On the accuracy of the ellipsoid norm approximation of the joint spectral radius (Q1765889)

The authors consider two computable approximations of the joint spectral radius for a finite set \({\mathcal U}\) of matrices. The first approximation satisfies \(\frac{1}{\sqrt{n}}\hat{\rho}\leq \rho\leq \hat{\rho}\) based on ellipsoid norms, where \(\rho\) is the joint spectral radius of \({\mathcal U}\), \(\hat{\rho}=\inf_{P\succ 0}\max_{A_i\in {\mathcal U}}\| A_i\| _p\), \(\| A\| _p\) is induced by the vector norm \(\| x\| _p=\sqrt{x^TPx}\) and \(n\) is the dimension of the matrices. Moreover, for the special case of symmetric matrices, triangular matrices, or for sets of matrices that have a solvable Lie algebra, the equality \(\rho=\hat{\rho}\) is satisfied. The other approximation is for the set of nonnegative matrices. In this case the approximation is proved to be within a factor at most \(m\) of the exact value, where \(m\) is the number of matrices in \({\mathcal U}\).