Studies Of The Eigenvalues Of Square Tensors And The Singular Values Of Rectangular Tensors

In this paper,we study relevant problems of the eigenvalues of square tensors and the singular values of rectangular tensors.In terms of square tensors,we obtain the minimum H~+-eigenvalue of strictly diagonally dominant Z-tensors with positive diago-nal entries,and establish the upper and lower bounds for this eigenvalue.Moreover,we study the bounds of eigenvalues of a strictly diagonally dominant tensor with positive diagonal entries but with arbitrary off-diagonal entries.Furthermore,other new bound-s for the minimum H~+-eigenvalue of nonsingular M-tensors are obtained.In terms of rectangular tensors,we obtain the largest H-singular value of a partially symmetric non-negative rectangular tensor,and establish some bounds for this singular value.Then we give the definition of copositive rectangular tensors.This concept extends from the con-cept of copositive square tensors.Partially symmetric nonnegative rectangular tensors and positive semi-definite rectangular tensors are examples of copositive rectangular tensors.We present two conditions of(strictly)copositivity of a partially symmetric rectangular tensor.Moreover,some further properties of copositive rectangular tensors are discussed.