A general method for investigating the eigenvalue problems of a rotor system with uncertain parameters is presented in this paper. The recurrence perturbation formulas based on the Riccati transfer matrix method are derived and used for calculating the first- and secondorder perturbations of eigenvalues and their respective eigenvectors for the rotor system with uncertain parameters. In addition, these formulas can be used for investigating the independent, and repeated, as well as the complex eigenvalue problems. The general method is called the Riccati perturbation transfer matrix method (Riccati-PTMM). The formulas for calculating the mean value, variance, and covariance of the eigenvalues and eigenvectors of the rotor system with random parameters are also given. Riccati-PTMM is used for calculating the random eigenvalues of a simply supported Timoshenko beam and a test rotor supported by two oil bearings. The results show that the method is accurate and efficient.