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	~~In [[mathematics]], a '''Hurwitz matrix''', or '''Routh~~-~~Hurwitz matrix~~'~~'', in [[engineering]] '''stability matrix''', is~~ a ~~structured real [[square matrix]] constructed with coefficients~~		Last week I woke up and realised - Now I've been solitary for a little while and after much intimidation from pals I now locate myself opted for online dating. They promised me that there are lots of pleasant, normal and interesting visitors to meet up, therefore the pitch is gone by here!<br>I make an effort to keep as toned as possible coming to the fitness center several-times a week. I enjoy my athletics and try to play or see since many a potential. I am going to frequently at [http://www.Adobe.com/cfusion/search/index.cfm?term=&Hawthorn+suits&loc=en_us&siteSection=home Hawthorn suits] being winter. Note: In case that you really considered shopping a hobby I really don't mind, I've seen the carnage of wrestling matches at stocktake revenue.<br>My pals and household are awe-inspiring and spending some time [http://lukebryantickets.asiapak.net tickets to luke bryan concert] together at pub gigabytes or dinners is constantly imperative. I haven't ever been into clubs as I realize that one can never have a nice dialogue together with the noise. [http://lukebryantickets.hamedanshahr.com buy luke bryan tickets] In addition, I got 2 quite adorable and undoubtedly cheeky dogs who are always excited to meet fresh people.<br><br>Also visit my web page ... [http://lukebryantickets.citizenswebcasting.com meet luke bryan]
	of ~~a real polynomial.~~

	~~==Hurwitz matrix~~ and the ~~Hurwitz stability criterion==~~
	~~Namely, given a real polynomial~~
	:<~~math~~>~~p(z)=a_{0}z^n+a_{1}z^{n-1}+\cdots+a_{n~~-~~1}z+a_n</math>~~
	~~the <math>n\~~times ~~n</math>~~ [~~[square matrix]]~~
	:~~<math> 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}<~~/~~math>~~
	~~is called '''Hurwitz matrix''' corresponding to the polynomial <math>p<~~/~~math>~~. ~~It was established by [[Adolf Hurwitz]] in 1895 that a real polynomial is [[Stable polynomial\|stable]]~~
	~~(that is, all its roots have strictly negative real part) if and only if all the leading principal [[Minor (linear algebra)\|minors]] of the matrix <math>H(p)<~~/~~math> are positive:~~

	~~:<math>~~
	~~\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}~~
	~~</math>~~
	~~and so on~~. ~~The minors <math>\Delta_k(p)</math> are called the [[Hurwitz determinant]]s.~~

	~~==Hurwitz stable matrices==~~
	In [[engineering]] and [[stability theory]], a [[square matrix]] <math>A</math> is called ''' stable matrix''' (or sometimes '''Hurwitz matrix''') if every [[eigenvalue]] of <math>A</math> has [[negative number\|strictly negative]] [[real part]], that ~~is,~~
	~~:<math>\mathop{\mathrm{Re}}[\lambda_i] < 0\,</math>~~
	~~for each eigenvalue <math>\lambda_i</math>. <math>A</math> is also called~~ a '~~''stability matrix''', because then the [[ordinary differential equation\|differential equation]]~~
	~~:<math>\dot x = A x</math>~~
	~~is [[stability theory\|asymptotically stable]], that is, <math>x(t)\to 0</math> as <math>~~t~~\to\infty.</math>~~

	~~If <math>G(s)</math> is a (matrix-valued) [[transfer function]]~~, ~~then <math>G</math> is called '''Hurwitz'~~'~~' if~~ the ~~[[pole (complex analysis)\|poles]]~~ of ~~all elements of <math>G</math> have negative real part~~. ~~Note that it is not necessary that~~ <~~math~~>~~G(s),</math> for a specific argument <math>s,</math> be a Hurwitz matrix — it need not even be square. The connection is that if <math>A</math> is a Hurwitz matrix, then the~~ [~~[dynamical system]]~~
	:~~<math>\dot x(t)=A x(t) + B u(t)<~~/~~math>~~
	~~:<!-- This is no typo~~. ~~There must be no dot on y --><math>y(t)=C x(t) + D u(t)\,</math>~~
	~~has a Hurwitz transfer function~~.

	~~Any hyperbolic [[Fixed point (mathematics)\|fixed point~~]~~] (~~or ~~[[equilibrium point]]) of a continuous [[Dynamical system (definition)\|dynamical system]]~~ is ~~locally [[Asymptotic stability\|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]].~~

	~~==See also==~~
	* [[Z-matrix (mathematics)\|Z-matrix]]
	* [[M-matrix]]
	* [[P-matrix]]
	* [[Perron–Frobenius theorem]]

	~~==References==~~
	* {{cite journal
	~~\| author = Hurwitz, A.~~
	~~\| year = 1895~~
	~~\| title = Ueber die Bedingungen, unter welchen eine Gleichung nur Wurzeln mit negativen reellen Teilen besitzt~~
	~~\| journal =~~ [http://~~www-gdz~~.~~sub~~.~~uni-goettingen.de/cgi-bin/digbib.cgi?PPN235181684_0046 ''Mathematische Annalen Nr. 46''~~], ~~Leipzig~~
	~~\| pages = 273–284~~
	}}
	*{{cite journal
	~~\| author = Gantmacher, F~~.R.
	~~\| year = 1959~~
	~~\| title = Applications of the Theory of Matrices~~
	~~\| journal = Interscience, New York~~
	~~\| volume = 641~~
	~~\| issue = 9~~
	~~\| pages = 1–8~~
	}}
	* Hassan K. ~~Khalil (2002)~~. ~~''Nonlinear Systems''. Prentice Hall.~~
	* Siegfried H. Lehnigk, [http://~~www~~.~~springerlink~~.com~~/content/h192106tq8nl2274/ ''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, [http://portal.acm.org/citation.cfm?id=1063186.1063190 ''Almost strict total positivity and a class of Hurwitz polynomials''], Journal of Approximation Theory, Volume 132, Issue 2 (February 2005)

	~~==External links==~~
	*{{planetmath reference\|id=5395\|title=Hurwitz matrix}}

	~~{{PlanetMath attribution\|id=5395\|title=Hurwitz matrix}}~~

	~~[[Category:Matrices]]~~
	~~[[Category:Differential equations]~~]

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