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It is shown that compact sets of complex matrices can always be brought, viasimilarity transformation, into a form where all matrix entries are bounded inabsolute value by the joint spectral radius (JSR). The key tool for this isthat every extremal norm of a matrix set admits an Auerbach basis; any suchbasis gives rise to a desired coordinate system. An immediate implication isthat all diagonal entries - equivalently, all one-dimensional principalsubmatrices - are uniformly bounded above by the JSR. It is shown that thecorresponding bounding property does not hold for higher dimensional principalsubmatrices. More precisely, we construct finite matrix sets for which, acrossthe entire similarity orbit, the JSRs of all higher-dimensional principalsubmatrices exceed that of the original set. This shows that the boundingresult does not extend to submatrices of dimension greater than one. Theconstructions rely on tools from the geometry of finite-dimensional Banachspaces, with projection constants of norms playing a key role. Additionalbounds of the JSR of principal submatrices are obtained using John'sellipsoidal approximation and known estimates for projection constants.

Auerbach bases, projection constants, and the joint spectral radius of principal submatrices