Hurwitz Matrix - Hurwitz Matrix and The Hurwitz Stability Criterion

Hurwitz Matrix and The Hurwitz Stability Criterion

Namely, given a real polynomial

the square matrix

H(p) := \begin{bmatrix} a_1 & a_3 & a_5 & a_7 & \ldots & 0\\ 1 & a_2 & a_4 & a_6& \ldots & 0\\ 0 & a_1 & a_3 & a_5& \ldots & 0\\ 0 & 1 & a_2 & a_4& \ldots & 0\\ 0 & 0 & a_1 & a_3& \ldots & 0\\ \vdots & \vdots & \vdots & \vdots& \ddots& \vdots\\ 0 & 0 & 0 & 0& \ldots& a_n\\ \end{bmatrix}

is called Hurwitz matrix corresponding to the polynomial . It was established by Adolf Hurwitz in 1895 that a real polynomial is stable (that is, all its roots have strictly negative real part) if and only if all the leading principal minors of the matrix are positive:

\begin{align} \Delta_1(p) &= \begin{vmatrix} a_{1} \end{vmatrix} &&=a_{1} > 0 \\ \Delta_2(p) &= \begin{vmatrix} a_{1} & a_{3} \\ 1 & a_{2} \\ \end{vmatrix} &&= a_2 a_1 - a_0 a_3 > 0\\ \Delta_3(p) &= \begin{vmatrix} a_{1} & a_{3} & a_{5} \\ 1 & a_{2} & a_{4} \\ 0 & a_{1} & a_{3} \\ \end{vmatrix} &&= a_3 \Delta_2 - a_1 (a_1 a_4 - a_0 a_5 ) > 0 \end{align}

and so on. The minors are called the Hurwitz determinants.

Read more about this topic:  Hurwitz Matrix

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