The Perron-Frobenius Theorem for Multi-homogeneous Maps
This provides a unified theoretical framework for spectral problems in nonnegative tensors, benefiting researchers in multilinear algebra and tensor computations.
The paper unifies Perron-Frobenius theorems for nonnegative tensors by introducing order-preserving multi-homogeneous maps, proving weak and strong Perron-Frobenius theorems and a Collatz-Wielandt principle, and generalizing the power method for computing maximal eigenvectors with convergence analysis.
We introduce the notion of order-preserving multi-homogeneous mapping which allows to study Perron-Frobenius type theorems and nonnegative tensors in unified fashion. We prove a weak and strong Perron-Frobenius theorem for these maps and provide a Collatz-Wielandt principle for the maximal eigenvalue. Additionally, we propose a generalization of the power method for the computation of the maximal eigenvector and analyse its convergence. We show that the general theory provides new results and strengthens existing results for various spectral problems for nonnegative tensors.