Routh–Hurwitz matrix

Namely, given a real polynomial

:$p(z)=a\_\{0\}z^n+a\_\{1\}z^\{n-1\}+\backslash cdots+a\_\{n-1\}z+a\_n$

the $n\backslash times\; n$ 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{align}

\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{align}

and so on. The minors $\backslash Delta\_k(p)$ are called the Hurwitz determinants. Similarly, if 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

:$$

M=

\begin{pmatrix}

-1 & -1 & 0 \\

1 & -1 & 0 \\

0 & 0 & -1

\end{pmatrix},

and let

:$$

\begin{align}

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{align}

be the characteristic polynomial of $M$. The Routh–Hurwitz matrix{{NoteTag|name=confusion|Both Routh–Hurwitz and Hurwitz-stable matrices are more commonly referred to simply as Hurwitz matrices. To reduce the risk of confusion, this section avoids that terminology.}} 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{align}

\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{align}

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.{{NoteTag|name=confusion}}

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

{{NoteFoot}}

{{reflist}}

• {{cite journal

|last1=Asner |first1=Bernard A. Jr.

|year=1970

|title=On the Total Nonnegativity of the Hurwitz Matrix

|journal=SIAM Journal on Applied Mathematics

|volume=18 |issue=2 |pages=407–414

|doi=10.1137/0118035

|jstor=2099475

}}

• {{cite journal

|last1=Dimitrov |first1=Dimitar K.

|last2=Peña |first2=Juan Manuel

|year=2005

|title=Almost strict total positivity and a class of Hurwitz polynomials

|journal=Journal of Approximation Theory

|volume=132 |issue=2 |pages=212–223

|doi=10.1016/j.jat.2004.10.010

|doi-access=free

|hdl=11449/21728

|hdl-access=free

}}

• {{cite book

|last=Gantmacher |first=F. R.

|year=1959

|title=Applications of the Theory of Matrices

|publisher=Interscience

|location=New York

}}

• {{cite journal

|last=Hurwitz |first=A.

|year=1895

|title=Ueber die Bedingungen, unter welchen eine Gleichung nur Wurzeln mit negativen reellen Teilen besitzt

|journal=Mathematische Annalen

|url=http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=GDZPPN002255472

|volume=46

|issue=2

|pages=273–284

|doi=10.1007/BF01446812

|s2cid=121036103

}}

• {{cite journal

|last1=Lehnigk |first1=Siegfried H.

|year=1970

|title=On the Hurwitz matrix

|journal=Zeitschrift für Angewandte Mathematik und Physik

|volume=21 |issue=3 |pages=498–500

|bibcode=1970ZaMP...21..498L

|doi=10.1007/BF01627957

|s2cid=123380473

}}

{{Matrix classes}}

Category:Matrices (mathematics)