Proof that the rank of $(A,B)$ is the same as $(A-BK,B)$
The image is taken from "Optimal Control Methods for Linear Discrete-Time Economic Systems" by Y. Murata
Could someone please explain why the sum of columns of $B, AB$ and so on do not affect the row rank of the matrix in the above proof?
The proof is based on the following fact (which is easy to show): If $w_1,\ldots,w_m\in\operatorname{span}\{v_1,\ldots,v_n\}$ then $$ \operatorname{span}\{v_1,\ldots,v_n,(u_1+w_1),\ldots,(u_m+w_m)\} = \operatorname{span}\{v_1,\ldots,v_n,u_1,\ldots,u_m\}. $$ In particular, $$ \dim\operatorname{span}\{v_1,\ldots,v_n,u_1+w_1,\ldots,u_m+w_m\} = \dim\operatorname{span}\{v_1,\ldots,v_n,u_1,\ldots,u_m\}, $$ which is the same as $$ \operatorname{rk}[V,U+W] = \operatorname{rk}[V,U], $$ where $V = [v_1,\ldots,v_n]$, $W = [w_1,\ldots,w_m]$, and $U = [u_1,\ldots,u_m]$. The assumption $w_1,\ldots,w_m\in\operatorname{span}\{v_1,\ldots,v_n\}$ now means that $W = VL$ with a matrix $L$ of appropriate dimensions.