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For a nonnegative weakly irreducible tensor, its spectral radius is aneigenvalue corresponding to a unique positive eigenvector up to a scalar calledthe Perron vector. But including the Perron vector, it may have more than oneeigenvector corresponding to the spectral radius. The projective eigenvarietyassociated with the spectral radius is the set of the eigenvectorscorresponding to the spectral radius considered in the complex projectivespace. In this paper we prove that the dimension of the above projectiveeigenvariety is zero, i.e. there are finite many eigenvectors associated withthe spectral radius up to a scalar. For a general nonnegative tensor, wecharacterize the nonnegative combinatorially symmetric tensor for which thedimension of projective eigenvariety associated with spectral radius is greaterthan zero. Finally we apply those results to the adjacency tensors of uniformhypergraphs.

The Dimension of Eigenvariety of Nonnegative Tensors Associated with Spectral Radius