Explicit and analytical solutions to Lyapunov algebraic matrix equations

Based on a recently proposed explicit and complete parametric solution to the Sylvester matrix equation in the form of AV+EVF=W, explicit and analytical solutions to the Lyapunov matrix equations E^TVF-V=-W and E^TV+VF=-W are deduced. Cases of multiple solutions are also treated. These solutions are in simple and compact forms, and may have important applications in control systems theory