In this paper, we extend some classes of structured matrices to higher order tensors. We discuss their relationships with positive semi-definite tensors and some other structured tensors. We show that every principal sub-tensor of such a structured tensor is still a structured tensor in the same class, with a lower dimension. The potential links of such structured tensors with optimization, nonlinear equations, nonlinear complementarity problems, variational inequalities and the nonnegative tensor theory are also discussed.

The Fredholm alternative type results are proved for eigenvalues (E-eigenvalues, H-eigenvalues, Z-eigenvalues) of a higher order tensor A. For the positively homogeneous operators F A and T A induced by a higher order tensor A, we show some relationship between the Gelfand formula and the spectral radius, and present the upper bound of their spectral radii. Furthermore, for a nonnegative tensor A, we obtain the practical relevance for the spectral radius of the operators F A and T A as well as the operator norms of F A and T A .

The tensor complementarity problem (q, A) is to find x ∈ R n such that x ≥ 0, q + Ax m−1 ≥ 0, and x ⊤ (q + Ax m−1 ) = 0.We prove that a real tensor A is a (strictly) semi-positive tensor if and only if the tensor complementarity problem (q, A) has a unique solution for q > 0 (q ≥ 0), and a symmetric real tensor is a (strictly) semi-positive tensor if and only if it is (strictly) copositive. That is, for a strictly copositive symmetric tensor A, the tensor complementarity problem (q, A) has a solution for all q ∈ R n .

It is easily checkable if a given tensor is a B tensor, or a B 0 tensor or not. In this paper, we show that a symmetric B tensor can always be decomposed to the sum of a strictly diagonally dominated symmetric M tensor and several positive multiples of partially all one tensors, and a symmetric B 0 tensor can always be decomposed to the sum of a diagonally dominated symmetric M tensor and several positive multiples of partially all one tensors. When the order is even, this implies that the corresponding B tensor is positive definite, and the corresponding B 0 tensor is positive semi-definite. This gives a checkable sufficient condition for positive definite and semi-definite tensors. This approach is different from the approach in the literature for proving a symmetric B matrix is positive definite, as that matrix approach cannot be extended to the tensor case.

The tensor complementarity problem is a specially structured nonlinear complementarity problem, then it has its particular and nice properties other than ones of the classical nonlinear complementarity problem. In this paper, it is proved that a tensor is an S-tensor if and only if the tensor complementarity problem is feasible, and each Q-tensor is an S-tensor. Furthermore, the boundedness of solution set of the tensor complementarity problem is equivalent to the uniqueness of solution for such a problem with zero vector. For the tensor complementarity problem with a strictly semi-positive tensor, we proved the global upper bounds for solution of such a problem. In particular, the upper bounds keep in close contact with the smallest Pareto H−(Z−)eigenvalue.