Hurwitz determinant
In mathematics, Hurwitz determinants were introduced by {{harvs|txt|authorlink=Adolf Hurwitz|first=Adolf|last= Hurwitz|year=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 , , are real.
The square Hurwitz matrix associated to P is given below:
:
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 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
References
- {{Citation | last1=Hurwitz | first1=A. | title=Ueber die Bedingungen, unter welchen eine Gleichung nur Wurzeln mit negativen reellen Theilen besitzt | doi= 10.1007/BF01446812 | year=1895 | journal= Mathematische Annalen | volume=46 | issue=2| pages=273–284 | s2cid=121036103 }}
- {{Citation | last1=Wall | first1=H. S. | title=Polynomials whose zeros have negative real parts | jstor=2305291 | mr=0012709 | year=1945 | journal=The American Mathematical Monthly | issn=0002-9890 | volume=52 | issue=6 | pages=308–322| doi=10.1080/00029890.1945.11991574 }}