## Poles of a State Space Filter

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,

*controllable*and

*observable*[37].) By Cramer's rule for matrix inversion, the denominator polynomial for is given by the

*determinant*

*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).

**Next Section:**

Difference Equations to State Space

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Transposition of a State Space Filter