Abstract:

Eigenvalues for tensors were introduced in 2005. Since then, a rich

theory on eigenvalues of tensors and special tensors has been developed.

There are two main definitions of eigenvalues of tensors. One

definition is based upon homogeneous polynomial systems. This

definition extends the definition of eigenvalues of matrices. A complex

number is an eigenvalue of a ``square'' tensor if and only if it is a

root of the characteristic polynomial of that tensor. The sum of all

the eigenvalues of that tensor is equal to the sum of the diagonal

entries of that tensor, while the product of all the eigenvalues of that

tensor is equal to the determinant of that tensor. An eigenvalue of a

real tensor with a real eigenvector is called an H-eigenvalue. An even

order real symmetric tensor is positive semi-definite if and only if

all of its H-eigenvalues are nonnegative.

Various tensor eigenvalue inclusion theorems have been developed.

Another tensor eigenvalue definition has the merit that the eigenvalues

under this definition are invariant under orthonormal transformations.

This makes eigenvalues under this definition found their applications in

engineering, physics and mechanics. Based upon the tensor eigenvalue

theory, theories on special tensors have also been developed. In

particular, the Perron-Frobenius theory for irreducible nonnegative

tensors and weakly irreducible nonnegative tensors has been developed,

and found applications in spectral hypergraph theory and higher order

Markov chains. Other special tensors include positive semi-definite

tensors, sum-of-squares tensors, completely positive tensors and

copositive tensors, etc. Eigenvalues of tensors also found their

applications in magnetic resonance imaging, elastic mechanics, liquid

crystal study, quantum entanglement and classicality problems, etc. A

book “Tensor Analysis: Spectral Theory and Special Tensors'' is

published by SIAM in April this year.