Hurwitz-stable 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.

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

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

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

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.