If the full state is available, a linear controller can feed it back directly:

$$u=-Kx+r_u,$$

where $K$ is the state-feedback gain and $r_u$ represents any separate reference input term.

Substituting this law into

$$\dot x=Ax+Bu$$

gives

$$\dot x=(A-BK)x+Br_u.$$

State feedback therefore replaces the open-loop state matrix $A$ by the closed-loop matrix

$$A_{cl}=A-BK.$$

Its eigenvalues determine the natural modes and stability of the controlled state dynamics.

This is the state-space counterpart of changing a closed-loop characteristic equation with feedback. The controller gain does not merely scale the state: it changes the matrix governing its evolution.

How $K$ should be chosen is a separate design question. Pole placement and optimal control are two different answers.