Perron-Frobenius theorem

Perron-Frobenius theorem

[pe′rōn frō′bā·nē·u̇s ‚thir·əm]
(mathematics)
If M is a matrix with positive entries, then its largest eigenvalue λ is positive and simple; moreover, there exist vectors v and w with positive components such that vM = λ v and Mw = λ w, if the inner product of v with w is 1, then the limit of λ -n times the i,j th entry of M n as n goes to infinity is the product of the i th component of w and the j th component of v.
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References in periodicals archive ?
This is enough for a general version of the Perron-Frobenius Theorem to state that a positive real eigenvector exists with a positive real eigenvalue.
Now the Perron-Frobenius Theorem for irreducible matrices gives us that the equilibrium vector is unique.
By the Perron-Frobenius theorem on nonnegative matrices, the adjacency matrix A of a connected network has an eigenvector each of whose entries is positive.
By Perron-Frobenius theorem for a nonnegative matrix A, the absolute value of each eigenvalue of A does not exceed the maximum eigenvalue [[lambda].
the set of eigenvalues) of each of the Leslie matrices obtained by using both sets of data, with the goal to compare our results with the conclusions of the Perron-Frobenius Theorem (see e.
In [3, 5], an approach to this problem using the Perron-Frobenius Theorem [1, 8], known as Perron cluster analysis, is detailed.
One first shows that the eigenvalue is zero using a weaker version of the Perron-Frobenius theorem.
Or, since the equation involves matrix multiplication in finite dimensions, one may instead appeal to the Perron-Frobenius theorem (Berman and Plemmons, 1994; Radjavi, 1999; Meyer, 2000).
Remarkably, a relatively obscure mathematical result known as the Perron-Frobenius theorem furnishes a recipe for calculating just such a ranking.
T] is irreducible and from the Perron-Frobenius Theorem 2.
beta]] being an irreducible positive matrix, then according to the Perron-Frobenius Theorem
In this analysis Potron made use of the Perron-Frobenius theorems, which at that time had been proved only for positive and irreducible non-negative matrices, but he immediately made attempts to cover also the more plausible case of decomposable input-output matrices.