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{aligned}\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{aligned)) 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 stable matrix (or sometimes 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 stability matrix, because then the differential equation

${\dot {x))=Ax$ 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

${\dot {x))(t)=Ax(t)+Bu(t)$ $y(t)=Cx(t)+Du(t)\,$ 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.