The Perron-Frobenius theorem for multi-homogeneous mappings

Abstract

The Perron-Frobenius theory for nonnegative matrices has been generalized to order-preserving homogeneous mappings on a cone and more recently to nonnegative multilinear forms. We unify both approaches by introducing the concept of order-preserving multi-homogeneous mappings, their associated nonlinear spectral problems and spectral radii. We show several Perron-Frobenius type results for these mappings addressing existence, uniqueness and maximality of nonnegative and positive eigenpairs. We prove a Collatz-Wielandt principle and other characterizations of the spectral radius and analyze the convergence of iterates of these mappings towards their unique positive eigenvectors. On top of providing a new extension of the nonlinear Perron-Frobenius theory to the multi-dimensional case, our contribution poses the basis for several improvements and a deeper understanding of the current spectral theory for nonnegative tensors. In fact, in recent years, important results have been obtained by recasting certain spectral equations for multilinear forms in terms of homogeneous maps, however as our approach is more adapted to such problems, these results can be further refined and improved by employing our new multi-homogeneous setting.

arXiv commentauthor's accepted version. arXiv admin note: text overlap with arXiv:1702.03230

References in corpus (7)

• Perron-Frobenius theorem for nonnegative multilinear forms and extensions
• Tropical polyhedra are equivalent to mean payoff games
• Node and layer eigenvector centralities for multiplex networks
• A maximin characterization of the escape rate of nonexpansive mappings in metrically convex spaces
• On the Hilbert Geometry of Convex Polytopes
• A unifying Perron-Frobenius theorem for nonnegative tensors via multi-homogeneous maps
• Non-linear eigenvalue problems arising from growth maximization of positive linear dynamical systems

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