Consider the linear time invariant system: $$\tag{1}\label{eq1} \dot{x}(t) = Ax(t) + Bu(t), \ \ x(0)=x_0\in\mathbb{R}^n, $$ where $A\in\mathbb{R}^{n\times n}$, $B\in\mathbb{R}^{n\times m}$. Let $p_M(s)$ denote the characteristic polynomial of a matrix $M\in\mathbb{R}^{n\times n}$. The following result is classic in linear control theory$^1$.
Theorem 1. Let $q(s)$ be any monic real polynomial of degree $n$. If the system \eqref{eq1} is controllable, then there exists a matrix $K\in\mathbb{R}^{m\times n}$ such that $$ p_{A+BK}(s) = q(s). $$
The standard proof of this result is constructive and makes use of a canonical form of \eqref{eq1}, known as the controller canonical form.
My question. Is there any known (not necessarily constructive) proof of Theorem 1 that does not use the controller canonical form?
$^1$ See e.g. Chapter 7 of T. Kailath. Linear systems. Prentice-Hall, 1980.
