On the existence of diagonal solutions to the Lyapunov equation for a third order system

An algebraic characterization of necessary and sufficient conditions for the existence of a diagonal solution P>0 to the Lyapunov equation for third order matrices has been derived. In this paper we show an alternative way of obtaining the same conditions. This is based on necessary and sufficient conditions that we derive for the existence of a common diagonal solution to the Lyapunov equation for two stable matrices in R^2×2. The importance of the new approach is that it appears to be possible to extend it to determine a common diagonal solution P>0 to the Lyapunov equation for two matrices A,B ∈ R^3×3 and even to more general cases.