Abstract: It is very interesting to study eigenvalues of a linear system by analyzing the properties of its mass matrix, stiffness matrix, damping matrix and so on. After a brief review on the complex mode theory, this paper developed a new set equations about corresponding matrices by recalling the orthogonality between the complex modes of a damped gyroscopic linear system. Based upon this, the applicability of two popular expressions about complex eigenvalues was investigated. The emphasis was given to the questio...