On $Q$-Tensors
Abstract
One of the central problems in the theory of linear complementarity problems (LCPs) is to study the class of -matrices since it characterizes the solvability of LCP. Recently, the concept of -matrix has been extended to the case of tensor, called -tensor, which characterizes the solvability of the corresponding tensor complementarity problem -- a generalization of LCP; and some basic results related to -tensors have been obtained in the literature. In this paper, we extend two famous results related to -matrices to the tensor space, i.e., we show that within the class of strong -tensors or nonnegative tensors, four classes of tensors, i.e., -tensors, -tensors, -tensors and -tensors, are all equivalent. We also construct several examples to show that three famous results related to -matrices cannot be extended to the tensor space; and one of which gives a negative answer to a question raised recently by Song and Qi.
Keywords
Cite
@article{arxiv.1509.03088,
title = {On $Q$-Tensors},
author = {Zheng-Hai Huang and Yun-Yang Suo and Jie Wang},
journal= {arXiv preprint arXiv:1509.03088},
year = {2015}
}