Numerous state estimation problems (e.g., under linear or nonlinear equality constraints, with correlated or singular measurement noise) can be formulated as the problem with noisy and noise-free measurements. Under the assumption that the initial state and the noise are jointly Gaussian distributed and the coefficient matrix for linear equality constraint has full row rank, some optimal state estimation algorithms were obtained in the literature. Given only the first two moments and without any assumption on the rank of the measurement matrix for noise-free measurement, optimal batch and two sequential forms of the state estimation algorithms in the sense of best linear unbiased estimation (BLUE) are obtained in this paper. Extension to nonlinear measurements is also discussed.