Abstract: The transition matrix method is used to solve non-uniform elastic mecha-nics problems widely.For examble,we can solve the stregth, vibration nd stability problems of the beam and cylindrical shell, as well as the crit-ical speeds of a rotating multiple span axis by the method.The transition ma-trix method's charac-teristic is that a non-uniform mechanics problem can be transformed a uniform problem on d:screte inteval by it. In this paper, the method's convergence is proved,and convergent condition as well as err-or estimate also is given out. Two numerical examples are given at the end of the paper which indicate that, using this paqptr's theory, the right mo-del can be obtain for the transition matrix method.