Over the last 20 years, linearized rotor-bearing models have been used in lateral vibration analyses of numerous rotating machines: i.e., prediction of critical speeds, unbalance response, non-synchronous response, and instability threshold speed. Advances in computers and computing techniques over this period have had their effect by expanding the complexity of rotor-bearing systems which can be comprehensively analyzed. However, certain fundamental physical and mathematical properties of these systems have naturally remained unchanged. Some of these fundamental properties have not been given the attention or clarification they deserve, in contrast to the coverage given computing techniques and results. This paper has been written to relate various physical properties of linearized rotor-bearing systems to certain unique mathematical properties they possess.