I am working with some linear multi input dynamical systems. There is a result here which reduces the problem to single input linear systems. Given the following linear system: $$ \dot{X} = A\cdot X + B\cdot U$$ where $A \in \mathbb{R}^{m \times m}$ and $B \in \mathbb{R}^{m \times l}$ and $U \in \mathbb{R}^{l\times 1}$. Let $B = \begin{bmatrix} B_1 &... &B_m \end{bmatrix}$ and $U = \begin{bmatrix} u_1 \\ \vdots \\ u_l \end{bmatrix}$
Problem:
I am not able to complete the proof for Lemma 3.3.4 (page 93 written, or 24 on the linked pdf). It says that if $$Co(A,B) = \begin{bmatrix} B &A\cdot B &... & A^{m-1}\cdot B\end{bmatrix} \in \mathbb{R}^{m\times (m\cdot l)}$$ has full rank, then for all $i\in\{1, ..., l\}$ $\exists K_i$ such that the single input system $$ \dot{X} = (A + B\cdot K_i) \cdot X + B_i \cdot u_i$$ is controllable
The author considers the case $i=1$ for a convenient presentation, then proceeds to form a matrix $M$ with $m$ linear columns of $Co(A,B)$ like so: $$ M = \begin{bmatrix} B_1 &... &A^{r_1-1}B_1 &B_2 &... &A^{r_2-1}B_2 &...\end{bmatrix}$$ with $r_1 + ... + r_l = m$. Then it simply forms the matrix $N \in \mathbb{R}^{l \times m}$ like so: $$ N = \begin{bmatrix} 0_{1 \times m}\\ e_{r_1}\\ e_{r_1+r_2}\\ \vdots \\ e_{r_1 + ... + r_{l-1}}\end{bmatrix}$$ (I give a slightly different definition for $N$ that I think is equivalent. However, the definition given in the presented document is somehow intuitive ... it implies that the column $r_1 + ... + r_l$ should be $e_{l+1}$ which is impossible since $N$ should have only $l$ lines ) However, it is then claimed that it is not difficult to show that $$ K_1 = N\cdot M^{-1}$$
My attempt
I think I should show that $$ Co2(A+BK_1,B1) = \begin{bmatrix} B_1 &(A+BK_1)B_1 &... &(A+BK_1)^{m-1}B_1\end{bmatrix}$$ has full rank ... but this seems ... not obvious