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.

Questo articolo esplora i giri e le loro implicazioni.

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