Determinants And Characteristic Polynomials Of Tensors

Higher order tensors are higher order generalizations of the matrices. The study of the determinants and characteristic polynomials of the tensors has become a significant topic in applied mathematics and numerical multilinear algebra. In recent years, rect-angular tensors arise from the strong ellipticity condition problem in solid mechanics and the entanglement problem in quantum physics. According to the specific properties of resultants, research on the determinants of the square tensors and E-haracteristic polynomials of the rectangular tensors has been developed. The main results:Theorem3.1For an mth-order n-dimensional tensor A and given i,j∈{1,...,n}, if we simultaneously interchange the ith. and jth entry of all its mode-1fibers, we will get a new tensor denoted by A’, i.e., where A’i and Ai denotes the subtensor of A’ and A, respectively, for i∈{1,..., n}. Then, we have Det(.A)=(-1)(m-1)nDet(A’).Theorem3.2Let A∈T(Cn,m). For the fixed index i G{1,...,n}, if we permute the other m-1indices of the ith entry of all its mode-1fibers, in other words, we permute all but the first index i of the subtensor Ai of A, the determinant of A will be invariant.Theorem4.2Suppose that p, q≥3and A is a real (p, q)th order (m x n)dimensional rectangular tensor. If A is regular, then a root of the E-characteristic polynomial Φfi (λ) is an E-singular value of A, for any j=1,..., n.