Routh–Hurwitz matrix explained

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.

Hurwitz matrix and the Hurwitz stability criterion

Namely, given a real polynomial

p(z) = a _{0} z ^{n + a} _{1} z ^{n-1} + … + a _{n-1} z + a _{n} the n x n square matrixH = \begin{pmatrix} a _{1} & a _{3} & a _{5} & ... & ... & ... & 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 & ... & ... & ... & 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 \\[ 2 mm] \Delta _{2(p)} & = \begin{vmatrix} a _{1} & a _{3} \\ a _{0} & a _{2} \\ \end{vmatrix} && = a _{2} a _{1} - a _{0} a _{3} > 0 \\[ 2 mm] \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 \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{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 associated to p is thenH = \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 \\[ 2 mm] \Delta _{2(p)} & = \begin{vmatrix} 3 & 2 \\ 1 & 4 \\ \end{vmatrix} && = 12 - 2 = 10 > 0 \\[ 2 mm] \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.

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 . . 18 . 2 . 407–414 . 10.1137/0118035 . 2099475.
• Dimitrov . Dimitar K. . Peña . Juan Manuel . 2005 . Almost strict total positivity and a class of Hurwitz polynomials . . 132 . 2 . 212–223 . 10.1016/j.jat.2004.10.010. free . 11449/21728 . free .
• Book: Gantmacher, F. R. . 1959 . Applications of the Theory of Matrices . . New York.
• Hurwitz . A. . 1895 . Ueber die Bedingungen, unter welchen eine Gleichung nur Wurzeln mit negativen reellen Teilen besitzt . . 46 . 2 . 273–284 . 10.1007/BF01446812. 121036103 .
• Lehnigk . Siegfried H. . 1970 . On the Hurwitz matrix . . 21 . 3 . 498–500 . 1970ZaMP...21..498L . 10.1007/BF01627957. 123380473 .

Notes and References

1. Book: Horn . Roger . Johnson . Charles . Topics in matrix analysis . 1991 . 0-521-30587-X . 101.