An approach is put forward to find the linealy independent vector of the observability matrix of completely observable linear multi-input multi-output(MIMO) system. Based on this approach，a method is advanced to obtain the transformation matrix which can be used for transforming the linear MIMO system to its Luenberger observable canonical form. Through the proving of several theormes，the relation between the transformation matrix and Luenberger observable canonical form is exposited and the Luenberger observal form is divided into two classes，the generalized Luenberger observable canonical form and the special Luenberger observable canonical form，according to its structure difference. The necessary and sufficient condition for the realization of generalized and special Luenberger observable canonical form of completed observable linear MIMO system are given，and three examples are used to verify the correctness and feasibility of the above viewpoint and method. Meanwhile，another method is put forward to transform a class of linear MIMO system，under the condition of unchanging its physical structure，which does not meet the above necessary and sufficient condition, to its Luenberger observable canonical form. Two examples are analyzed and compared to elaborate on the method.