We distinguish a subclass of switched linear systems that we call pairwise connected. We show that the dynamics of such systems can be described by Lur’e systems. For pairwise connected systems, we obtain a sufficient frequency-domain condition for the existence of a quadratic Lyapunov function. The well-known Aizerman problem is reformulated for switched linear systems. We show an example of a system with switchings between three linear third order subsystems for which Aizerman’s problem has a positive solution.