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In many regulation control problems it is desired to design the controller so that various system states have acceptable root mean squared values. This is a multiple objective problem. One way to satisfy these objectives is to assign a specified state covariance to the system. This paper introduces and solves the following problem: 1) characterize the entire set of state covariances which may be assigned to a linear discrete-time system by state feedback; and 2) find the set of all state feedback gains which will assign an admissible state covariance to the system. Extensions are also presented for state estimate feedback.