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.

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.

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.