Hurwitz matrix

Ní ìmọ̀ ìṣirò, Hurwitz matrix, tàbí Routh-Hurwitz matrix, ní ìmọ̀ ẹ̀rọ, ìdúróṣinṣin matrix, jẹ́ ètò square matrix gidi tí wọ́n ṣètò ẹ̀ pẹ̀lú àwọn coeficient polynomial gidi.

Hurwitz matrix àti ti àmì ìdúróṣinṣin Hurwitz

Ìdárúkọ, tí wọ́n bá fún wa ní polynomial gidi

$p(z)=a_{0}z^{n}+a_{1}z^{n-1}+\cdots +a_{n-1}z+a_{n}$

ti $n\times n$ square matrix

$H={\begin{pmatrix}a_{1}&a_{3}&a_{5}&\dots &\dots &\dots &0&0&0\\a_{0}&a_{2}&a_{4}&&&&\vdots &\vdots &\vdots \\0&a_{1}&a_{3}&&&&\vdots &\vdots &\vdots \\\vdots &a_{0}&a_{2}&\ddots &&&0&\vdots &\vdots \\\vdots &0&a_{1}&&\ddots &&a_{n}&\vdots &\vdots \\\vdots &\vdots &a_{0}&&&\ddots &a_{n-1}&0&\vdots \\\vdots &\vdots &0&&&&a_{n-2}&a_{n}&\vdots \\\vdots &\vdots &\vdots &&&&a_{n-3}&a_{n-1}&0\\0&0&0&\dots &\dots &\dots &a_{n-4}&a_{n-2}&a_{n}\end{pmatrix}}.$

ń pèé ní  Hurwitz matrix tí ó bá polynomial $p$ mu. Olùdásílẹ̀ rẹ̀ ní  Adolf Hurwitz ní ọdún 1895 pé polynomial gidi dúróṣinṣin (leyí tójẹ́ pé, gbogbo root wọn ní ní apá òdì gidi) tí ó bá jẹ́ pé, tí ó sì jẹ́ pé àwọn ipò lábébé iwájú ti matrix $H(p)$ jẹ́ dájú:

${\begin{aligned}\Delta _{1}(p)&={\begin{vmatrix}a_{1}\end{vmatrix}}&&=a_{1}>0\\[2mm]\Delta _{2}(p)&={\begin{vmatrix}a_{1}&a_{3}\\a_{0}&a_{2}\\\end{vmatrix}}&&=a_{2}a_{1}-a_{0}a_{3}>0\\[2mm]\Delta _{3}(p)&={\begin{vmatrix}a_{1}&a_{3}&a_{5}\\a_{0}&a_{2}&a_{4}\\0&a_{1}&a_{3}\\\end{vmatrix}}&&=a_{3}\Delta _{2}-a_{1}(a_{1}a_{4}-a_{0}a_{5})>0\end{aligned}}$

àti bẹ́ẹ̀ bẹ́ẹ̀ lọ. Tí à ń pe bàwọn lábébé $\Delta _{k}(p)$ rẹ̀ ní Hurwitz determinants.

Ìdúróṣinsin Hurwitz matrices

Ní ìmọ̀ ẹ̀rọ, àti àlàyé ìdúróṣinṣin, à ń pe square matrix $A$ ní ìdúróṣinṣin matrix (tàbí nígbàmíràn ní Hurwitz matrix) tí gbogbo eigenvalue ti $A$ ní apá òdì,

$\mathop {\mathrm {Re} } [\lambda _{i}]<0\,$

fún ìkọ̀ọ̀kan eigenvalue $\lambda _{i}$. $A$ maa ń jẹ́ ìdúróṣinṣin matrix, nítorí differential equation

${\dot {x}}=Ax$

jẹ́ asymptotically stable, that is, $x(t)\to 0$ sí $t\to \infty .$

Tí $G(s)$ jẹ́ (matrix-valued) ìrékọjá iṣẹ́ nígbà náà $G$ maa ń jẹ́ Hurwitz tí àwọn òpó ìdá ipilẹ̀ $G$. Mọ̀ wípé kò ṣe pàtàkì kí $G(s),$ fún àríyànjiyàn kan pàtó $s,$ jẹ́ Hurwitz matrix — kò ti lẹ̀ ní lati jẹ́ square. Ìlọ́pọ̀ yẹn ní pé t́ $A$ bá jẹ/ Hurwitz matrix, kí  dynamical system

${\dot {x}}(t)=Ax(t)+Bu(t)$

$y(t)=Cx(t)+Du(t)\,$

sì ní ìrékọjá iṣẹ́ Hurwitz .

Àwọn ìtọ́kasí

• Hurwitz, A. (1895). "Ueber die Bedingungen, unter welchen eine Gleichung nur Wurzeln mit negativen reellen Teilen besitzt". Mathematische Annalen, Leipzig (Nr. 46): 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)

Àwọn ìjápọ̀ látìta

• Hurwitz matrix at PlanetMath.org.