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Perron's Theorem

If mu=(mu_1,mu_2,...,mu_n) is an arbitrary set of positive numbers, then all eigenvalues lambda of the n×n matrix a=a_(ij) lie on the disk |z|<=m_mu, where

m_mu=max_(1<=i<=n)sum_(j=1)^n(mu_j)/(mu_i)|a_(ij)|.

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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. 1121, 2000.MacCluer, C. R. "The Many Proofs and Applications of Perron's Theorem." SIAM Rev. 42, 487-498, 2000.Perron, O. "Grundlagen für eine Theorie des Jacobischen Kettenbruchalgorithmus." Math. Ann. 64, 11-76, 1907.

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Perron's Theorem

Cite this as:

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

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