In mathematics, a Hurwitz matrix, or Routh–Hurwitz matrix, in engineering stability matrix, is a structured real square matrix constructed with coefficients of a real polynomial.

Hurwitz matrix and the Hurwitz stability criterion

Namely, given a real polynomial

the square matrix

is called Hurwitz matrix corresponding to the polynomial . It was established by Adolf Hurwitz in 1895 that a real polynomial with is stable (that is, all its roots have strictly negative real part) if and only if all the leading principal minors of the matrix are positive:

and so on. The minors are called the Hurwitz determinants. Similarly, if then the polynomial is stable if and only if the principal minors have alternating signs starting with a negative one.

Hurwitz stable matrices

In engineering and stability 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.

See also


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