Transfer function for controllable part of a linear system
Let's try a numerical example:

$$ A = \begin{bmatrix} 1&amp;1 \\ 4&amp;-2 \end{bmatrix} $$
$$ B = \begin{bmatrix} 1&amp;-1 \\ 1&amp;-1 \end{bmatrix} $$
$$ C = \begin{bmatrix} 1&amp;0 \\ 0&amp;1 \end{bmatrix} $$

This system has a controllable mode at 2 and an uncontrollable mode at -3. The transfer matrix goes:

$$ C(sI-A)^{-1}B = \frac{1}{(s-2)(s+3)} \begin{bmatrix} (s+3) &amp; -(s+3) \\ (s+3) &amp; -(s+3)\end{bmatrix} $$

As can be seen, the uncontrollable mode shows up to be canceled in all terms of the transfer matrix. Only controllable and observable modes do not cancel out in the transfer matrix, because the transfer matrix describes an input to output relation. Controllable modes are the ones that are influenced by inputs, and observable modes are the ones that influence the outputs.