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:

$P(\lambda )=a_0{\lambda }^n+a_1{\lambda }^{n-1}+\cdots +a_{n-1}\lambda +a_n$

where $a_i$, $i=0,1,\dots ,n$, are real. The square Hurwitz matrix associated to P is given below:

$H=\left(\begin{array}{ccccccccc}{a}_1& a_3& a_5& \dots & \dots & \dots & 0& 0& 0\\ a_0& a_2& a_4& & & & ⋮& ⋮& ⋮\\ 0& a_1& a_3& & & & ⋮& ⋮& ⋮\\ ⋮& a_0& a_2& \ddots & & & 0& ⋮& ⋮\\ ⋮& 0& a_1& & \ddots & & a_n& ⋮& ⋮\\ ⋮& ⋮& a_0& & & \ddots & a_{n-1}& 0& ⋮\\ ⋮& ⋮& 0& & & & a_{n-2}& a_n& ⋮\\ ⋮& ⋮& ⋮& & & & a_{n-3}& a_{n-1}& 0\\ 0& 0& 0& \dots & \dots & \dots & a_{n-4}& a_{n-2}& a_n\end{array}\right).$

The i-th Hurwitz determinant is the i-th leading principal minor (minor is a determinant) 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