Sala Riunioni (Dip. Matematica).
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.