A Criterion for ${\rm Q}$-tensors

Abstract: A tensor ${\mathcal A}$ of order $m$ and dimension $n$ is called a ${\rm Q}$-tensor if the tensor complementarity problem has a solution for all ${\bf q} \in {\mathbb R}^{n}$. This means that for every vector ${\bf q}$, there exists a vector ${\bf u}$ such that ${\bf u} \geq {\bf 0},{\bf w} = {\mathcal A}{\bf u}^{m-1}+{\bf q} \geq {\bf 0},~\text{and}~ {\bf u}^{T}{\bf w} = 0$. In this paper, we prove that within the class of rank one symmetric tensors, the ${\rm Q}$-tensors are precisely the positive tensors. Additionally, for a symmetric ${\mathrm Q}$-tensor ${\mathcal A}$ with $rank({\mathcal A})=2$, we show that ${\mathcal A}$ is an ${\mathrm R}_{0}$-tensor. The idea is inspired by the recent work of Parthasarathy et al. \cite{Parthasarathy} and Sivakumar et al. \cite{Sivakumar} on ${\rm Q}$-matrices.