Based on the feasibility of applying trajectory eigenvalues analysis for large disturbances， the paper tries to analyze and predict the transient stability of a power system. Along with the system trajectory after faults being cleared， the Taylor expansion model is established. Then the system is approximated into a large number of linear systems with disturbance quantity. Based on stability theory of linear systems and the characteristics of power system instability， the criterion of power system transient instability is provided. The on-line transient stability prediction algorithm is proposed based on the criterion of losing stability. Case studies of New England system and an actual power grid in China are provided to demonstrate the proposed criterion and algorithm， and the results show that the proposed methods are efficient.