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

Let us consider a characteristic polynomial P in the variable λ of the form:

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

where $a_{i}$ for $i\in \{1,\dots ,k\}$ are rational functions in $\mathbb {R} (Z)$ with coefficients in $Z$.

The square Hurwitz matrix associated to P is given below:

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

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

References

• Hurwitz, A. (1895), "Ueber die Bedingungen, unter welchen eine Gleichung nur Wurzeln mit negativen reellen Theilen besitzt", Mathematische Annalen, 46 (2), doi:10.1007/BF01446812
• Wall, H. S. (1945), "Polynomials whose zeros have negative real parts", The American Mathematical Monthly, 52: 308–322, ISSN 0002-9890, MR0012709