This paper deals with the minimization of a scalar measure of the matrix which solves the Lyapunov equationP = [A − HC]P[A − HC]′ + BWB′ + HVH′with respect to the pair (H, C). This problem arises in the prediction of the state of a discrete-time, stochastic linear system, when one is concerned with minimizing the prediction error covariance both with respect to the predictor gain and with respect to the output matrix, which accounts for the physical device by which the state of the system is observed. By duality, the problem can be interpreted as the optimization of a suitable performance index for a linear regulator with respect both to the input matrix and to the feedback regulator gain. First, the index to be minimized is carefully chosen, in order to obtain a meaningful optimization problem, and the optimization with respect to the predictor gain or with respect to the output matrix, when the other matrix is fixed, is considered. The related results suggest an algorithm which generates a sequence of pairs (output matrix, predictor gain) which ensure stability of the matrix [A − HC], and to which a decreasing sequence of values of the index there corresponds. Results about boundedness and convergence of the sequence thus obtained are proved. The problem and the related solution algorithm is also extended to deal with the singular case of no measurement noise. Numerical experiments, in which comparison is made with the behaviour of a standard gradient-based method, confirm the robustness of the proposed algorithm, anticipated on the basis of its theoretical properties.