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Routh-Hurwitz Theorem

Consider the characteristic equation

|lambdaI-A|=lambda^n+b_1lambda^(n-1)+...+b_(n-1)lambda+b_n=0
(1)

determining the n eigenvalues lambda of a real n×n square matrix A, where I is the identity matrix. Then the eigenvalues lambda all have negative real parts if

Delta_1>0,Delta_2>0,...,Delta_n>0,
(2)

where

Delta_k=|b_1 1 0 0 0 0 ... 0; b_3 b_2 b_1 1 0 0 ... 0; b_5 b_4 b_3 b_2 b_1 1 ... 0; | | | | | | ... |; b_(2k-1) b_(2k-2) b_(2k-3) b_(2k-4) b_(2k-5) b_(2k-6) ... b_k|
(3)

(Gradshteyn and Ryzhik 2000, p. 1076).

See also

Stable Polynomial

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References

Gantmacher, F. R. Applications of the Theory of Matrices. New York: Wiley, p. 230, 1959.Gradshteyn, I. S. and Ryzhik, I. M. "Routh-Hurwitz Theorem." §15.715 in Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1076, 2000.Séroul, R. "Stable Polynomials." §10.13 in Programming for Mathematicians. Berlin: Springer-Verlag, pp. 280-286, 2000.

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Routh-Hurwitz Theorem

Cite this as:

Weisstein, Eric W. "Routh-Hurwitz Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Routh-HurwitzTheorem.html

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