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.