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In mathematics, Hurwitz matrix is a structured real square matrix constructed with coefficients of a real polynomial.

Hurwitz matrix and the Hurwitz stability criterion

Namely, given a real polynomial

$$p(z)=a_0z^n+a_{1}z^{n-1}+\cdots+a_{n-1}z+a_n$$

the $$n\times n$$ square matrix

$$H(p) := \begin{bmatrix} a_1 & a_3 & a_5 & a_7 & \ldots & 0\\ a_0 & a_2 & a_4 & a_6& \ldots & 0\\ 0 & a_1 & a_3 & a_5& \ldots & 0\\ 0 & a_0 & a_2 & a_4& \ldots & 0\\ 0 & 0 & a_1 & a_3& \ldots & 0\\ \vdots & \vdots & \vdots & \vdots& \ddots& \vdots\\ 0 & 0 & 0 & 0& \ldots& a_n\\ \end{bmatrix}$$

is called Hurwitz matrix corresponding to the polynomial p. It was established by Adolf Hurwitz in 1895 that a real polynomial is stable (that is, all its roots lie in the open left half-plane of the complex plane) 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 $$\Delta_k(p)$$ are called the Hurwitz determinants.
Hurwitz stable matrices

In engineering and stability theory, a square matrix A is called stable matrix (or sometimes Hurwitz matrix) if every eigenvalue of A has strictly negative real part, that is,

$$\mathop{\mathrm{Re}}[\lambda_i] < 0\,$$

for each eigenvalue $$\lambda_i$$. A is also called a stability matrix, because then the differential equation

$$\dot x = A x$$

is asymptotically stable, that is, x(t)\to 0 as $$t\to\infty$$. Hurwitz matrix is named after Adolf Hurwitz.

If G(s) is a (matrix-valued) transfer function, then G is called Hurwitz if the poles of all elements of G have negative real part. Note that it is not necessary that G(s), for a specific argument s, be a Hurwitz matrix — it need not even be square. The connection is that if A is a Hurwitz matrix, then the dynamical system

$$\dot x(t)=A x(t) + B u(t)$$
$$y(t)=C x(t) + D u(t)\,$$

has a Hurwitz transfer function.

Any hyperbolic fixed point (or equilibrium point) of a continuous dynamical system is locally asymptotically stable if and only if the Jacobian of the dynamical system is Hurwitz stable at the fixed point.

The Hurwitz stability matrix is in crucial part on control theory. A system is stable if its control matrix is a Hurwitz matrix. The negative real components of the eigenvalues of the matrix represent negative feedback. Similarly, a system is inherently unstable if any of the eigenvalues have positive real components, representing positive feedback.

Hurwitz Stability Criterion | How to form Hurwitz Determinant

References

Hurwitz, A. (1895). "Ueber die Bedingungen, unter welchen eine Gleichung nur Wurzeln mit negativen reellen Teilen besitzt". Mathematische Annalen Nr. 46, Leipzig: 273–284.
Gantmacher, F.R. (1959). "Applications of the Theory of Matrices". Interscience, New York 641 (9): 1–8.
Hassan K. Khalil (2002). Nonlinear Systems. Prentice Hall.
Siegfried H. Lehnigk, On the Hurwitz matrix, Zeitschrift für Angewandte Mathematik und Physik (ZAMP), May 1970
Bernard A. Asner, Jr., On the Total Nonnegativity of the Hurwitz Matrix, SIAM Journal on Applied Mathematics, Vol. 18, No. 2 (Mar., 1970)
Dimitar K. Dimitrov and Juan Manuel Peña, Almost strict total positivity and a class of Hurwitz polynomials, Journal of Approximation Theory, Volume 132, Issue 2 (February 2005)