Abstract: The matrix perturbation theory for the distinct eigenvalues of the real symmetric matrix was well developed. If some of eigenvalues are multiple, the nature of the problem changes and difficulties arise. To avoid such difficulties, it is usually assumed that the system has a set of complete eigenvectors to span the space, i.e. the system is non defective. However, in actual engineering problems, such as general damping systems, flutter analysis of aero elasticity, and so on, the defective system, that do no...