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Ballieu's Theorem

Let the characteristic polynomial of an n×n complex matrix A be written in the form

P(lambda)=|lambdaI-A|
(1)
=lambda^n+b_1lambda^(n-1)+b_2lambda^(n-2)+...+b_(n-1)lambda+b_n.
(2)

Then for any set mu=(mu_1,mu_2,...,mu_n) of positive numbers with mu_0=0 and

M_mu=max_(0<=k<=n-1)(mu_k+mu_n|b_(n-k)|)/(mu_(k+1)),
(3)

all the eigenvalues lambda_i (for i=1, ..., n) lie on the closed disk |z|<=M_mu in the complex plane.

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References

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1153, 2000.

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Ballieu's Theorem

Cite this as:

Weisstein, Eric W. "Ballieu's Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BallieusTheorem.html

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