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

Abstract

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