Hurwitz Determinants

In mathematics, Hurwitz determinants were introduced by , 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:

:$P(\lambda)= a_0 \lambda^n + a_1 \lambda^ + \cdots + a_ \lambda + a_n$

where $a_i$, $i=0,1,\ldots,n$, are real.

The square Hurwitz matrix associated to ''P'' is given below:
:$H= \begin a_1 & a_3 & a_5 & \dots & \dots & \dots & 0 & 0 & 0 \\ a_0 & a_2 & a_4 & & & & \vdots & \vdots & \vdots \\ 0 & a_1 & a_3 & & & & \vdots & \vdots & \vdots \\ \vdots & a_0 & a_2 & \ddots & & & 0 & \vdots & \vdots \\ \vdots & 0 & a_1 & & \ddots & & a_n & \vdots & \vdots \\ \vdots & \vdots & a_0 & & & \ddots & a_ & 0 & \vdots \\ \vdots & \vdots & 0 & & & & a_ & a_n & \vdots \\ \vdots & \vdots & \vdots & & & & a_ & a_ & 0 \\ 0 & 0 & 0 & \dots & \dots & \dots & a_ & a_ & a_n \end.$

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

*
*{{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

Linear algebra
Determinants

de:Hurwitzpolynom#Hurwitz-Kriterium