Hurwitz determinant

In mathematics, Hurwitz determinants were introduced by Adolf Hurwitz (1895), who used them to give a criterion for all roots of a polynomial to have negative real part.

Definition

Consider a characteristic polynomial P in the variable λ of the form:

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

where ${\displaystyle a_{i}}$, ${\displaystyle i=0,1,\ldots ,n}$, are real.

The square Hurwitz matrix associated to P is given below:

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

The ith Hurwitz determinant is the determinant of the ith leading principal minor of the above Hurwitz matrix H. There are n Hurwitz determinants for a characteristic polynomial of degree n.

See also

• Transfer matrix

References

• Hurwitz, A. (1895), "Ueber die Bedingungen, unter welchen eine Gleichung nur Wurzeln mit negativen reellen Theilen besitzt", Mathematische Annalen, 46 (2): 273–284, doi:10.1007/BF01446812, S2CID 121036103
• Wall, H. S. (1945), "Polynomials whose zeros have negative real parts", The American Mathematical Monthly, 52 (6): 308–322, doi:10.1080/00029890.1945.11991574, ISSN 0002-9890, JSTOR 2305291, MR 0012709