Common solution to the Lyapunov equation for 2 × 2 complex matrices

In this work we solve the problem of a common solution to the Lyapunov equation for 2 × 2 complex matrices. We show that necessary and sufficient conditions for the existence of a common solution to the Lyapunov equation for 2 × 2 complex matrices A and B is that matrices (A + iαI)(B + iβI) and (A + iαI)−1(B + iβI) have no negative real eigenvalues for all α,β∈R. We show how these results relate to a special class of 4 × 4 real matrices.