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

In this work we solve the problem of a common Solution to the Lyapunov equation for 2 x 2 complex matrices. We show that necessary and sufficient conditions for the existence of a common solution to the Lyapunov equation for 2 x 2 complex matrices A and B is that matrices (A + i alpha I)(B + i beta I) and (A + i alpha I)(-1) (B + i beta I) have no negative real eigenvalues for all alpha, beta is an element of R. We show how these results relate to a special class of 4 x 4 real matrices. (c) 2006 Elsevier Inc. All rights reserved.