Routh–Hurwitz matrix

In mathematics, the Routh–Hurwitz matrix,^[1] or more commonly just Hurwitz matrix, corresponding to a polynomial is a particular matrix whose nonzero entries are coefficients of the polynomial.

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.

As an example, consider the matrix

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

and let

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

be the characteristic polynomial of ${\displaystyle M}$. The Routh–Hurwitz matrix^{[note 1]} associated to ${\displaystyle p}$ is then

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

The leading principal minors of ${\displaystyle H}$ are

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

Since the leading principal minors are all positive, all of the roots of ${\displaystyle p}$ have negative real part. Moreover, since ${\displaystyle p}$ is the characteristic polynomial of ${\displaystyle M}$, it follows that all the eigenvalues of ${\displaystyle M}$ have negative real part, and hence ${\displaystyle M}$ is a Hurwitz-stable matrix.^{[note 1]}

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

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• Gantmacher, F. R. (1959). Applications of the Theory of Matrices. New York: Interscience.
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