In this paper, a methodology is presented for the identification of the complete mass, damping, and stiffness matrices of a dynamical system using a limited number of time histories of the input excitation and of the response output. Usually, in this type of inverse problems, the common assumption is that the excitation and the response are recorded at a sufficiently large number of locations so that the full-order mass, damping, and stiffness matrices can be estimated. However, in most applications, an incomplete set of recorded time histories is available and this impairs the possibility of a complete identification of a second-order model. In this proposed approach, all the complex eigenvectors are correctly identified at the instrumented locations (either at a sensor or at an actuator location). The remaining eigenvector components are instead obtained through a nonlinear least-squares optimization process that minimizes the output error between the measured and predicted responses at the instrumented locations. The effectiveness of this approach is shown through numerical examples and issues related to its robustness to noise polluted measurements and to uniqueness of the solution are addressed.