Some Problems On Rank And Eigenvalue Of Tensors

The concept of tensors was introduced by Gauss, Riemann and Christoffel, etc., in the 19 th century in the study of differential geometry. In the very beginning of the 20 th century, Ricci, Levi-Civita, etc., further developed tensor analysis as a mathematical discipline. It was Einstein who applied tensor analysis in his study of general relativity in 1916. This made tensor analysis be an important tool in theoretical physics, continuum mechanics and many other areas of science and engineering.Tensor is a natural generalization of a matrix. Symmetric tensor, anti-symmetric tensor, tensor’s eigenvalue and tensor’s rank are both generalization of general symmetric matrix, anti-symmetric matrix, matrix’s eigenvalue and matrix’s rank respectively. They are powerful tools that can be quite useful in studying quantum mechanics and many other modern techniques.In this thesis, we do some research about tensor’s eigenvalue and tensor’s rank. As a result, our work launched in many aspects as follows:First, we study eigenvalues of some special kinds of tensor, we get the result that: if anti-symmetric tensor’s order greater than two, the eigenvalue of anti-symmetric tensor is zero. In the complex field, the spectrum of the pyramid is equal to the set of diagonal elements. In the real field, if tensor’s order is even, then the spectrum of the pyramid is equal to the set of diagonal elements; if tensor’s order is odd, then the spectrum of the pyramid belongs to the set of diagonal elements, and we can find a pyramid that it’s spectrum isn’t equal to the set of diagonal elements. We prove that:if λ is the eigenvalue of Hermitian tensor A, then (?) is the eigenvalue of Hermitian tensor A. At last, we use the relationship between the symmetric tensor’s slices and the polynomial to study the problem of how to reconstruct symmetric tensor by eigenvalues and eigenvectors.Second, we use tensor decomposition, offset diagonalization and Kruskal theorem to give some sufficient conditions for the equation of generalized rank of matrix rank( A(?)B) =rank( A) +rank(B).