  In 1907, Oskar Perron showed that a positive square matrix has a unique
largest positive eigenvalue with a positive eigenvector. This result was
extended to irreducible nonnegative matrices by Geog Frobenius in 1912, and to
irreducible nonnegative tensors and weakly irreducible nonnegative tensors
recently. This result is a fundamental result in matrix theory and has found
wide applications in probability theory, internet search engines, spectral
graph and hypergraph theory, etc. In this paper, we give a necessary and
sufficient condition for the existence of such a positive eigenvector, i.e., a
positive Perron vector, for a nonnegative tensor. We show that every
nonnegative tensor has a canonical nonnegative partition form, from which we
introduce strongly nonnegative tensors. A tensor is called strongly
nonnegative, if the spectral radius of each genuine weakly irreducible block is
equal to the spectral radius of the tensor, which is strictly larger than the
spectral radius of any other block. We prove that a nonnegative tensor has a
positive Perron vector if and only if it is strongly nonnegative. The proof is
nontrivial. Numerical results for finding a positive Perron vector are
reported.
