The spectral theory of tensors and its applications

Abstract

Over the last few years, there has been a realization that there is a reasonably complete and consistent theory of eigenvalues and singular values for tensors of higher order. This theory, which generalizes the theory of matrix eigenvalues and singular values in various manners and extent, was proposed by Lim 1 and Qi 2 independently. We will use the term ‘spectral theory of tensors’ to refer to this body of work. This special issue contains nine papers with new results on the spectral theory of tensors and its applications. The first paper 3 by Chang et al. is a survey on some recent developments in the spectral theory of nonnegative tensors: H-eigenvalue and Z-eigenvalue problems, Perron–Frobenius theory, applications to higher order Markov chains, spectral theory of hypergraphs, and quantum entanglement. The next two papers are about the spectral theory of nonnegative tensors. In 4, Zhou et al. proposed new spectral characterizations of symmetric nonnegative tensors, while in 5, Zhang et al. extended the concept of essential nonnegativity from matrices to higher order tensors and established the convexity and log convexity of dominant eigenvalues for such a class of tensors. The next three papers concern the numerical computations of tensor eigenvalues or singular values. In 6, Friedland et al. proposed an algorithm called the alternating singular value decomposition for computing a best rank-one approximation of tensors based on computations of maximal singular values and singular vectors of matrices. In 7, Dupont and Scott studied the angular orientations of convergent iterates generated by Newton's method in multiple spatial dimensions for solving a tensor eigenproblem. In 8, Hu et al. studied tensor conic linear programming and employed it to solve the extreme Z-eigenvalue problem. In 9, Li et al. studied the perturbation bound for the Perron vector of a transition probability tensor P, and derived a new perturbation bound for the Perron vector of a transition probability matrix. Finally, the last papers are related to the applications of the spectral theory of tensors to hypergraphs. In 10, Li et al. examined some fundamental analytic properties of Z-eigenvalues of even order symmetric tensors; they showed that the maximum Z-eigenvalue function of the associated characteristic tensor provides a link between the combinatorial and analytic structures of the underlying hypergraph. In 11, Xie and Chang generalized the signless Laplacian matrices for graphs to an analogous notion for even uniform hypergraphs and deduced some of its fundamental properties. We hope that the reader will find these articles interesting and be spurred to either commence or continue working in this exciting new area.