A sequential decomposition method is developed for solving the stationary discrete-time algebraic matrix Riccati equation. This decomposition is applied to the partially singular stationary filtering problem where some of the system outputs are free of measurement noise. Explicit expressions are obtained for the minimum error covariance matrices in the case where the number of the noise-free measurements is equal to the number of independent inputs. The method is generalized also to the case where the singular subsystem is non-square.