Hurwitz matrix

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
 * $$p(z)=a_{0}z^n+a_{1}z^{n-1}+\cdots+a_{n-1}z+a_n$$

the $$n\times n$$ square matrix

H= \begin{pmatrix} a_1 & a_3 & a_5 & \dots & \dots & \dots & 0 & 0 & 0 \\ a_0 & a_2 & a_4 & & & & \vdots & \vdots & \vdots \\ 0 & a_1 & a_3 & & & & \vdots & \vdots & \vdots \\ \vdots & a_0 & a_2 & \ddots & & & 0 & \vdots & \vdots \\ \vdots & 0 & a_1 & & \ddots & & a_n & \vdots & \vdots \\ \vdots & \vdots & a_0 & & & \ddots &  a_{n-1} & 0 & \vdots \\ \vdots & \vdots & 0 & & & & a_{n-2} & a_n & \vdots \\ \vdots & \vdots & \vdots & & & & a_{n-3} & a_{n-1} & 0 \\ 0 & 0 & 0 & \dots & \dots & \dots & a_{n-4} & a_{n-2} & a_n \end{pmatrix}. $$ is called Hurwitz matrix corresponding to the polynomial $$p$$. It was established by Adolf Hurwitz in 1895 that a real polynomial with $$a_0 > 0$$ is stable (that is, all its roots have strictly negative real part) if and only if all the leading principal minors of the matrix $$H(p)$$ are positive:



\begin{align} \Delta_1(p) &= \begin{vmatrix} a_{1} \end{vmatrix} &&=a_{1} > 0 \\[2mm] \Delta_2(p) &= \begin{vmatrix} a_{1} & a_{3} \\ a_{0} & a_{2} \\ \end{vmatrix} &&= a_2 a_1 - a_0 a_3 > 0\\[2mm] \Delta_3(p) &= \begin{vmatrix} a_{1} & a_{3} & a_{5} \\ a_{0} & a_{2} & a_{4} \\ 0    & a_{1} & a_{3} \\ \end{vmatrix} &&= a_3 \Delta_2 - a_1 (a_1 a_4 - a_0 a_5 ) > 0 \end{align} $$ and so on. The minors $$\Delta_k(p)$$ are called the Hurwitz determinants. Similarly, if $$a_0 < 0$$ 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 $$A$$ is called a Hurwitz matrix if every eigenvalue of $$A$$ has strictly negative real part, that is,
 * $$\operatorname{Re}[\lambda_i] < 0\,$$

for each eigenvalue $$\lambda_i$$. $$A$$ is also called a stable matrix, because then the differential equation
 * $$\dot x = A x$$

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

If $$G(s)$$ is a (matrix-valued) transfer function, then $$G$$ is called Hurwitz if the poles of all elements of $$G$$ have negative real part. Note that it is not necessary that $$G(s),$$ for a specific argument $$s,$$ be a Hurwitz matrix — it need not even be square. The connection is that if $$A$$ is a Hurwitz matrix, then the dynamical system
 * $$\dot x(t)=A x(t) + B u(t)$$
 * $$y(t)=C x(t) + D u(t)\,$$

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.