In this section, we show that the poles of a state-space model are given by the eigenvalues of the state-transition matrix .
Beginning again with the transfer function of the general state-space model,
we may first observe that the poles of are either the same as or some subset of the poles of
(They are the same when all modes are controllable and observable [37].) By Cramer's rule for matrix inversion, the denominator polynomial for is given by the determinant
where denotes the determinant of the square matrix . (The determinant of is also often written .) In linear algebra, the polynomial is called the characteristic polynomial for the matrix . The roots of the characteristic polynomial are called the eigenvalues of .
Thus, the eigenvalues of the state transition matrix are the poles of the corresponding linear time-invariant system. In particular, note that the poles of the system do not depend on the matrices , although these matrices, by placing system zeros, can cause pole-zero cancellations (unobservable or uncontrollable modes).