Abstract: This paper is a study on the inverse problem of LQ optimal regulators. With the control weight given, the state weighting matrix satisfying the closed-loop eigenvalue requirements is parametrized in terms of a set of free variables. Based on the parametrization, an analytic procedure and a matrix transformation method are proposed to determine the state weighting matrix, as well as the free variables. As a result, by using the solved free variables, the optimal controller gain matrix can be determined without solving the algebraic Riccati matrix equation.