Routh–Hurwitz matrix

This article is about the matrices used to check stability of polynomials. For matrices whose eigenvalues have negative real part, see Hurwitz-stable matrix.

In mathematics, the Routh–Hurwitz matrix, or more commonly just Hurwitz matrix, corresponding to a polynomial is a particular matrix whose nonzero entries are coefficients of the 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.

Example

As an example, consider the matrix

$M={\begin{pmatrix}-1&-1&0\\1&-1&0\\0&0&-1\end{pmatrix}},$

and let

${\begin{aligned}p(x)&=\det(xI-M)\\&={\begin{vmatrix}x+1&1&0\\-1&x+1&0\\0&0&x+1\end{vmatrix}}\\&=(x+1)^{3}-(1)(-1)(x+1)\\&=x^{3}+3x^{2}+4x+2\end{aligned}}$

be the characteristic polynomial of $M$. The Routh–Hurwitz matrix associated to $p$ is then

$H={\begin{pmatrix}3&2&0\\1&4&0\\0&3&2\end{pmatrix}}.$

The leading principal minors of $H$ are

${\begin{aligned}\Delta _{1}(p)&={\begin{vmatrix}3\end{vmatrix}}&&=3>0\\[2mm]\Delta _{2}(p)&={\begin{vmatrix}3&2\\1&4\\\end{vmatrix}}&&=12-2=10>0\\[2mm]\Delta _{3}(p)&={\begin{vmatrix}3&2&0\\1&4&0\\0&3&2\\\end{vmatrix}}&&=2\Delta _{2}(p)=20>0.\end{aligned}}$

Since the leading principal minors are all positive, all of the roots of $p$ have negative real part. Moreover, since $p$ is the characteristic polynomial of $M$, it follows that all the eigenvalues of $M$ have negative real part, and hence $M$ is a Hurwitz-stable matrix.

See also

• Routh–Hurwitz stability criterion
• Liénard–Chipart criterion
• P-matrix

References

• 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. hdl:11449/21728.
• 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. S2CID 121036103.
• 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. S2CID 123380473.