Hurwitz determinant

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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:

$P(\lambda)= a_0 \lambda^n + a_1 \lambda^{n-1} + \cdots + a_{n-1} \lambda + a_n$

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