Summary: This paper shows that diagonal Lyapunov functions play an essential role in the robust stability problem for linear autonomous continuous and discrete-time systems in state-space form subjected to a broad class of perturbations. The results obtained generalize a well-known result of Persidskij and are related to a variety of similar results in the literature. Techniques are suggested for applying the results to the problem of evaluating robust stability bounds in different contexts. Several examples are given.