Eigenvalues of Higher-Order Tensors and Applications

In this project, the aim is to develop the theory of eigenvalues and invariants for higher-order tensors. The construction of eigenvalue methods are planned for the positive definiteness identification problem for tensors of order 4 and 6, and dimension 3 and 4, to establish characteristic polynomials for higher-order tensors such that their coefficients are invariants, and to investigate mathematical properties and physical meanings of such invariants. The plan is to use the resultant theory as a tool in the research. It is innovative to combine tensor analysis and the resultant theory.The success of the project will result in a new theory of eigenvalues and invariants of higher-order tensors, and some practical eigenvalue methods for the positive definiteness identification problem in the case of order 4 or 6 and dimension 3 or 4. The success of the project will have a significant impact on tensor analysis and its application areas, including theoretical physics and continuum mechanics.