Hurwitz-stable matrix

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

${\displaystyle \operatorname {Re} [\lambda _{i}]<0\,}$ 

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

${\displaystyle {\dot {x}}=Ax}$ 

is asymptotically stable, that is, ${\displaystyle x(t)\to 0}$  as ${\displaystyle t\to \infty .}$ 

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

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

${\displaystyle y(t)=Cx(t)+Du(t)\,}$ 

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.

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

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

• "Hurwitz matrix". PlanetMath.