# 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.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
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They cover basic facts on vector spaces, matrices, and linear transformations; canonical forms; applications of the Jordan canonical form; inner product spaces; Perron-Frobenius theorem; and tensor products.
In order to reveal the dynamics generated by (3) and (4), we assume in both cases solutions of the form [[bar.x].sub.t] = [[lambda].sup.t][bar.w] and apply the Perron-Frobenius theorem which may be formulated as follows.
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.
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.
If G [member of] [T.sub.[omega]+1,[omega]], then, from the Perron-Frobenius theorem, the first [omega] - 1 smallest values of the spectral radius of [T.sub.[omega]+1[omega]] are [PK.sub.1[omega];i] (0 [less than or equal to] i [less than or equal to] [omega] - 2), respectively, where [PK.sub.1,[omega];i] is the graph obtained from PK1w by adding i (0 [less than or equal to] i [less than or equal to] [omega] - 2) edges.
Since [A.sub.[epsilon]] has positive off-diagonal elements, by the Perron-Frobenius theorem, there is a positive eigenvector [eta] for the maximum eigenvalue [mu] of [A.sub.[epsilon]].
The block map [[PHI].sup.(L)] plays an important role in this method and one of the advantages of this method is that we can prove that (3) holds for N [member of] [[GAMMA].sup.H.sub.+](S) by using the matrix theory argument (Perron-Frobenius Theorem ).
For the largest H-eigenvalue of a nonnegative tensor, the Perron-Frobenius theorem was proved by Chang et al.
Moreover, we also analyze the spectrum (i.e., 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.g.
By Perron-Frobenius Theorem, [bar.[alpha]] > 1 or [bar.[alpha]] < 0, that is a is a Sturm number.
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. The second step is to show that the remaining eigenvalue problem is non degenerate, namely that the dimension of the eigenspace must be one.

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