  For a nonnegative weakly irreducible tensor, its spectral radius is an
eigenvalue corresponding to a unique positive eigenvector up to a scalar called
the Perron vector. But including the Perron vector, it may have more than one
eigenvector corresponding to the spectral radius. The projective eigenvariety
associated with the spectral radius is the set of the eigenvectors
corresponding to the spectral radius considered in the complex projective
space. In this paper we prove that the dimension of the above projective
eigenvariety is zero, i.e. there are finite many eigenvectors associated with
the spectral radius up to a scalar. For a general nonnegative tensor, we
characterize the nonnegative combinatorially symmetric tensor for which the
dimension of projective eigenvariety associated with spectral radius is greater
than zero. Finally we apply those results to the adjacency tensors of uniform
hypergraphs.
