# 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 $a_{i}$ , $i=0,1,\ldots ,n$ , 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.