A complete solution is given for the inverse problem of LQ optimal regulators for the discrete-time multi-input case. The Kalman equation for discrete-time systems is derived to characterize the optimal regulators. It is then shown that this equation holds for some quadratic weighting if and only if the feedback satisfies a condition corresponding to the circle criterion for continuous-time systems. The optimality criterion given here consists of this condition and an additional condition to ensure detectability. An example is given to illustrate the results.