Q-tensors And The Tensor Complementarity Problem

In this paper,we mainly discuss Q-tensors and the tensor complementarity problem.There are R-tensors,strictly semi-positive tensors and many other tensors as subclasses of Q-tensors.Study about Q-tensors mainly focus on properties of a Q-tensor itself,relationships with other structured tensors,solutions of the tensor complementarity problem,and entries of a Q-tensor.These properties contribute to the study of eigenvalues and the rank of a tensor,and there are many results.The tensor complementarity problem(TCP(q,A))is a special complementary problem.Due to the particularity of a tensor,it has some good and unique properties in solving the TCP(q,A).The genetic property and the relationship between Q-tensors and strictly semi-positive ten-sors are both helpful in solving the problem.However,achievements in these two respects are not enough.This paper makes a further study in view of this issue.We give an equivalent con-dition of Q-tensors,prove the genetic property and make a conclusion about this property.We also study solutions of the TCP(q,A).Our main work are carried out in following two aspects:1.Properties of tensors:firstly,an equivalent condition of positive semi-definite tensors is given.We study the principal minors of(strictly)diagonally dominant tensors and prove that a strictly diagonally dominant tensor is a wP-tensor.We also prove that R0-tensors and strictly semi-positive tensors are equivalent to Q-tensors under a nonnegative assumption.The genetic property of an R-tensor is proved.In addition,we construct a counterexample to show that not all completely Q-tensors are strictly semi-positive tensors by utilizing properties of R-tensors.2.Properties of the solution set of the TCP(q,A):We demonstrate that there is no nonzero solution of the tensor complementarity problem for q ? 0 if A is an ER-tensor.We consider an even order symmetric tensor,then the TCP(q,A)has a unique solution if the tensor is diagonaliz-able and strictly copositive.Furthermore,the TCP(q,A)has a solution when the corresponding nonnegative tensor is also a strictly semi-positive tensor or an R0-tensor.The solution set of the TCP(q,A)with a nonnegative and strictly diagonally dominant tensor is nonempty and compact.