A criterion for Q-tensors

Abstract

A tensor ${\mathcal {A}}$ of order m and dimension n is called a ${\mathrm Q}$-tensor if the tensor complementarity problem has a solution for all $\mathbf{{q}} \in {{\mathbb {R}}^n}$. This means that for every vector $\mathbf{{q}}$, there exists a vector ${\mathbf{{u}}}$ such that ${\mathbf{{u}}} \ge \textbf{0},{\textbf{w}} = {\mathcal {A}}{\mathbf{{u}}}^{m-1}+\mathbf{{q}} \ge \textbf{0},~\text {and}~ {\mathbf{{u}}}^{T}{\textbf{w}} = 0$. In this paper, we prove that within the class of rank-one symmetric tensors, the ${\mathrm Q}$-tensors are precisely the positive tensors. Additionally, for a symmetric ${\mathrm Q}$-tensor ${\mathcal {A}}$ of order m and dimension 2 with ${ rank({\mathcal {A}})=2}$, we show that ${\mathcal {A}}$ is an $\textrm{R}_{0}$-tensor. The idea is inspired by the recent work of Parthasarathy et al. (J Optim Theory Appl 195:131–147, 2022) and Sivakumar et al. (Linear Multilinear Algebra 70:6947–6964, 2021) on ${\mathrm Q}$-matrices.