A quadratic performance index together with a set of prescribed closed-loop eigenvalues are considered as criteria for designing linear multi-variable control systems. A method is developed for finding the weighting matrix elements that correspond to the given eigenvalues. The method is completely general as to system structure and is easy to implement computationally. It is also shown how the method may be used to design optimal estimators with prescribed eigenvalues. Several examples are presented to illustrate the use of the method.