The Perron-Frobenius theorem for multi-homogeneous maps

Gautier Antoine, Francesco Tudisco, Matthias Hein

Research output: Working paper

Abstract

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.
LanguageEnglish
Place of PublicationIthaca, NY
Number of pages54
Publication statusPublished - 10 Feb 2017

Fingerprint

Perron-Frobenius Theorem
Tensor
Non-negative
Perron-Frobenius
Power Method
Spectral Problem
Strong Theorems
Eigenvector
Eigenvalue
Theorem
Generalization

Keywords

  • functional analysis
  • numerical analysis
  • Perron-Frobenius theorem

Cite this

Antoine, G., Tudisco, F., & Hein, M. (2017). The Perron-Frobenius theorem for multi-homogeneous maps. Ithaca, NY.
Antoine, Gautier ; Tudisco, Francesco ; Hein, Matthias. / The Perron-Frobenius theorem for multi-homogeneous maps. Ithaca, NY, 2017.
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Antoine, G, Tudisco, F & Hein, M 2017 'The Perron-Frobenius theorem for multi-homogeneous maps' Ithaca, NY.

The Perron-Frobenius theorem for multi-homogeneous maps. / Antoine, Gautier; Tudisco, Francesco; Hein, Matthias.

Ithaca, NY, 2017.

Research output: Working paper

TY - UNPB

T1 - The Perron-Frobenius theorem for multi-homogeneous maps

AU - Antoine, Gautier

AU - Tudisco, Francesco

AU - Hein, Matthias

PY - 2017/2/10

Y1 - 2017/2/10

N2 - 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.

AB - 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.

KW - functional analysis

KW - numerical analysis

KW - Perron-Frobenius theorem

UR - https://arxiv.org/abs/1702.03230

M3 - Working paper

BT - The Perron-Frobenius theorem for multi-homogeneous maps

CY - Ithaca, NY

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Antoine G, Tudisco F, Hein M. The Perron-Frobenius theorem for multi-homogeneous maps. Ithaca, NY. 2017 Feb 10.