Some properties of generalized Lyapunov equations

By means of the spectral analysis method, this paper studies a class of generalized Lyapunov equations (GLEs) arising from stochastic stability. A necessary and sufficient condition is given for the existence and uniqueness of the symmetric/skew-symmetric solutions of such GLEs. A sufficient condition for the GLEs have no and have more than one symmetric/skew-symmetric solution is also presented, respectively.