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 [6]).

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.