On the robustness of symmetric systems

This note concerns the robustness of LTI symmetric systems under symmetric or diagonal perturbations. It is shown that for symmetric systems the robust stability condition for dynamic perturbations given by the small gain theorem is also necessary if the perturbation matrix is assumed to be diagonal, but the robust stability condition for parametric perturbations given by the real stability radius is no longer necessary if the perturbation matrix is merely assumed to be symmetric