Some Questions on Kalman Standard Decomposition of Linear Time-Invariable System

It is shown that for a linear time-invariable system,its Kalman canonical form can not be obtained by decomposing the system into its controllable canonical form,and then decomposing its controllable and uncontrollable subsystems into their observable canonical form,respectively.The possibility is discussed to construct a non-singular transformation which transforms the system to its Kalman canonical form through choosing a basis in < A|β >∩η,< A|β >∩η^⊥,< A|β >^⊥∩η and < A|β >^⊥∩η^⊥ .Finally ,it is proved that the system can be transformed to its Kalman canonical form using an orthogonal transformation under appropriate conditions.