In mathematics, a Hurwitz-stable matrix,^[1] or more commonly simply Hurwitz matrix,^[2] is a square matrix whose eigenvalues all have strictly negative real part. Some authors also use the term stability matrix.^[2] Such matrices play an important role in control theory.

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

for each eigenvalue . is also called a stable matrix, because then the differential equation

is asymptotically stable, that is, as

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

• 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.