# 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

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

ti ${\displaystyle n\times n}$ square matrix

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

ń pèé ní  Hurwitz matrix tí ó bá polynomial ${\displaystyle p}$ 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 ${\displaystyle H(p)}$ jẹ́ dájú:

{\displaystyle {\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 ${\displaystyle \Delta _{k}(p)}$ rẹ̀ ní Hurwitz determinants.

## Ìdúróṣinsin Hurwitz matrices

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

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

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

${\displaystyle {\dot {x}}=Ax}$

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

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

${\displaystyle {\dot {x}}(t)=Ax(t)+Bu(t)}$
${\displaystyle y(t)=Cx(t)+Du(t)\,}$

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)