A unifying Perron-Frobenius theorem for nonnegative tensors via multi-homogeneous maps

Inspired by the definition of symmetric decomposition, we introduce the concept of shape partition of a tensor and formulate a general tensor spectral problem that includes all the relevant spectral 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 spectral 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 general and unifying Perron-Frobenius theorem for nonnegative tensors that either implies previous results of this kind or improves them by weakening the assumptions there considered. We introduce a general power method for the computation of the dominant tensor eigenpair, and provide a detailed convergence analysis.