  We introduce the concept of shape partition of a tensor and formulate a
general tensor eigenvalue problem that includes all previously studied
eigenvalue problems as special cases. We formulate irreducibility and symmetry
properties of a nonnegative tensor $T$ in terms of the associated shape
partition. We recast the eigenvalue problem for $T$ as a fixed point problem on
a suitable product of projective spaces. This allows us to use the theory of
multi-homogeneous order-preserving maps to derive a new and unifying
Perron-Frobenius theorem for nonnegative tensors which either implies earlier
results of this kind or improves them, as weaker assumptions are required. We
introduce a general power method for the computation of the dominant tensor
eigenpair, and provide a detailed convergence analysis.
