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 TheoremIn 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.