ON Q-TENSORS

摘要

abstract_textpOne of the central problems in the theory of linear complementarity problems (LCPs) is to study the class of Q-matrices since it characterizes the solvability of LCP. Recently, Song and Qi [Ann. of Appl. Math. 33 (2017) 308-323] extended the concept of Q-matrix to the case of tensor, called Q-tensor, which characterizes the solvability of the corresponding tensor complementarity problem a generalization of LCP. They investigated properties of Q-tensors, and proposed the following question: Whether or not a nonzero solution of the tensor complementarity problem contains at least two nonzero components if the involved tensor is a semi-positive Q-tensor. In this paper, we make further studies for Q-tensors. We extend two famous results related to Q-matrices to the tensor space, i.e., we show that within the class of strong Po-tensors or nonnegative tensors, four classes of tensors, i.e., Ro-tensors, R-tensors, ER-tensors and Q-tensors, are all equivalent. We also construct several examples to show that three famous results related to Q-matrices cannot be extended to the tensor space; and one of which gives a negative answer to the question mentioned above raised by Song and Qi./p/abstract_text