A transfer function and a state-space model can describe the same linear input-output behavior from different viewpoints.

For

$$\dot x=Ax+Bu,$$

$$y=Cx+Du,$$

with zero initial state, taking Laplace transforms gives

$$sX=AX+BU.$$

Therefore

$$X=(sI-A)^{-1}BU,$$

and the transfer function is

$$G(s)=C(sI-A)^{-1}B+D.$$

A state-space model that produces a given transfer function is called a realization of that transfer function.

The realization is not unique: changing state coordinates can produce different matrices while preserving the same external input-output map.

Transfer functions emphasize external behavior. State-space models expose internal variables and their evolution. Moving between the two representations shows which conclusions concern observable input-output behavior and which concern a particular internal model.