Abstract: A large amount of parallel multi- conductor systems is used in power industry. The basic parameters of such a system are the per-unit-length series impedance matrix Z and shunt admittance matrix Y. Similarity transformation matrixes Tv and Ti diagonalize ZY and YZ,respectively. By applying linear matrix algebraic theory, this paper discusses the inherent relationship between Tv and Ti through strict mathematic deduction. The equation T_i= T_v^-T,which has been cited in a variety of literature,is just a rule made purposely to facilitate the mathematic process,but not the necessary condition for the diagonalization of ZY and YZ. It is proved that T_v and Ti can also diagonalize Z and Y,which has been otherwise stated in some reference. Surge impedances, parameters resulted from the basic ones,are clearly defined for the phase domain. Methods and formulae to evaluate them are presented. The surge impedances in the modal domain change with the tranforamtion matrix,but those in the phase domain do not. The difference between the surge impedances of a single conductor and those of a multi-conductor system is clarified,and the formulae for the former should not be adopted directly for the latter. At the end,the diagnolizing of ZY and YZ and the calculating of surge impedances are demonstrated.