Control theory: when does $G(s) = \frac{1}{P_\lambda(A)}$

Question

In other words, under what condition is the system transfer function G(s) = Y(s)/U(s) equivalent to the reciprocal of the characteristic equation of the $A$ matrix in state space realization?

Answer 1

The characteristic function of the transfer function is equal to the characteristic function of the system matrix if and only if the state space realization is minimal, i.e. it is controllable and observable. If you also want that the nominator of $G(s)$ to be 1, $C ~ \text{adj} (\lambda I - A) B = 1$ should be satisfied, because

$$ G(s) = C (\lambda I - A)^{-1} B = \frac{C ~ \text{adj} (\lambda I - A) B}{\text{det} (\lambda I - A)} $$