55 The transfer function of SISO $(A,B,C,D)$

The transfer function of a SISO system is a proper rational function, and all the poles are eigenvalues of $A$. Proof. The characteristic polynomial $det(sI-A)$ has degree $n$, and leading term $s^n$. A cofactor of $sI-A$ is the determinant of a $(n-1)\times (n-1)$ submatrix of $sI-A$ and hence is a polynomial of degree less than or equal to $n-1$. Now

where $\text{adj}(sI-A)$ is the transpose of the matrix of cofactors. Hence the entries of ${(sI-A)}^{-1}$ are strictly proper rational functions. The eigenvalues of $A$ are precisely the zeros of $det(sI-A)$, hence are the only possible poles of entries of ${(sI-A)}^{-1}$. Since $C,B$ and $D$ are constant matrices, they do not introduce any more factors involving $s$, so $T(s)$ is a proper rational function.