Hurwitz-stable matrix

This article is about matrices whose eigenvalues have negative real part. For the Hurwitz matrices used to check stability of polynomials, see Routh–Hurwitz matrix.

In mathematics, a Hurwitz-stable matrix, or more commonly simply Hurwitz matrix, is a square matrix whose eigenvalues all have strictly negative real part. Some authors also use the term stability matrix. Such matrices play an important role in control theory.

Definition

A square matrix $A$ is called a Hurwitz matrix if every eigenvalue of $A$ has strictly negative real part, that is,

$\mathrm{Re}[{\lambda }_i]<0$

for each eigenvalue ${\lambda }_i$. $A$ is also called a stable matrix, because then the differential equation

$\dot{x}=Ax$

is asymptotically stable, that is, $x(t)\to 0$ as $t\to \mathrm{\infty }.$ 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. Any hyperbolic fixed point (or equilibrium point) of a continuous dynamical system is locally asymptotically stable if and only if the Jacobian of the dynamical system is Hurwitz stable at the fixed point. The Hurwitz stability matrix is a crucial part of control theory. A system is stable if its control matrix is a Hurwitz matrix. The negative real components of the eigenvalues of the matrix represent negative feedback. Similarly, a system is inherently unstable if any of the eigenvalues have positive real components, representing positive feedback.

See also

• M-matrix
• Perron–Frobenius theorem, which shows that any Hurwitz matrix must have at least one negative entry
• Z-matrix

References

This article incorporates material from Hurwitz matrix on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

External links

• "Hurwitz matrix". PlanetMath.