For discrete stochastic Markov jump systems, the properties of solutions to generalized Lyapunov equations(GLEs) are investigated based on the ?-representation method. Firstly, it is proved that the GLE has a unique solution in the form of a sequence of real symmetric matrices, if and only if the spectrum of the system does not contain zero eigenvalue. Moreover, the structure of solutions to GLEs is also discussed when there exist zero eigenvalue in the spectrum of the system. Finally, a numerical example is given to show the validity of the obtained results.