How to divide an uncontrollable LTI system into controllable and uncontrollable parts? Consider this linear system $\frac{dx}{dt}=Ax+Bu$
Assume that $B\neq 0$ and the system is uncontrollable. It’s easy to show the existence of an invertible state transform $x=Ty$ satisfying
$$\frac{dy}{dt}=
\begin{pmatrix}
  A_{1}  & A_{2} \\ 0 & A_{3}
\end{pmatrix}y+
\begin{pmatrix}
  B_{1} \\ 0
\end{pmatrix}u$$
by choosing the irrelevant column vector group of the controllability matrix and expand that to a basis of the whole vector space.
But I need help to prove that the subsystem $[A_{1},B_{1}]$ is controllable. So could anyone can tell me how to prove it?
 A: The rank of the controllability matrix remains unchanged under such a transformation since
$$\bar{\mathcal{C}}=T^{-1}\mathcal{C}$$
where $\mathcal{C}$, $\bar{\mathcal{C}}$ are the controllability matrices of the initial and the transformed system. Thus, $rank(\bar{\mathcal{C}})=r$ with $r$ the dimension of $A_1$. Now if you carry out the calculations to derive $\bar{\mathcal{C}}$ you will obtain
$$\bar{\mathcal{C}}=\left[\matrix{B_1 & A_1B_1 &\cdots & A_1^{n-1}B_1\\0 & 0& \cdots &0}\right]\qquad\qquad\qquad\qquad\qquad  (1)$$ From  $rank(\bar{\mathcal{C}})=r$ we have that
$$rank\left[\matrix{B_1 & A_1B_1 &\cdots & A_1^{n-1}B_1}\right]=r\qquad\qquad\qquad\qquad  (2)$$
which yields
$$rank\left[\matrix{B_1 & A_1B_1 &\cdots & A_1^{r-1}B_1}\right]=r\qquad\qquad\qquad\qquad  (3)$$
since all matrices $A_1^{j-1}$ with $j>r$ can be written as a linear combination of $\mathbb{I}_{r}$, $A_1$, $\cdots$, $A_1^{r-1}$ due to Cayley-Hamilton theorem. From (3) we obtain the controllability of $(A_1,B_1)$.
