C25 #Combinatorics (math.CO) #FOS: Mathematics #Functional Analysis (math.FA) #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.1709.08967

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 K₆ - e (respectively, K₇) is the smallest minimally rigid graph for the class of 2 x 2 symmetric (respectively, hermitian) matrices with the trace norm.

Cited by

• Dilworth truncations and Hadamard products of linear spaces
• Algebraic connectivity in normed spaces

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