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Routh-Hurwitz Theorem

Consider the Characteristic Equation

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

determining the $n$ 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

\Delta_1>0, \Delta_2>0, \ldots, \Delta_n>0,


b_1 & 1 & 0 & 0 & 0 & 0 & \cdots...
... b_{2k-4} & b_{2k-5} & b_{2k-6} & \cdots & b_k\cr}\right\vert.


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

© 1996-9 Eric W. Weisstein