Routh–Hurwitz 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

$H=\left(\begin{array}{ccccccccc}{a}_1& a_3& a_5& \dots & \dots & \dots & 0& 0& 0\\ a_0& a_2& a_4& & & & ⋮& ⋮& ⋮\\ 0& a_1& a_3& & & & ⋮& ⋮& ⋮\\ ⋮& a_0& a_2& \ddots & & & 0& ⋮& ⋮\\ ⋮& 0& a_1& & \ddots & & a_n& ⋮& ⋮\\ ⋮& ⋮& a_0& & & \ddots & a_{n-1}& 0& ⋮\\ ⋮& ⋮& 0& & & & a_{n-2}& a_n& ⋮\\ ⋮& ⋮& ⋮& & & & a_{n-3}& a_{n-1}& 0\\ 0& 0& 0& \dots & \dots & \dots & a_{n-4}& a_{n-2}& a_n\end{array}\right).$

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:

$\begin{array}{cccc}{\mathrm{\Delta }}_1(p)& =\left|\begin{array}{c}{a}_1\end{array}\right|& & =a_1>0\\ {\mathrm{\Delta }}_2(p)& =\left|\begin{array}{cc}{a}_1& a_3\\ a_0& a_2\end{array}\right|& & =a_2{a}_1-a_0{a}_3>0\\ {\mathrm{\Delta }}_3(p)& =\left|\begin{array}{ccc}{a}_1& a_3& a_5\\ a_0& a_2& a_4\\ 0& a_1& a_3\end{array}\right|& & =a_3{\mathrm{\Delta }}_2-a_1(a_1{a}_4-a_0{a}_5)>0\end{array}$

and so on. The minors ${\mathrm{\Delta }}_k(p)$ 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

$M=\left(\begin{array}{ccc}-1& -1& 0\\ 1& -1& 0\\ 0& 0& -1\end{array}\right),$

and let

$\begin{array}{cc}p(x)& =\det (xI-M)\\ & =\left|\begin{array}{ccc}x+1& 1& 0\\ -1& x+1& 0\\ 0& 0& x+1\end{array}\right|\\ & =(x+1{)}^3-(1)(-1)(x+1)\\ & =x^3+3x^2+4x+2\end{array}$

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

$H=\left(\begin{array}{ccc}3& 2& 0\\ 1& 4& 0\\ 0& 3& 2\end{array}\right).$

The leading principal minors of $H$ are

$\begin{array}{cccc}{\mathrm{\Delta }}_1(p)& =\left|\begin{array}{c}3\end{array}\right|& & =3>0\\ {\mathrm{\Delta }}_2(p)& =\left|\begin{array}{cc}3& 2\\ 1& 4\end{array}\right|& & =12-2=10>0\\ {\mathrm{\Delta }}_3(p)& =\left|\begin{array}{ccc}3& 2& 0\\ 1& 4& 0\\ 0& 3& 2\end{array}\right|& & =2{\mathrm{\Delta }}_2(p)=20>0.\end{array}$

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.Template:NoteTag

See also

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

References

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