Hidden Poles

In the transfer function of a state space model

$$G(s)=c(sI-A{)}^{-1}b+d=\frac{c\text{adj}(sI-A)}{\det (sI-A)}+d$$

where $\text{adj}$ is the adjugate of a matrix.

If there is no common factor on the denominator and numerator, then $a(s)=\text{det}(sI-A)$, i.e., $a(s)$ is the characteristic polynomial of matrix $A$, and the poles of the system are the eigen values of $A$.

Otherwise, $a(s)$ is only a factor of $\det (sI-A)$. Hence there exists hidden poles.