Hurwitz matrix
From Wikipedia, the free encyclopedia
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\,](../../../../math/8/0/3/80385a6ca3c1c823309aa1805222e390.png)
for each eigenvalue λi. A is also called a stability matrix, because then the differential equation
is stable, that is, 
as 
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
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)
[edit] External links
This article incorporates material from Hurwitz matrix on PlanetMath, which is licensed under the GFDL.
