The result given by the Kalman filter is the BLUE (best linear unbiased estimate) provided that the mean and variance of the initial state are known. The same state estimation problem is reconsidered for multi-input multi-output (MIMO) systems when the statistics knowledge about the initial state is not known. The algorithm presented in this note gives the BLUE of system states without the requirement bf any a priori knowledge about the initial state. The concept of complete reconstructibility of stochastic systems is established, and the necessary and sufficient condition for complete reconstructibility is given. When applied to a completely reconstructible deterministic system, the proposed algorithms give the deadbeat state estimates even if the system is not observable.