Perron-Frobenius theorem

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Perron-Frobenius theorem [pe′rōn frō′bā·nē·u̇s ‚thir·əm]

(mathematics)

IfMis a matrix with positive entries, then its largest eigenvalue λ is positive and simple; moreover, there exist vectorsvandwwith positive components such thatvM= λvandMw= λw, if the inner product ofvwithwis 1, then the limit of λ -ntimes thei,jth entry ofMⁿasngoes to infinity is the product of theith component ofwand thejth component ofv.

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