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

Understand

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.