On the necessary and sufficient conditions for the existence of classical normal modes in damped linear dynamic systems

General theorems concerning the simultaneous diagonalization of two real symmetric matrices using a real orthogonal transformation matrix are presented to place the Caughey and O?Kelly?s theorem on classical normal modes in damped linear dynamic systems in context. It was also brought to attention that equivalent theorems similar to Caughey and O?Kelly?s existed in linear algebra since the works of Weirestrass. The examples given by Liang et al. are shown to be inadequate to modify Caughey and O?Kelly?s classification of damped systems. The classification of damped systems into proportional/non-proportional systems based on Caughey and O?Kelly?s criterion still remains valid if proportional damping is meant to convey that all the modes of the damped system are real, not just one or a few and that all the three system matrices are simultaneously diagonalised by the same transformation matrix.