Hurwitz matrix

A square matrix $A$ is called a Hurwitz matrix if all eigenvalues of $A$ have strictly negative real part, ; $A$ is also called a stability matrix, because the feedback system

$$\dot{x}=Ax$$

is stable.

If $G(s)$ is a (matrix-valued) transfer function, then $G$ is called Hurwitz if the poles of all elements of $G$ have negative real part. Note that it is not necessary that $G(s)$, for a specific argument $s$, be a Hurwitz matrix — it need not even be square. The connection is that if $A$ is a Hurwitz matrix, then the dynamical system

$\dot{x}(t)$$=$$Ax(t)+Bu(t)$

$y(t)$$=$$Cx(t)+Du(t)$

has a Hurwitz transfer function.

Reference: Hassan K. Khalil, Nonlinear Systems, Prentice Hall, 2002