Hurwitz matrix

Namely, given a real polynomial

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

the ${\displaystyle n\times n}$ square matrix

${\displaystyle H={\begin{pmatrix}a_{1}&a_{3}&a_{5}&\dots &\dots &\dots &0&0&0\\a_{0}&a_{2}&a_{4}&&&&\vdots &\vdots &\vdots \\0&a_{1}&a_{3}&&&&\vdots &\vdots &\vdots \\\vdots &a_{0}&a_{2}&\ddots &&&0&\vdots &\vdots \\\vdots &0&a_{1}&&\ddots &&a_{n}&\vdots &\vdots \\\vdots &\vdots &a_{0}&&&\ddots &a_{n-1}&0&\vdots \\\vdots &\vdots &0&&&&a_{n-2}&a_{n}&\vdots \\\vdots &\vdots &\vdots &&&&a_{n-3}&a_{n-1}&0\\0&0&0&\dots &\dots &\dots &a_{n-4}&a_{n-2}&a_{n}\end{pmatrix}}.}$

is called Hurwitz matrix corresponding to the polynomial ${\displaystyle p}$. It was established by Adolf Hurwitz in 1895 that a real polynomial with ${\displaystyle a_{0}>0}$ is stable (that is, all its roots have strictly negative real part) if and only if all the leading principal minors of the matrix ${\displaystyle H(p)}$ are positive:

${\displaystyle {\begin{aligned}\Delta _{1}(p)&={\begin{vmatrix}a_{1}\end{vmatrix}}&&=a_{1}>0\\[2mm]\Delta _{2}(p)&={\begin{vmatrix}a_{1}&a_{3}\\a_{0}&a_{2}\\\end{vmatrix}}&&=a_{2}a_{1}-a_{0}a_{3}>0\\[2mm]\Delta _{3}(p)&={\begin{vmatrix}a_{1}&a_{3}&a_{5}\\a_{0}&a_{2}&a_{4}\\0&a_{1}&a_{3}\\\end{vmatrix}}&&=a_{3}\Delta _{2}-a_{1}(a_{1}a_{4}-a_{0}a_{5})>0\end{aligned}}}$

and so on. The minors ${\displaystyle \Delta _{k}(p)}$ are called the Hurwitz determinants. Similarly, if ${\displaystyle a_{0}<0}$ 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 ${\displaystyle A}$ is called a stable matrix (or sometimes a Hurwitz matrix) if every eigenvalue of ${\displaystyle A}$ has strictly negative real part, that is,

${\displaystyle \operatorname {Re} [\lambda _{i}]<0\,}$

for each eigenvalue ${\displaystyle \lambda _{i}}$. ${\displaystyle A}$ is also called a stability matrix, because then the differential equation

${\displaystyle {\dot {x}}=Ax}$

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

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

${\displaystyle {\dot {x}}(t)=Ax(t)+Bu(t)}$

${\displaystyle y(t)=Cx(t)+Du(t)\,}$

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.

• M-matrix
• P-matrix
• Perron–Frobenius theorem
• Z-matrix

• Asner, Bernard A., Jr. (1970). "On the Total Nonnegativity of the Hurwitz Matrix". SIAM Journal on Applied Mathematics. 18 (2): 407–414. doi:10.1137/0118035. JSTOR 2099475.
• Dimitrov, Dimitar K.; Peña, Juan Manuel (2005). "Almost strict total positivity and a class of Hurwitz polynomials". Journal of Approximation Theory. 132 (2): 212–223. doi:10.1016/j.jat.2004.10.010.
• Gantmacher, F. R. (1959). Applications of the Theory of Matrices. New York: Interscience.
• Hurwitz, A. (1895). "Ueber die Bedingungen, unter welchen eine Gleichung nur Wurzeln mit negativen reellen Teilen besitzt". Mathematische Annalen. 46 (2): 273–284. doi:10.1007/BF01446812.
• Khalil, Hassan K. (2002). Nonlinear Systems. Prentice Hall.
• Lehnigk, Siegfried H. (1970). "On the Hurwitz matrix". Zeitschrift für Angewandte Mathematik und Physik. 21 (3): 498–500. Bibcode:1970ZaMP...21..498L. doi:10.1007/BF01627957.

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• "Hurwitz matrix". PlanetMath.