A survey on the spectral theory of nonnegative tensors

Abstract

SUMMARY This is a survey paper on the recent development of the spectral theory of nonnegative tensors and its applications. After a brief review of the basic definitions on tensors, the H ‐eigenvalue problem and the Z ‐eigenvalue problem for tensors are studied separately. To the H ‐eigenvalue problem for nonnegative tensors, the whole Perron–Frobenius theory for nonnegative matrices is completely extended, while to the Z ‐eigenvalue problem, there are many distinctions and are studied carefully in details. Numerical methods are also discussed. Three kinds of applications are studied: higher order Markov chains, spectral theory of hypergraphs, and the quantum entanglement. Copyright © 2013 John Wiley & Sons, Ltd.