Determinant de Hurwitz

Considerem un polinomi característic ${\displaystyle P}$ en la variable ${\displaystyle \lambda }$ de la manera:

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

on ${\displaystyle a_{i}}$ , ${\displaystyle i=0,1,\ldots ,n}$ son reals.

La matriu de Hurwitz quadrada associada a ${\displaystyle P}$ ve donada com:

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

El determinant de Hurwitz nombre ${\displaystyle i}$ és el determinant del primer menor nombre ${\displaystyle i}$ de la matriu de Hurwitz ${\displaystyle H}$ . Els tres primers determinants de Hurwitz són:

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

Existeixen ${\displaystyle n}$ determinants de Hurwitz per a un polinomi característic de grau ${\displaystyle n}$ .

• Hurwitz, Adolf «Ueber die Bedingungen, unter welchen eine Gleichung nur Wurzeln mit negativen reellen Teilen besitzt» (en alemany). Mathematische Annalen [Saxònia], 46, 1895, pàg. 273-284. DOI: 10.1007/BF01446812. ISSN: 0025-5831.
• Wall, H. S «Polynomials whose zeros have negative real parts» (en anglès). The American Mathematical Monthly. Mathematical Association of America (MAA) [Washington D. C], 52(6), 1945, pàg. 308-322. DOI: 10.2307/2305291. ISSN: 0002-9890.