Routh-Hurwitz Theorem

Consider the Characteristic Equation

$\begin{displaymath} \vert\lambda{\hbox{\sf I}}-{\hbox{\sf A}}\vert=\lambda^n+b_1\lambda^{n-1}+\ldots+b_{n-1}\lambda+b_n=0 \end{displaymath}$

determining the

Eigenvalues $\lambda$ of a Real $n\times n$ Matrix ${\hbox{\sf A}}$ , where I is the Identity Matrix. Then the Eigenvalues $\lambda$ all have Negative Real Parts if

$\begin{displaymath} \Delta_1>0, \Delta_2>0, \ldots, \Delta_n>0, \end{displaymath}$

where

$\begin{displaymath} \Delta_k=\left\vert\matrix{ b_1 & 1 & 0 & 0 & 0 & 0 & \cdots... ... b_{2k-4} & b_{2k-5} & b_{2k-6} & \cdots & b_k\cr}\right\vert. \end{displaymath}$

References

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 5th ed. San Diego, CA: Academic Press, p. 1119, 1979.