Francesco Tudisco

Associate Professor (Reader) in Machine Learning

School of Mathematics, The University of Edinburgh
The Maxwell Institute for Mathematical Sciences
School of Mathematics, Gran Sasso Science Institute JCMB, King’s Buildings, Edinburgh EH93FD UK
email: f dot tudisco at

A unifying Perron-Frobenius theorem for nonnegative tensors via multihomogeneous maps

Antoine Gautier, Francesco Tudisco, Matthias Hein,
SIAM J. Matrix Analysis Appl., 40 : 1206--1231 (2019)


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 multihomogeneous 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. We introduce a general power method for the computation of the dominant tensor eigenpair and provide a detailed convergence analysis.

Please cite this paper as:

  title={A unifying {P}erron-{F}robenius theorem for nonnegative tensors via multihomogeneous maps},
  author={Gautier, Antoine and Tudisco, Francesco and Hein, Matthias},
  journal={SIAM Journal on Matrix Analysis and Applications},

Links: doi arxiv

Keywords: Perron-Frobenius theory nonlinear eigenvalues nonnegative tensors Hilbert metric Thompson metric power method multi-homogeneous maps