Given the following system: $$G(s)=G_2(s)G_1(s)=\begin{bmatrix} > \begin{array}{c|c} A_2&B_2\\ \hline C_2&D_2 \end{array} > \end{bmatrix} \begin{bmatrix} \begin{array}{c|c} A_1&B_1\\ \hline > C_1&D_1 \end{array} \end{bmatrix}$$ I need to show that $pdir_o(G,p_i)=G_2(p_i)pdir_o(G_1,p_i)$ knowing that $p_i\in > spec(A_1)$ for all $p_i\notin spec(A_2)$
(We can not assume invertibility or minimality of $G_i$)
My way of approaching this problem is a such: we know that we can't assume invertibility or minimality of the system, but i do know that $$pdir_o(G,p_i)=Cker(p_iI-A)$$ which helps me with $pdir_o(G_1,p_i)$ but im still stuck with the $G_2(p_i)$ part
help will be appreciated