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
:$p(z)=a_z^n+a_z^+\cdots+a_z+a_n$
the $n\times n$ square matrix
:$H= \begin a_1 & a_3 & a_5 & \dots & \dots & \dots & 0 & 0 & 0 \\ a_0 & a_2 & a_4 & & & & \vdots & \vdots & \vdots \\ 0 & a_1 & a_3 & & & & \vdots & \vdots & \vdots \\ \vdots & a_0 & a_2 & \ddots & & & 0 & \vdots & \vdots \\ \vdots & 0 & a_1 & & \ddots & & a_n & \vdots & \vdots \\ \vdots & \vdots & a_0 & & & \ddots & a_ & 0 & \vdots \\ \vdots & \vdots & 0 & & & & a_ & a_n & \vdots \\ \vdots & \vdots & \vdots & & & & a_ & a_ & 0 \\ 0 & 0 & 0 & \dots & \dots & \dots & a_ & a_ & a_n \end.$
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
\Delta_1(p) &= \begin a_ \end &&=a_ > 0 \\ mm\Delta_2(p) &= \begin
a_ & a_ \\
a_ & a_ \\
\end &&= a_2 a_1 - a_0 a_3 > 0\\ mm\Delta_3(p) &= \begin
a_ & a_ & a_ \\
a_ & a_ & a_ \\
0 & a_ & a_ \\
\end &&= a_3 \Delta_2 - a_1 (a_1 a_4 - a_0 a_5 ) > 0
\end

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
:$M= \begin -1 & -1 & 0 \\ 1 & -1 & 0 \\ 0 & 0 & -1 \end,$
and let
:$\begin p(x)&=\det(xI-M)\\ &= \begin x+1 & 1 & 0 \\ -1 & x+1 & 0 \\ 0 & 0 & x+1 \end\\ &=(x+1)^3-(1)(-1)(x+1)\\ &=x^3+3x^2+4x+2 \end$
be the characteristic polynomial of $M$. The Routh–Hurwitz matrix associated to $p$ is then
:$H= \begin 3 & 2 & 0 \\ 1 & 4 & 0 \\ 0 & 3 & 2 \end.$
The leading principal minors of $H$ are
:
\begin
\Delta_1(p) &= \begin 3\end &&=3>0\\ mm\Delta_2(p) &= \begin
3 & 2 \\
1 & 4 \\
\end &&= 12 - 2 = 10 > 0\\ mm\Delta_3(p) &= \begin
3 & 2 & 0 \\
1 & 4 & 0 \\
0 & 3 & 2 \\
\end &&= 2\Delta_2(p)=20 > 0.
\end

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

Notes

References

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{{Matrix classes

Matrices (mathematics)