Hurwitz matrix

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In mathematics, a square matrix $A$ is called a Hurwitz matrix if every eigenvalue of $A$ has strictly negative real part, that is,

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

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

$\dot x = A x$

is stable, that is, $x(t)\to 0$ as $t\to\infty.$

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.

[edit] References

Hassan K. Khalil (2002). Nonlinear Systems. Prentice Hall.
Siegfried H. Lehnigk, On the Hurwitz matrix, Zeitschrift für Angewandte Mathematik und Physik (ZAMP), May 1970
Hurwitz-Radon matrices revisited: From effective solution of the Hurwitz matrix equations to Bott periodicity, in Mathematical Survey Lectures 1943–2004, Springer Berlin Heidelberg, 2006
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)