Hurwitz determinant

Hurwitz determinants are named after the German mathematician Adolf Hurwitz.

Let us consider a characteristic polynomial $P$ in the variable $\lambda$ of the form:

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

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

The square Hurwitz matrix associated to $P$ is given below:

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

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