Characterization of joint spectral radius via trace
For a bounded set \(\Sigma\) of square matrices the joint spectral radius \(\rho(\Sigma)\) is defined as the limit superior of the supremum over the set \({\mathcal {L}}_n\) of all matrix products of length \(n\) from the collection \(\Sigma\) of the \(n^{th}\) root of the product matrix norm, while the joint generalized spectral radius \(\widetilde \rho(\Sigma)\) is the same expression involving the spectral radii of the matrix products instead. Inequalities such as \(\sup \rho (A)^{1/n} \leq \widetilde \rho(\Sigma) \leq \rho(\Sigma) \leq \sup \|A\|^{1/n}\) are known. For a finite set \(\Sigma\) of matrices this paper establishes that \(\rho(\Sigma) = \limsup\limits _{n \to \infty} \sup \limits_{A \in {\mathcal {L}}_n} |{trace}(A)|^{1/n}\) and gives estimates for the joint spectral radius for sets generated by two matrices in terms of the trace. These concepts apply to cascade algorithms for wavelets with compact support.
scientific article; zbMATH DE number 1523269
scientific article