Auerbach bases, projection constants, and the joint spectral radius of principal submatrices
Jeremias Epperlein, Fabian Wirth
TL;DR
This work analyzes whether a compact matrix set can be similarity-transformed so that every entry is bounded by its joint spectral radius $\rho(\mathcal{M})$, and whether this bound extends to principal submatrices. It proves, via extremal norms and Auerbach bases, that an entrywise bound by $\rho(\mathcal{M})$ can be achieved when $\rho(\mathcal{M})>0$. However, it shows that higher-dimensional principal submatrices resist such normalization, providing counterexamples where submatrix JSRs remain above $\rho(\mathcal{M})$ across the similarity orbit; this obstruction is tied to projection constants and the geometry of finite-dimensional Banach spaces. On the positive side, the authors derive universal bounds for submatrix JSRs using John’s ellipsoid and projection-constant techniques, yielding bounds like $\rho((T^{-1}\mathcal{M}T)_{J,J})\le \sqrt{d}\,\rho(\mathcal{M})$ (and sharper for fixed submatrix size $m$ with constants $\delta_{\mathbb{K}}(m)$), thereby linking matrix analysis to convex geometry and spacelike properties of normed spaces. The paper also discusses special cases for pairs of matrices and outlines directions for refining the bounds and exploring optimal similarity transformations in broader norm contexts, underscoring the deep connections to Banach-space geometry and projection constants.
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.