An optimal multistage Kalman estimator (OMSKE) is proposed as a generalization of the optimal two-stage Kalman estimator for the reduction of the computational burden of the Kalman estimator (KE) for discrete-time linear time-varying systems with triangular transition matrices. This new filter is obtained by applying a multistage U - V transformation to decouple the covariances of the KE. It is shown analytically that the computational complexity of the OMSKE is less than that of the KE and is minimum when the system transition matrix has the maximum stage number.