Reference for linear system theory result Exercise 13.2 of Wilson J. Rugh's book Linear System Theory reads
Consider the $n$-dimensional linear state equation
$$\dot{x}(t) = \begin{bmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{bmatrix} x(t) + \begin{bmatrix} B_{11} \\ 0 \end{bmatrix} u(t)$$
where $A_{11}$ is $q \times q$ and $B_{11}$ is $q \times m$ with rank $q$. Prove that this state equation is controllable if and only if the $(n-q)$-dimensional linear state equation
$$\dot{z}(t) = A_{22} z(t) + A_{21} v(t)$$
is controllable.
I would like to use this result in some application but am unable to come up with a proof. I fear it's not enough to cite this exercise in my work, so does someone know a reference?
 A: I haven't come across such problem before, so I don't know a reference for it. However, deriving a proof for it is not too complicated. Namely, because the rank of $B_{11}$ is $q$ implies that $B_{11} B_{11}^\top$ is invertible, such that one can always apply the following change of coordinates
$$
u(t) = B_{11}^\top \left(B_{11}\,B_{11}^\top\right)^{-1} \left(w(t) - 
\begin{bmatrix}
A_{11} & A_{12}
\end{bmatrix} x(t)\right). \tag{1}
$$
Substituting $(1)$ into the original system yields
$$
\dot{x}(t) = 
\begin{bmatrix}
0 & 0 \\
A_{21} & A_{22}
\end{bmatrix} x(t) + 
\begin{bmatrix}
I \\ 0
\end{bmatrix} w(t). \tag{2}
$$
The controllability matrix associated with $(2)$ system can be shown to be
$$
\mathcal{C} = 
\begin{bmatrix}
I & 0 & 0 & \cdots & 0 \\
0 & A_{21} & A_{22} A_{21} & \cdots & A_{22}^{n-q-1} A_{21}
\end{bmatrix}, \tag{3}
$$
from which it follows that $\mathcal{C}$ is full rank if and only if $(A_{22},A_{21})$ is controllable. If the controllability matrix from $(3)$ is full rank then $(2)$ is controllable, which would also imply that the original system would be controllable.
