Research On Eigenvalues And Complementarity Problems Of Several Structured Tensors

This thesis studies the theoretical properties of several structured tensors and com-plementarity problem,mainly discusses the specific upper and lower bounds of solution set of tensor complementarity problem for nonnegative Q-tensors;the upper bounds of the eigenvalues of Cauchy-Hankel tensors;the properties of rectangular Z-tensors and rectangular P-tensors,and the existence of solutions for the corresponding complemen-tarity problems.And calculate the eigenvalues of tensors by related algorithms.The thesis is organized as follows.First,we present some new results on Q-tensors,which are defined by the solvabil-ity of the corresponding tensor complementarity problem.For such structured tensors,we give a sufficient condition to guarantee the nonzero solution of the corresponding tensor complementarity problem with a vector containing at least two nonzero compo-nents,and discuss their relationships with some other structured tensors.Furthermore,with respect to the tensor complementarity problem with a non-negative Q-tensor,we obtain the upper and lower bounds of its solution set,and by the way,we show that the eigenvalues of such a tensor are closely related to this solution set.Next,we present upper bounds of eigenvalues for finite and infinite dimension-al Cauchy-Hankel tensors.It is proved that an m-order infinite dimensional Cauchy-Hankel tensor defines a bounded and positively(m-1)-homogeneous operator from l~1into l~p(1<p<∞),and two upper bounds of corresponding positively homogeneous op-erator norms are given.Moreover,for a fourth-order real partially symmetric Cauchy-Hankel tensor,sufficient and necessary conditions of M-positive definiteness are ob-tained,and an upper bound of M-eigenvalue is also given.And numerical experiments show that the upper bound is very close to that calculated by power method.Last,we discuss some properties of rectangular Z-tensors.It is proved that a rect-angular Z-tensor is a rectangular M-tensor if and only if all of its V~+-singular value are nonnegative,we present that the maximal diagonal element of rectangular M-tensors is nonnegative.Moreover,some corresponding results of strong rectangular M-tensors are also obtained.In addition,we prove that an even order strictly diagonally dominat-ed rectangular tensor is a rectangular P-tensor,and also establish that rectangular tensor complementarity problem,corresponding to a rectangular P-tensor,has only zero vec-tor solution for any positive vectors.Besides,some sufficient conditions are given for the rectangular tensor complementarity problem with nonnegative rectangular tensors to have no solution.At last,numerical experiments are showed to test the efficiency of the proposed algorithm.