I need to show that the following system is always controllable:
\begin{align}A &= \begin{bmatrix} -\alpha_1I_{k\times k}& -\alpha_2I_{k\times k}& \cdots &-\alpha_{n-1}I_{k\times k}&-\alpha_nI_{k\times k}\\ I_{k\times k}&0_{k\times k}&\cdots&0_{k\times k}&0_{k\times k}\\ 0_{k\times k}&I_{k\times k}&\cdots&0_{k\times k}&0_{k\times k}\\ \vdots & \vdots & \ddots& \vdots&\vdots\\ 0_{k\times k}&0_{k\times k}&\cdots&I_{k\times k}&0_{k\times k}\end{bmatrix}_{nk\times nk}\\ B&=\begin{bmatrix} I_{k\times k}\\ 0_{k\times k}\\ \vdots\\ 0_{k\times k}\end{bmatrix}_{nk\times n} \quad C=\begin{bmatrix}N_1&N_2&\cdots&N_n\end{bmatrix}_{m\times nk}. \end{align}
Now, I think this is actually the controllable canonical form, but I'm really confused about how to show this is always controllable. Can anyone help? Sorry if this is a dumb question.