Uniqueness of State Transformation Matrix for Controllable Systems

[System X]

Homework Statement

Two systems are given (both are completely controllable):
x-dot = Ax + bu
z-dot = A*z + b*u

They are related by the state transformation:

z=Tx

prove that the transformation matrix T is unique.

The Attempt at a Solution

Since the systems are completely controllable, we the kalman matrix (k=(b,Ab,A^2b,...) is non-singular. If T is unique, there is only one possible chose for the coefficients. I'm lost from there.

 

[Mark44]

Please fix your LaTeX, particularly this bit
(k=(b,Ab,[ tex]^{}[/ tex]b,...)

It's causing a large, very wide blank area to be rendered on the page.